%------------------------------------------------------------------------------
% File     : Vampire-SAT---4.8
% Problem  : ITP001_1 : TPTP v8.1.2. Released v8.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : vampire --mode casc_sat -m 16384 --cores 7 -t %d %s

% Computer : n013.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 : Sun May  5 07:14:25 EDT 2024

% Result   : Theorem 3.92s 1.02s
% Output   : Refutation 3.92s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   13
%            Number of leaves      : 1770
% Syntax   : Number of formulae    : 4778 ( 854 unt; 397 typ;   0 def)
%            Number of atoms       : 15932 (1761 equ)
%            Maximal formula atoms :   49 (   3 avg)
%            Number of connectives : 18050 (6499   ~;6962   |;2323   &)
%                                         ( 996 <=>;1269  =>;   0  <=;   1 <~>)
%            Maximal formula depth :   14 (   6 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of FOOLs       :    2 (   2 fml;   0 var)
%            Number arithmetic     : 19241 (6302 atm;7093 fun;3157 num;2689 var)
%            Number of types       :   29 (  27 usr;   1 ari)
%            Number of type conns  :  615 ( 356   >; 259   *;   0   +;   0  <<)
%            Number of predicates  :  944 ( 940 usr; 842 prp; 0-5 aty)
%            Number of functors    :  278 ( 271 usr;  18 con; 0-3 aty)
%            Number of variables   : 10944 (10247   !; 697   ?;10944   :)

% Comments : 
%------------------------------------------------------------------------------
tff(type_def_5,type,
    'Int_bool_fun$': $tType ).

tff(type_def_6,type,
    'Nat_nat_bool_fun_fun$': $tType ).

tff(type_def_7,type,
    'Int_set_set_set$': $tType ).

tff(type_def_8,type,
    'Int_set$': $tType ).

tff(type_def_9,type,
    'Int_int_fun$': $tType ).

tff(type_def_10,type,
    'Int_set_set$': $tType ).

tff(type_def_11,type,
    'N_set$': $tType ).

tff(type_def_12,type,
    'Nat_n_fun$': $tType ).

tff(type_def_13,type,
    'Int_set_int_fun$': $tType ).

tff(type_def_14,type,
    'N_int_set_fun$': $tType ).

tff(type_def_15,type,
    'Nat_nat_fun$': $tType ).

tff(type_def_16,type,
    'Nat$': $tType ).

tff(type_def_17,type,
    'N_int_fun$': $tType ).

tff(type_def_18,type,
    'N$': $tType ).

tff(type_def_19,type,
    tlbool: $tType ).

tff(type_def_20,type,
    'Int_set_int_set_fun$': $tType ).

tff(type_def_21,type,
    'Int_n_fun$': $tType ).

tff(type_def_22,type,
    'Int_int_set_fun$': $tType ).

tff(type_def_23,type,
    'Int_set_n_fun$': $tType ).

tff(type_def_24,type,
    'N_n_fun$': $tType ).

tff(type_def_25,type,
    'Int_set_bool_fun$': $tType ).

tff(type_def_26,type,
    'Int_int_bool_fun_fun$': $tType ).

tff(type_def_27,type,
    'Nat_bool_fun$': $tType ).

tff(type_def_28,type,
    'N_set_set$': $tType ).

tff(type_def_29,type,
    'N_n_bool_fun_fun$': $tType ).

tff(type_def_30,type,
    'Nat_int_fun$': $tType ).

tff(type_def_31,type,
    'N_bool_fun$': $tType ).

tff(func_def_0,type,
    'fun_app$p': ( 'Int_n_fun$' * $int ) > 'N$' ).

tff(func_def_1,type,
    'plus$e': ( 'Int_set_set_set$' * 'Int_set_set_set$' ) > 'Int_set_set_set$' ).

tff(func_def_2,type,
    'fun_app$r': ( 'Nat_nat_fun$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_3,type,
    'fun_app$l': ( 'Int_int_set_fun$' * $int ) > 'Int_set$' ).

tff(func_def_4,type,
    'less$': 'N_n_bool_fun_fun$' ).

tff(func_def_5,type,
    'uua$': 'Int_set_set$' > 'Int_set_bool_fun$' ).

tff(func_def_6,type,
    'fun_app$g': ( 'N_n_bool_fun_fun$' * 'N$' ) > 'N_bool_fun$' ).

tff(func_def_7,type,
    'triangle$': 'Nat_nat_fun$' ).

tff(func_def_8,type,
    'from_nat$': 'Nat_n_fun$' ).

tff(func_def_9,type,
    'collect$a': 'Int_set_bool_fun$' > 'Int_set_set$' ).

tff(func_def_10,type,
    'fun_app$o': ( 'N_int_fun$' * 'N$' ) > $int ).

tff(func_def_11,type,
    'fun_app$m': ( 'Int_set_n_fun$' * 'Int_set$' ) > 'N$' ).

tff(func_def_12,type,
    'uuc$': 'Int_int_bool_fun_fun$' ).

tff(func_def_13,type,
    'plus$c': ( 'N_set$' * 'N_set$' ) > 'N_set$' ).

tff(func_def_14,type,
    'fun_app$e': ( 'Nat_int_fun$' * 'Nat$' ) > $int ).

tff(func_def_15,type,
    'fun_app$b': ( 'Int_int_fun$' * $int ) > $int ).

tff(func_def_16,type,
    'ja$': 'Nat$' ).

tff(func_def_17,type,
    'uud$': $int > 'Int_int_fun$' ).

tff(func_def_18,type,
    'a$': 'Nat$' ).

tff(func_def_19,type,
    'i$': 'N$' ).

tff(func_def_20,type,
    'zero$': 'Int_set$' ).

tff(func_def_21,type,
    'of_nat$': 'Nat_int_fun$' ).

tff(func_def_22,type,
    'fun_app$a': ( 'Int_int_bool_fun_fun$' * $int ) > 'Int_bool_fun$' ).

tff(func_def_23,type,
    'plus$a': ( 'Int_set_set$' * 'Int_set_set$' ) > 'Int_set_set$' ).

tff(func_def_24,type,
    'member$a': $int > 'Int_set_bool_fun$' ).

tff(func_def_25,type,
    tltrue: tlbool ).

tff(func_def_26,type,
    'plus$': 'N$' > 'N_n_fun$' ).

tff(func_def_27,type,
    'less_eq$': 'N_n_bool_fun_fun$' ).

tff(func_def_28,type,
    'minus$': 'N$' > 'N_n_fun$' ).

tff(func_def_29,type,
    'fun_app$f': ( 'N_n_fun$' * 'N$' ) > 'N$' ).

tff(func_def_30,type,
    'fun_app$n': ( 'Int_set_int_fun$' * 'Int_set$' ) > $int ).

tff(func_def_31,type,
    'fun_app$i': ( 'Nat_n_fun$' * 'Nat$' ) > 'N$' ).

tff(func_def_32,type,
    'iterate_add$': 'Nat$' > 'Int_int_fun$' ).

tff(func_def_33,type,
    'collect$b': 'Int_bool_fun$' > 'Int_set$' ).

tff(func_def_34,type,
    'plus$d': ( 'N_set_set$' * 'N_set_set$' ) > 'N_set_set$' ).

tff(func_def_35,type,
    'uu$': 'N_set$' > 'N_bool_fun$' ).

tff(func_def_36,type,
    'collect$': 'N_bool_fun$' > 'N_set$' ).

tff(func_def_37,type,
    tlfalse: tlbool ).

tff(func_def_38,type,
    'nat$': $int > 'Nat$' ).

tff(func_def_39,type,
    'zero$a': 'N$' ).

tff(func_def_40,type,
    'plus$b': 'Int_set$' > 'Int_set_int_set_fun$' ).

tff(func_def_41,type,
    'minus$a': 'Int_set$' > 'Int_set_int_set_fun$' ).

tff(func_def_42,type,
    'fun_app$h': ( 'Int_set_int_set_fun$' * 'Int_set$' ) > 'Int_set$' ).

tff(func_def_43,type,
    'less$a': 'Int_set$' > 'Int_set_bool_fun$' ).

tff(func_def_44,type,
    'less_eq$a': 'Int_set$' > 'Int_set_bool_fun$' ).

tff(func_def_45,type,
    'fun_app$k': ( 'N_int_set_fun$' * 'N$' ) > 'Int_set$' ).

tff(func_def_46,type,
    'fun_app$q': ( 'Nat_nat_bool_fun_fun$' * 'Nat$' ) > 'Nat_bool_fun$' ).

tff(func_def_47,type,
    'uub$': 'Int_set$' > 'Int_bool_fun$' ).

tff(func_def_48,type,
    'x$': 'N$' ).

tff(func_def_49,type,
    'to_nat$': 'N$' > 'Nat$' ).

tff(func_def_55,type,
    sK89: 'N_bool_fun$' > 'N$' ).

tff(func_def_56,type,
    sK90: ( 'N_bool_fun$' * 'N$' ) > 'N$' ).

tff(func_def_57,type,
    sK91: 'N_bool_fun$' > 'N$' ).

tff(func_def_58,type,
    sK92: 'N_bool_fun$' > 'N$' ).

tff(func_def_59,type,
    sK93: ( 'N_bool_fun$' * 'N$' ) > 'N$' ).

tff(func_def_60,type,
    sK94: 'N_bool_fun$' > 'N$' ).

tff(func_def_61,type,
    sK95: $int > $int ).

tff(func_def_62,type,
    sK96: $int > $int ).

tff(func_def_63,type,
    sK97: ( 'Int_set$' * 'Int_set$' ) > $int ).

tff(func_def_64,type,
    sK98: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_65,type,
    sK99: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_66,type,
    sK100: 'Nat$' > 'Nat$' ).

tff(func_def_67,type,
    sK101: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_68,type,
    sK102: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_69,type,
    sK103: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_70,type,
    sK104: 'Nat_bool_fun$' > 'Nat$' ).

tff(func_def_71,type,
    sK105: ( 'Int_set$' * 'Int_set$' ) > $int ).

tff(func_def_72,type,
    sK106: 'N_bool_fun$' > 'N$' ).

tff(func_def_73,type,
    sK107: 'N_bool_fun$' > 'N$' ).

tff(func_def_74,type,
    sK108: 'Nat_bool_fun$' > 'Nat$' ).

tff(func_def_75,type,
    sK109: 'Nat_bool_fun$' > 'Nat$' ).

tff(func_def_76,type,
    sK110: ( 'Int_bool_fun$' * 'Int_bool_fun$' ) > $int ).

tff(func_def_77,type,
    sK111: ( 'Int_set$' * 'Int_set$' ) > $int ).

tff(func_def_78,type,
    sK112: ( 'Int_set$' * 'Int_set$' ) > $int ).

tff(func_def_79,type,
    sK113: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_80,type,
    sK114: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_81,type,
    sK115: ( 'Nat$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_82,type,
    sK116: ( 'Nat_bool_fun$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_83,type,
    sK117: ( 'Nat$' * 'Nat_bool_fun$' ) > 'Nat$' ).

tff(func_def_84,type,
    sK118: ( 'Nat_bool_fun$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_85,type,
    sK119: ( 'Nat$' * 'Nat_bool_fun$' ) > 'Nat$' ).

tff(func_def_86,type,
    sK120: 'Nat_bool_fun$' > 'Nat$' ).

tff(func_def_87,type,
    sK121: ( 'Nat_bool_fun$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_88,type,
    sK122: 'Nat_bool_fun$' > 'Nat$' ).

tff(func_def_89,type,
    sK123: ( 'N$' * 'N_set$' * 'N_set$' ) > 'N$' ).

tff(func_def_90,type,
    sK124: ( 'N$' * 'N_set$' * 'N_set$' ) > 'N$' ).

tff(func_def_91,type,
    sK125: ( 'Int_set$' * 'Int_set_set$' * 'Int_set_set$' ) > 'Int_set$' ).

tff(func_def_92,type,
    sK126: ( 'Int_set$' * 'Int_set_set$' * 'Int_set_set$' ) > 'Int_set$' ).

tff(func_def_93,type,
    sK127: ( 'Nat$' * 'Nat_bool_fun$' ) > 'Nat$' ).

tff(func_def_94,type,
    sK128: ( 'Nat$' * 'Nat$' * 'Nat_bool_fun$' ) > 'Nat$' ).

tff(func_def_95,type,
    sK129: ( 'Nat$' * 'Nat$' * 'Nat_bool_fun$' ) > 'Nat$' ).

tff(func_def_96,type,
    sK130: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_97,type,
    sK131: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_98,type,
    sK132: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_99,type,
    sK133: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_100,type,
    sK134: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_101,type,
    sK135: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_102,type,
    sK136: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_103,type,
    sK137: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_104,type,
    sK138: 'Nat_nat_bool_fun_fun$' > 'Nat$' ).

tff(func_def_105,type,
    sK139: ( 'Nat_bool_fun$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_106,type,
    sK140: ( 'Nat$' * 'Nat_bool_fun$' ) > 'Nat$' ).

tff(func_def_107,type,
    sK141: ( $int * 'Int_set$' * 'Int_set$' ) > $int ).

tff(func_def_108,type,
    sK142: ( $int * 'Int_set$' * 'Int_set$' ) > $int ).

tff(func_def_109,type,
    sK143: 'Nat_int_fun$' > 'Nat$' ).

tff(func_def_110,type,
    sK144: 'Nat_int_fun$' > 'Nat$' ).

tff(func_def_111,type,
    sK145: 'Nat_int_fun$' > 'Nat$' ).

tff(func_def_112,type,
    sK146: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_113,type,
    sK147: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_114,type,
    sK148: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_115,type,
    sK149: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_116,type,
    sK150: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_117,type,
    sK151: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_118,type,
    sK152: ( 'Nat_bool_fun$' * 'Nat$' ) > 'Nat$' ).

tff(func_def_119,type,
    sK153: ( 'Nat$' * 'Nat_bool_fun$' ) > 'Nat$' ).

tff(func_def_120,type,
    sK154: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_121,type,
    sK155: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_122,type,
    sK156: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_123,type,
    sK157: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_124,type,
    sK158: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_125,type,
    sK159: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_126,type,
    sK160: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_127,type,
    sK161: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_128,type,
    sK162: 'Int_int_bool_fun_fun$' > $int ).

tff(func_def_129,type,
    sK163: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_130,type,
    sK164: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_131,type,
    sK165: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_132,type,
    sK166: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_133,type,
    sK167: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_134,type,
    sK168: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_135,type,
    sK169: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_136,type,
    sK170: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_137,type,
    sK171: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_138,type,
    sK172: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_139,type,
    sK173: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_140,type,
    sK174: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_141,type,
    sK175: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_142,type,
    sK176: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_143,type,
    sK177: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_144,type,
    sK178: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_145,type,
    sK179: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_146,type,
    sK180: 'N_n_bool_fun_fun$' > 'N$' ).

tff(func_def_147,type,
    sK181: 'Nat_nat_fun$' > 'Nat$' ).

tff(func_def_148,type,
    sK182: 'Nat_nat_fun$' > 'Nat$' ).

tff(func_def_149,type,
    sK183: 'Nat_int_fun$' > 'Nat$' ).

tff(func_def_150,type,
    sK184: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_151,type,
    sK185: 'Nat_n_fun$' > 'Nat$' ).

tff(func_def_152,type,
    sK186: 'Nat_nat_fun$' > 'Nat$' ).

tff(func_def_153,type,
    sK187: 'Nat_nat_fun$' > 'Nat$' ).

tff(func_def_154,type,
    sK188: 'Int_int_set_fun$' > $int ).

tff(func_def_155,type,
    sK189: 'Int_int_set_fun$' > $int ).

tff(func_def_156,type,
    sK190: 'Int_n_fun$' > $int ).

tff(func_def_157,type,
    sK191: 'Int_n_fun$' > $int ).

tff(func_def_158,type,
    sK192: 'Int_int_fun$' > $int ).

tff(func_def_159,type,
    sK193: 'Int_int_fun$' > $int ).

tff(func_def_160,type,
    sK194: 'Int_int_fun$' > $int ).

tff(func_def_161,type,
    sK195: 'Int_int_fun$' > $int ).

tff(func_def_162,type,
    sK196: 'Int_int_fun$' > $int ).

tff(func_def_163,type,
    sK197: 'Int_int_fun$' > $int ).

tff(func_def_164,type,
    sK198: 'Int_n_fun$' > $int ).

tff(func_def_165,type,
    sK199: 'Int_n_fun$' > $int ).

tff(func_def_166,type,
    sK200: 'Int_n_fun$' > $int ).

tff(func_def_167,type,
    sK201: 'Int_n_fun$' > $int ).

tff(func_def_168,type,
    sK202: 'Int_n_fun$' > $int ).

tff(func_def_169,type,
    sK203: 'Int_n_fun$' > $int ).

tff(func_def_170,type,
    sK204: 'Int_int_fun$' > $int ).

tff(func_def_171,type,
    sK205: 'Int_int_fun$' > $int ).

tff(func_def_172,type,
    sK206: 'Int_int_fun$' > $int ).

tff(func_def_173,type,
    sK207: 'Int_int_fun$' > $int ).

tff(func_def_174,type,
    sK208: 'Int_int_fun$' > $int ).

tff(func_def_175,type,
    sK209: 'Int_int_fun$' > $int ).

tff(func_def_176,type,
    sK210: 'Int_n_fun$' > $int ).

tff(func_def_177,type,
    sK211: 'Int_n_fun$' > $int ).

tff(func_def_178,type,
    sK212: 'Int_n_fun$' > $int ).

tff(func_def_179,type,
    sK213: 'Int_n_fun$' > $int ).

tff(func_def_180,type,
    sK214: 'Int_set_int_fun$' > 'Int_set$' ).

tff(func_def_181,type,
    sK215: 'Int_set_int_fun$' > 'Int_set$' ).

tff(func_def_182,type,
    sK216: 'Int_set_n_fun$' > 'Int_set$' ).

tff(func_def_183,type,
    sK217: 'Int_set_n_fun$' > 'Int_set$' ).

tff(func_def_184,type,
    sK218: 'Int_set_int_set_fun$' > 'Int_set$' ).

tff(func_def_185,type,
    sK219: 'Int_set_int_set_fun$' > 'Int_set$' ).

tff(func_def_186,type,
    sK220: 'Int_set_int_fun$' > 'Int_set$' ).

tff(func_def_187,type,
    sK221: 'Int_set_int_fun$' > 'Int_set$' ).

tff(func_def_188,type,
    sK222: 'Int_set_int_set_fun$' > 'Int_set$' ).

tff(func_def_189,type,
    sK223: 'Int_set_int_set_fun$' > 'Int_set$' ).

tff(func_def_190,type,
    sK224: 'Int_set_n_fun$' > 'Int_set$' ).

tff(func_def_191,type,
    sK225: 'Int_set_n_fun$' > 'Int_set$' ).

tff(func_def_192,type,
    sK226: 'Int_int_set_fun$' > $int ).

tff(func_def_193,type,
    sK227: 'Int_int_set_fun$' > $int ).

tff(func_def_194,type,
    sK228: 'N_int_set_fun$' > 'N$' ).

tff(func_def_195,type,
    sK229: 'N_int_set_fun$' > 'N$' ).

tff(func_def_196,type,
    sK230: 'Int_set_int_fun$' > 'Int_set$' ).

tff(func_def_197,type,
    sK231: 'Int_set_int_fun$' > 'Int_set$' ).

tff(func_def_198,type,
    sK232: 'Int_set_n_fun$' > 'Int_set$' ).

tff(func_def_199,type,
    sK233: 'Int_set_n_fun$' > 'Int_set$' ).

tff(func_def_200,type,
    sK234: 'Int_n_fun$' > $int ).

tff(func_def_201,type,
    sK235: 'Int_n_fun$' > $int ).

tff(func_def_202,type,
    sK236: 'Int_n_fun$' > $int ).

tff(func_def_203,type,
    sK237: 'Int_n_fun$' > $int ).

tff(func_def_204,type,
    sK238: 'N_int_fun$' > 'N$' ).

tff(func_def_205,type,
    sK239: 'N_int_fun$' > 'N$' ).

tff(func_def_206,type,
    sK240: 'N_int_fun$' > 'N$' ).

tff(func_def_207,type,
    sK241: 'N_int_fun$' > 'N$' ).

tff(func_def_208,type,
    sK242: 'Int_set_int_set_fun$' > 'Int_set$' ).

tff(func_def_209,type,
    sK243: 'Int_set_int_set_fun$' > 'Int_set$' ).

tff(func_def_210,type,
    sK244: 'Int_int_fun$' > $int ).

tff(func_def_211,type,
    sK245: 'Int_int_fun$' > $int ).

tff(func_def_212,type,
    sK246: 'Int_int_fun$' > $int ).

tff(func_def_213,type,
    sK247: 'Int_int_fun$' > $int ).

tff(func_def_214,type,
    sK248: 'N_n_fun$' > 'N$' ).

tff(func_def_215,type,
    sK249: 'N_n_fun$' > 'N$' ).

tff(func_def_216,type,
    sK250: 'N_n_fun$' > 'N$' ).

tff(func_def_217,type,
    sK251: 'N_n_fun$' > 'N$' ).

tff(func_def_218,type,
    sK252: 'N_int_set_fun$' > 'N$' ).

tff(func_def_219,type,
    sK253: 'N_int_set_fun$' > 'N$' ).

tff(func_def_220,type,
    sK254: 'N_int_fun$' > 'N$' ).

tff(func_def_221,type,
    sK255: 'N_int_fun$' > 'N$' ).

tff(func_def_222,type,
    sK256: 'N_int_fun$' > 'N$' ).

tff(func_def_223,type,
    sK257: 'N_int_fun$' > 'N$' ).

tff(func_def_224,type,
    sK258: 'N_n_fun$' > 'N$' ).

tff(func_def_225,type,
    sK259: 'N_n_fun$' > 'N$' ).

tff(func_def_226,type,
    sK260: 'N_n_fun$' > 'N$' ).

tff(func_def_227,type,
    sK261: 'N_n_fun$' > 'N$' ).

tff(func_def_228,type,
    sK262: 'N_int_fun$' > 'N$' ).

tff(func_def_229,type,
    sK263: 'N_int_fun$' > 'N$' ).

tff(func_def_230,type,
    sK264: 'N_int_fun$' > 'N$' ).

tff(func_def_231,type,
    sK265: 'N_int_fun$' > 'N$' ).

tff(func_def_232,type,
    sK266: 'N_int_fun$' > 'N$' ).

tff(func_def_233,type,
    sK267: 'N_int_fun$' > 'N$' ).

tff(func_def_234,type,
    sK268: 'N_int_fun$' > 'N$' ).

tff(func_def_235,type,
    sK269: 'N_int_fun$' > 'N$' ).

tff(func_def_236,type,
    sK270: 'N_n_fun$' > 'N$' ).

tff(func_def_237,type,
    sK271: 'N_n_fun$' > 'N$' ).

tff(func_def_238,type,
    sK272: 'N_n_fun$' > 'N$' ).

tff(func_def_239,type,
    sK273: 'N_n_fun$' > 'N$' ).

tff(func_def_240,type,
    sK274: 'N_n_fun$' > 'N$' ).

tff(func_def_241,type,
    sK275: 'N_n_fun$' > 'N$' ).

tff(func_def_242,type,
    sK276: 'N_n_fun$' > 'N$' ).

tff(func_def_243,type,
    sK277: 'N_n_fun$' > 'N$' ).

tff(func_def_244,type,
    sK278: 'N_int_fun$' > 'N$' ).

tff(func_def_245,type,
    sK279: 'N_int_fun$' > 'N$' ).

tff(func_def_246,type,
    sK280: 'N_int_fun$' > 'N$' ).

tff(func_def_247,type,
    sK281: 'N_int_fun$' > 'N$' ).

tff(func_def_248,type,
    sK282: 'Int_int_fun$' > $int ).

tff(func_def_249,type,
    sK283: 'Int_int_fun$' > $int ).

tff(func_def_250,type,
    sK284: 'Int_int_fun$' > $int ).

tff(func_def_251,type,
    sK285: 'Int_int_fun$' > $int ).

tff(func_def_252,type,
    sK286: 'N_int_fun$' > 'N$' ).

tff(func_def_253,type,
    sK287: 'N_int_fun$' > 'N$' ).

tff(func_def_254,type,
    sK288: 'N_int_fun$' > 'N$' ).

tff(func_def_255,type,
    sK289: 'N_int_fun$' > 'N$' ).

tff(func_def_256,type,
    sK290: 'Int_int_fun$' > $int ).

tff(func_def_257,type,
    sK291: 'Int_int_fun$' > $int ).

tff(func_def_258,type,
    sK292: 'Int_int_fun$' > $int ).

tff(func_def_259,type,
    sK293: 'Int_int_fun$' > $int ).

tff(func_def_260,type,
    sK294: 'Int_n_fun$' > $int ).

tff(func_def_261,type,
    sK295: 'Int_n_fun$' > $int ).

tff(func_def_262,type,
    sK296: 'Int_n_fun$' > $int ).

tff(func_def_263,type,
    sK297: 'Int_n_fun$' > $int ).

tff(func_def_264,type,
    sK298: 'Int_n_fun$' > $int ).

tff(func_def_265,type,
    sK299: 'Int_n_fun$' > $int ).

tff(func_def_266,type,
    sK300: 'Int_n_fun$' > $int ).

tff(func_def_267,type,
    sK301: 'Int_n_fun$' > $int ).

tff(func_def_268,type,
    sK302: 'N_n_fun$' > 'N$' ).

tff(func_def_269,type,
    sK303: 'N_n_fun$' > 'N$' ).

tff(func_def_270,type,
    sK304: 'N_n_fun$' > 'N$' ).

tff(func_def_271,type,
    sK305: 'N_n_fun$' > 'N$' ).

tff(func_def_272,type,
    sK306: 'N_n_fun$' > 'N$' ).

tff(func_def_273,type,
    sK307: 'N_n_fun$' > 'N$' ).

tff(func_def_274,type,
    sK308: 'N_n_fun$' > 'N$' ).

tff(func_def_275,type,
    sK309: 'N_n_fun$' > 'N$' ).

tff(pred_def_1,type,
    'member$c': ( 'N_set$' * 'N_set_set$' ) > $o ).

tff(pred_def_2,type,
    'member$b': ( 'N$' * 'N_set$' ) > $o ).

tff(pred_def_3,type,
    'member$': ( 'Int_set$' * 'Int_set_set$' ) > $o ).

tff(pred_def_4,type,
    'less_eq$c': ( 'Int_set_set$' * 'Int_set_set$' ) > $o ).

tff(pred_def_5,type,
    'fun_app$c': ( 'Int_set_bool_fun$' * 'Int_set$' ) > $o ).

tff(pred_def_6,type,
    'less_eq$b': ( 'N_set$' * 'N_set$' ) > $o ).

tff(pred_def_7,type,
    'fun_app$': ( 'Int_bool_fun$' * $int ) > $o ).

tff(pred_def_8,type,
    'member$d': ( 'Int_set_set$' * 'Int_set_set_set$' ) > $o ).

tff(pred_def_9,type,
    'fun_app$d': ( 'N_bool_fun$' * 'N$' ) > $o ).

tff(pred_def_10,type,
    'fun_app$j': ( 'Nat_bool_fun$' * 'Nat$' ) > $o ).

tff(pred_def_13,type,
    sP0: ( 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_14,type,
    sP1: ( 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_15,type,
    sP2: ( 'Nat_bool_fun$' * 'Nat$' ) > $o ).

tff(pred_def_16,type,
    sP3: ( 'Nat_bool_fun$' * 'Nat$' ) > $o ).

tff(pred_def_17,type,
    sP4: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_18,type,
    sP5: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_19,type,
    sP6: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_20,type,
    sP7: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_21,type,
    sP8: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_22,type,
    sP9: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_23,type,
    sP10: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_24,type,
    sP11: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_25,type,
    sP12: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_26,type,
    sP13: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_27,type,
    sP14: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_28,type,
    sP15: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_29,type,
    sP16: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_30,type,
    sP17: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_31,type,
    sP18: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_32,type,
    sP19: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_33,type,
    sP20: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_34,type,
    sP21: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_35,type,
    sP22: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_36,type,
    sP23: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_37,type,
    sP24: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_38,type,
    sP25: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_39,type,
    sP26: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_40,type,
    sP27: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_41,type,
    sP28: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_42,type,
    sP29: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_43,type,
    sP30: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_44,type,
    sP31: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_45,type,
    sP32: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_46,type,
    sP33: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_47,type,
    sP34: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_48,type,
    sP35: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_49,type,
    sP36: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_50,type,
    sP37: ( 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_51,type,
    sP38: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_52,type,
    sP39: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_53,type,
    sP40: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_54,type,
    sP41: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_55,type,
    sP42: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_56,type,
    sP43: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_57,type,
    sP44: ( 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_58,type,
    sP45: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_59,type,
    sP46: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_60,type,
    sP47: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_61,type,
    sP48: ( 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_62,type,
    sP49: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_63,type,
    sP50: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_64,type,
    sP51: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_65,type,
    sP52: ( 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_66,type,
    sP53: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_67,type,
    sP54: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_68,type,
    sP55: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_69,type,
    sP56: ( 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_70,type,
    sP57: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_71,type,
    sP58: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_72,type,
    sP59: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_73,type,
    sP60: 'Nat_nat_bool_fun_fun$' > $o ).

tff(pred_def_74,type,
    sP61: ( 'Nat_bool_fun$' * 'Nat$' ) > $o ).

tff(pred_def_75,type,
    sP62: ( $int * $int * $int ) > $o ).

tff(pred_def_76,type,
    sP63: ( $int * $int * $int ) > $o ).

tff(pred_def_77,type,
    sP64: ( $int * $int * $int ) > $o ).

tff(pred_def_78,type,
    sP65: ( $int * $int * $int ) > $o ).

tff(pred_def_79,type,
    sP66: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_80,type,
    sP67: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_81,type,
    sP68: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_82,type,
    sP69: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_83,type,
    sP70: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_84,type,
    sP71: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_85,type,
    sP72: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_86,type,
    sP73: ( 'N$' * 'N$' * 'N$' ) > $o ).

tff(pred_def_87,type,
    sP74: ( 'Nat_bool_fun$' * 'Nat$' ) > $o ).

tff(pred_def_88,type,
    sP75: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_89,type,
    sP76: ( 'Nat$' * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_90,type,
    sP77: ( $int * $int * $int * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_91,type,
    sP78: ( 'Nat$' * 'Nat$' * $int * $int * $int ) > $o ).

tff(pred_def_92,type,
    sP79: ( $int * $int * $int * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_93,type,
    sP80: ( 'Nat$' * 'Nat$' * $int * $int * $int ) > $o ).

tff(pred_def_94,type,
    sP81: ( $int * $int * $int * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_95,type,
    sP82: ( 'Nat$' * 'Nat$' * $int * $int * $int ) > $o ).

tff(pred_def_96,type,
    sP83: ( $int * $int * 'Nat$' * 'Nat$' * $int ) > $o ).

tff(pred_def_97,type,
    sP84: ( $int * 'Nat$' * 'Nat$' * $int * $int ) > $o ).

tff(pred_def_98,type,
    sP85: ( $int * $int * 'Nat$' * 'Nat$' * $int ) > $o ).

tff(pred_def_99,type,
    sP86: ( $int * 'Nat$' * 'Nat$' * $int * $int ) > $o ).

tff(pred_def_100,type,
    sP87: ( $int * $int * $int * 'Nat$' * 'Nat$' ) > $o ).

tff(pred_def_101,type,
    sP88: ( 'Nat$' * 'Nat$' * $int * $int * $int ) > $o ).

tff(f17751,plain,
    $false,
    inference(avatar_sat_refutation,[],[f4234,f4238,f4243,f4247,f4251,f4255,f4259,f4263,f4268,f4272,f4276,f4280,f4284,f4288,f4292,f4296,f4300,f4304,f4308,f4312,f4316,f4320,f4324,f4328,f4332,f4336,f4345,f4350,f4355,f4360,f4364,f4368,f4372,f4376,f4380,f4384,f4388,f4392,f4396,f4400,f4404,f4408,f4412,f4416,f4420,f4424,f4428,f4432,f4436,f4440,f4444,f4448,f4452,f4456,f4460,f4464,f4468,f4497,f4501,f4505,f4510,f4514,f4518,f4522,f4526,f4530,f4534,f4538,f4542,f4546,f4550,f4554,f4558,f4562,f4566,f4570,f4574,f4628,f4644,f4654,f4658,f4662,f4666,f4670,f4674,f4678,f4682,f4686,f4690,f4694,f4698,f4702,f4706,f4710,f4714,f4718,f4722,f4726,f4730,f4734,f4738,f4742,f4746,f4750,f4754,f4758,f4762,f4766,f4770,f4774,f4778,f4782,f4786,f4790,f4794,f4841,f4867,f4882,f4886,f4890,f4894,f4898,f4902,f4906,f4910,f4914,f4918,f4922,f4926,f4930,f4937,f4941,f4945,f4949,f4953,f4957,f4961,f4965,f4969,f4973,f4978,f4984,f4988,f4992,f5202,f5207,f5211,f5215,f5219,f5223,f5227,f5231,f5235,f5239,f5243,f5247,f5252,f5263,f5267,f5271,f5275,f5279,f5283,f5287,f5291,f5295,f5300,f5304,f5308,f5312,f5316,f5320,f5324,f5328,f5332,f5336,f5340,f5344,f5348,f5352,f5356,f5360,f5364,f5368,f5372,f5376,f5380,f5385,f5389,f5502,f5708,f5712,f5716,f5720,f5724,f5728,f5732,f5736,f5740,f5744,f5748,f5752,f5756,f5760,f5764,f5768,f5772,f5777,f5844,f5909,f5913,f5920,f5924,f5928,f5932,f5936,f5940,f5944,f5948,f5952,f5956,f5961,f5965,f5969,f5973,f5977,f5981,f5985,f5989,f5993,f5997,f6001,f6007,f6011,f6015,f6019,f6023,f6027,f6031,f6035,f6302,f6472,f6492,f6496,f6500,f6504,f6508,f6512,f6516,f6520,f6524,f6528,f6532,f6537,f6541,f6545,f6549,f6553,f6557,f6561,f6565,f6569,f6573,f6577,f6591,f6595,f6599,f6603,f6607,f6611,f6615,f6978,f6982,f6986,f6990,f6994,f7018,f7022,f7026,f7030,f7034,f7038,f7042,f7046,f7050,f7054,f7059,f7063,f7067,f7071,f7075,f7079,f7083,f7087,f7091,f7095,f7099,f7106,f7110,f7114,f7118,f7122,f7126,f7130,f7134,f7138,f7142,f7147,f7151,f7155,f7159,f7163,f7167,f7171,f7175,f7179,f7183,f7187,f7196,f7200,f7204,f7208,f7212,f7216,f7220,f7224,f7228,f7232,f7237,f7427,f7528,f7566,f7732,f7804,f7808,f7812,f7816,f7820,f7824,f7828,f7832,f7836,f7840,f7844,f7848,f7852,f7856,f7860,f7864,f7868,f7872,f7876,f7880,f8046,f8050,f8055,f8059,f8063,f8067,f8071,f8075,f8079,f8083,f8087,f8091,f8095,f8105,f8109,f8113,f8117,f8121,f8125,f8216,f8240,f8244,f8248,f8252,f8256,f8260,f8265,f8269,f8274,f8280,f8284,f8288,f8292,f8296,f8300,f8301,f8305,f8309,f8313,f8317,f8321,f8325,f8329,f8333,f8337,f8341,f8349,f8353,f8357,f8361,f8365,f8369,f8373,f8377,f8384,f8521,f8724,f8731,f8735,f8739,f8743,f8747,f8751,f8755,f8759,f8763,f8767,f8772,f8776,f8780,f8784,f8788,f8792,f8796,f8800,f8804,f8808,f8812,f8817,f8821,f8825,f8829,f8833,f8837,f8841,f8845,f8849,f8853,f8857,f8869,f8873,f8877,f8881,f8885,f9187,f9197,f9201,f9205,f9209,f9213,f9217,f9221,f9225,f9230,f9234,f9238,f9242,f9246,f9250,f9254,f9258,f9262,f9266,f9316,f9458,f9462,f9466,f9470,f9474,f9478,f9482,f9486,f9490,f9494,f9498,f9502,f9506,f9510,f9514,f9518,f9522,f9526,f9530,f9534,f9538,f9542,f9546,f9550,f9554,f9558,f9562,f9566,f9570,f9676,f9747,f9910,f9914,f9918,f9922,f9926,f9930,f9934,f9938,f9942,f9946,f9950,f9954,f9958,f9962,f9966,f9970,f9974,f9978,f9982,f9986,f9990,f9994,f9998,f10002,f10006,f10010,f10014,f10018,f10022,f10026,f10030,f10034,f10038,f10042,f10046,f10050,f10054,f10058,f10062,f10066,f10070,f10074,f10078,f10082,f10086,f10092,f10096,f10100,f10104,f10108,f10112,f10642,f10721,f10725,f10729,f10733,f10739,f10743,f10747,f10751,f10755,f10759,f10763,f10767,f10771,f10775,f10779,f10783,f10787,f10791,f10795,f10799,f10803,f10807,f10811,f10815,f10819,f10823,f10827,f10831,f10835,f10839,f10843,f10862,f10906,f10949,f10968,f10972,f10976,f10980,f10984,f10988,f10992,f10996,f11000,f11004,f11008,f11012,f11016,f11020,f11024,f11028,f11032,f11036,f11040,f11044,f11048,f11052,f11084,f11149,f11171,f11175,f11179,f11183,f11187,f11191,f11195,f11199,f11203,f11207,f11211,f11215,f11219,f11223,f11227,f11231,f11237,f11241,f11245,f11249,f11253,f11257,f11261,f11700,f11704,f11708,f11712,f11716,f11720,f11724,f11728,f11732,f11736,f11740,f11744,f11748,f11752,f11756,f11760,f11764,f11768,f11772,f11776,f11782,f11786,f11790,f11794,f11798,f11802,f12282,f12289,f12293,f12297,f12301,f12330,f12334,f12338,f12342,f12346,f12350,f12354,f12358,f12362,f12366,f12370,f12403,f12407,f12411,f12415,f12419,f12423,f12427,f12431,f12435,f12439,f12444,f12448,f12452,f12456,f12460,f12464,f12621,f12691,f12695,f12699,f12703,f12707,f12711,f12715,f12719,f12723,f12727,f12759,f12763,f12767,f12771,f12775,f12779,f12803,f12935,f12952,f12956,f12960,f12964,f12968,f12972,f12976,f12980,f12984,f12988,f12992,f13316,f13345,f13349,f13353,f13357,f13361,f13365,f13369,f13373,f13377,f13381,f13385,f13389,f13393,f13461,f13547,f13551,f13555,f13559,f13582,f13586,f13590,f13594,f13598,f13602,f13606,f13610,f13629,f13692,f13702,f13707,f13717,f13721,f13757,f13762,f13766,f13770,f13774,f13807,f13812,f13825,f13829,f13843,f13847,f13851,f13856,f13866,f13898,f13902,f13906,f13910,f14069,f14075,f14080,f14096,f14100,f14300,f14304,f14308,f14312,f14336,f14341,f14357,f14381,f14385,f14398,f14411,f14415,f14419,f14432,f14460,f14473,f14487,f14749,f14755,f14762,f14767,f14960,f14967,f14972,f14977,f14981,f14985,f15751,f15755,f15967,f15971,f15975,f15979,f15984,f15988,f15992,f15997,f16002,f16006,f16011,f17058,f17065,f17070,f17074,f17079,f17083,f17087,f17091,f17095,f17099,f17103,f17107,f17111,f17115,f17119,f17123,f17127,f17131,f17135,f17139,f17143,f17147,f17151,f17155,f17159,f17163,f17167,f17172,f17176,f17193,f17731]) ).

tff(f17731,plain,
    ( spl310_59
    | ~ spl310_823 ),
    inference(avatar_contradiction_clause,[],[f17730]) ).

tff(f17730,plain,
    ( $false
    | spl310_59
    | ~ spl310_823 ),
    inference(trivial_inequality_removal,[],[f17691]) ).

tff(f17691,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != 'fun_app$e'('of_nat$','to_nat$'('x$')) )
    | spl310_59
    | ~ spl310_823 ),
    inference(superposition,[],[f4496,f17106]) ).

tff(f17106,plain,
    ( ! [X0: $int,X1: $int] : ( $sum(X1,$sum(X0,$uminus(X1))) = X0 )
    | ~ spl310_823 ),
    inference(avatar_component_clause,[],[f17105]) ).

tff(f17105,plain,
    ( spl310_823
  <=> ! [X0: $int,X1: $int] : ( $sum(X1,$sum(X0,$uminus(X1))) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_823])]) ).

tff(f4496,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) )
    | spl310_59 ),
    inference(avatar_component_clause,[],[f4494]) ).

tff(f4494,plain,
    ( spl310_59
  <=> ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_59])]) ).

tff(f17193,plain,
    ( spl310_841
    | ~ spl310_24
    | ~ spl310_773 ),
    inference(avatar_split_clause,[],[f14377,f14302,f4326,f17191]) ).

tff(f17191,plain,
    ( spl310_841
  <=> ! [X0: 'Nat$'] : ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_841])]) ).

tff(f4326,plain,
    ( spl310_24
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | ~ $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_24])]) ).

tff(f14302,plain,
    ( spl310_773
  <=> ! [X0: 'Nat$'] : $less(0,$sum(1,'fun_app$e'('of_nat$',X0))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_773])]) ).

tff(f14377,plain,
    ( ! [X0: 'Nat$'] : ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),0)
    | ~ spl310_24
    | ~ spl310_773 ),
    inference(resolution,[],[f14303,f4327]) ).

tff(f4327,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ $less(X0,X1) )
    | ~ spl310_24 ),
    inference(avatar_component_clause,[],[f4326]) ).

tff(f14303,plain,
    ( ! [X0: 'Nat$'] : $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
    | ~ spl310_773 ),
    inference(avatar_component_clause,[],[f14302]) ).

tff(f17176,plain,
    ( spl310_840
    | ~ spl310_295 ),
    inference(avatar_split_clause,[],[f7519,f7089,f17174]) ).

tff(f17174,plain,
    ( spl310_840
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP46(X0,X0,X1)
        | ~ sP47(X0,X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_840])]) ).

tff(f7089,plain,
    ( spl310_295
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP46(X1,X0,X2)
        | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) )
        | ~ sP47(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_295])]) ).

tff(f7519,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP46(X0,X0,X1)
        | ~ sP47(X0,X0,X1) )
    | ~ spl310_295 ),
    inference(equality_resolution,[],[f7090]) ).

tff(f7090,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) )
        | sP46(X1,X0,X2)
        | ~ sP47(X0,X1,X2) )
    | ~ spl310_295 ),
    inference(avatar_component_clause,[],[f7089]) ).

tff(f17172,plain,
    ( spl310_839
    | ~ spl310_29
    | ~ spl310_272 ),
    inference(avatar_split_clause,[],[f7331,f6976,f4352,f17169]) ).

tff(f17169,plain,
    ( spl310_839
  <=> $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','a$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_839])]) ).

tff(f4352,plain,
    ( spl310_29
  <=> 'fun_app$d'('fun_app$g'('less$','x$'),'fun_app$i'('from_nat$','a$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_29])]) ).

tff(f6976,plain,
    ( spl310_272
  <=> ! [X0: 'N$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$',X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$i'('from_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_272])]) ).

tff(f7331,plain,
    ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','a$'))
    | ~ spl310_29
    | ~ spl310_272 ),
    inference(resolution,[],[f6977,f4354]) ).

tff(f4354,plain,
    ( 'fun_app$d'('fun_app$g'('less$','x$'),'fun_app$i'('from_nat$','a$'))
    | ~ spl310_29 ),
    inference(avatar_component_clause,[],[f4352]) ).

tff(f6977,plain,
    ( ! [X0: 'N$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$i'('from_nat$',X1))
        | $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$',X1)) )
    | ~ spl310_272 ),
    inference(avatar_component_clause,[],[f6976]) ).

tff(f17167,plain,
    ( spl310_838
    | ~ spl310_137
    | ~ spl310_201 ),
    inference(avatar_split_clause,[],[f5849,f5742,f4955,f17165]) ).

tff(f17165,plain,
    ( spl310_838
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | $less($uminus(X1),$uminus(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_838])]) ).

tff(f4955,plain,
    ( spl310_137
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X2,X1),$sum(X2,X0))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_137])]) ).

tff(f5742,plain,
    ( spl310_201
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( $less($sum(X2,$uminus(X1)),$sum(X2,$uminus(X0)))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_201])]) ).

tff(f5849,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | $less($uminus(X1),$uminus(X0)) )
    | ~ spl310_137
    | ~ spl310_201 ),
    inference(resolution,[],[f5743,f4956]) ).

tff(f4956,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X2,X1),$sum(X2,X0))
        | $less(X1,X0) )
    | ~ spl310_137 ),
    inference(avatar_component_clause,[],[f4955]) ).

tff(f5743,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( $less($sum(X2,$uminus(X1)),$sum(X2,$uminus(X0)))
        | ~ $less(X0,X1) )
    | ~ spl310_201 ),
    inference(avatar_component_clause,[],[f5742]) ).

tff(f17163,plain,
    ( spl310_837
    | ~ spl310_138
    | ~ spl310_200 ),
    inference(avatar_split_clause,[],[f5809,f5738,f4959,f17161]) ).

tff(f17161,plain,
    ( spl310_837
  <=> ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | ~ $less($uminus(X1),$uminus(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_837])]) ).

tff(f4959,plain,
    ( spl310_138
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( $less($sum(X2,X0),$sum(X2,X1))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_138])]) ).

tff(f5738,plain,
    ( spl310_200
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X2,$uminus(X0)),$sum(X2,$uminus(X1)))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_200])]) ).

tff(f5809,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | ~ $less($uminus(X1),$uminus(X0)) )
    | ~ spl310_138
    | ~ spl310_200 ),
    inference(resolution,[],[f5739,f4960]) ).

tff(f4960,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( $less($sum(X2,X0),$sum(X2,X1))
        | ~ $less(X0,X1) )
    | ~ spl310_138 ),
    inference(avatar_component_clause,[],[f4959]) ).

tff(f5739,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X2,$uminus(X0)),$sum(X2,$uminus(X1)))
        | $less(X1,X0) )
    | ~ spl310_200 ),
    inference(avatar_component_clause,[],[f5738]) ).

tff(f17159,plain,
    ( spl310_836
    | ~ spl310_16
    | ~ spl310_187 ),
    inference(avatar_split_clause,[],[f5683,f5374,f4294,f17157]) ).

tff(f17157,plain,
    ( spl310_836
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X1,0)
        | $less($sum(X1,X0),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_836])]) ).

tff(f4294,plain,
    ( spl310_16
  <=> ! [X0: $int] : ( 0 = $sum(X0,$uminus(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_16])]) ).

tff(f5374,plain,
    ( spl310_187
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X2,$sum(X0,$uminus(X1)))
        | $less($sum(X2,X1),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_187])]) ).

tff(f5683,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X1,0)
        | $less($sum(X1,X0),X0) )
    | ~ spl310_16
    | ~ spl310_187 ),
    inference(superposition,[],[f5375,f4295]) ).

tff(f4295,plain,
    ( ! [X0: $int] : ( 0 = $sum(X0,$uminus(X0)) )
    | ~ spl310_16 ),
    inference(avatar_component_clause,[],[f4294]) ).

tff(f5375,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X2,$sum(X0,$uminus(X1)))
        | $less($sum(X2,X1),X0) )
    | ~ spl310_187 ),
    inference(avatar_component_clause,[],[f5374]) ).

tff(f17155,plain,
    ( spl310_835
    | ~ spl310_16
    | ~ spl310_186 ),
    inference(avatar_split_clause,[],[f5659,f5370,f4294,f17153]) ).

tff(f17153,plain,
    ( spl310_835
  <=> ! [X0: $int,X1: $int] :
        ( $less(X1,0)
        | ~ $less($sum(X1,X0),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_835])]) ).

tff(f5370,plain,
    ( spl310_186
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X2,X1),X0)
        | $less(X2,$sum(X0,$uminus(X1))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_186])]) ).

tff(f5659,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X1,0)
        | ~ $less($sum(X1,X0),X0) )
    | ~ spl310_16
    | ~ spl310_186 ),
    inference(superposition,[],[f5371,f4295]) ).

tff(f5371,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( $less(X2,$sum(X0,$uminus(X1)))
        | ~ $less($sum(X2,X1),X0) )
    | ~ spl310_186 ),
    inference(avatar_component_clause,[],[f5370]) ).

tff(f17151,plain,
    ( spl310_834
    | ~ spl310_16
    | ~ spl310_185 ),
    inference(avatar_split_clause,[],[f5634,f5366,f4294,f17149]) ).

tff(f17149,plain,
    ( spl310_834
  <=> ! [X0: $int,X1: $int] :
        ( $less(0,X1)
        | ~ $less(X0,$sum(X1,X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_834])]) ).

tff(f5366,plain,
    ( spl310_185
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X1,$sum(X0,X2))
        | $less($sum(X1,$uminus(X2)),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_185])]) ).

tff(f5634,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(0,X1)
        | ~ $less(X0,$sum(X1,X0)) )
    | ~ spl310_16
    | ~ spl310_185 ),
    inference(superposition,[],[f5367,f4295]) ).

tff(f5367,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( $less($sum(X1,$uminus(X2)),X0)
        | ~ $less(X1,$sum(X0,X2)) )
    | ~ spl310_185 ),
    inference(avatar_component_clause,[],[f5366]) ).

tff(f17147,plain,
    ( spl310_833
    | ~ spl310_16
    | ~ spl310_160 ),
    inference(avatar_split_clause,[],[f5597,f5265,f4294,f17145]) ).

tff(f17145,plain,
    ( spl310_833
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(0,X1)
        | $less(X0,$sum(X1,X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_833])]) ).

tff(f5265,plain,
    ( spl310_160
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X2,$uminus(X1)),X0)
        | $less(X2,$sum(X0,X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_160])]) ).

tff(f5597,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(0,X1)
        | $less(X0,$sum(X1,X0)) )
    | ~ spl310_16
    | ~ spl310_160 ),
    inference(superposition,[],[f5266,f4295]) ).

tff(f5266,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X2,$uminus(X1)),X0)
        | $less(X2,$sum(X0,X1)) )
    | ~ spl310_160 ),
    inference(avatar_component_clause,[],[f5265]) ).

tff(f17143,plain,
    ( spl310_832
    | ~ spl310_16
    | ~ spl310_146 ),
    inference(avatar_split_clause,[],[f5432,f5200,f4294,f17141]) ).

tff(f17141,plain,
    ( spl310_832
  <=> ! [X0: $int,X1: $int] : ( $sum(X0,$sum($uminus(X0),X1)) = X1 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_832])]) ).

tff(f5200,plain,
    ( spl310_146
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$sum(X1,X2)) = $sum($sum(X0,X1),X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_146])]) ).

tff(f5432,plain,
    ( ! [X0: $int,X1: $int] : ( $sum(X0,$sum($uminus(X0),X1)) = X1 )
    | ~ spl310_16
    | ~ spl310_146 ),
    inference(evaluation,[],[f5390]) ).

tff(f5390,plain,
    ( ! [X0: $int,X1: $int] : ( $sum(0,X1) = $sum(X0,$sum($uminus(X0),X1)) )
    | ~ spl310_16
    | ~ spl310_146 ),
    inference(superposition,[],[f5201,f4295]) ).

tff(f5201,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$sum(X1,X2)) = $sum($sum(X0,X1),X2) )
    | ~ spl310_146 ),
    inference(avatar_component_clause,[],[f5200]) ).

tff(f17139,plain,
    ( spl310_831
    | ~ spl310_61
    | ~ spl310_138 ),
    inference(avatar_split_clause,[],[f5132,f4959,f4503,f17137]) ).

tff(f17137,plain,
    ( spl310_831
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X0,1)
        | ~ $less(X1,$sum(X1,X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_831])]) ).

tff(f4503,plain,
    ( spl310_61
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | ~ $less(X1,$sum(X0,1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_61])]) ).

tff(f5132,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X0,1)
        | ~ $less(X1,$sum(X1,X0)) )
    | ~ spl310_61
    | ~ spl310_138 ),
    inference(resolution,[],[f4960,f4504]) ).

tff(f4504,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X1,$sum(X0,1))
        | ~ $less(X0,X1) )
    | ~ spl310_61 ),
    inference(avatar_component_clause,[],[f4503]) ).

tff(f17135,plain,
    ( spl310_830
    | ~ spl310_60
    | ~ spl310_137 ),
    inference(avatar_split_clause,[],[f5107,f4955,f4499,f17133]) ).

tff(f17133,plain,
    ( spl310_830
  <=> ! [X0: $int,X1: $int] :
        ( $less(X0,1)
        | $less(X1,$sum(X1,X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_830])]) ).

tff(f4499,plain,
    ( spl310_60
  <=> ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | $less(X1,$sum(X0,1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_60])]) ).

tff(f5107,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X0,1)
        | $less(X1,$sum(X1,X0)) )
    | ~ spl310_60
    | ~ spl310_137 ),
    inference(resolution,[],[f4956,f4500]) ).

tff(f4500,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X1,$sum(X0,1))
        | $less(X0,X1) )
    | ~ spl310_60 ),
    inference(avatar_component_clause,[],[f4499]) ).

tff(f17131,plain,
    ( spl310_829
    | ~ spl310_65
    | ~ spl310_116 ),
    inference(avatar_split_clause,[],[f4876,f4792,f4520,f17129]) ).

tff(f17129,plain,
    ( spl310_829
  <=> ! [X0: $int,X1: $int] :
        ( ( 0 != X1 )
        | ( $sum(X0,X1) = X0 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_829])]) ).

tff(f4520,plain,
    ( spl310_65
  <=> ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X0)) = X1 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_65])]) ).

tff(f4792,plain,
    ( spl310_116
  <=> ! [X0: $int,X1: $int] :
        ( ( 0 != $sum(X0,$uminus(X1)) )
        | ( X0 = X1 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_116])]) ).

tff(f4876,plain,
    ( ! [X0: $int,X1: $int] :
        ( ( 0 != X1 )
        | ( $sum(X0,X1) = X0 ) )
    | ~ spl310_65
    | ~ spl310_116 ),
    inference(superposition,[],[f4793,f4521]) ).

tff(f4521,plain,
    ( ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X0)) = X1 )
    | ~ spl310_65 ),
    inference(avatar_component_clause,[],[f4520]) ).

tff(f4793,plain,
    ( ! [X0: $int,X1: $int] :
        ( ( 0 != $sum(X0,$uminus(X1)) )
        | ( X0 = X1 ) )
    | ~ spl310_116 ),
    inference(avatar_component_clause,[],[f4792]) ).

tff(f17127,plain,
    ( spl310_828
    | ~ spl310_64
    | ~ spl310_116 ),
    inference(avatar_split_clause,[],[f4875,f4792,f4516,f17125]) ).

tff(f17125,plain,
    ( spl310_828
  <=> ! [X0: $int,X1: $int] :
        ( ( 0 != X0 )
        | ( $sum(X0,X1) = X1 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_828])]) ).

tff(f4516,plain,
    ( spl310_64
  <=> ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X1)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_64])]) ).

tff(f4875,plain,
    ( ! [X0: $int,X1: $int] :
        ( ( 0 != X0 )
        | ( $sum(X0,X1) = X1 ) )
    | ~ spl310_64
    | ~ spl310_116 ),
    inference(superposition,[],[f4793,f4517]) ).

tff(f4517,plain,
    ( ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X1)) = X0 )
    | ~ spl310_64 ),
    inference(avatar_component_clause,[],[f4516]) ).

tff(f17123,plain,
    ( spl310_827
    | ~ spl310_26
    | ~ spl310_76 ),
    inference(avatar_split_clause,[],[f4640,f4564,f4334,f17121]) ).

tff(f17121,plain,
    ( spl310_827
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( ~ sP3(X0,X1)
        | sP3(X0,sK118(X0,X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_827])]) ).

tff(f4334,plain,
    ( spl310_26
  <=> ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( sP3(X1,X0)
        | ~ 'fun_app$j'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_26])]) ).

tff(f4564,plain,
    ( spl310_76
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,sK118(X0,X1))
        | ~ sP3(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_76])]) ).

tff(f4640,plain,
    ( ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( ~ sP3(X0,X1)
        | sP3(X0,sK118(X0,X1)) )
    | ~ spl310_26
    | ~ spl310_76 ),
    inference(resolution,[],[f4565,f4335]) ).

tff(f4335,plain,
    ( ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( ~ 'fun_app$j'(X1,X0)
        | sP3(X1,X0) )
    | ~ spl310_26 ),
    inference(avatar_component_clause,[],[f4334]) ).

tff(f4565,plain,
    ( ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,sK118(X0,X1))
        | ~ sP3(X0,X1) )
    | ~ spl310_76 ),
    inference(avatar_component_clause,[],[f4564]) ).

tff(f17119,plain,
    ( spl310_826
    | ~ spl310_68
    | ~ spl310_74 ),
    inference(avatar_split_clause,[],[f4638,f4556,f4532,f17117]) ).

tff(f17117,plain,
    ( spl310_826
  <=> ! [X0: 'Int_bool_fun$',X1: $int] :
        ( ~ 'fun_app$'(X0,X1)
        | 'fun_app$'('uub$'('collect$b'(X0)),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_826])]) ).

tff(f4532,plain,
    ( spl310_68
  <=> ! [X0: 'Int_set$',X1: $int] :
        ( 'fun_app$'('uub$'(X0),X1)
        | ~ 'fun_app$c'('member$a'(X1),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_68])]) ).

tff(f4556,plain,
    ( spl310_74
  <=> ! [X0: $int,X1: 'Int_bool_fun$'] :
        ( 'fun_app$c'('member$a'(X0),'collect$b'(X1))
        | ~ 'fun_app$'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_74])]) ).

tff(f4638,plain,
    ( ! [X0: 'Int_bool_fun$',X1: $int] :
        ( ~ 'fun_app$'(X0,X1)
        | 'fun_app$'('uub$'('collect$b'(X0)),X1) )
    | ~ spl310_68
    | ~ spl310_74 ),
    inference(resolution,[],[f4557,f4533]) ).

tff(f4533,plain,
    ( ! [X0: 'Int_set$',X1: $int] :
        ( ~ 'fun_app$c'('member$a'(X1),X0)
        | 'fun_app$'('uub$'(X0),X1) )
    | ~ spl310_68 ),
    inference(avatar_component_clause,[],[f4532]) ).

tff(f4557,plain,
    ( ! [X0: $int,X1: 'Int_bool_fun$'] :
        ( 'fun_app$c'('member$a'(X0),'collect$b'(X1))
        | ~ 'fun_app$'(X1,X0) )
    | ~ spl310_74 ),
    inference(avatar_component_clause,[],[f4556]) ).

tff(f17115,plain,
    ( spl310_825
    | ~ spl310_61
    | ~ spl310_66 ),
    inference(avatar_split_clause,[],[f4618,f4524,f4503,f17113]) ).

tff(f17113,plain,
    ( spl310_825
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ $less($sum(X0,-1),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_825])]) ).

tff(f4524,plain,
    ( spl310_66
  <=> ! [X0: $int,X1: $int] : ( $sum($sum(X0,$uminus(X1)),X1) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_66])]) ).

tff(f4618,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ $less($sum(X0,-1),X1) )
    | ~ spl310_61
    | ~ spl310_66 ),
    inference(evaluation,[],[f4615]) ).

tff(f4615,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ $less($sum(X0,$uminus(1)),X1) )
    | ~ spl310_61
    | ~ spl310_66 ),
    inference(superposition,[],[f4504,f4525]) ).

tff(f4525,plain,
    ( ! [X0: $int,X1: $int] : ( $sum($sum(X0,$uminus(X1)),X1) = X0 )
    | ~ spl310_66 ),
    inference(avatar_component_clause,[],[f4524]) ).

tff(f17111,plain,
    ( spl310_824
    | ~ spl310_60
    | ~ spl310_66 ),
    inference(avatar_split_clause,[],[f4617,f4524,f4499,f17109]) ).

tff(f17109,plain,
    ( spl310_824
  <=> ! [X0: $int,X1: $int] :
        ( $less(X1,X0)
        | $less($sum(X0,-1),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_824])]) ).

tff(f4617,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X1,X0)
        | $less($sum(X0,-1),X1) )
    | ~ spl310_60
    | ~ spl310_66 ),
    inference(evaluation,[],[f4616]) ).

tff(f4616,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X1,X0)
        | $less($sum(X0,$uminus(1)),X1) )
    | ~ spl310_60
    | ~ spl310_66 ),
    inference(superposition,[],[f4500,f4525]) ).

tff(f17107,plain,
    ( spl310_823
    | ~ spl310_27
    | ~ spl310_66 ),
    inference(avatar_split_clause,[],[f4609,f4524,f4343,f17105]) ).

tff(f4343,plain,
    ( spl310_27
  <=> ! [X0: $int,X1: $int] : ( $sum(X0,X1) = $sum(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_27])]) ).

tff(f4609,plain,
    ( ! [X0: $int,X1: $int] : ( $sum(X1,$sum(X0,$uminus(X1))) = X0 )
    | ~ spl310_27
    | ~ spl310_66 ),
    inference(superposition,[],[f4525,f4344]) ).

tff(f4344,plain,
    ( ! [X0: $int,X1: $int] : ( $sum(X0,X1) = $sum(X1,X0) )
    | ~ spl310_27 ),
    inference(avatar_component_clause,[],[f4343]) ).

tff(f17103,plain,
    ( spl310_822
    | ~ spl310_27
    | ~ spl310_66 ),
    inference(avatar_split_clause,[],[f4604,f4524,f4343,f17101]) ).

tff(f17101,plain,
    ( spl310_822
  <=> ! [X0: $int,X1: $int] : ( $sum($sum($uminus(X1),X0),X1) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_822])]) ).

tff(f4604,plain,
    ( ! [X0: $int,X1: $int] : ( $sum($sum($uminus(X1),X0),X1) = X0 )
    | ~ spl310_27
    | ~ spl310_66 ),
    inference(superposition,[],[f4525,f4344]) ).

tff(f17099,plain,
    ( spl310_821
    | ~ spl310_27
    | ~ spl310_65 ),
    inference(avatar_split_clause,[],[f4597,f4520,f4343,f17097]) ).

tff(f17097,plain,
    ( spl310_821
  <=> ! [X0: $int,X1: $int] : ( $sum($uminus(X0),$sum(X0,X1)) = X1 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_821])]) ).

tff(f4597,plain,
    ( ! [X0: $int,X1: $int] : ( $sum($uminus(X0),$sum(X0,X1)) = X1 )
    | ~ spl310_27
    | ~ spl310_65 ),
    inference(superposition,[],[f4521,f4344]) ).

tff(f17095,plain,
    ( spl310_820
    | ~ spl310_27
    | ~ spl310_64 ),
    inference(avatar_split_clause,[],[f4588,f4516,f4343,f17093]) ).

tff(f17093,plain,
    ( spl310_820
  <=> ! [X0: $int,X1: $int] : ( $sum($uminus(X1),$sum(X0,X1)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_820])]) ).

tff(f4588,plain,
    ( ! [X0: $int,X1: $int] : ( $sum($uminus(X1),$sum(X0,X1)) = X0 )
    | ~ spl310_27
    | ~ spl310_64 ),
    inference(superposition,[],[f4517,f4344]) ).

tff(f17091,plain,
    ( spl310_819
    | ~ spl310_27
    | ~ spl310_61 ),
    inference(avatar_split_clause,[],[f4582,f4503,f4343,f17089]) ).

tff(f17089,plain,
    ( spl310_819
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X1,$sum(1,X0))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_819])]) ).

tff(f4582,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X1,$sum(1,X0))
        | ~ $less(X0,X1) )
    | ~ spl310_27
    | ~ spl310_61 ),
    inference(superposition,[],[f4504,f4344]) ).

tff(f17087,plain,
    ( spl310_818
    | ~ spl310_27
    | ~ spl310_60 ),
    inference(avatar_split_clause,[],[f4578,f4499,f4343,f17085]) ).

tff(f17085,plain,
    ( spl310_818
  <=> ! [X0: $int,X1: $int] :
        ( $less(X1,$sum(1,X0))
        | $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_818])]) ).

tff(f4578,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X1,$sum(1,X0))
        | $less(X0,X1) )
    | ~ spl310_27
    | ~ spl310_60 ),
    inference(superposition,[],[f4500,f4344]) ).

tff(f17083,plain,
    ( spl310_817
    | ~ spl310_24
    | ~ spl310_60 ),
    inference(avatar_split_clause,[],[f4576,f4499,f4326,f17081]) ).

tff(f17081,plain,
    ( spl310_817
  <=> ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | ~ $less($sum(X0,1),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_817])]) ).

tff(f4576,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | ~ $less($sum(X0,1),X1) )
    | ~ spl310_24
    | ~ spl310_60 ),
    inference(resolution,[],[f4500,f4327]) ).

tff(f17079,plain,
    ( spl310_816
    | ~ spl310_29
    | ~ spl310_32 ),
    inference(avatar_split_clause,[],[f4476,f4366,f4352,f17076]) ).

tff(f17076,plain,
    ( spl310_816
  <=> 'fun_app$d'('fun_app$g'('less$','x$'),sK94('fun_app$g'('less$','x$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_816])]) ).

tff(f4366,plain,
    ( spl310_32
  <=> ! [X6: 'N$',X0: 'N_bool_fun$'] :
        ( 'fun_app$d'(X0,sK94(X0))
        | ~ 'fun_app$d'(X0,X6) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_32])]) ).

tff(f4476,plain,
    ( 'fun_app$d'('fun_app$g'('less$','x$'),sK94('fun_app$g'('less$','x$')))
    | ~ spl310_29
    | ~ spl310_32 ),
    inference(resolution,[],[f4367,f4354]) ).

tff(f4367,plain,
    ( ! [X0: 'N_bool_fun$',X6: 'N$'] :
        ( ~ 'fun_app$d'(X0,X6)
        | 'fun_app$d'(X0,sK94(X0)) )
    | ~ spl310_32 ),
    inference(avatar_component_clause,[],[f4366]) ).

tff(f17074,plain,
    ( spl310_815
    | ~ spl310_15
    | ~ spl310_32 ),
    inference(avatar_split_clause,[],[f4474,f4366,f4290,f17072]) ).

tff(f17072,plain,
    ( spl310_815
  <=> ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),sK94('fun_app$g'('less_eq$',X0))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_815])]) ).

tff(f4290,plain,
    ( spl310_15
  <=> ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_15])]) ).

tff(f4474,plain,
    ( ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),sK94('fun_app$g'('less_eq$',X0)))
    | ~ spl310_15
    | ~ spl310_32 ),
    inference(resolution,[],[f4367,f4291]) ).

tff(f4291,plain,
    ( ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),X0)
    | ~ spl310_15 ),
    inference(avatar_component_clause,[],[f4290]) ).

tff(f17070,plain,
    ( spl310_814
    | ~ spl310_29
    | ~ spl310_31 ),
    inference(avatar_split_clause,[],[f4472,f4362,f4352,f17067]) ).

tff(f17067,plain,
    ( spl310_814
  <=> 'fun_app$d'('fun_app$g'('less$','x$'),sK91('fun_app$g'('less$','x$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_814])]) ).

tff(f4362,plain,
    ( spl310_31
  <=> ! [X6: 'N$',X0: 'N_bool_fun$'] :
        ( 'fun_app$d'(X0,sK91(X0))
        | ~ 'fun_app$d'(X0,X6) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_31])]) ).

tff(f4472,plain,
    ( 'fun_app$d'('fun_app$g'('less$','x$'),sK91('fun_app$g'('less$','x$')))
    | ~ spl310_29
    | ~ spl310_31 ),
    inference(resolution,[],[f4363,f4354]) ).

tff(f4363,plain,
    ( ! [X0: 'N_bool_fun$',X6: 'N$'] :
        ( ~ 'fun_app$d'(X0,X6)
        | 'fun_app$d'(X0,sK91(X0)) )
    | ~ spl310_31 ),
    inference(avatar_component_clause,[],[f4362]) ).

tff(f17065,plain,
    ( spl310_813
    | ~ spl310_15
    | ~ spl310_31 ),
    inference(avatar_split_clause,[],[f4470,f4362,f4290,f17063]) ).

tff(f17063,plain,
    ( spl310_813
  <=> ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),sK91('fun_app$g'('less_eq$',X0))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_813])]) ).

tff(f4470,plain,
    ( ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),sK91('fun_app$g'('less_eq$',X0)))
    | ~ spl310_15
    | ~ spl310_31 ),
    inference(resolution,[],[f4363,f4291]) ).

tff(f17058,plain,
    ( spl310_812
    | ~ spl310_89
    | ~ spl310_795 ),
    inference(avatar_split_clause,[],[f15850,f14969,f4684,f17056]) ).

tff(f17056,plain,
    ( spl310_812
  <=> ! [X0: 'Nat$',X1: 'Nat$'] : ~ sP4('to_nat$'('i$'),X0,X1) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_812])]) ).

tff(f4684,plain,
    ( spl310_89
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP4(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_89])]) ).

tff(f14969,plain,
    ( spl310_795
  <=> $less(0,'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_795])]) ).

tff(f15850,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] : ~ sP4('to_nat$'('i$'),X0,X1)
    | ~ spl310_89
    | ~ spl310_795 ),
    inference(resolution,[],[f14971,f4685]) ).

tff(f4685,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP4(X0,X1,X2) )
    | ~ spl310_89 ),
    inference(avatar_component_clause,[],[f4684]) ).

tff(f14971,plain,
    ( $less(0,'fun_app$e'('of_nat$','to_nat$'('i$')))
    | ~ spl310_795 ),
    inference(avatar_component_clause,[],[f14969]) ).

tff(f16011,plain,
    ( ~ spl310_811
    | ~ spl310_29
    | ~ spl310_126 ),
    inference(avatar_split_clause,[],[f5066,f4908,f4352,f16008]) ).

tff(f16008,plain,
    ( spl310_811
  <=> 'fun_app$d'('fun_app$g'('less$','fun_app$i'('from_nat$','a$')),'x$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_811])]) ).

tff(f4908,plain,
    ( spl310_126
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$',X1),X0)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_126])]) ).

tff(f5066,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'('from_nat$','a$')),'x$')
    | ~ spl310_29
    | ~ spl310_126 ),
    inference(resolution,[],[f4909,f4354]) ).

tff(f4909,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$',X1),X0)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) )
    | ~ spl310_126 ),
    inference(avatar_component_clause,[],[f4908]) ).

tff(f16006,plain,
    ( spl310_810
    | ~ spl310_90
    | ~ spl310_795 ),
    inference(avatar_split_clause,[],[f15849,f14969,f4688,f16004]) ).

tff(f16004,plain,
    ( spl310_810
  <=> ! [X0: 'Nat$',X1: 'Nat$'] : ~ sP8('to_nat$'('i$'),X0,X1) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_810])]) ).

tff(f4688,plain,
    ( spl310_90
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP8(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_90])]) ).

tff(f15849,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] : ~ sP8('to_nat$'('i$'),X0,X1)
    | ~ spl310_90
    | ~ spl310_795 ),
    inference(resolution,[],[f14971,f4689]) ).

tff(f4689,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP8(X0,X1,X2) )
    | ~ spl310_90 ),
    inference(avatar_component_clause,[],[f4688]) ).

tff(f16002,plain,
    ( ~ spl310_809
    | ~ spl310_30
    | ~ spl310_125 ),
    inference(avatar_split_clause,[],[f5061,f4904,f4357,f15999]) ).

tff(f15999,plain,
    ( spl310_809
  <=> 'fun_app$d'('fun_app$g'('less$','fun_app$i'('from_nat$','a$')),'i$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_809])]) ).

tff(f4357,plain,
    ( spl310_30
  <=> 'fun_app$d'('fun_app$g'('less_eq$','i$'),'fun_app$i'('from_nat$','a$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_30])]) ).

tff(f4904,plain,
    ( spl310_125
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$',X1),X0)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_125])]) ).

tff(f5061,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'('from_nat$','a$')),'i$')
    | ~ spl310_30
    | ~ spl310_125 ),
    inference(resolution,[],[f4905,f4359]) ).

tff(f4359,plain,
    ( 'fun_app$d'('fun_app$g'('less_eq$','i$'),'fun_app$i'('from_nat$','a$'))
    | ~ spl310_30 ),
    inference(avatar_component_clause,[],[f4357]) ).

tff(f4905,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        | ~ 'fun_app$d'('fun_app$g'('less$',X1),X0) )
    | ~ spl310_125 ),
    inference(avatar_component_clause,[],[f4904]) ).

tff(f15997,plain,
    ( spl310_808
    | spl310_28
    | ~ spl310_124 ),
    inference(avatar_split_clause,[],[f5046,f4900,f4347,f15994]) ).

tff(f15994,plain,
    ( spl310_808
  <=> 'fun_app$d'('fun_app$g'('less_eq$','x$'),'fun_app$i'('from_nat$','a$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_808])]) ).

tff(f4347,plain,
    ( spl310_28
  <=> 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'('from_nat$','a$')),'x$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_28])]) ).

tff(f4900,plain,
    ( spl310_124
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_124])]) ).

tff(f5046,plain,
    ( 'fun_app$d'('fun_app$g'('less_eq$','x$'),'fun_app$i'('from_nat$','a$'))
    | spl310_28
    | ~ spl310_124 ),
    inference(resolution,[],[f4901,f4349]) ).

tff(f4349,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'('from_nat$','a$')),'x$')
    | spl310_28 ),
    inference(avatar_component_clause,[],[f4347]) ).

tff(f4901,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
    | ~ spl310_124 ),
    inference(avatar_component_clause,[],[f4900]) ).

tff(f15992,plain,
    ( spl310_807
    | ~ spl310_6
    | ~ spl310_92 ),
    inference(avatar_split_clause,[],[f4858,f4696,f4253,f15990]) ).

tff(f15990,plain,
    ( spl310_807
  <=> ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | $less(X1,sK96(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_807])]) ).

tff(f4253,plain,
    ( spl310_6
  <=> ! [X0: $int] : $less(X0,sK96(X0)) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_6])]) ).

tff(f4696,plain,
    ( spl310_92
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X2,X0)
        | $less(X2,X1)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_92])]) ).

tff(f4858,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | $less(X1,sK96(X0)) )
    | ~ spl310_6
    | ~ spl310_92 ),
    inference(resolution,[],[f4697,f4254]) ).

tff(f4254,plain,
    ( ! [X0: $int] : $less(X0,sK96(X0))
    | ~ spl310_6 ),
    inference(avatar_component_clause,[],[f4253]) ).

tff(f4697,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X2,X0)
        | $less(X2,X1)
        | $less(X1,X0) )
    | ~ spl310_92 ),
    inference(avatar_component_clause,[],[f4696]) ).

tff(f15988,plain,
    ( spl310_806
    | ~ spl310_5
    | ~ spl310_92 ),
    inference(avatar_split_clause,[],[f4852,f4696,f4249,f15986]) ).

tff(f15986,plain,
    ( spl310_806
  <=> ! [X0: $int,X1: $int] :
        ( $less(sK95(X0),X1)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_806])]) ).

tff(f4249,plain,
    ( spl310_5
  <=> ! [X0: $int] : $less(sK95(X0),X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_5])]) ).

tff(f4852,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(sK95(X0),X1)
        | $less(X1,X0) )
    | ~ spl310_5
    | ~ spl310_92 ),
    inference(resolution,[],[f4697,f4250]) ).

tff(f4250,plain,
    ( ! [X0: $int] : $less(sK95(X0),X0)
    | ~ spl310_5 ),
    inference(avatar_component_clause,[],[f4249]) ).

tff(f15984,plain,
    ( spl310_805
    | ~ spl310_5
    | ~ spl310_84 ),
    inference(avatar_split_clause,[],[f4825,f4664,f4249,f15981]) ).

tff(f15981,plain,
    ( spl310_805
  <=> ( 0 = 'fun_app$e'('of_nat$','nat$'(sK95(0))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_805])]) ).

tff(f4664,plain,
    ( spl310_84
  <=> ! [X0: $int] :
        ( ( 0 = 'fun_app$e'('of_nat$','nat$'(X0)) )
        | ~ $less(X0,0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_84])]) ).

tff(f4825,plain,
    ( ( 0 = 'fun_app$e'('of_nat$','nat$'(sK95(0))) )
    | ~ spl310_5
    | ~ spl310_84 ),
    inference(resolution,[],[f4665,f4250]) ).

tff(f4665,plain,
    ( ! [X0: $int] :
        ( ~ $less(X0,0)
        | ( 0 = 'fun_app$e'('of_nat$','nat$'(X0)) ) )
    | ~ spl310_84 ),
    inference(avatar_component_clause,[],[f4664]) ).

tff(f15979,plain,
    ( spl310_804
    | ~ spl310_6
    | ~ spl310_80 ),
    inference(avatar_split_clause,[],[f4796,f4642,f4253,f15977]) ).

tff(f15977,plain,
    ( spl310_804
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | $less(X0,sK96(X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_804])]) ).

tff(f4642,plain,
    ( spl310_80
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | ~ $less(X1,X2)
        | $less(X0,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_80])]) ).

tff(f4796,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | $less(X0,sK96(X1)) )
    | ~ spl310_6
    | ~ spl310_80 ),
    inference(resolution,[],[f4643,f4254]) ).

tff(f4643,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X1,X2)
        | ~ $less(X0,X1)
        | $less(X0,X2) )
    | ~ spl310_80 ),
    inference(avatar_component_clause,[],[f4642]) ).

tff(f15975,plain,
    ( spl310_803
    | ~ spl310_5
    | ~ spl310_80 ),
    inference(avatar_split_clause,[],[f4795,f4642,f4249,f15973]) ).

tff(f15973,plain,
    ( spl310_803
  <=> ! [X0: $int,X1: $int] :
        ( ~ $less(X0,sK95(X1))
        | $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_803])]) ).

tff(f4795,plain,
    ( ! [X0: $int,X1: $int] :
        ( ~ $less(X0,sK95(X1))
        | $less(X0,X1) )
    | ~ spl310_5
    | ~ spl310_80 ),
    inference(resolution,[],[f4643,f4250]) ).

tff(f15971,plain,
    ( spl310_802
    | ~ spl310_16
    | ~ spl310_70 ),
    inference(avatar_split_clause,[],[f4629,f4540,f4294,f15969]) ).

tff(f15969,plain,
    ( spl310_802
  <=> ! [X0: $int] :
        ( ( 0 != $uminus(X0) )
        | ( 0 = X0 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_802])]) ).

tff(f4540,plain,
    ( spl310_70
  <=> ! [X0: $int,X1: $int] :
        ( ( 0 = X0 )
        | ( $sum(X0,X1) != X1 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_70])]) ).

tff(f4629,plain,
    ( ! [X0: $int] :
        ( ( 0 != $uminus(X0) )
        | ( 0 = X0 ) )
    | ~ spl310_16
    | ~ spl310_70 ),
    inference(superposition,[],[f4541,f4295]) ).

tff(f4541,plain,
    ( ! [X0: $int,X1: $int] :
        ( ( $sum(X0,X1) != X1 )
        | ( 0 = X0 ) )
    | ~ spl310_70 ),
    inference(avatar_component_clause,[],[f4540]) ).

tff(f15967,plain,
    ( spl310_801
    | ~ spl310_16
    | ~ spl310_69 ),
    inference(avatar_split_clause,[],[f4619,f4536,f4294,f15965]) ).

tff(f15965,plain,
    ( spl310_801
  <=> ! [X0: $int] :
        ( ( 0 != X0 )
        | ( 0 = $uminus(X0) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_801])]) ).

tff(f4536,plain,
    ( spl310_69
  <=> ! [X0: $int,X1: $int] :
        ( ( 0 = X1 )
        | ( $sum(X0,X1) != X0 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_69])]) ).

tff(f4619,plain,
    ( ! [X0: $int] :
        ( ( 0 != X0 )
        | ( 0 = $uminus(X0) ) )
    | ~ spl310_16
    | ~ spl310_69 ),
    inference(superposition,[],[f4537,f4295]) ).

tff(f4537,plain,
    ( ! [X0: $int,X1: $int] :
        ( ( $sum(X0,X1) != X0 )
        | ( 0 = X1 ) )
    | ~ spl310_69 ),
    inference(avatar_component_clause,[],[f4536]) ).

tff(f15755,plain,
    ( spl310_800
    | ~ spl310_17
    | spl310_58
    | ~ spl310_546
    | ~ spl310_771 ),
    inference(avatar_split_clause,[],[f14294,f14098,f10639,f4490,f4298,f15753]) ).

tff(f15753,plain,
    ( spl310_800
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X0),$sum(X1,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X0),X2)))) = $sum('fun_app$b'('iterate_add$'('ja$'),X0),$sum(X1,$uminus(X2))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_800])]) ).

tff(f4298,plain,
    ( spl310_17
  <=> ! [X0: 'Nat$'] : ( 'nat$'('fun_app$e'('of_nat$',X0)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_17])]) ).

tff(f4490,plain,
    ( spl310_58
  <=> $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_58])]) ).

tff(f10639,plain,
    ( spl310_546
  <=> ( 'fun_app$e'('of_nat$','ja$') = $sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$')))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_546])]) ).

tff(f14098,plain,
    ( spl310_771
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),$sum(X3,$uminus(X4))) = $sum('fun_app$b'('iterate_add$'(X1),X2),$sum(X3,$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4)))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_771])]) ).

tff(f14294,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X0),$sum(X1,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X0),X2)))) = $sum('fun_app$b'('iterate_add$'('ja$'),X0),$sum(X1,$uminus(X2))) )
    | ~ spl310_17
    | spl310_58
    | ~ spl310_546
    | ~ spl310_771 ),
    inference(forward_demodulation,[],[f14293,f4299]) ).

tff(f4299,plain,
    ( ! [X0: 'Nat$'] : ( 'nat$'('fun_app$e'('of_nat$',X0)) = X0 )
    | ~ spl310_17 ),
    inference(avatar_component_clause,[],[f4298]) ).

tff(f14293,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X0),$sum(X1,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X0),X2)))) = $sum('fun_app$b'('iterate_add$'('nat$'('fun_app$e'('of_nat$','ja$'))),X0),$sum(X1,$uminus(X2))) )
    | spl310_58
    | ~ spl310_546
    | ~ spl310_771 ),
    inference(forward_demodulation,[],[f14275,f10641]) ).

tff(f10641,plain,
    ( ( 'fun_app$e'('of_nat$','ja$') = $sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$')))) )
    | ~ spl310_546 ),
    inference(avatar_component_clause,[],[f10639]) ).

tff(f14275,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$')))))),X0),$sum(X1,$uminus(X2))) = $sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X0),$sum(X1,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X0),X2)))) )
    | spl310_58
    | ~ spl310_771 ),
    inference(resolution,[],[f14099,f4491]) ).

tff(f4491,plain,
    ( ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
    | spl310_58 ),
    inference(avatar_component_clause,[],[f4490]) ).

tff(f14099,plain,
    ( ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
        ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),$sum(X3,$uminus(X4))) = $sum('fun_app$b'('iterate_add$'(X1),X2),$sum(X3,$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4)))) ) )
    | ~ spl310_771 ),
    inference(avatar_component_clause,[],[f14098]) ).

tff(f15751,plain,
    ( spl310_799
    | ~ spl310_17
    | spl310_58
    | ~ spl310_546
    | ~ spl310_770 ),
    inference(avatar_split_clause,[],[f14196,f14094,f10639,f4490,f4298,f15749]) ).

tff(f15749,plain,
    ( spl310_799
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X1),$sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X1),X2)))) = $sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('ja$'),X1),X2))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_799])]) ).

tff(f14094,plain,
    ( spl310_770
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( ( $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))) = $sum('fun_app$b'('iterate_add$'(X0),X2),$sum(X3,$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4)))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_770])]) ).

tff(f14196,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X1),$sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X1),X2)))) = $sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('ja$'),X1),X2))) )
    | ~ spl310_17
    | spl310_58
    | ~ spl310_546
    | ~ spl310_770 ),
    inference(forward_demodulation,[],[f14195,f4299]) ).

tff(f14195,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X1),$sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X1),X2)))) = $sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('nat$'('fun_app$e'('of_nat$','ja$'))),X1),X2))) )
    | spl310_58
    | ~ spl310_546
    | ~ spl310_770 ),
    inference(forward_demodulation,[],[f14177,f10641]) ).

tff(f14177,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$')))))),X1),X2))) = $sum('fun_app$b'('iterate_add$'('to_nat$'('i$')),X1),$sum(X0,$uminus($sum('fun_app$b'('iterate_add$'('to_nat$'('x$')),X1),X2)))) )
    | spl310_58
    | ~ spl310_770 ),
    inference(resolution,[],[f14095,f4491]) ).

tff(f14095,plain,
    ( ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
        ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))) = $sum('fun_app$b'('iterate_add$'(X0),X2),$sum(X3,$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4)))) ) )
    | ~ spl310_770 ),
    inference(avatar_component_clause,[],[f14094]) ).

tff(f14985,plain,
    ( spl310_798
    | ~ spl310_22
    | ~ spl310_87 ),
    inference(avatar_split_clause,[],[f4845,f4676,f4318,f14983]) ).

tff(f14983,plain,
    ( spl310_798
  <=> ! [X0: 'N$'] : ( 'fun_app$f'('minus$'(X0),'zero$a') = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_798])]) ).

tff(f4318,plain,
    ( spl310_22
  <=> ! [X0: 'N$'] : ( 'fun_app$f'('plus$'(X0),'zero$a') = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_22])]) ).

tff(f4676,plain,
    ( spl310_87
  <=> ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X1)),X1) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_87])]) ).

tff(f4845,plain,
    ( ! [X0: 'N$'] : ( 'fun_app$f'('minus$'(X0),'zero$a') = X0 )
    | ~ spl310_22
    | ~ spl310_87 ),
    inference(superposition,[],[f4677,f4319]) ).

tff(f4319,plain,
    ( ! [X0: 'N$'] : ( 'fun_app$f'('plus$'(X0),'zero$a') = X0 )
    | ~ spl310_22 ),
    inference(avatar_component_clause,[],[f4318]) ).

tff(f4677,plain,
    ( ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X1)),X1) = X0 )
    | ~ spl310_87 ),
    inference(avatar_component_clause,[],[f4676]) ).

tff(f14981,plain,
    ( spl310_797
    | ~ spl310_21
    | ~ spl310_87 ),
    inference(avatar_split_clause,[],[f4844,f4676,f4314,f14979]) ).

tff(f14979,plain,
    ( spl310_797
  <=> ! [X0: 'N$'] : ( 'zero$a' = 'fun_app$f'('minus$'(X0),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_797])]) ).

tff(f4314,plain,
    ( spl310_21
  <=> ! [X0: 'N$'] : ( 'fun_app$f'('plus$'('zero$a'),X0) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_21])]) ).

tff(f4844,plain,
    ( ! [X0: 'N$'] : ( 'zero$a' = 'fun_app$f'('minus$'(X0),X0) )
    | ~ spl310_21
    | ~ spl310_87 ),
    inference(superposition,[],[f4677,f4315]) ).

tff(f4315,plain,
    ( ! [X0: 'N$'] : ( 'fun_app$f'('plus$'('zero$a'),X0) = X0 )
    | ~ spl310_21 ),
    inference(avatar_component_clause,[],[f4314]) ).

tff(f14977,plain,
    ( spl310_796
    | ~ spl310_2
    | ~ spl310_83 ),
    inference(avatar_split_clause,[],[f4820,f4660,f4236,f14974]) ).

tff(f14974,plain,
    ( spl310_796
  <=> ( 0 = 'fun_app$e'('of_nat$','nat$'(0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_796])]) ).

tff(f4236,plain,
    ( spl310_2
  <=> ! [X0: $int] : ~ $less(X0,X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_2])]) ).

tff(f4660,plain,
    ( spl310_83
  <=> ! [X0: $int] :
        ( ( 'fun_app$e'('of_nat$','nat$'(X0)) = X0 )
        | $less(X0,0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_83])]) ).

tff(f4820,plain,
    ( ( 0 = 'fun_app$e'('of_nat$','nat$'(0)) )
    | ~ spl310_2
    | ~ spl310_83 ),
    inference(resolution,[],[f4661,f4237]) ).

tff(f4237,plain,
    ( ! [X0: $int] : ~ $less(X0,X0)
    | ~ spl310_2 ),
    inference(avatar_component_clause,[],[f4236]) ).

tff(f4661,plain,
    ( ! [X0: $int] :
        ( $less(X0,0)
        | ( 'fun_app$e'('of_nat$','nat$'(X0)) = X0 ) )
    | ~ spl310_83 ),
    inference(avatar_component_clause,[],[f4660]) ).

tff(f14972,plain,
    ( spl310_795
    | ~ spl310_14
    | ~ spl310_121 ),
    inference(avatar_split_clause,[],[f14568,f4888,f4286,f14969]) ).

tff(f4286,plain,
    ( spl310_14
  <=> ! [X0: 'Nat$'] : ~ $less('fun_app$e'('of_nat$',X0),0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_14])]) ).

tff(f4888,plain,
    ( spl310_121
  <=> ! [X0: $int] :
        ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),X0)
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_121])]) ).

tff(f14568,plain,
    ( $less(0,'fun_app$e'('of_nat$','to_nat$'('i$')))
    | ~ spl310_14
    | ~ spl310_121 ),
    inference(resolution,[],[f4889,f4287]) ).

tff(f4287,plain,
    ( ! [X0: 'Nat$'] : ~ $less('fun_app$e'('of_nat$',X0),0)
    | ~ spl310_14 ),
    inference(avatar_component_clause,[],[f4286]) ).

tff(f4889,plain,
    ( ! [X0: $int] :
        ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),X0)
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    | ~ spl310_121 ),
    inference(avatar_component_clause,[],[f4888]) ).

tff(f14967,plain,
    ( spl310_794
    | ~ spl310_5
    | ~ spl310_61 ),
    inference(avatar_split_clause,[],[f4580,f4503,f4249,f14965]) ).

tff(f14965,plain,
    ( spl310_794
  <=> ! [X0: $int] : ~ $less(X0,sK95($sum(X0,1))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_794])]) ).

tff(f4580,plain,
    ( ! [X0: $int] : ~ $less(X0,sK95($sum(X0,1)))
    | ~ spl310_5
    | ~ spl310_61 ),
    inference(resolution,[],[f4504,f4250]) ).

tff(f14960,plain,
    ( ~ spl310_793
    | ~ spl310_9
    | ~ spl310_125 ),
    inference(avatar_split_clause,[],[f5062,f4904,f4265,f14957]) ).

tff(f14957,plain,
    ( spl310_793
  <=> 'fun_app$d'('fun_app$g'('less$','x$'),'i$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_793])]) ).

tff(f4265,plain,
    ( spl310_9
  <=> 'fun_app$d'('fun_app$g'('less_eq$','i$'),'x$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_9])]) ).

tff(f5062,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less$','x$'),'i$')
    | ~ spl310_9
    | ~ spl310_125 ),
    inference(resolution,[],[f4905,f4267]) ).

tff(f4267,plain,
    ( 'fun_app$d'('fun_app$g'('less_eq$','i$'),'x$')
    | ~ spl310_9 ),
    inference(avatar_component_clause,[],[f4265]) ).

tff(f14767,plain,
    ( spl310_792
    | spl310_58
    | ~ spl310_447 ),
    inference(avatar_split_clause,[],[f9356,f9199,f4490,f14764]) ).

tff(f14764,plain,
    ( spl310_792
  <=> ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$',sK113('to_nat$'('i$'),'to_nat$'('x$')))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_792])]) ).

tff(f9199,plain,
    ( spl310_447
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK113(X0,X1))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_447])]) ).

tff(f9356,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$',sK113('to_nat$'('i$'),'to_nat$'('x$')))) )
    | spl310_58
    | ~ spl310_447 ),
    inference(resolution,[],[f9200,f4491]) ).

tff(f9200,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK113(X0,X1))) ) )
    | ~ spl310_447 ),
    inference(avatar_component_clause,[],[f9199]) ).

tff(f14762,plain,
    ( spl310_791
    | spl310_58
    | ~ spl310_446 ),
    inference(avatar_split_clause,[],[f9306,f9195,f4490,f14759]) ).

tff(f14759,plain,
    ( spl310_791
  <=> ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$',sK98('to_nat$'('i$'),'to_nat$'('x$')))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_791])]) ).

tff(f9195,plain,
    ( spl310_446
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK98(X0,X1))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_446])]) ).

tff(f9306,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$',sK98('to_nat$'('i$'),'to_nat$'('x$')))) )
    | spl310_58
    | ~ spl310_446 ),
    inference(resolution,[],[f9196,f4491]) ).

tff(f9196,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK98(X0,X1))) ) )
    | ~ spl310_446 ),
    inference(avatar_component_clause,[],[f9195]) ).

tff(f14755,plain,
    ( ~ spl310_790
    | spl310_59
    | ~ spl310_546 ),
    inference(avatar_split_clause,[],[f10643,f10639,f4494,f14752]) ).

tff(f14752,plain,
    ( spl310_790
  <=> ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','ja$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_790])]) ).

tff(f10643,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','ja$')) )
    | spl310_59
    | ~ spl310_546 ),
    inference(superposition,[],[f4496,f10641]) ).

tff(f14749,plain,
    ( spl310_789
    | ~ spl310_16
    | ~ spl310_236 ),
    inference(avatar_split_clause,[],[f6476,f6017,f4294,f14747]) ).

tff(f14747,plain,
    ( spl310_789
  <=> ! [X0: 'Nat$'] : sP44(X0,X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_789])]) ).

tff(f6017,plain,
    ( spl310_236
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP44(X0,X1)
        | ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_236])]) ).

tff(f6476,plain,
    ( ! [X0: 'Nat$'] : sP44(X0,X0)
    | ~ spl310_16
    | ~ spl310_236 ),
    inference(trivial_inequality_removal,[],[f6473]) ).

tff(f6473,plain,
    ( ! [X0: 'Nat$'] :
        ( ( 0 != 0 )
        | sP44(X0,X0) )
    | ~ spl310_16
    | ~ spl310_236 ),
    inference(superposition,[],[f6018,f4295]) ).

tff(f6018,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | sP44(X0,X1) )
    | ~ spl310_236 ),
    inference(avatar_component_clause,[],[f6017]) ).

tff(f14487,plain,
    spl310_788,
    inference(avatar_split_clause,[],[f4178,f14485]) ).

tff(f14485,plain,
    ( spl310_788
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ( $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP29(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_788])]) ).

tff(f4178,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4177,f864]) ).

tff(f864,plain,
    ! [X0: $int,X1: $int] : ( $sum(X0,X1) = $sum(X1,X0) ),
    introduced(theory_axiom_135,[]) ).

tff(f4177,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4176,f864]) ).

tff(f4176,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),1)))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4175,f865]) ).

tff(f865,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$sum(X1,X2)) = $sum($sum(X0,X1),X2) ),
    introduced(theory_axiom_136,[]) ).

tff(f4175,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3073,f864]) ).

tff(f3073,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1979]) ).

tff(f1979,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP29(X0,X1,X2) ),
    inference(nnf_transformation,[],[f1624]) ).

tff(f1624,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP29(X0,X1,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP29])]) ).

tff(f14473,plain,
    spl310_787,
    inference(avatar_split_clause,[],[f4147,f14471]) ).

tff(f14471,plain,
    ( spl310_787
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1)))),'fun_app$e'('of_nat$',X0))
        | $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
        | sP25(X0,X1,X2)
        | ~ sP27(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_787])]) ).

tff(f4147,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1)))),'fun_app$e'('of_nat$',X0))
      | $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4146,f864]) ).

tff(f4146,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | ~ $less($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),'fun_app$e'('of_nat$',X0))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4145,f864]) ).

tff(f4145,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),'fun_app$e'('of_nat$',X0))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4144,f2892]) ).

tff(f2892,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$sum(X1,X2)) = $sum(X1,$sum(X0,X2)) ),
    inference(cnf_transformation,[],[f47]) ).

tff(f47,axiom,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum(X0,$sum(X1,X2)) = $sum(X1,$sum(X0,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom45) ).

tff(f4144,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum(1,$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),'fun_app$e'('of_nat$',X0))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3056,f865]) ).

tff(f3056,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),1),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),'fun_app$e'('of_nat$',X0))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1972]) ).

tff(f1972,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),1),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) )
          | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) )
      | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(rectify,[],[f1971]) ).

tff(f1971,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) )
          | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
      | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP25(X2,X0,X1)
      | ~ sP27(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1621]) ).

tff(f1621,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) )
          | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
      | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP25(X2,X0,X1)
      | ~ sP27(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP27])]) ).

tff(f14460,plain,
    spl310_786,
    inference(avatar_split_clause,[],[f4165,f14458]) ).

tff(f14458,plain,
    ( spl310_786
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ( $sum(1,$uminus('fun_app$e'('of_nat$',X0))) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP26(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_786])]) ).

tff(f4165,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ( $sum(1,$uminus('fun_app$e'('of_nat$',X0))) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4164,f864]) ).

tff(f4164,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(1,$uminus('fun_app$e'('of_nat$',X0))) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4163,f2892]) ).

tff(f4163,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(1,$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$sum(1,$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3921,f865]) ).

tff(f3921,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),1),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) = $sum(1,$uminus('fun_app$e'('of_nat$',X0))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(evaluation,[],[f3058]) ).

tff(f3058,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),1),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1974]) ).

tff(f1974,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),1),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) = $sum($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),$uminus('fun_app$e'('of_nat$',X0))) )
              | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
            & ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),$uminus('fun_app$e'('of_nat$',X0))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),1),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X0))) )
              | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
            & ( ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X0))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP26(X0,X1,X2) ),
    inference(rectify,[],[f1973]) ).

tff(f1973,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP26(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1620]) ).

tff(f1620,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP26(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP26])]) ).

tff(f14432,plain,
    ( spl310_785
    | ~ spl310_190
    | ~ spl310_773 ),
    inference(avatar_split_clause,[],[f14372,f14302,f5387,f14430]) ).

tff(f14430,plain,
    ( spl310_785
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] : ~ sP16(X0,X1,X2) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_785])]) ).

tff(f5387,plain,
    ( spl310_190
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
        | ~ sP16(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_190])]) ).

tff(f14372,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] : ~ sP16(X0,X1,X2)
    | ~ spl310_190
    | ~ spl310_773 ),
    inference(resolution,[],[f14303,f5388]) ).

tff(f5388,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
        | ~ sP16(X0,X1,X2) )
    | ~ spl310_190 ),
    inference(avatar_component_clause,[],[f5387]) ).

tff(f14419,plain,
    spl310_784,
    inference(avatar_split_clause,[],[f4184,f14417]) ).

tff(f14417,plain,
    ( spl310_784
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ( $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) = $sum(-1,'fun_app$e'('of_nat$',X2)) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP29(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_784])]) ).

tff(f4184,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) = $sum(-1,'fun_app$e'('of_nat$',X2)) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4183,f864]) ).

tff(f4183,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) = $sum(-1,'fun_app$e'('of_nat$',X2)) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4182,f864]) ).

tff(f4182,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),1)))) = $sum(-1,'fun_app$e'('of_nat$',X2)) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4181,f865]) ).

tff(f4181,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum(-1,'fun_app$e'('of_nat$',X2)) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3917,f864]) ).

tff(f3917,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),-1) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(evaluation,[],[f3071]) ).

tff(f3071,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1979]) ).

tff(f14415,plain,
    spl310_783,
    inference(avatar_split_clause,[],[f4174,f14413]) ).

tff(f14413,plain,
    ( spl310_783
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$sum(1,'fun_app$e'('of_nat$',X0)))
        | ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus($sum(1,'fun_app$e'('of_nat$',X0))))) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | sP28(X2,X1,X0)
        | ~ sP30(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_783])]) ).

tff(f4174,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$sum(1,'fun_app$e'('of_nat$',X0)))
      | ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus($sum(1,'fun_app$e'('of_nat$',X0))))) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP28(X2,X1,X0)
      | ~ sP30(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4173,f864]) ).

tff(f4173,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus($sum(1,'fun_app$e'('of_nat$',X0))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X0),1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP28(X2,X1,X0)
      | ~ sP30(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4172,f864]) ).

tff(f4172,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus($sum('fun_app$e'('of_nat$',X0),1)))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X0),1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP28(X2,X1,X0)
      | ~ sP30(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3068,f865]) ).

tff(f3068,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X0),1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP28(X2,X1,X0)
      | ~ sP30(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1978]) ).

tff(f1978,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
          | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X0),1)) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X0),1)) ) )
      | ( ~ $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),1))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | sP28(X2,X1,X0)
      | ~ sP30(X0,X1,X2) ),
    inference(rectify,[],[f1977]) ).

tff(f1977,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) )
          | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
      | ( ~ $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP28(X2,X0,X1)
      | ~ sP30(X1,X0,X2) ),
    inference(nnf_transformation,[],[f1625]) ).

tff(f1625,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) )
          | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
      | ( ~ $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP28(X2,X0,X1)
      | ~ sP30(X1,X0,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP30])]) ).

tff(f14411,plain,
    spl310_782,
    inference(avatar_split_clause,[],[f4150,f14409]) ).

tff(f14409,plain,
    ( spl310_782
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | sP25(X0,X1,X2)
        | ~ sP27(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_782])]) ).

tff(f4150,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4149,f864]) ).

tff(f4149,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4148,f2892]) ).

tff(f4148,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum(1,$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3055,f865]) ).

tff(f3055,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),1),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) )
      | $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | sP25(X0,X1,X2)
      | ~ sP27(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1972]) ).

tff(f14398,plain,
    spl310_781,
    inference(avatar_split_clause,[],[f4198,f14396]) ).

tff(f14396,plain,
    ( spl310_781
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP41(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_781])]) ).

tff(f4198,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4197,f867]) ).

tff(f867,plain,
    ! [X0: $int,X1: $int] : ( $uminus($sum(X0,X1)) = $sum($uminus(X1),$uminus(X0)) ),
    introduced(theory_axiom_139,[]) ).

tff(f4197,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4196,f865]) ).

tff(f4196,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus('fun_app$e'('of_nat$',X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4195,f2892]) ).

tff(f4195,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3202,f4057]) ).

tff(f4057,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$uminus($sum(X1,$uminus(X2)))) = $sum(X0,$sum(X2,$uminus(X1))) ),
    inference(forward_demodulation,[],[f2890,f865]) ).

tff(f2890,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum($sum(X0,X2),$uminus(X1)) = $sum(X0,$uminus($sum(X1,$uminus(X2)))) ),
    inference(cnf_transformation,[],[f880]) ).

tff(f880,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum($sum(X0,X2),$uminus(X1)) = $sum(X0,$uminus($sum(X1,$uminus(X2)))) ),
    inference(theory_normalization,[],[f726]) ).

tff(f726,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum($sum(X0,X2),$uminus(X1)) = $sum(X0,$uminus($difference(X1,X2))) ),
    inference(theory_normalization,[],[f87]) ).

tff(f87,axiom,
    ! [X0: $int,X1: $int,X2: $int] : ( $difference($sum(X0,X2),X1) = $difference(X0,$difference(X1,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom85) ).

tff(f3202,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2017]) ).

tff(f2017,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(rectify,[],[f2016]) ).

tff(f2016,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1640]) ).

tff(f1640,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP41])]) ).

tff(f14385,plain,
    spl310_780,
    inference(avatar_split_clause,[],[f4114,f14383]) ).

tff(f14383,plain,
    ( spl310_780
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus($sum(1,'fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ sP18(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_780])]) ).

tff(f4114,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus($sum(1,'fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4113,f864]) ).

tff(f4113,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus($sum('fun_app$e'('of_nat$',X0),1)) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4112,f867]) ).

tff(f4112,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus($sum('fun_app$e'('of_nat$',X0),1)) = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3942,f865]) ).

tff(f3942,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus($sum('fun_app$e'('of_nat$',X0),1)) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(evaluation,[],[f3941]) ).

tff(f3941,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2977]) ).

tff(f2977,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP18(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1957]) ).

tff(f1957,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X0))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP18(X0,X1,X2) ),
    inference(rectify,[],[f1956]) ).

tff(f1956,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP18(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1610]) ).

tff(f1610,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP18(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP18])]) ).

tff(f14381,plain,
    spl310_779,
    inference(avatar_split_clause,[],[f3912,f14379]) ).

tff(f14379,plain,
    ( spl310_779
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) )
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP41(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_779])]) ).

tff(f3912,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(evaluation,[],[f3201]) ).

tff(f3201,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP41(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2017]) ).

tff(f14357,plain,
    spl310_778,
    inference(avatar_split_clause,[],[f4180,f14355]) ).

tff(f14355,plain,
    ( spl310_778
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP29(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_778])]) ).

tff(f4180,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4179,f864]) ).

tff(f4179,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3072,f864]) ).

tff(f3072,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1979]) ).

tff(f14341,plain,
    ( spl310_777
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_776 ),
    inference(avatar_split_clause,[],[f14337,f14334,f5938,f5200,f14339]) ).

tff(f14339,plain,
    ( spl310_777
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP19(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_777])]) ).

tff(f5938,plain,
    ( spl310_217
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum(X1,$uminus(X2)) = $sum($sum(X0,X1),$uminus($sum(X0,X2))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_217])]) ).

tff(f14334,plain,
    ( spl310_776
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP19(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_776])]) ).

tff(f14337,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP19(X0,X1,X2) )
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_776 ),
    inference(forward_demodulation,[],[f14335,f6116]) ).

tff(f6116,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum(X1,$uminus(X2)) = $sum(X0,$sum(X1,$uminus($sum(X0,X2)))) )
    | ~ spl310_146
    | ~ spl310_217 ),
    inference(superposition,[],[f5939,f5201]) ).

tff(f5939,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum(X1,$uminus(X2)) = $sum($sum(X0,X1),$uminus($sum(X0,X2))) )
    | ~ spl310_217 ),
    inference(avatar_component_clause,[],[f5938]) ).

tff(f14335,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP19(X0,X1,X2) )
    | ~ spl310_776 ),
    inference(avatar_component_clause,[],[f14334]) ).

tff(f14336,plain,
    spl310_776,
    inference(avatar_split_clause,[],[f4105,f14334]) ).

tff(f4105,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4104,f2892]) ).

tff(f4104,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4103,f865]) ).

tff(f4103,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4102,f867]) ).

tff(f4102,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4101,f2892]) ).

tff(f4101,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum(1,$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4100,f865]) ).

tff(f4100,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),1),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2972,f865]) ).

tff(f2972,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP19(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1955]) ).

tff(f1955,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP19(X0,X1,X2) ),
    inference(rectify,[],[f1954]) ).

tff(f1954,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP19(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1611]) ).

tff(f1611,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP19(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP19])]) ).

tff(f14312,plain,
    spl310_775,
    inference(avatar_split_clause,[],[f4162,f14310]) ).

tff(f14310,plain,
    ( spl310_775
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))))) )
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP26(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_775])]) ).

tff(f4162,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4161,f864]) ).

tff(f4161,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4160,f867]) ).

tff(f4160,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4159,f865]) ).

tff(f4159,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4158,f2892]) ).

tff(f4158,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$sum(1,$uminus('fun_app$e'('of_nat$',X0)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3059,f865]) ).

tff(f3059,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),1),$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1974]) ).

tff(f14308,plain,
    spl310_774,
    inference(avatar_split_clause,[],[f4140,f14306]) ).

tff(f14306,plain,
    ( spl310_774
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
        | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | sP22(X2,X1,X0)
        | ~ sP24(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_774])]) ).

tff(f4140,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP22(X2,X1,X0)
      | ~ sP24(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3042,f865]) ).

tff(f3042,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP22(X2,X1,X0)
      | ~ sP24(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1967]) ).

tff(f1967,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ) )
      | ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | sP22(X2,X1,X0)
      | ~ sP24(X0,X1,X2) ),
    inference(rectify,[],[f1966]) ).

tff(f1966,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
      | ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP22(X2,X0,X1)
      | ~ sP24(X1,X0,X2) ),
    inference(nnf_transformation,[],[f1617]) ).

tff(f1617,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
        & ( ( 0 = 0 )
          | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
      | ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP22(X2,X0,X1)
      | ~ sP24(X1,X0,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP24])]) ).

tff(f14304,plain,
    ( spl310_773
    | ~ spl310_268
    | ~ spl310_713 ),
    inference(avatar_split_clause,[],[f14064,f12982,f6601,f14302]) ).

tff(f6601,plain,
    ( spl310_268
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_268])]) ).

tff(f12982,plain,
    ( spl310_713
  <=> ! [X0: $int] :
        ( $less(0,$sum(1,X0))
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_713])]) ).

tff(f14064,plain,
    ( ! [X0: 'Nat$'] : $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
    | ~ spl310_268
    | ~ spl310_713 ),
    inference(duplicate_literal_removal,[],[f13952]) ).

tff(f13952,plain,
    ( ! [X0: 'Nat$'] :
        ( $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
        | $less(0,$sum(1,'fun_app$e'('of_nat$',X0))) )
    | ~ spl310_268
    | ~ spl310_713 ),
    inference(resolution,[],[f12983,f6602]) ).

tff(f6602,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | $less(0,$sum(1,'fun_app$e'('of_nat$',X0))) )
    | ~ spl310_268 ),
    inference(avatar_component_clause,[],[f6601]) ).

tff(f12983,plain,
    ( ! [X0: $int] :
        ( $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$')))
        | $less(0,$sum(1,X0)) )
    | ~ spl310_713 ),
    inference(avatar_component_clause,[],[f12982]) ).

tff(f14300,plain,
    spl310_772,
    inference(avatar_split_clause,[],[f4073,f14298]) ).

tff(f14298,plain,
    ( spl310_772
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X1)) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ sP10(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_772])]) ).

tff(f4073,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X1)) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP10(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4072,f867]) ).

tff(f4072,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X1)) = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP10(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3955,f865]) ).

tff(f3955,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X1)) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP10(X0,X1,X2) ),
    inference(evaluation,[],[f3954]) ).

tff(f3954,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP10(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2941]) ).

tff(f2941,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP10(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1942]) ).

tff(f1942,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X0))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP10(X0,X1,X2) ),
    inference(rectify,[],[f1941]) ).

tff(f1941,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( $less(0,'fun_app$e'('of_nat$',X1))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ sP10(X1,X2,X0) ),
    inference(nnf_transformation,[],[f1601]) ).

tff(f1601,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( $less(0,'fun_app$e'('of_nat$',X1))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ sP10(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP10])]) ).

tff(f14100,plain,
    spl310_771,
    inference(avatar_split_clause,[],[f4227,f14098]) ).

tff(f4227,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),$sum(X3,$uminus(X4))) = $sum('fun_app$b'('iterate_add$'(X1),X2),$sum(X3,$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4)))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f4226,f865]) ).

tff(f4226,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),$sum(X3,$uminus(X4))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3892,f865]) ).

tff(f3892,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),$uminus(X4)) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(duplicate_literal_removal,[],[f3651]) ).

tff(f3651,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),$uminus(X4)) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1574]) ).

tff(f1574,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),$uminus(X4)) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),$uminus(X4)) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f889]) ).

tff(f889,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),$uminus(X4)) ) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),$uminus(X4)) ) ) ) ),
    inference(theory_normalization,[],[f857]) ).

tff(f857,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X3),$uminus(X4)) ) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X0),X2),X4))) = $sum($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),$uminus(X4)) ) ) ) ),
    inference(theory_normalization,[],[f248]) ).

tff(f248,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $difference($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4)) = $difference($sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X3),X4) ) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $difference($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4)) = $difference($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom246) ).

tff(f14096,plain,
    spl310_770,
    inference(avatar_split_clause,[],[f4225,f14094]) ).

tff(f4225,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))) = $sum('fun_app$b'('iterate_add$'(X0),X2),$sum(X3,$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4)))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3893,f865]) ).

tff(f3893,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(duplicate_literal_removal,[],[f3649]) ).

tff(f3649,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1573]) ).

tff(f1573,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f888]) ).

tff(f888,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))) ) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))) ) ) ) ),
    inference(theory_normalization,[],[f856]) ).

tff(f856,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X4))) ) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $sum($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$uminus($sum('fun_app$b'('iterate_add$'(X1),X2),X4))) = $sum(X3,$uminus($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))) ) ) ) ),
    inference(theory_normalization,[],[f249]) ).

tff(f249,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $difference($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4)) = $difference(X3,$sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X4)) ) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
         => ( $difference($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4)) = $difference(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4)) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom247) ).

tff(f14080,plain,
    ( spl310_769
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_768 ),
    inference(avatar_split_clause,[],[f14076,f14073,f5938,f5200,f14078]) ).

tff(f14078,plain,
    ( spl310_769
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_769])]) ).

tff(f14073,plain,
    ( spl310_768
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_768])]) ).

tff(f14076,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) )
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_768 ),
    inference(forward_demodulation,[],[f14074,f6116]) ).

tff(f14074,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) )
    | ~ spl310_768 ),
    inference(avatar_component_clause,[],[f14073]) ).

tff(f14075,plain,
    spl310_768,
    inference(avatar_split_clause,[],[f4099,f14073]) ).

tff(f4099,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4098,f2892]) ).

tff(f4098,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4097,f865]) ).

tff(f4097,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4096,f867]) ).

tff(f4096,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4095,f2892]) ).

tff(f4095,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum(1,$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4094,f865]) ).

tff(f4094,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),1),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2973,f865]) ).

tff(f2973,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1955]) ).

tff(f14069,plain,
    spl310_767,
    inference(avatar_split_clause,[],[f3992,f14067]) ).

tff(f14067,plain,
    ( spl310_767
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) )
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_767])]) ).

tff(f3992,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3991,f864]) ).

tff(f3991,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3990,f864]) ).

tff(f3990,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus($sum(1,'fun_app$e'('of_nat$',X1))))) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1)) ),
    inference(forward_demodulation,[],[f3989,f864]) ).

tff(f3989,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus($sum('fun_app$e'('of_nat$',X1),1)))) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1)) ),
    inference(forward_demodulation,[],[f2465,f865]) ).

tff(f2465,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),1),$uminus($sum('fun_app$e'('of_nat$',X1),1))) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1)) ),
    inference(cnf_transformation,[],[f992]) ).

tff(f992,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),1),$uminus($sum('fun_app$e'('of_nat$',X1),1))) )
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),1),$uminus($sum('fun_app$e'('of_nat$',X1),1))) )
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1)) )
      & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) )
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ( 0 = 0 )
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | ~ $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1)) ) ),
    inference(ennf_transformation,[],[f650]) ).

tff(f650,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1))
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),1),$uminus($sum('fun_app$e'('of_nat$',X1),1))) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),1),$uminus($sum('fun_app$e'('of_nat$',X1),1))) ) ) ) )
      & ( $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1))
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( 0 = 0 ) ) ) ) ),
    inference(theory_normalization,[],[f261]) ).

tff(f261,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1))
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( $difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) = $difference($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1)) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( 0 = $difference($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1)) ) ) ) )
      & ( $less($sum('fun_app$e'('of_nat$',X0),1),$sum('fun_app$e'('of_nat$',X1),1))
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( 0 = $difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( 0 = 0 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom259) ).

tff(f13910,plain,
    spl310_766,
    inference(avatar_split_clause,[],[f4202,f13908]) ).

tff(f13908,plain,
    ( spl310_766
  <=> ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK130(X2)))))
        | 'fun_app$j'('fun_app$q'(X2,X0),X1)
        | 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK132(X2))
        | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_766])]) ).

tff(f4202,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK130(X2)))))
      | 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK132(X2))
      | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3305,f864]) ).

tff(f3305,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK130(X2)),1)))
      | 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK132(X2))
      | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2071]) ).

tff(f2071,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK130(X2)),1)))
      | ( ~ 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK133(X2))
        & 'fun_app$j'('fun_app$q'(X2,sK132(X2)),sK133(X2))
        & 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK132(X2)) )
      | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK130,sK131,sK132,sK133,sK134])],[f1328,f2070,f2069,f2068]) ).

tff(f2068,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
     => ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK130(X2)),1))) ),
    introduced(choice_axiom,[]) ).

tff(f2069,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X4: 'Nat$',X5: 'Nat$',X6: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X2,X4),X6)
          & 'fun_app$j'('fun_app$q'(X2,X5),X6)
          & 'fun_app$j'('fun_app$q'(X2,X4),X5) )
     => ( ~ 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK133(X2))
        & 'fun_app$j'('fun_app$q'(X2,sK132(X2)),sK133(X2))
        & 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK132(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2070,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X7: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X7),X7)
     => ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2)) ),
    introduced(choice_axiom,[]) ).

tff(f1328,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ? [X3: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
      | ? [X4: 'Nat$',X5: 'Nat$',X6: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X2,X4),X6)
          & 'fun_app$j'('fun_app$q'(X2,X5),X6)
          & 'fun_app$j'('fun_app$q'(X2,X4),X5) )
      | ? [X7: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X7),X7)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1327]) ).

tff(f1327,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ? [X3: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
      | ? [X4: 'Nat$',X5: 'Nat$',X6: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X2,X4),X6)
          & 'fun_app$j'('fun_app$q'(X2,X5),X6)
          & 'fun_app$j'('fun_app$q'(X2,X4),X5) )
      | ? [X7: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X7),X7)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f965]) ).

tff(f965,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( ( ! [X3: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
        & ! [X4: 'Nat$',X5: 'Nat$',X6: 'Nat$'] :
            ( ( 'fun_app$j'('fun_app$q'(X2,X5),X6)
              & 'fun_app$j'('fun_app$q'(X2,X4),X5) )
           => 'fun_app$j'('fun_app$q'(X2,X4),X6) )
        & ! [X7: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X7),X7)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => 'fun_app$j'('fun_app$q'(X2,X0),X1) ),
    inference(rectify,[],[f786]) ).

tff(f786,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( ( ! [X3: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
        & ! [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
            ( ( 'fun_app$j'('fun_app$q'(X2,X4),X5)
              & 'fun_app$j'('fun_app$q'(X2,X3),X4) )
           => 'fun_app$j'('fun_app$q'(X2,X3),X5) )
        & ! [X3: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X3),X3)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => 'fun_app$j'('fun_app$q'(X2,X0),X1) ),
    inference(theory_normalization,[],[f515]) ).

tff(f515,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( ( ! [X3: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
        & ! [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
            ( ( 'fun_app$j'('fun_app$q'(X2,X4),X5)
              & 'fun_app$j'('fun_app$q'(X2,X3),X4) )
           => 'fun_app$j'('fun_app$q'(X2,X3),X5) )
        & ! [X3: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X3),X3)
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'('fun_app$q'(X2,X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom513) ).

tff(f13906,plain,
    spl310_765,
    inference(avatar_split_clause,[],[f4201,f13904]) ).

tff(f13904,plain,
    ( spl310_765
  <=> ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK130(X2)))))
        | 'fun_app$j'('fun_app$q'(X2,X0),X1)
        | 'fun_app$j'('fun_app$q'(X2,sK132(X2)),sK133(X2))
        | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_765])]) ).

tff(f4201,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK130(X2)))))
      | 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | 'fun_app$j'('fun_app$q'(X2,sK132(X2)),sK133(X2))
      | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3306,f864]) ).

tff(f3306,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK130(X2)),1)))
      | 'fun_app$j'('fun_app$q'(X2,sK132(X2)),sK133(X2))
      | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2071]) ).

tff(f13902,plain,
    spl310_764,
    inference(avatar_split_clause,[],[f4200,f13900]) ).

tff(f13900,plain,
    ( spl310_764
  <=> ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK130(X2)))))
        | 'fun_app$j'('fun_app$q'(X2,X0),X1)
        | ~ 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK133(X2))
        | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_764])]) ).

tff(f4200,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK130(X2)))))
      | 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ~ 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK133(X2))
      | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3307,f864]) ).

tff(f3307,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ~ 'fun_app$j'('fun_app$q'(X2,sK130(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK130(X2)),1)))
      | ~ 'fun_app$j'('fun_app$q'(X2,sK131(X2)),sK133(X2))
      | ~ 'fun_app$j'('fun_app$q'(X2,sK134(X2)),sK134(X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2071]) ).

tff(f13898,plain,
    spl310_763,
    inference(avatar_split_clause,[],[f4143,f13896]) ).

tff(f13896,plain,
    ( spl310_763
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))) )
        | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP23(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_763])]) ).

tff(f4143,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP23(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3924,f865]) ).

tff(f3924,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X2) = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP23(X0,X1,X2) ),
    inference(evaluation,[],[f3045]) ).

tff(f3045,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP23(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1968]) ).

tff(f1968,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP23(X0,X1,X2) ),
    inference(nnf_transformation,[],[f1616]) ).

tff(f1616,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP23(X0,X1,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP23])]) ).

tff(f13866,plain,
    spl310_762,
    inference(avatar_split_clause,[],[f4122,f13864]) ).

tff(f13864,plain,
    ( spl310_762
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ sP15(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_762])]) ).

tff(f4122,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP15(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4121,f867]) ).

tff(f4121,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP15(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3935,f865]) ).

tff(f3935,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP15(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2987]) ).

tff(f2987,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP15(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1963]) ).

tff(f1963,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X0))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP15(X0,X1,X2) ),
    inference(rectify,[],[f1962]) ).

tff(f1962,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP15(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1607]) ).

tff(f1607,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP15(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP15])]) ).

tff(f13856,plain,
    ( spl310_761
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_760 ),
    inference(avatar_split_clause,[],[f13852,f13849,f5938,f5200,f13854]) ).

tff(f13854,plain,
    ( spl310_761
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP20(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_761])]) ).

tff(f13849,plain,
    ( spl310_760
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP20(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_760])]) ).

tff(f13852,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP20(X0,X1,X2) )
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_760 ),
    inference(forward_demodulation,[],[f13850,f6116]) ).

tff(f13850,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP20(X0,X1,X2) )
    | ~ spl310_760 ),
    inference(avatar_component_clause,[],[f13849]) ).

tff(f13851,plain,
    spl310_760,
    inference(avatar_split_clause,[],[f4093,f13849]) ).

tff(f4093,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4092,f2892]) ).

tff(f4092,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4091,f865]) ).

tff(f4091,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4090,f867]) ).

tff(f4090,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum(1,$sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4089,f2892]) ).

tff(f4089,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum('fun_app$e'('of_nat$',X2),$sum(1,$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4088,f865]) ).

tff(f4088,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum($sum('fun_app$e'('of_nat$',X2),1),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3944,f865]) ).

tff(f3944,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(evaluation,[],[f3943]) ).

tff(f3943,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP20(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2966]) ).

tff(f2966,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP20(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1953]) ).

tff(f1953,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X0))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP20(X0,X1,X2) ),
    inference(rectify,[],[f1952]) ).

tff(f1952,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP20(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1612]) ).

tff(f1612,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP20(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP20])]) ).

tff(f13847,plain,
    spl310_759,
    inference(avatar_split_clause,[],[f4075,f13845]) ).

tff(f13845,plain,
    ( spl310_759
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ sP7(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_759])]) ).

tff(f4075,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP7(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4074,f867]) ).

tff(f4074,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP7(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3948,f865]) ).

tff(f3948,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ sP7(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2951]) ).

tff(f2951,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP7(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1948]) ).

tff(f1948,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X0))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP7(X0,X1,X2) ),
    inference(rectify,[],[f1947]) ).

tff(f1947,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( $less(0,'fun_app$e'('of_nat$',X1))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ sP7(X1,X2,X0) ),
    inference(nnf_transformation,[],[f1598]) ).

tff(f1598,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( $less(0,'fun_app$e'('of_nat$',X1))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ sP7(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP7])]) ).

tff(f13843,plain,
    spl310_758,
    inference(avatar_split_clause,[],[f4069,f13841]) ).

tff(f13841,plain,
    ( spl310_758
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP11(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_758])]) ).

tff(f4069,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP11(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4068,f867]) ).

tff(f4068,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP11(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2936,f865]) ).

tff(f2936,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP11(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1940]) ).

tff(f1940,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ~ sP11(X0,X1,X2) ),
    inference(rectify,[],[f1939]) ).

tff(f1939,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ~ sP11(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1602]) ).

tff(f1602,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ~ sP11(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP11])]) ).

tff(f13829,plain,
    spl310_757,
    inference(avatar_split_clause,[],[f4142,f13827]) ).

tff(f13827,plain,
    ( spl310_757
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP23(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_757])]) ).

tff(f4142,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP23(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3046,f4057]) ).

tff(f3046,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP23(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1968]) ).

tff(f13825,plain,
    spl310_756,
    inference(avatar_split_clause,[],[f4062,f13823]) ).

tff(f13823,plain,
    ( spl310_756
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP5(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_756])]) ).

tff(f4062,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP5(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4061,f867]) ).

tff(f4061,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP5(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2916,f865]) ).

tff(f2916,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP5(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1932]) ).

tff(f1932,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
              | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
              | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) )
              | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
            & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
              | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP5(X0,X1,X2) ),
    inference(rectify,[],[f1931]) ).

tff(f1931,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
              | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP5(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1595]) ).

tff(f1595,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
              | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
              | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP5(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP5])]) ).

tff(f13812,plain,
    ( spl310_755
    | ~ spl310_16
    | ~ spl310_218
    | ~ spl310_754 ),
    inference(avatar_split_clause,[],[f13808,f13805,f5942,f4294,f13810]) ).

tff(f13810,plain,
    ( spl310_755
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP40(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_755])]) ).

tff(f5942,plain,
    ( spl310_218
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus($sum(X2,X1))) = $sum(X0,$uminus(X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_218])]) ).

tff(f13805,plain,
    ( spl310_754
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP40(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_754])]) ).

tff(f13808,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP40(X0,X1,X2) )
    | ~ spl310_16
    | ~ spl310_218
    | ~ spl310_754 ),
    inference(forward_demodulation,[],[f13806,f6259]) ).

tff(f6259,plain,
    ( ! [X0: $int,X1: $int] : ( $sum(X0,$uminus(X1)) = $uminus($sum(X1,$uminus(X0))) )
    | ~ spl310_16
    | ~ spl310_218 ),
    inference(evaluation,[],[f6189]) ).

tff(f6189,plain,
    ( ! [X0: $int,X1: $int] : ( $sum(X0,$uminus(X1)) = $sum(0,$uminus($sum(X1,$uminus(X0)))) )
    | ~ spl310_16
    | ~ spl310_218 ),
    inference(superposition,[],[f5943,f4295]) ).

tff(f5943,plain,
    ( ! [X2: $int,X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus($sum(X2,X1))) = $sum(X0,$uminus(X2)) )
    | ~ spl310_218 ),
    inference(avatar_component_clause,[],[f5942]) ).

tff(f13806,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP40(X0,X1,X2) )
    | ~ spl310_754 ),
    inference(avatar_component_clause,[],[f13805]) ).

tff(f13807,plain,
    spl310_754,
    inference(avatar_split_clause,[],[f3909,f13805]) ).

tff(f3909,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP40(X0,X1,X2) ),
    inference(evaluation,[],[f3206]) ).

tff(f3206,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP40(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2019]) ).

tff(f2019,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) = $sum(0,$uminus(0)) )
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = $sum(0,$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP40(X0,X1,X2) ),
    inference(rectify,[],[f2018]) ).

tff(f2018,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus(0)) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum(0,$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP40(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1639]) ).

tff(f1639,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus(0)) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = $sum(0,$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP40(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP40])]) ).

tff(f13774,plain,
    spl310_753,
    inference(avatar_split_clause,[],[f4067,f13772]) ).

tff(f13772,plain,
    ( spl310_753
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X1))
        | ~ sP11(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_753])]) ).

tff(f4067,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP11(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4066,f867]) ).

tff(f4066,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP11(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2937,f865]) ).

tff(f2937,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP11(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1940]) ).

tff(f13770,plain,
    spl310_752,
    inference(avatar_split_clause,[],[f3908,f13768]) ).

tff(f13768,plain,
    ( spl310_752
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) )
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP40(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_752])]) ).

tff(f3908,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP40(X0,X1,X2) ),
    inference(evaluation,[],[f3205]) ).

tff(f3205,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))) = $sum(0,$uminus(0)) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP40(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2019]) ).

tff(f13766,plain,
    spl310_751,
    inference(avatar_split_clause,[],[f3419,f13764]) ).

tff(f13764,plain,
    ( spl310_751
  <=> ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))))),$sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2)))
        | ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK186(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK187(X0)))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_751])]) ).

tff(f3419,plain,
    ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))))),$sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2)))
      | ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK186(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK187(X0)))) ),
    inference(cnf_transformation,[],[f2168]) ).

tff(f2168,plain,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))))),$sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2)))
      | ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK186(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK187(X0))))
        & $less('fun_app$e'('of_nat$',sK186(X0)),'fun_app$e'('of_nat$',sK187(X0))) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK186,sK187])],[f1383,f2167]) ).

tff(f2167,plain,
    ! [X0: 'Nat_nat_fun$'] :
      ( ? [X3: 'Nat$',X4: 'Nat$'] :
          ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
     => ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK186(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK187(X0))))
        & $less('fun_app$e'('of_nat$',sK186(X0)),'fun_app$e'('of_nat$',sK187(X0))) ) ),
    introduced(choice_axiom,[]) ).

tff(f1383,plain,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))))),$sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2)))
      | ? [X3: 'Nat$',X4: 'Nat$'] :
          ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) ) ),
    inference(ennf_transformation,[],[f800]) ).

tff(f800,plain,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ! [X3: 'Nat$',X4: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
         => $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4))) )
     => ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))))),$sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2))) ),
    inference(theory_normalization,[],[f561]) ).

tff(f561,axiom,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ! [X3: 'Nat$',X4: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
         => $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4))) )
     => $lesseq($sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom559) ).

tff(f13762,plain,
    ( ~ spl310_750
    | ~ spl310_24
    | ~ spl310_743 ),
    inference(avatar_split_clause,[],[f13688,f13626,f4326,f13759]) ).

tff(f13759,plain,
    ( spl310_750
  <=> $less($sum(2,'fun_app$e'('of_nat$','to_nat$'('i$'))),0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_750])]) ).

tff(f13626,plain,
    ( spl310_743
  <=> $less(0,$sum(2,'fun_app$e'('of_nat$','to_nat$'('i$')))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_743])]) ).

tff(f13688,plain,
    ( ~ $less($sum(2,'fun_app$e'('of_nat$','to_nat$'('i$'))),0)
    | ~ spl310_24
    | ~ spl310_743 ),
    inference(resolution,[],[f13628,f4327]) ).

tff(f13628,plain,
    ( $less(0,$sum(2,'fun_app$e'('of_nat$','to_nat$'('i$'))))
    | ~ spl310_743 ),
    inference(avatar_component_clause,[],[f13626]) ).

tff(f13757,plain,
    spl310_749,
    inference(avatar_split_clause,[],[f3199,f13755]) ).

tff(f13755,plain,
    ( spl310_749
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP42(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_749])]) ).

tff(f3199,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP42(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2015]) ).

tff(f2015,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less(0,0)
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP42(X0,X1,X2) ),
    inference(rectify,[],[f2014]) ).

tff(f2014,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
            & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
            & ( $less(0,0)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP42(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1641]) ).

tff(f1641,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
            & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
            & ( $less(0,0)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP42(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP42])]) ).

tff(f13721,plain,
    spl310_748,
    inference(avatar_split_clause,[],[f4229,f13719]) ).

tff(f13719,plain,
    ( spl310_748
  <=> ! [X4: 'Nat$',X3: 'Nat$',X2: $int,X1: $int] :
        ( sP79($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
        | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_748])]) ).

tff(f4229,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP79($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) ) ),
    inference(inner_rewriting,[],[f3889]) ).

tff(f3889,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP79($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) ) ),
    inference(equality_resolution,[],[f3668]) ).

tff(f3668,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP79(X0,X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != X0 )
      | ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) != X0 ) ),
    inference(cnf_transformation,[],[f2336]) ).

tff(f2336,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: 'Nat$',X4: 'Nat$'] :
      ( ( sP79(X0,X1,X2,X3,X4)
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != X0 )
          & ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) != X0 )
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
      & ( ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) = X0 )
            | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) = X0 )
            | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
        | ~ sP79(X0,X1,X2,X3,X4) ) ),
    inference(rectify,[],[f2335]) ).

tff(f2335,plain,
    ! [X3: $int,X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP79(X3,X4,X2,X0,X1)
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4) != X3 )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) != X3 )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4) = X3 )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) = X3 )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP79(X3,X4,X2,X0,X1) ) ),
    inference(flattening,[],[f2334]) ).

tff(f2334,plain,
    ! [X3: $int,X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP79(X3,X4,X2,X0,X1)
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4) != X3 )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) != X3 )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4) = X3 )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) = X3 )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP79(X3,X4,X2,X0,X1) ) ),
    inference(nnf_transformation,[],[f1692]) ).

tff(f1692,plain,
    ! [X3: $int,X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( sP79(X3,X4,X2,X0,X1)
    <=> ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4) = X3 )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) = X3 )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP79])]) ).

tff(f13717,plain,
    spl310_747,
    inference(avatar_split_clause,[],[f4228,f13715]) ).

tff(f13715,plain,
    ( spl310_747
  <=> ! [X4: 'Nat$',X3: 'Nat$',X2: $int,X1: $int] :
        ( sP77($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
        | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_747])]) ).

tff(f4228,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP77($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) ) ),
    inference(inner_rewriting,[],[f3886]) ).

tff(f3886,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP77($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) ) ),
    inference(equality_resolution,[],[f3659]) ).

tff(f3659,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP77(X0,X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != X0 )
      | ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) != X0 ) ),
    inference(cnf_transformation,[],[f2331]) ).

tff(f2331,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: 'Nat$',X4: 'Nat$'] :
      ( ( sP77(X0,X1,X2,X3,X4)
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != X0 )
          & ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) != X0 )
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
      & ( ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) = X0 )
            | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) = X0 )
            | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
        | ~ sP77(X0,X1,X2,X3,X4) ) ),
    inference(rectify,[],[f2330]) ).

tff(f2330,plain,
    ! [X4: $int,X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP77(X4,X3,X2,X0,X1)
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3) != X4 )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) != X4 )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3) = X4 )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) = X4 )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP77(X4,X3,X2,X0,X1) ) ),
    inference(flattening,[],[f2329]) ).

tff(f2329,plain,
    ! [X4: $int,X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP77(X4,X3,X2,X0,X1)
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3) != X4 )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) != X4 )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3) = X4 )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) = X4 )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP77(X4,X3,X2,X0,X1) ) ),
    inference(nnf_transformation,[],[f1689]) ).

tff(f1689,plain,
    ! [X4: $int,X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( sP77(X4,X3,X2,X0,X1)
    <=> ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3) = X4 )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) = X4 )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP77])]) ).

tff(f13707,plain,
    ( spl310_746
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_745 ),
    inference(avatar_split_clause,[],[f13703,f13700,f5938,f5200,f13705]) ).

tff(f13705,plain,
    ( spl310_746
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_746])]) ).

tff(f13700,plain,
    ( spl310_745
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_745])]) ).

tff(f13703,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) )
    | ~ spl310_146
    | ~ spl310_217
    | ~ spl310_745 ),
    inference(forward_demodulation,[],[f13701,f6116]) ).

tff(f13701,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP19(X0,X1,X2) )
    | ~ spl310_745 ),
    inference(avatar_component_clause,[],[f13700]) ).

tff(f13702,plain,
    spl310_745,
    inference(avatar_split_clause,[],[f4111,f13700]) ).

tff(f4111,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4110,f2892]) ).

tff(f4110,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4109,f865]) ).

tff(f4109,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4108,f867]) ).

tff(f4108,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4107,f2892]) ).

tff(f4107,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum(1,$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1))))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4106,f865]) ).

tff(f4106,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),1),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X0),1)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2971,f865]) ).

tff(f2971,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X2),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP19(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1955]) ).

tff(f13692,plain,
    spl310_744,
    inference(avatar_split_clause,[],[f4065,f13690]) ).

tff(f13690,plain,
    ( spl310_744
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP12(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_744])]) ).

tff(f4065,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP12(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4064,f867]) ).

tff(f4064,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP12(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3957,f865]) ).

tff(f3957,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus('fun_app$e'('of_nat$',X1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP12(X0,X1,X2) ),
    inference(evaluation,[],[f3956]) ).

tff(f3956,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus('fun_app$e'('of_nat$',X1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP12(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2930]) ).

tff(f2930,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus('fun_app$e'('of_nat$',X1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP12(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1938]) ).

tff(f1938,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$uminus('fun_app$e'('of_nat$',X1))) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less(0,'fun_app$e'('of_nat$',X0))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP12(X0,X1,X2) ),
    inference(rectify,[],[f1937]) ).

tff(f1937,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( $less(0,'fun_app$e'('of_nat$',X1))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ sP12(X1,X2,X0) ),
    inference(nnf_transformation,[],[f1603]) ).

tff(f1603,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( $less(0,'fun_app$e'('of_nat$',X1))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ sP12(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP12])]) ).

tff(f13629,plain,
    ( spl310_743
    | ~ spl310_79
    | ~ spl310_703 ),
    inference(avatar_split_clause,[],[f13576,f12801,f4626,f13626]) ).

tff(f4626,plain,
    ( spl310_79
  <=> ! [X0: $int] : $less(X0,$sum(X0,1)) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_79])]) ).

tff(f12801,plain,
    ( spl310_703
  <=> ! [X0: $int] :
        ( $less(0,X0)
        | ~ $less($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_703])]) ).

tff(f13576,plain,
    ( $less(0,$sum(2,'fun_app$e'('of_nat$','to_nat$'('i$'))))
    | ~ spl310_79
    | ~ spl310_703 ),
    inference(evaluation,[],[f13570]) ).

tff(f13570,plain,
    ( $less(0,$sum($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),1))
    | ~ spl310_79
    | ~ spl310_703 ),
    inference(resolution,[],[f12802,f4627]) ).

tff(f4627,plain,
    ( ! [X0: $int] : $less(X0,$sum(X0,1))
    | ~ spl310_79 ),
    inference(avatar_component_clause,[],[f4626]) ).

tff(f12802,plain,
    ( ! [X0: $int] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),X0)
        | $less(0,X0) )
    | ~ spl310_703 ),
    inference(avatar_component_clause,[],[f12801]) ).

tff(f13610,plain,
    spl310_742,
    inference(avatar_split_clause,[],[f4221,f13608]) ).

tff(f13608,plain,
    ( spl310_742
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK185(X0)))))),'fun_app$i'(X0,sK185(X0)))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_742])]) ).

tff(f4221,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK185(X0)))))),'fun_app$i'(X0,sK185(X0)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ),
    inference(forward_demodulation,[],[f3416,f864]) ).

tff(f3416,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK185(X0)),1)))),'fun_app$i'(X0,sK185(X0))) ),
    inference(cnf_transformation,[],[f2166]) ).

tff(f2166,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
          | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK185(X0)),1)))),'fun_app$i'(X0,sK185(X0))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK185])],[f2164,f2165]) ).

tff(f2165,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3))
     => ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK185(X0)),1)))),'fun_app$i'(X0,sK185(X0))) ),
    introduced(choice_axiom,[]) ).

tff(f2164,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
          | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ) )
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(nnf_transformation,[],[f1382]) ).

tff(f1382,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      <=> $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(ennf_transformation,[],[f320]) ).

tff(f320,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3))
     => ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      <=> $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom318) ).

tff(f13606,plain,
    spl310_741,
    inference(avatar_split_clause,[],[f4220,f13604]) ).

tff(f13604,plain,
    ( spl310_741
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK185(X0)))))),'fun_app$i'(X0,sK185(X0)))
        | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_741])]) ).

tff(f4220,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK185(X0)))))),'fun_app$i'(X0,sK185(X0)))
      | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3417,f864]) ).

tff(f3417,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK185(X0)),1)))),'fun_app$i'(X0,sK185(X0))) ),
    inference(cnf_transformation,[],[f2166]) ).

tff(f13602,plain,
    spl310_740,
    inference(avatar_split_clause,[],[f4219,f13600]) ).

tff(f13600,plain,
    ( spl310_740
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK184(X0))))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_740])]) ).

tff(f4219,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK184(X0))))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ),
    inference(forward_demodulation,[],[f3414,f864]) ).

tff(f3414,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK184(X0)),1)))) ),
    inference(cnf_transformation,[],[f2163]) ).

tff(f2163,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
          | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK184(X0)),1)))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK184])],[f2161,f2162]) ).

tff(f2162,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK184(X0)),1)))) ),
    introduced(choice_axiom,[]) ).

tff(f2161,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
          | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ) )
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(nnf_transformation,[],[f1381]) ).

tff(f1381,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(ennf_transformation,[],[f405]) ).

tff(f405,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom403) ).

tff(f13598,plain,
    spl310_739,
    inference(avatar_split_clause,[],[f4218,f13596]) ).

tff(f13596,plain,
    ( spl310_739
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK184(X0))))))
        | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_739])]) ).

tff(f4218,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK184(X0))))))
      | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(forward_demodulation,[],[f3415,f864]) ).

tff(f3415,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK184(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK184(X0)),1)))) ),
    inference(cnf_transformation,[],[f2163]) ).

tff(f13594,plain,
    spl310_738,
    inference(avatar_split_clause,[],[f4215,f13592]) ).

tff(f13592,plain,
    ( spl310_738
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK151(X0)))))),'fun_app$i'(X0,sK151(X0)))
        | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_738])]) ).

tff(f4215,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK151(X0)))))),'fun_app$i'(X0,sK151(X0)))
      | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(forward_demodulation,[],[f3380,f864]) ).

tff(f3380,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK151(X0)),1)))),'fun_app$i'(X0,sK151(X0))) ),
    inference(cnf_transformation,[],[f2127]) ).

tff(f2127,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK151(X0)),1)))),'fun_app$i'(X0,sK151(X0))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK151])],[f1363,f2126]) ).

tff(f2126,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3))
     => ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK151(X0)),1)))),'fun_app$i'(X0,sK151(X0))) ),
    introduced(choice_axiom,[]) ).

tff(f1363,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(flattening,[],[f1362]) ).

tff(f1362,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(ennf_transformation,[],[f325]) ).

tff(f325,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom323) ).

tff(f13590,plain,
    spl310_737,
    inference(avatar_split_clause,[],[f4214,f13588]) ).

tff(f13588,plain,
    ( spl310_737
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK150(X0)))))),'fun_app$i'(X0,sK150(X0)))
        | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_737])]) ).

tff(f4214,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK150(X0)))))),'fun_app$i'(X0,sK150(X0)))
      | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3379,f864]) ).

tff(f3379,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK150(X0)),1)))),'fun_app$i'(X0,sK150(X0))) ),
    inference(cnf_transformation,[],[f2125]) ).

tff(f2125,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK150(X0)),1)))),'fun_app$i'(X0,sK150(X0))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK150])],[f1361,f2124]) ).

tff(f2124,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3))
     => ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK150(X0)),1)))),'fun_app$i'(X0,sK150(X0))) ),
    introduced(choice_axiom,[]) ).

tff(f1361,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(flattening,[],[f1360]) ).

tff(f1360,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(ennf_transformation,[],[f796]) ).

tff(f796,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1)) ),
    inference(theory_normalization,[],[f553]) ).

tff(f553,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom551) ).

tff(f13586,plain,
    spl310_736,
    inference(avatar_split_clause,[],[f4213,f13584]) ).

tff(f13584,plain,
    ( spl310_736
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK149(X0)))))),'fun_app$i'(X0,sK149(X0)))
        | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_736])]) ).

tff(f4213,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK149(X0)))))),'fun_app$i'(X0,sK149(X0)))
      | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3378,f864]) ).

tff(f3378,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK149(X0)),1)))),'fun_app$i'(X0,sK149(X0))) ),
    inference(cnf_transformation,[],[f2123]) ).

tff(f2123,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK149(X0)),1)))),'fun_app$i'(X0,sK149(X0))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK149])],[f1359,f2122]) ).

tff(f2122,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3))
     => ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK149(X0)),1)))),'fun_app$i'(X0,sK149(X0))) ),
    introduced(choice_axiom,[]) ).

tff(f1359,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(flattening,[],[f1358]) ).

tff(f1358,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) ),
    inference(ennf_transformation,[],[f795]) ).

tff(f795,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1)) ),
    inference(theory_normalization,[],[f555]) ).

tff(f555,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))),'fun_app$i'(X0,X3)) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X2)),'fun_app$i'(X0,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom553) ).

tff(f13582,plain,
    spl310_735,
    inference(avatar_split_clause,[],[f4212,f13580]) ).

tff(f13580,plain,
    ( spl310_735
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK148(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK148(X0))))))
        | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_735])]) ).

tff(f4212,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK148(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK148(X0))))))
      | 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(forward_demodulation,[],[f3377,f864]) ).

tff(f3377,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK148(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK148(X0)),1)))) ),
    inference(cnf_transformation,[],[f2121]) ).

tff(f2121,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK148(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK148(X0)),1)))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK148])],[f1357,f2120]) ).

tff(f2120,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,sK148(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK148(X0)),1)))) ),
    introduced(choice_axiom,[]) ).

tff(f1357,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(flattening,[],[f1356]) ).

tff(f1356,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(ennf_transformation,[],[f407]) ).

tff(f407,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom405) ).

tff(f13559,plain,
    spl310_734,
    inference(avatar_split_clause,[],[f4211,f13557]) ).

tff(f13557,plain,
    ( spl310_734
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK147(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK147(X0))))))
        | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_734])]) ).

tff(f4211,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK147(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK147(X0))))))
      | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3376,f864]) ).

tff(f3376,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK147(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK147(X0)),1)))) ),
    inference(cnf_transformation,[],[f2119]) ).

tff(f2119,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK147(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK147(X0)),1)))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK147])],[f1355,f2118]) ).

tff(f2118,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK147(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK147(X0)),1)))) ),
    introduced(choice_axiom,[]) ).

tff(f1355,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(flattening,[],[f1354]) ).

tff(f1354,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(ennf_transformation,[],[f794]) ).

tff(f794,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ),
    inference(theory_normalization,[],[f551]) ).

tff(f551,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom549) ).

tff(f13555,plain,
    spl310_733,
    inference(avatar_split_clause,[],[f4210,f13553]) ).

tff(f13553,plain,
    ( spl310_733
  <=> ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK146(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK146(X0))))))
        | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_733])]) ).

tff(f4210,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK146(X0))),'fun_app$i'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK146(X0))))))
      | 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3375,f864]) ).

tff(f3375,plain,
    ! [X2: 'Nat$',X0: 'Nat_n_fun$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK146(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK146(X0)),1)))) ),
    inference(cnf_transformation,[],[f2117]) ).

tff(f2117,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK146(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK146(X0)),1)))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK146])],[f1353,f2116]) ).

tff(f2116,plain,
    ! [X0: 'Nat_n_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,sK146(X0))),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK146(X0)),1)))) ),
    introduced(choice_axiom,[]) ).

tff(f1353,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(flattening,[],[f1352]) ).

tff(f1352,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(ennf_transformation,[],[f793]) ).

tff(f793,plain,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ),
    inference(theory_normalization,[],[f556]) ).

tff(f556,axiom,
    ! [X0: 'Nat_n_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X3)),'fun_app$i'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'(X0,X1)),'fun_app$i'(X0,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom554) ).

tff(f13551,plain,
    spl310_732,
    inference(avatar_split_clause,[],[f4166,f13549]) ).

tff(f13549,plain,
    ( spl310_732
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ( 0 = $sum(1,$uminus('fun_app$e'('of_nat$',X0))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP26(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_732])]) ).

tff(f4166,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X2)),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ( 0 = $sum(1,$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3920,f864]) ).

tff(f3920,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(1,$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(evaluation,[],[f3057]) ).

tff(f3057,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),1),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP26(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1974]) ).

tff(f13547,plain,
    spl310_731,
    inference(avatar_split_clause,[],[f3194,f13545]) ).

tff(f13545,plain,
    ( spl310_731
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | sP39(X0,X2,X1)
        | ~ sP43(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_731])]) ).

tff(f3194,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | sP39(X0,X2,X1)
      | ~ sP43(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2013]) ).

tff(f2013,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( 0 = 0 )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ( sP38(X2,X0,X1)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | sP39(X0,X2,X1)
      | ~ sP43(X0,X1,X2) ),
    inference(rectify,[],[f2012]) ).

tff(f2012,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( ( 0 = 0 )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      | ( sP38(X0,X2,X1)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP39(X2,X0,X1)
      | ~ sP43(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1642]) ).

tff(f1642,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( ( 0 = 0 )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      | ( sP38(X0,X2,X1)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | sP39(X2,X0,X1)
      | ~ sP43(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP43])]) ).

tff(f13461,plain,
    ( spl310_730
    | ~ spl310_137
    | ~ spl310_659 ),
    inference(avatar_split_clause,[],[f12748,f12328,f4955,f13459]) ).

tff(f13459,plain,
    ( spl310_730
  <=> ! [X0: $int] :
        ( ~ $less($sum(1,X0),0)
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_730])]) ).

tff(f12328,plain,
    ( spl310_659
  <=> ! [X0: $int] :
        ( ~ $less(X0,0)
        | $less(X0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_659])]) ).

tff(f12748,plain,
    ( ! [X0: $int] :
        ( ~ $less($sum(1,X0),0)
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    | ~ spl310_137
    | ~ spl310_659 ),
    inference(resolution,[],[f12329,f4956]) ).

tff(f12329,plain,
    ( ! [X0: $int] :
        ( $less(X0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))))
        | ~ $less(X0,0) )
    | ~ spl310_659 ),
    inference(avatar_component_clause,[],[f12328]) ).

tff(f13393,plain,
    spl310_729,
    inference(avatar_split_clause,[],[f4186,f13391]) ).

tff(f13391,plain,
    ( spl310_729
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ( 0 = $sum(-1,'fun_app$e'('of_nat$',X2)) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP29(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_729])]) ).

tff(f4186,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( 0 = $sum(-1,'fun_app$e'('of_nat$',X2)) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4185,f864]) ).

tff(f4185,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(-1,'fun_app$e'('of_nat$',X2)) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3916,f864]) ).

tff(f3916,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),-1) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(evaluation,[],[f3070]) ).

tff(f3070,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP29(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1979]) ).

tff(f13389,plain,
    spl310_728,
    inference(avatar_split_clause,[],[f4117,f13387]) ).

tff(f13387,plain,
    ( spl310_728
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP17(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_728])]) ).

tff(f4117,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP17(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3937,f864]) ).

tff(f3937,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus($sum('fun_app$e'('of_nat$',X0),1)) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP17(X0,X1,X2) ),
    inference(evaluation,[],[f2980]) ).

tff(f2980,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP17(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1959]) ).

tff(f1959,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP17(X0,X1,X2) ),
    inference(rectify,[],[f1958]) ).

tff(f1958,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP17(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1609]) ).

tff(f1609,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP17(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP17])]) ).

tff(f13385,plain,
    spl310_727,
    inference(avatar_split_clause,[],[f3704,f13383]) ).

tff(f13383,plain,
    ( spl310_727
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( sP87(X0,X1,X2,X3,X4)
        | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
        | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_727])]) ).

tff(f3704,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP87(X0,X1,X2,X3,X4)
      | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
      | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ),
    inference(cnf_transformation,[],[f2356]) ).

tff(f2356,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: 'Nat$',X4: 'Nat$'] :
      ( ( sP87(X0,X1,X2,X3,X4)
        | ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
          & ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
        | ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
      & ( ( ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
            | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
          & ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
            | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
        | ~ sP87(X0,X1,X2,X3,X4) ) ),
    inference(rectify,[],[f2355]) ).

tff(f2355,plain,
    ! [X4: $int,X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP87(X4,X3,X2,X0,X1)
        | ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),X4)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),X4)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP87(X4,X3,X2,X0,X1) ) ),
    inference(flattening,[],[f2354]) ).

tff(f2354,plain,
    ! [X4: $int,X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP87(X4,X3,X2,X0,X1)
        | ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),X4)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),X4)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP87(X4,X3,X2,X0,X1) ) ),
    inference(nnf_transformation,[],[f1704]) ).

tff(f1704,plain,
    ! [X4: $int,X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( sP87(X4,X3,X2,X0,X1)
    <=> ( ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),X4)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP87])]) ).

tff(f13381,plain,
    spl310_726,
    inference(avatar_split_clause,[],[f3695,f13379]) ).

tff(f13379,plain,
    ( spl310_726
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( sP85(X0,X1,X2,X3,X4)
        | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
        | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_726])]) ).

tff(f3695,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( sP85(X0,X1,X2,X3,X4)
      | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
      | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ),
    inference(cnf_transformation,[],[f2351]) ).

tff(f2351,plain,
    ! [X0: $int,X1: $int,X2: 'Nat$',X3: 'Nat$',X4: $int] :
      ( ( sP85(X0,X1,X2,X3,X4)
        | ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) )
        | ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
            | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) )
          & ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP85(X0,X1,X2,X3,X4) ) ),
    inference(rectify,[],[f2350]) ).

tff(f2350,plain,
    ! [X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$',X3: $int] :
      ( ( sP85(X4,X2,X0,X1,X3)
        | ( ~ $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP85(X4,X2,X0,X1,X3) ) ),
    inference(flattening,[],[f2349]) ).

tff(f2349,plain,
    ! [X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$',X3: $int] :
      ( ( sP85(X4,X2,X0,X1,X3)
        | ( ~ $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP85(X4,X2,X0,X1,X3) ) ),
    inference(nnf_transformation,[],[f1701]) ).

tff(f1701,plain,
    ! [X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$',X3: $int] :
      ( sP85(X4,X2,X0,X1,X3)
    <=> ( ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP85])]) ).

tff(f13377,plain,
    ( spl310_725
    | ~ spl310_136
    | ~ spl310_659 ),
    inference(avatar_split_clause,[],[f12746,f12328,f4951,f13375]) ).

tff(f13375,plain,
    ( spl310_725
  <=> ! [X0: $int] :
        ( ~ $less($sum(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))),0)
        | $less(X0,1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_725])]) ).

tff(f4951,plain,
    ( spl310_136
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X1,X2),$sum(X0,X2))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_136])]) ).

tff(f12746,plain,
    ( ! [X0: $int] :
        ( ~ $less($sum(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))),0)
        | $less(X0,1) )
    | ~ spl310_136
    | ~ spl310_659 ),
    inference(resolution,[],[f12329,f4952]) ).

tff(f4952,plain,
    ( ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less($sum(X1,X2),$sum(X0,X2))
        | $less(X1,X0) )
    | ~ spl310_136 ),
    inference(avatar_component_clause,[],[f4951]) ).

tff(f13373,plain,
    spl310_724,
    inference(avatar_split_clause,[],[f3686,f13371]) ).

tff(f13371,plain,
    ( spl310_724
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( sP83(X0,X1,X2,X3,X4)
        | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
        | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_724])]) ).

tff(f3686,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( sP83(X0,X1,X2,X3,X4)
      | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
      | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ),
    inference(cnf_transformation,[],[f2346]) ).

tff(f2346,plain,
    ! [X0: $int,X1: $int,X2: 'Nat$',X3: 'Nat$',X4: $int] :
      ( ( sP83(X0,X1,X2,X3,X4)
        | ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) )
        | ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
            | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) )
          & ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP83(X0,X1,X2,X3,X4) ) ),
    inference(rectify,[],[f2345]) ).

tff(f2345,plain,
    ! [X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( sP83(X3,X2,X0,X1,X4)
        | ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP83(X3,X2,X0,X1,X4) ) ),
    inference(flattening,[],[f2344]) ).

tff(f2344,plain,
    ! [X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( sP83(X3,X2,X0,X1,X4)
        | ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP83(X3,X2,X0,X1,X4) ) ),
    inference(nnf_transformation,[],[f1698]) ).

tff(f1698,plain,
    ! [X3: $int,X2: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP83(X3,X2,X0,X1,X4)
    <=> ( ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP83])]) ).

tff(f13369,plain,
    spl310_723,
    inference(avatar_split_clause,[],[f3677,f13367]) ).

tff(f13367,plain,
    ( spl310_723
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( sP81(X0,X1,X2,X3,X4)
        | $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
        | $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_723])]) ).

tff(f3677,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP81(X0,X1,X2,X3,X4)
      | $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
      | $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ),
    inference(cnf_transformation,[],[f2341]) ).

tff(f2341,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: 'Nat$',X4: 'Nat$'] :
      ( ( sP81(X0,X1,X2,X3,X4)
        | ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
          & ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
        | ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
      & ( ( ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
            | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
          & ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
            | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) )
        | ~ sP81(X0,X1,X2,X3,X4) ) ),
    inference(rectify,[],[f2340]) ).

tff(f2340,plain,
    ! [X3: $int,X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP81(X3,X4,X2,X0,X1)
        | ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4),X3)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4),X3)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4),X3)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4),X3)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP81(X3,X4,X2,X0,X1) ) ),
    inference(flattening,[],[f2339]) ).

tff(f2339,plain,
    ! [X3: $int,X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP81(X3,X4,X2,X0,X1)
        | ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4),X3)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4),X3)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4),X3)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4),X3)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP81(X3,X4,X2,X0,X1) ) ),
    inference(nnf_transformation,[],[f1695]) ).

tff(f1695,plain,
    ! [X3: $int,X4: $int,X2: $int,X0: 'Nat$',X1: 'Nat$'] :
      ( sP81(X3,X4,X2,X0,X1)
    <=> ( ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4),X3)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4),X3)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP81])]) ).

tff(f13365,plain,
    spl310_722,
    inference(avatar_split_clause,[],[f3418,f13363]) ).

tff(f13363,plain,
    ( spl310_722
  <=> ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))))),$sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2)))
        | $less('fun_app$e'('of_nat$',sK186(X0)),'fun_app$e'('of_nat$',sK187(X0))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_722])]) ).

tff(f3418,plain,
    ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))))),$sum('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$',X2)))
      | $less('fun_app$e'('of_nat$',sK186(X0)),'fun_app$e'('of_nat$',sK187(X0))) ),
    inference(cnf_transformation,[],[f2168]) ).

tff(f13361,plain,
    spl310_721,
    inference(avatar_split_clause,[],[f3411,f13359]) ).

tff(f13359,plain,
    ( spl310_721
  <=> ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1)))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK181(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK182(X0)))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_721])]) ).

tff(f3411,plain,
    ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK181(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK182(X0)))) ),
    inference(cnf_transformation,[],[f2157]) ).

tff(f2157,plain,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK181(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK182(X0))))
        & $less('fun_app$e'('of_nat$',sK181(X0)),'fun_app$e'('of_nat$',sK182(X0))) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK181,sK182])],[f1379,f2156]) ).

tff(f2156,plain,
    ! [X0: 'Nat_nat_fun$'] :
      ( ? [X3: 'Nat$',X4: 'Nat$'] :
          ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
     => ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,sK181(X0))),'fun_app$e'('of_nat$','fun_app$r'(X0,sK182(X0))))
        & $less('fun_app$e'('of_nat$',sK181(X0)),'fun_app$e'('of_nat$',sK182(X0))) ) ),
    introduced(choice_axiom,[]) ).

tff(f1379,plain,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$',X4: 'Nat$'] :
          ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) ) ),
    inference(flattening,[],[f1378]) ).

tff(f1378,plain,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$',X4: 'Nat$'] :
          ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) ) ),
    inference(ennf_transformation,[],[f799]) ).

tff(f799,plain,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & ! [X3: 'Nat$',X4: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
           => $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4))) ) )
     => ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1))) ),
    inference(theory_normalization,[],[f531]) ).

tff(f531,axiom,
    ! [X0: 'Nat_nat_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$',X4: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
           => $less('fun_app$e'('of_nat$','fun_app$r'(X0,X3)),'fun_app$e'('of_nat$','fun_app$r'(X0,X4))) ) )
     => $lesseq('fun_app$e'('of_nat$','fun_app$r'(X0,X1)),'fun_app$e'('of_nat$','fun_app$r'(X0,X2))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom529) ).

tff(f13357,plain,
    spl310_720,
    inference(avatar_split_clause,[],[f3297,f13355]) ).

tff(f13355,plain,
    ( spl310_720
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP57(X0,X1,X2)
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_720])]) ).

tff(f3297,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP57(X0,X1,X2)
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0) ),
    inference(cnf_transformation,[],[f2064]) ).

tff(f2064,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP57(X0,X1,X2)
        | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP57(X0,X1,X2) ) ),
    inference(rectify,[],[f2063]) ).

tff(f2063,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP57(X0,X2,X1)
        | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP57(X0,X2,X1) ) ),
    inference(flattening,[],[f2062]) ).

tff(f2062,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP57(X0,X2,X1)
        | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP57(X0,X2,X1) ) ),
    inference(nnf_transformation,[],[f1660]) ).

tff(f1660,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( sP57(X0,X2,X1)
    <=> ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP57])]) ).

tff(f13353,plain,
    spl310_719,
    inference(avatar_split_clause,[],[f3276,f13351]) ).

tff(f13351,plain,
    ( spl310_719
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP53(X0,X1,X2)
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_719])]) ).

tff(f3276,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP53(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ),
    inference(cnf_transformation,[],[f2053]) ).

tff(f2053,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP53(X0,X1,X2)
        | ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP53(X0,X1,X2) ) ),
    inference(flattening,[],[f2052]) ).

tff(f2052,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP53(X0,X1,X2)
        | ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP53(X0,X1,X2) ) ),
    inference(nnf_transformation,[],[f1655]) ).

tff(f1655,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP53(X0,X1,X2)
    <=> ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP53])]) ).

tff(f13349,plain,
    spl310_718,
    inference(avatar_split_clause,[],[f3255,f13347]) ).

tff(f13347,plain,
    ( spl310_718
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP49(X0,X1,X2)
        | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_718])]) ).

tff(f3255,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP49(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ),
    inference(cnf_transformation,[],[f2043]) ).

tff(f2043,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP49(X0,X1,X2)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP49(X0,X1,X2) ) ),
    inference(flattening,[],[f2042]) ).

tff(f2042,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP49(X0,X1,X2)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP49(X0,X1,X2) ) ),
    inference(nnf_transformation,[],[f1650]) ).

tff(f1650,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP49(X0,X1,X2)
    <=> ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP49])]) ).

tff(f13345,plain,
    spl310_717,
    inference(avatar_split_clause,[],[f3213,f13343]) ).

tff(f13343,plain,
    ( spl310_717
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
        | ~ sP38(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_717])]) ).

tff(f3213,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
      | ~ sP38(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2023]) ).

tff(f2023,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP38(X0,X1,X2) ),
    inference(rectify,[],[f2022]) ).

tff(f2022,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ~ sP38(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1637]) ).

tff(f1637,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ~ sP38(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP38])]) ).

tff(f13316,plain,
    ( spl310_716
    | ~ spl310_24
    | ~ spl310_659 ),
    inference(avatar_split_clause,[],[f12733,f12328,f4326,f13314]) ).

tff(f13314,plain,
    ( spl310_716
  <=> ! [X0: $int] :
        ( ~ $less(X0,0)
        | ~ $less($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_716])]) ).

tff(f12733,plain,
    ( ! [X0: $int] :
        ( ~ $less(X0,0)
        | ~ $less($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),X0) )
    | ~ spl310_24
    | ~ spl310_659 ),
    inference(resolution,[],[f12329,f4327]) ).

tff(f12992,plain,
    spl310_715,
    inference(avatar_split_clause,[],[f4190,f12990]) ).

tff(f12990,plain,
    ( spl310_715
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK122(X0)))))
        | 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_715])]) ).

tff(f4190,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK122(X0)))))
      | 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(forward_demodulation,[],[f3134,f864]) ).

tff(f3134,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK122(X0)),1)))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f2001]) ).

tff(f2001,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( 'fun_app$j'(X0,'nat$'(0))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      | ( ~ 'fun_app$j'(X0,sK122(X0))
        & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK122(X0)),1))) )
      | ~ 'fun_app$j'(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK122])],[f1230,f2000]) ).

tff(f2000,plain,
    ! [X0: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X3)
          & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))) )
     => ( ~ 'fun_app$j'(X0,sK122(X0))
        & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK122(X0)),1))) ) ),
    introduced(choice_axiom,[]) ).

tff(f1230,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( 'fun_app$j'(X0,'nat$'(0))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X3)
          & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))) )
      | ~ 'fun_app$j'(X0,X1) ),
    inference(flattening,[],[f1229]) ).

tff(f1229,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( 'fun_app$j'(X0,'nat$'(0))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X3)
          & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))) )
      | ~ 'fun_app$j'(X0,X1) ),
    inference(ennf_transformation,[],[f766]) ).

tff(f766,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
           => 'fun_app$j'(X0,X3) )
        & 'fun_app$j'(X0,X1) )
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))))) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => 'fun_app$j'(X0,'nat$'(0)) ) ) ),
    inference(theory_normalization,[],[f530]) ).

tff(f530,axiom,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
           => 'fun_app$j'(X0,X3) )
        & 'fun_app$j'(X0,X1) )
     => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => 'fun_app$j'(X0,'nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => 'fun_app$j'(X0,'nat$'(0)) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom528) ).

tff(f12988,plain,
    spl310_714,
    inference(avatar_split_clause,[],[f4071,f12986]) ).

tff(f12986,plain,
    ( spl310_714
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X1))
        | ~ sP11(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_714])]) ).

tff(f4071,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP11(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4070,f867]) ).

tff(f4070,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$sum($uminus('fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X0)))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP11(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2935,f865]) ).

tff(f2935,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP11(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1940]) ).

tff(f12984,plain,
    ( spl310_713
    | ~ spl310_137
    | ~ spl310_654 ),
    inference(avatar_split_clause,[],[f12391,f12280,f4955,f12982]) ).

tff(f12280,plain,
    ( spl310_654
  <=> ! [X0: $int] :
        ( $less(0,X0)
        | $less(X0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_654])]) ).

tff(f12391,plain,
    ( ! [X0: $int] :
        ( $less(0,$sum(1,X0))
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    | ~ spl310_137
    | ~ spl310_654 ),
    inference(resolution,[],[f12281,f4956]) ).

tff(f12281,plain,
    ( ! [X0: $int] :
        ( $less(X0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))))
        | $less(0,X0) )
    | ~ spl310_654 ),
    inference(avatar_component_clause,[],[f12280]) ).

tff(f12980,plain,
    spl310_712,
    inference(avatar_split_clause,[],[f3647,f12978]) ).

tff(f12978,plain,
    ( spl310_712
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK308(X1))),'fun_app$f'(X1,sK309(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_712])]) ).

tff(f3647,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK308(X1))),'fun_app$f'(X1,sK309(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2326]) ).

tff(f2326,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK308(X1))),'fun_app$f'(X1,sK309(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK308(X1)),sK309(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK308,sK309])],[f1572,f2325]) ).

tff(f2325,plain,
    ! [X1: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK308(X1))),'fun_app$f'(X1,sK309(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK308(X1)),sK309(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1572,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(flattening,[],[f1571]) ).

tff(f1571,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(ennf_transformation,[],[f292]) ).

tff(f292,axiom,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less$',X2),X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom290) ).

tff(f12976,plain,
    spl310_711,
    inference(avatar_split_clause,[],[f3645,f12974]) ).

tff(f12974,plain,
    ( spl310_711
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK306(X1))),'fun_app$f'(X1,sK307(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_711])]) ).

tff(f3645,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK306(X1))),'fun_app$f'(X1,sK307(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2324]) ).

tff(f2324,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK306(X1))),'fun_app$f'(X1,sK307(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK306(X1)),sK307(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK306,sK307])],[f1570,f2323]) ).

tff(f2323,plain,
    ! [X1: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK306(X1))),'fun_app$f'(X1,sK307(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK306(X1)),sK307(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1570,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(flattening,[],[f1569]) ).

tff(f1569,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(ennf_transformation,[],[f197]) ).

tff(f197,axiom,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom195) ).

tff(f12972,plain,
    spl310_710,
    inference(avatar_split_clause,[],[f3643,f12970]) ).

tff(f12970,plain,
    ( spl310_710
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK304(X1))),'fun_app$f'(X1,sK305(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_710])]) ).

tff(f3643,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK304(X1))),'fun_app$f'(X1,sK305(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2322]) ).

tff(f2322,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK304(X1))),'fun_app$f'(X1,sK305(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK304(X1)),sK305(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK304,sK305])],[f1568,f2321]) ).

tff(f2321,plain,
    ! [X1: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK304(X1))),'fun_app$f'(X1,sK305(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK304(X1)),sK305(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1568,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(flattening,[],[f1567]) ).

tff(f1567,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(ennf_transformation,[],[f491]) ).

tff(f491,axiom,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom489) ).

tff(f12968,plain,
    spl310_709,
    inference(avatar_split_clause,[],[f3641,f12966]) ).

tff(f12966,plain,
    ( spl310_709
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK302(X1))),'fun_app$f'(X1,sK303(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_709])]) ).

tff(f3641,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK302(X1))),'fun_app$f'(X1,sK303(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2320]) ).

tff(f2320,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK302(X1))),'fun_app$f'(X1,sK303(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK302(X1)),sK303(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK302,sK303])],[f1566,f2319]) ).

tff(f2319,plain,
    ! [X1: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK302(X1))),'fun_app$f'(X1,sK303(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK302(X1)),sK303(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1566,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(flattening,[],[f1565]) ).

tff(f1565,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(ennf_transformation,[],[f484]) ).

tff(f484,axiom,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less$',X2),X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom482) ).

tff(f12964,plain,
    spl310_708,
    inference(avatar_split_clause,[],[f3611,f12962]) ).

tff(f12962,plain,
    ( spl310_708
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK276(X2))),'fun_app$f'(X2,sK277(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_708])]) ).

tff(f3611,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK276(X2))),'fun_app$f'(X2,sK277(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2294]) ).

tff(f2294,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK276(X2))),'fun_app$f'(X2,sK277(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK276(X2)),sK277(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK276,sK277])],[f1532,f2293]) ).

tff(f2293,plain,
    ! [X2: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK276(X2))),'fun_app$f'(X2,sK277(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK276(X2)),sK277(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1532,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1531]) ).

tff(f1531,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f193]) ).

tff(f193,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom191) ).

tff(f12960,plain,
    spl310_707,
    inference(avatar_split_clause,[],[f3609,f12958]) ).

tff(f12958,plain,
    ( spl310_707
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK274(X2))),'fun_app$f'(X2,sK275(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_707])]) ).

tff(f3609,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK274(X2))),'fun_app$f'(X2,sK275(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2292]) ).

tff(f2292,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK274(X2))),'fun_app$f'(X2,sK275(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK274(X2)),sK275(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK274,sK275])],[f1530,f2291]) ).

tff(f2291,plain,
    ! [X2: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK274(X2))),'fun_app$f'(X2,sK275(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK274(X2)),sK275(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1530,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1529]) ).

tff(f1529,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f288]) ).

tff(f288,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom286) ).

tff(f12956,plain,
    spl310_706,
    inference(avatar_split_clause,[],[f3607,f12954]) ).

tff(f12954,plain,
    ( spl310_706
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK272(X2))),'fun_app$f'(X2,sK273(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_706])]) ).

tff(f3607,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK272(X2))),'fun_app$f'(X2,sK273(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2290]) ).

tff(f2290,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK272(X2))),'fun_app$f'(X2,sK273(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK272(X2)),sK273(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK272,sK273])],[f1528,f2289]) ).

tff(f2289,plain,
    ! [X2: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK272(X2))),'fun_app$f'(X2,sK273(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK272(X2)),sK273(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1528,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1527]) ).

tff(f1527,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f496]) ).

tff(f496,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom494) ).

tff(f12952,plain,
    spl310_705,
    inference(avatar_split_clause,[],[f3605,f12950]) ).

tff(f12950,plain,
    ( spl310_705
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK270(X2))),'fun_app$f'(X2,sK271(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_705])]) ).

tff(f3605,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK270(X2))),'fun_app$f'(X2,sK271(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2288]) ).

tff(f2288,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK270(X2))),'fun_app$f'(X2,sK271(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK270(X2)),sK271(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK270,sK271])],[f1526,f2287]) ).

tff(f2287,plain,
    ! [X2: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK270(X2))),'fun_app$f'(X2,sK271(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK270(X2)),sK271(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1526,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1525]) ).

tff(f1525,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f487]) ).

tff(f487,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom485) ).

tff(f12935,plain,
    ( spl310_704
    | ~ spl310_136
    | ~ spl310_654 ),
    inference(avatar_split_clause,[],[f12389,f12280,f4951,f12933]) ).

tff(f12933,plain,
    ( spl310_704
  <=> ! [X0: $int] :
        ( $less(0,$sum(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))))
        | $less(X0,1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_704])]) ).

tff(f12389,plain,
    ( ! [X0: $int] :
        ( $less(0,$sum(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))))
        | $less(X0,1) )
    | ~ spl310_136
    | ~ spl310_654 ),
    inference(resolution,[],[f12281,f4952]) ).

tff(f12803,plain,
    ( spl310_703
    | ~ spl310_24
    | ~ spl310_654 ),
    inference(avatar_split_clause,[],[f12376,f12280,f4326,f12801]) ).

tff(f12376,plain,
    ( ! [X0: $int] :
        ( $less(0,X0)
        | ~ $less($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),X0) )
    | ~ spl310_24
    | ~ spl310_654 ),
    inference(resolution,[],[f12281,f4327]) ).

tff(f12779,plain,
    spl310_702,
    inference(avatar_split_clause,[],[f4126,f12777]) ).

tff(f12777,plain,
    ( spl310_702
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | $less($sum(1,$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
        | sP21(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_702])]) ).

tff(f4126,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | $less($sum(1,$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP21(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4125,f864]) ).

tff(f4125,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP21(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f4124,f2892]) ).

tff(f4124,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP21(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f4123,f865]) ).

tff(f4123,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP21(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2997,f864]) ).

tff(f2997,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP21(X0,X1,X2)
      | $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
      | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1614]) ).

tff(f1614,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP21(X0,X1,X2)
        | ( ( $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
            | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,$sum('fun_app$e'('of_nat$',X2),1))
            | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( sP15(X2,X1,X0)
          & sP14(X2,X1,X0) )
        | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
          & ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
        | sP16(X2,X1,X0) ) ),
    inference(definition_folding,[],[f1163,f1613,f1612,f1611,f1610,f1609,f1608,f1607,f1606]) ).

tff(f1606,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = 0 )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP14(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP14])]) ).

tff(f1608,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),1))
        & $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
      | ~ sP16(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP16])]) ).

tff(f1613,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( sP20(X2,X1,X0)
            & sP19(X2,X1,X0) )
          | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
        & ( ( sP18(X2,X1,X0)
            & sP17(X2,X1,X0) )
          | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP21(X0,X1,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP21])]) ).

tff(f1163,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X2))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
              & ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
            | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
          & ( ( ( ( ( ( $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X2))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
              & ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
            | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) )
        | ( ( $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
            | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,$sum('fun_app$e'('of_nat$',X2),1))
            | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
            | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
              & ( $less(0,'fun_app$e'('of_nat$',X2))
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
          & ( ( 0 = 0 )
            | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
              & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
            | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
        | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
          & ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
        | ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),1))
          & $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(flattening,[],[f1162]) ).

tff(f1162,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X2))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
              & ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
            | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
          & ( ( ( ( ( ( $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X2))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
              & ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
            | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) )
        | ( ( $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
            | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,$sum('fun_app$e'('of_nat$',X2),1))
            | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
            | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
              & ( $less(0,'fun_app$e'('of_nat$',X2))
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
          & ( ( 0 = 0 )
            | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
              & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
            | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
        | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
          & ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
        | ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),1))
          & $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(ennf_transformation,[],[f883]) ).

tff(f883,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
             => $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1)) )
            & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
             => $less(0,$sum('fun_app$e'('of_nat$',X2),1)) ) )
       => ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) )
          & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) ) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1)) )
          & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => $less(0,$sum('fun_app$e'('of_nat$',X2),1)) ) )
       => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                 => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                 => $less(0,'fun_app$e'('of_nat$',X2)) ) )
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
          & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => $less(0,'fun_app$e'('of_nat$',X2)) ) )
           => ( 0 = 0 ) ) ) ) ),
    inference(theory_normalization,[],[f735]) ).

tff(f735,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
             => $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1)) )
            & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
             => $less(0,$sum('fun_app$e'('of_nat$',X2),1)) ) )
       => ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) = $sum($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( 0 = $sum($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) )
          & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),1))) ) ) ) ) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1)) )
          & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => $less(0,$sum('fun_app$e'('of_nat$',X2),1)) ) )
       => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                 => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
                & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                 => $less(0,'fun_app$e'('of_nat$',X2)) ) )
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => ( 0 = $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
          & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => $less(0,'fun_app$e'('of_nat$',X2)) ) )
           => ( 0 = 0 ) ) ) ) ),
    inference(theory_normalization,[],[f260]) ).

tff(f260,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
             => $less($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X2),1)) )
            & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
             => $less(0,$sum('fun_app$e'('of_nat$',X2),1)) ) )
       => ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) = $difference($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X2),1)) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $difference(0,'fun_app$e'('of_nat$',X2)) = $difference($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X2),1)) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( 0 = $difference($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X2),1)) ) ) ) )
          & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                     => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) = $difference(0,$sum('fun_app$e'('of_nat$',X2),1)) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => ( $difference(0,'fun_app$e'('of_nat$',X2)) = $difference(0,$sum('fun_app$e'('of_nat$',X2),1)) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                   => $less(0,'fun_app$e'('of_nat$',X2)) ) )
               => ( 0 = $difference(0,$sum('fun_app$e'('of_nat$',X2),1)) ) ) ) ) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => $less($difference($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X2),1)) )
          & ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
           => $less(0,$sum('fun_app$e'('of_nat$',X2),1)) ) )
       => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                 => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
                & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
                 => $less(0,'fun_app$e'('of_nat$',X2)) ) )
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => ( 0 = $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) ) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => ( 0 = $difference(0,'fun_app$e'('of_nat$',X2)) ) ) ) )
          & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
               => $less(0,'fun_app$e'('of_nat$',X2)) ) )
           => ( 0 = 0 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom258) ).

tff(f12775,plain,
    spl310_701,
    inference(avatar_split_clause,[],[f4116,f12773]) ).

tff(f12773,plain,
    ( spl310_701
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP17(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_701])]) ).

tff(f4116,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP17(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3938,f864]) ).

tff(f3938,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus($sum('fun_app$e'('of_nat$',X0),1)) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP17(X0,X1,X2) ),
    inference(evaluation,[],[f2981]) ).

tff(f2981,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP17(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1959]) ).

tff(f12771,plain,
    spl310_700,
    inference(avatar_split_clause,[],[f3950,f12769]) ).

tff(f12769,plain,
    ( spl310_700
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP9(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_700])]) ).

tff(f3950,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP9(X0,X1,X2) ),
    inference(evaluation,[],[f2944]) ).

tff(f2944,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP9(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1944]) ).

tff(f1944,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ~ sP9(X0,X1,X2) ),
    inference(rectify,[],[f1943]) ).

tff(f1943,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ~ sP9(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1600]) ).

tff(f1600,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ~ sP9(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP9])]) ).

tff(f12767,plain,
    spl310_699,
    inference(avatar_split_clause,[],[f3703,f12765]) ).

tff(f12765,plain,
    ( spl310_699
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( sP87(X0,X1,X2,X3,X4)
        | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_699])]) ).

tff(f3703,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP87(X0,X1,X2,X3,X4)
      | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(cnf_transformation,[],[f2356]) ).

tff(f12763,plain,
    spl310_698,
    inference(avatar_split_clause,[],[f3700,f12761]) ).

tff(f12761,plain,
    ( spl310_698
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP87(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_698])]) ).

tff(f3700,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP87(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2356]) ).

tff(f12759,plain,
    spl310_697,
    inference(avatar_split_clause,[],[f3694,f12757]) ).

tff(f12757,plain,
    ( spl310_697
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( sP85(X0,X1,X2,X3,X4)
        | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
        | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_697])]) ).

tff(f3694,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( sP85(X0,X1,X2,X3,X4)
      | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
      | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ),
    inference(cnf_transformation,[],[f2351]) ).

tff(f12727,plain,
    spl310_696,
    inference(avatar_split_clause,[],[f3691,f12725]) ).

tff(f12725,plain,
    ( spl310_696
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
        | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
        | ~ sP85(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_696])]) ).

tff(f3691,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
      | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
      | ~ sP85(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2351]) ).

tff(f12723,plain,
    spl310_695,
    inference(avatar_split_clause,[],[f3685,f12721]) ).

tff(f12721,plain,
    ( spl310_695
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( sP83(X0,X1,X2,X3,X4)
        | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
        | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_695])]) ).

tff(f3685,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( sP83(X0,X1,X2,X3,X4)
      | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
      | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ),
    inference(cnf_transformation,[],[f2346]) ).

tff(f12719,plain,
    spl310_694,
    inference(avatar_split_clause,[],[f3682,f12717]) ).

tff(f12717,plain,
    ( spl310_694
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
        | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
        | ~ sP83(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_694])]) ).

tff(f3682,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X3),$uminus('fun_app$e'('of_nat$',X2))))),X1),X0))
      | $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
      | ~ sP83(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2346]) ).

tff(f12715,plain,
    spl310_693,
    inference(avatar_split_clause,[],[f3676,f12713]) ).

tff(f12713,plain,
    ( spl310_693
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( sP81(X0,X1,X2,X3,X4)
        | $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_693])]) ).

tff(f3676,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP81(X0,X1,X2,X3,X4)
      | $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(cnf_transformation,[],[f2341]) ).

tff(f12711,plain,
    spl310_692,
    inference(avatar_split_clause,[],[f3673,f12709]) ).

tff(f12709,plain,
    ( spl310_692
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP81(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_692])]) ).

tff(f3673,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X0)
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP81(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2341]) ).

tff(f12707,plain,
    spl310_691,
    inference(avatar_split_clause,[],[f3664,f12705]) ).

tff(f12705,plain,
    ( spl310_691
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) = X0 )
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP79(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_691])]) ).

tff(f3664,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) = X0 )
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP79(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2336]) ).

tff(f12703,plain,
    spl310_690,
    inference(avatar_split_clause,[],[f3655,f12701]) ).

tff(f12701,plain,
    ( spl310_690
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) = X0 )
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP77(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_690])]) ).

tff(f3655,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) = X0 )
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP77(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2331]) ).

tff(f12699,plain,
    spl310_689,
    inference(avatar_split_clause,[],[f3631,f12697]) ).

tff(f12697,plain,
    ( spl310_689
  <=> ! [X0: 'Nat$',X3: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
        | ( $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_689])]) ).

tff(f3631,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
      | ( $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1556]) ).

tff(f1556,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$',X3: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
      | ( $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(flattening,[],[f1555]) ).

tff(f1555,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$',X3: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
      | ( $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f537]) ).

tff(f537,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$',X3: 'Nat$'] :
      ( ( ( $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom535) ).

tff(f12695,plain,
    spl310_688,
    inference(avatar_split_clause,[],[f3559,f12693]) ).

tff(f12693,plain,
    ( spl310_688
  <=> ! [X0: 'Int_set$',X3: 'N$',X2: 'Int_set_n_fun$',X1: 'Int_set$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK224(X2))),'fun_app$m'(X2,sK225(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_688])]) ).

tff(f3559,plain,
    ! [X2: 'Int_set_n_fun$',X3: 'N$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK224(X2))),'fun_app$m'(X2,sK225(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2242]) ).

tff(f2242,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK224(X2))),'fun_app$m'(X2,sK225(X2)))
        & 'fun_app$c'('less_eq$a'(sK224(X2)),sK225(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK224,sK225])],[f1480,f2241]) ).

tff(f2241,plain,
    ! [X2: 'Int_set_n_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK224(X2))),'fun_app$m'(X2,sK225(X2)))
        & 'fun_app$c'('less_eq$a'(sK224(X2)),sK225(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1480,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1479]) ).

tff(f1479,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f190]) ).

tff(f190,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom188) ).

tff(f12691,plain,
    spl310_687,
    inference(avatar_split_clause,[],[f3195,f12689]) ).

tff(f12689,plain,
    ( spl310_687
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | sP38(X2,X0,X1)
        | sP39(X0,X2,X1)
        | ~ sP43(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_687])]) ).

tff(f3195,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | sP38(X2,X0,X1)
      | sP39(X0,X2,X1)
      | ~ sP43(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2013]) ).

tff(f12621,plain,
    ( ~ spl310_686
    | ~ spl310_24
    | ~ spl310_680 ),
    inference(avatar_split_clause,[],[f12513,f12441,f4326,f12618]) ).

tff(f12618,plain,
    ( spl310_686
  <=> $less(sK96($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))),0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_686])]) ).

tff(f12441,plain,
    ( spl310_680
  <=> $less(0,sK96($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_680])]) ).

tff(f12513,plain,
    ( ~ $less(sK96($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))),0)
    | ~ spl310_24
    | ~ spl310_680 ),
    inference(resolution,[],[f12443,f4327]) ).

tff(f12443,plain,
    ( $less(0,sK96($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))))
    | ~ spl310_680 ),
    inference(avatar_component_clause,[],[f12441]) ).

tff(f12464,plain,
    spl310_685,
    inference(avatar_split_clause,[],[f4217,f12462]) ).

tff(f12462,plain,
    ( spl310_685
  <=> ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK183(X0))))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_685])]) ).

tff(f4217,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK183(X0))))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2)) ),
    inference(forward_demodulation,[],[f3412,f864]) ).

tff(f3412,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK183(X0)),1)))) ),
    inference(cnf_transformation,[],[f2160]) ).

tff(f2160,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2)) ) )
      | ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK183(X0)),1)))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK183])],[f2158,f2159]) ).

tff(f2159,plain,
    ! [X0: 'Nat_int_fun$'] :
      ( ? [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK183(X0)),1)))) ),
    introduced(choice_axiom,[]) ).

tff(f2158,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2)) ) )
      | ? [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(nnf_transformation,[],[f1380]) ).

tff(f1380,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | ? [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(ennf_transformation,[],[f404]) ).

tff(f404,axiom,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ! [X3: 'Nat$'] : $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom402) ).

tff(f12460,plain,
    spl310_684,
    inference(avatar_split_clause,[],[f4216,f12458]) ).

tff(f12458,plain,
    ( spl310_684
  <=> ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK183(X0))))))
        | $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_684])]) ).

tff(f4216,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK183(X0))))))
      | $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(forward_demodulation,[],[f3413,f864]) ).

tff(f3413,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'(X0,sK183(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK183(X0)),1)))) ),
    inference(cnf_transformation,[],[f2160]) ).

tff(f12456,plain,
    spl310_683,
    inference(avatar_split_clause,[],[f4209,f12454]) ).

tff(f12454,plain,
    ( spl310_683
  <=> ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'(X0,sK145(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK145(X0))))))
        | $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_683])]) ).

tff(f4209,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,sK145(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK145(X0))))))
      | $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(forward_demodulation,[],[f3374,f864]) ).

tff(f3374,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'(X0,sK145(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK145(X0)),1)))) ),
    inference(cnf_transformation,[],[f2115]) ).

tff(f2115,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'(X0,sK145(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK145(X0)),1)))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK145])],[f1351,f2114]) ).

tff(f2114,plain,
    ! [X0: 'Nat_int_fun$'] :
      ( ? [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => ~ $less('fun_app$e'(X0,sK145(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK145(X0)),1)))) ),
    introduced(choice_axiom,[]) ).

tff(f1351,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ? [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(flattening,[],[f1350]) ).

tff(f1350,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ? [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(ennf_transformation,[],[f406]) ).

tff(f406,axiom,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom404) ).

tff(f12452,plain,
    spl310_682,
    inference(avatar_split_clause,[],[f4208,f12450]) ).

tff(f12450,plain,
    ( spl310_682
  <=> ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
        ( $less('fun_app$e'(X0,sK144(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK144(X0))))))
        | ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_682])]) ).

tff(f4208,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'(X0,sK144(X0)),'fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK144(X0))))))
      | ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3373,f864]) ).

tff(f3373,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'(X0,sK144(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK144(X0)),1)))) ),
    inference(cnf_transformation,[],[f2113]) ).

tff(f2113,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'(X0,sK144(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK144(X0)),1)))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK144])],[f1349,f2112]) ).

tff(f2112,plain,
    ! [X0: 'Nat_int_fun$'] :
      ( ? [X3: 'Nat$'] : $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))))
     => $less('fun_app$e'(X0,sK144(X0)),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK144(X0)),1)))) ),
    introduced(choice_axiom,[]) ).

tff(f1349,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(flattening,[],[f1348]) ).

tff(f1348,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) ),
    inference(ennf_transformation,[],[f792]) ).

tff(f792,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & ! [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => ~ $less('fun_app$e'(X0,X1),'fun_app$e'(X0,X2)) ),
    inference(theory_normalization,[],[f554]) ).

tff(f554,axiom,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : $lesseq('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))),'fun_app$e'(X0,X3)) )
     => $lesseq('fun_app$e'(X0,X2),'fun_app$e'(X0,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom552) ).

tff(f12448,plain,
    spl310_681,
    inference(avatar_split_clause,[],[f4207,f12446]) ).

tff(f12446,plain,
    ( spl310_681
  <=> ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
        ( $less('fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK143(X0))))),'fun_app$e'(X0,sK143(X0)))
        | ~ $less('fun_app$e'(X0,X2),'fun_app$e'(X0,X1))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_681])]) ).

tff(f4207,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK143(X0))))),'fun_app$e'(X0,sK143(X0)))
      | ~ $less('fun_app$e'(X0,X2),'fun_app$e'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3372,f864]) ).

tff(f3372,plain,
    ! [X2: 'Nat$',X0: 'Nat_int_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X2),'fun_app$e'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK143(X0)),1))),'fun_app$e'(X0,sK143(X0))) ),
    inference(cnf_transformation,[],[f2111]) ).

tff(f2111,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X2),'fun_app$e'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK143(X0)),1))),'fun_app$e'(X0,sK143(X0))) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK143])],[f1347,f2110]) ).

tff(f2110,plain,
    ! [X0: 'Nat_int_fun$'] :
      ( ? [X3: 'Nat$'] : $less('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))),'fun_app$e'(X0,X3))
     => $less('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK143(X0)),1))),'fun_app$e'(X0,sK143(X0))) ),
    introduced(choice_axiom,[]) ).

tff(f1347,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X2),'fun_app$e'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : $less('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))),'fun_app$e'(X0,X3)) ),
    inference(flattening,[],[f1346]) ).

tff(f1346,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'(X0,X2),'fun_app$e'(X0,X1))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ? [X3: 'Nat$'] : $less('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))),'fun_app$e'(X0,X3)) ),
    inference(ennf_transformation,[],[f791]) ).

tff(f791,plain,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & ! [X3: 'Nat$'] : ~ $less('fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1))),'fun_app$e'(X0,X3)) )
     => ~ $less('fun_app$e'(X0,X2),'fun_app$e'(X0,X1)) ),
    inference(theory_normalization,[],[f552]) ).

tff(f552,axiom,
    ! [X0: 'Nat_int_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ! [X3: 'Nat$'] : $lesseq('fun_app$e'(X0,X3),'fun_app$e'(X0,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))) )
     => $lesseq('fun_app$e'(X0,X1),'fun_app$e'(X0,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom550) ).

tff(f12444,plain,
    ( spl310_680
    | ~ spl310_44
    | ~ spl310_654 ),
    inference(avatar_split_clause,[],[f12397,f12280,f4414,f12441]) ).

tff(f4414,plain,
    ( spl310_44
  <=> ! [X0: $int] : ~ $less(sK96(X0),X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_44])]) ).

tff(f12397,plain,
    ( $less(0,sK96($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))))
    | ~ spl310_44
    | ~ spl310_654 ),
    inference(resolution,[],[f12281,f4415]) ).

tff(f4415,plain,
    ( ! [X0: $int] : ~ $less(sK96(X0),X0)
    | ~ spl310_44 ),
    inference(avatar_component_clause,[],[f4414]) ).

tff(f12439,plain,
    spl310_679,
    inference(avatar_split_clause,[],[f3959,f12437]) ).

tff(f12437,plain,
    ( spl310_679
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
        | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP5(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_679])]) ).

tff(f3959,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP5(X0,X1,X2) ),
    inference(evaluation,[],[f2914]) ).

tff(f2914,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP5(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1932]) ).

tff(f12435,plain,
    spl310_678,
    inference(avatar_split_clause,[],[f3639,f12433]) ).

tff(f12433,plain,
    ( spl310_678
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK300(X1))),'fun_app$p'(X1,sK301(X1)))
        | ~ $less(X2,X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_678])]) ).

tff(f3639,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK300(X1))),'fun_app$p'(X1,sK301(X1)))
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2318]) ).

tff(f2318,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK300(X1))),'fun_app$p'(X1,sK301(X1)))
        & $less(sK300(X1),sK301(X1)) )
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK300,sK301])],[f1564,f2317]) ).

tff(f2317,plain,
    ! [X1: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK300(X1))),'fun_app$p'(X1,sK301(X1)))
        & $less(sK300(X1),sK301(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1564,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(flattening,[],[f1563]) ).

tff(f1563,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(ennf_transformation,[],[f483]) ).

tff(f483,axiom,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & $less(X2,X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom481) ).

tff(f12431,plain,
    spl310_677,
    inference(avatar_split_clause,[],[f3637,f12429]) ).

tff(f12429,plain,
    ( spl310_677
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK298(X1))),'fun_app$p'(X1,sK299(X1)))
        | ~ $less(X2,X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_677])]) ).

tff(f3637,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK298(X1))),'fun_app$p'(X1,sK299(X1)))
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2316]) ).

tff(f2316,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK298(X1))),'fun_app$p'(X1,sK299(X1)))
        & $less(sK298(X1),sK299(X1)) )
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK298,sK299])],[f1562,f2315]) ).

tff(f2315,plain,
    ! [X1: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK298(X1))),'fun_app$p'(X1,sK299(X1)))
        & $less(sK298(X1),sK299(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1562,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(flattening,[],[f1561]) ).

tff(f1561,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(ennf_transformation,[],[f291]) ).

tff(f291,axiom,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & $less(X2,X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom289) ).

tff(f12427,plain,
    spl310_676,
    inference(avatar_split_clause,[],[f3635,f12425]) ).

tff(f12425,plain,
    ( spl310_676
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK296(X1))),'fun_app$p'(X1,sK297(X1)))
        | $less(X3,X2)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_676])]) ).

tff(f3635,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK296(X1))),'fun_app$p'(X1,sK297(X1)))
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2314]) ).

tff(f2314,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK296(X1))),'fun_app$p'(X1,sK297(X1)))
        & ~ $less(sK297(X1),sK296(X1)) )
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK296,sK297])],[f1560,f2313]) ).

tff(f2313,plain,
    ! [X1: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK296(X1))),'fun_app$p'(X1,sK297(X1)))
        & ~ $less(sK297(X1),sK296(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1560,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(flattening,[],[f1559]) ).

tff(f1559,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(ennf_transformation,[],[f855]) ).

tff(f855,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & ~ $less(X3,X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3)) ),
    inference(theory_normalization,[],[f198]) ).

tff(f198,axiom,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & $lesseq(X2,X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom196) ).

tff(f12423,plain,
    spl310_675,
    inference(avatar_split_clause,[],[f3633,f12421]) ).

tff(f12421,plain,
    ( spl310_675
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK294(X1))),'fun_app$p'(X1,sK295(X1)))
        | $less(X3,X2)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_675])]) ).

tff(f3633,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK294(X1))),'fun_app$p'(X1,sK295(X1)))
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2312]) ).

tff(f2312,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK294(X1))),'fun_app$p'(X1,sK295(X1)))
        & ~ $less(sK295(X1),sK294(X1)) )
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK294,sK295])],[f1558,f2311]) ).

tff(f2311,plain,
    ! [X1: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK294(X1))),'fun_app$p'(X1,sK295(X1)))
        & ~ $less(sK295(X1),sK294(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1558,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(flattening,[],[f1557]) ).

tff(f1557,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(ennf_transformation,[],[f854]) ).

tff(f854,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & ~ $less(X3,X2)
        & 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3)) ),
    inference(theory_normalization,[],[f493]) ).

tff(f493,axiom,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & $lesseq(X2,X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom491) ).

tff(f12419,plain,
    spl310_674,
    inference(avatar_split_clause,[],[f3545,f12417]) ).

tff(f12417,plain,
    ( spl310_674
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK212(X2))),'fun_app$p'(X2,sK213(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_674])]) ).

tff(f3545,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK212(X2))),'fun_app$p'(X2,sK213(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2230]) ).

tff(f2230,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK212(X2))),'fun_app$p'(X2,sK213(X2)))
        & $less(sK212(X2),sK213(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK212,sK213])],[f1464,f2229]) ).

tff(f2229,plain,
    ! [X2: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK212(X2))),'fun_app$p'(X2,sK213(X2)))
        & $less(sK212(X2),sK213(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1464,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1463]) ).

tff(f1463,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f495]) ).

tff(f495,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
        & $less(X0,X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom493) ).

tff(f12415,plain,
    spl310_673,
    inference(avatar_split_clause,[],[f3543,f12413]) ).

tff(f12413,plain,
    ( spl310_673
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK210(X2))),'fun_app$p'(X2,sK211(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_673])]) ).

tff(f3543,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK210(X2))),'fun_app$p'(X2,sK211(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2228]) ).

tff(f2228,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK210(X2))),'fun_app$p'(X2,sK211(X2)))
        & $less(sK210(X2),sK211(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK210,sK211])],[f1462,f2227]) ).

tff(f2227,plain,
    ! [X2: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK210(X2))),'fun_app$p'(X2,sK211(X2)))
        & $less(sK210(X2),sK211(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1462,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1461]) ).

tff(f1461,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f286]) ).

tff(f286,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
        & $less(X0,X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom284) ).

tff(f12411,plain,
    spl310_672,
    inference(avatar_split_clause,[],[f3527,f12409]) ).

tff(f12409,plain,
    ( spl310_672
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK200(X2))),'fun_app$p'(X2,sK201(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_672])]) ).

tff(f3527,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK200(X2))),'fun_app$p'(X2,sK201(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2218]) ).

tff(f2218,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK200(X2))),'fun_app$p'(X2,sK201(X2)))
        & ~ $less(sK201(X2),sK200(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK200,sK201])],[f1440,f2217]) ).

tff(f2217,plain,
    ! [X2: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK200(X2))),'fun_app$p'(X2,sK201(X2)))
        & ~ $less(sK201(X2),sK200(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1440,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(flattening,[],[f1439]) ).

tff(f1439,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f830]) ).

tff(f830,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
        & ~ $less(X1,X0) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3) ),
    inference(theory_normalization,[],[f489]) ).

tff(f489,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
        & $lesseq(X0,X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom487) ).

tff(f12407,plain,
    spl310_671,
    inference(avatar_split_clause,[],[f3525,f12405]) ).

tff(f12405,plain,
    ( spl310_671
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK198(X2))),'fun_app$p'(X2,sK199(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_671])]) ).

tff(f3525,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK198(X2))),'fun_app$p'(X2,sK199(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2216]) ).

tff(f2216,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK198(X2))),'fun_app$p'(X2,sK199(X2)))
        & ~ $less(sK199(X2),sK198(X2)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK198,sK199])],[f1438,f2215]) ).

tff(f2215,plain,
    ! [X2: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK198(X2))),'fun_app$p'(X2,sK199(X2)))
        & ~ $less(sK199(X2),sK198(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1438,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(flattening,[],[f1437]) ).

tff(f1437,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f829]) ).

tff(f829,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
        & ~ $less(X1,X0) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3) ),
    inference(theory_normalization,[],[f195]) ).

tff(f195,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
        & $lesseq(X0,X1) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom193) ).

tff(f12403,plain,
    spl310_670,
    inference(avatar_split_clause,[],[f3296,f12401]) ).

tff(f12401,plain,
    ( spl310_670
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP57(X0,X1,X2)
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_670])]) ).

tff(f3296,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP57(X0,X1,X2)
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2064]) ).

tff(f12370,plain,
    spl310_669,
    inference(avatar_split_clause,[],[f3293,f12368]) ).

tff(f12368,plain,
    ( spl310_669
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP57(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_669])]) ).

tff(f3293,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP57(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2064]) ).

tff(f12366,plain,
    spl310_668,
    inference(avatar_split_clause,[],[f3275,f12364]) ).

tff(f12364,plain,
    ( spl310_668
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP53(X0,X1,X2)
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_668])]) ).

tff(f3275,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP53(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2053]) ).

tff(f12362,plain,
    spl310_667,
    inference(avatar_split_clause,[],[f3272,f12360]) ).

tff(f12360,plain,
    ( spl310_667
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP53(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_667])]) ).

tff(f3272,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP53(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2053]) ).

tff(f12358,plain,
    spl310_666,
    inference(avatar_split_clause,[],[f3254,f12356]) ).

tff(f12356,plain,
    ( spl310_666
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP49(X0,X1,X2)
        | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_666])]) ).

tff(f3254,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP49(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(cnf_transformation,[],[f2043]) ).

tff(f12354,plain,
    spl310_665,
    inference(avatar_split_clause,[],[f3251,f12352]) ).

tff(f12352,plain,
    ( spl310_665
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP49(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_665])]) ).

tff(f3251,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP49(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2043]) ).

tff(f12350,plain,
    spl310_664,
    inference(avatar_split_clause,[],[f3233,f12348]) ).

tff(f12348,plain,
    ( spl310_664
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP45(X0,X1,X2)
        | ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_664])]) ).

tff(f3233,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP45(X0,X1,X2)
      | ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2033]) ).

tff(f2033,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP45(X0,X1,X2)
        | ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP45(X0,X1,X2) ) ),
    inference(rectify,[],[f2032]) ).

tff(f2032,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP45(X0,X2,X1)
        | ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP45(X0,X2,X1) ) ),
    inference(flattening,[],[f2031]) ).

tff(f2031,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP45(X0,X2,X1)
        | ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP45(X0,X2,X1) ) ),
    inference(nnf_transformation,[],[f1645]) ).

tff(f1645,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( sP45(X0,X2,X1)
    <=> ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP45])]) ).

tff(f12346,plain,
    spl310_663,
    inference(avatar_split_clause,[],[f3230,f12344]) ).

tff(f12344,plain,
    ( spl310_663
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP45(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_663])]) ).

tff(f3230,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP45(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2033]) ).

tff(f12342,plain,
    spl310_662,
    inference(avatar_split_clause,[],[f3212,f12340]) ).

tff(f12340,plain,
    ( spl310_662
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP38(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_662])]) ).

tff(f3212,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP38(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2023]) ).

tff(f12338,plain,
    spl310_661,
    inference(avatar_split_clause,[],[f3198,f12336]) ).

tff(f12336,plain,
    ( spl310_661
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP42(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_661])]) ).

tff(f3198,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP42(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2015]) ).

tff(f12334,plain,
    spl310_660,
    inference(avatar_split_clause,[],[f3197,f12332]) ).

tff(f12332,plain,
    ( spl310_660
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP42(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_660])]) ).

tff(f3197,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP42(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2015]) ).

tff(f12330,plain,
    ( spl310_659
    | ~ spl310_80
    | ~ spl310_329 ),
    inference(avatar_split_clause,[],[f8344,f7234,f4642,f12328]) ).

tff(f7234,plain,
    ( spl310_329
  <=> $less(0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_329])]) ).

tff(f8344,plain,
    ( ! [X0: $int] :
        ( ~ $less(X0,0)
        | $less(X0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))) )
    | ~ spl310_80
    | ~ spl310_329 ),
    inference(resolution,[],[f7236,f4643]) ).

tff(f7236,plain,
    ( $less(0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))))
    | ~ spl310_329 ),
    inference(avatar_component_clause,[],[f7234]) ).

tff(f12301,plain,
    spl310_658,
    inference(avatar_split_clause,[],[f4115,f12299]) ).

tff(f12299,plain,
    ( spl310_658
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X0)) = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP18(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_658])]) ).

tff(f4115,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3940,f864]) ).

tff(f3940,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $uminus($sum('fun_app$e'('of_nat$',X0),1)) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(evaluation,[],[f3939]) ).

tff(f3939,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP18(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2974]) ).

tff(f2974,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP18(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1957]) ).

tff(f12297,plain,
    spl310_657,
    inference(avatar_split_clause,[],[f4020,f12295]) ).

tff(f12295,plain,
    ( spl310_657
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',X0)) )
        | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK101(X0,X1))) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_657])]) ).

tff(f4020,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',X0)) )
      | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK101(X0,X1))) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f4019,f864]) ).

tff(f4019,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK101(X0,X1))) )
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2632,f864]) ).

tff(f2632,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK101(X0,X1)),1) )
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1752]) ).

tff(f1752,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK101(X0,X1)),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK101(X0,X1))) )
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK101])],[f1134,f1751]) ).

tff(f1751,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
     => ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK101(X0,X1)),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK101(X0,X1))) ) ),
    introduced(choice_axiom,[]) ).

tff(f1134,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f937]) ).

tff(f937,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ~ ( ! [X2: 'Nat$'] :
            ~ ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X0),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(flattening,[],[f936]) ).

tff(f936,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ~ ( ! [X2: 'Nat$'] :
            ~ ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ~ 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(true_and_false_elimination,[],[f403]) ).

tff(f403,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ! [X2: 'Nat$'] :
            ( ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
           => $false )
        & ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
         => $false )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom401) ).

tff(f12293,plain,
    spl310_656,
    inference(avatar_split_clause,[],[f3951,f12291]) ).

tff(f12291,plain,
    ( spl310_656
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X1))
        | ~ sP9(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_656])]) ).

tff(f3951,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP9(X0,X1,X2) ),
    inference(evaluation,[],[f2945]) ).

tff(f2945,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP9(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1944]) ).

tff(f12289,plain,
    spl310_655,
    inference(avatar_split_clause,[],[f3410,f12287]) ).

tff(f12287,plain,
    ( spl310_655
  <=> ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1)))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',sK181(X0)),'fun_app$e'('of_nat$',sK182(X0))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_655])]) ).

tff(f3410,plain,
    ! [X2: 'Nat$',X0: 'Nat_nat_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$','fun_app$r'(X0,X2)),'fun_app$e'('of_nat$','fun_app$r'(X0,X1)))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',sK181(X0)),'fun_app$e'('of_nat$',sK182(X0))) ),
    inference(cnf_transformation,[],[f2157]) ).

tff(f12282,plain,
    ( spl310_654
    | ~ spl310_92
    | ~ spl310_329 ),
    inference(avatar_split_clause,[],[f8343,f7234,f4696,f12280]) ).

tff(f8343,plain,
    ( ! [X0: $int] :
        ( $less(0,X0)
        | $less(X0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$')))) )
    | ~ spl310_92
    | ~ spl310_329 ),
    inference(resolution,[],[f7236,f4697]) ).

tff(f11802,plain,
    spl310_653,
    inference(avatar_split_clause,[],[f4204,f11800]) ).

tff(f11800,plain,
    ( spl310_653
  <=> ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'('fun_app$q'(X2,sK138(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK138(X2)))))
        | 'fun_app$j'('fun_app$q'(X2,X0),X1)
        | sP60(X2)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_653])]) ).

tff(f4204,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'('fun_app$q'(X2,sK138(X2)),'nat$'($sum(1,'fun_app$e'('of_nat$',sK138(X2)))))
      | 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | sP60(X2)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3322,f864]) ).

tff(f3322,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | sP60(X2)
      | ~ 'fun_app$j'('fun_app$q'(X2,sK138(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK138(X2)),1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f2078]) ).

tff(f2078,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | sP60(X2)
      | ~ 'fun_app$j'('fun_app$q'(X2,sK138(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK138(X2)),1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK138])],[f2076,f2077]) ).

tff(f2077,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X3: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
     => ~ 'fun_app$j'('fun_app$q'(X2,sK138(X2)),'nat$'($sum('fun_app$e'('of_nat$',sK138(X2)),1))) ),
    introduced(choice_axiom,[]) ).

tff(f2076,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | sP60(X2)
      | ? [X3: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(rectify,[],[f1665]) ).

tff(f1665,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | sP60(X2)
      | ? [X6: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X6),'nat$'($sum('fun_app$e'('of_nat$',X6),1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(definition_folding,[],[f1336,f1664]) ).

tff(f1664,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X2,X3),X5)
          & 'fun_app$j'('fun_app$q'(X2,X4),X5)
          & 'fun_app$j'('fun_app$q'(X2,X3),X4)
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
      | ~ sP60(X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP60])]) ).

tff(f1336,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ? [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X2,X3),X5)
          & 'fun_app$j'('fun_app$q'(X2,X4),X5)
          & 'fun_app$j'('fun_app$q'(X2,X3),X4)
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
      | ? [X6: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X6),'nat$'($sum('fun_app$e'('of_nat$',X6),1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(flattening,[],[f1335]) ).

tff(f1335,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X2,X0),X1)
      | ? [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X2,X3),X5)
          & 'fun_app$j'('fun_app$q'(X2,X4),X5)
          & 'fun_app$j'('fun_app$q'(X2,X3),X4)
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
      | ? [X6: 'Nat$'] : ~ 'fun_app$j'('fun_app$q'(X2,X6),'nat$'($sum('fun_app$e'('of_nat$',X6),1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f966]) ).

tff(f966,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( ( ! [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
            ( ( 'fun_app$j'('fun_app$q'(X2,X4),X5)
              & 'fun_app$j'('fun_app$q'(X2,X3),X4)
              & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
           => 'fun_app$j'('fun_app$q'(X2,X3),X5) )
        & ! [X6: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X6),'nat$'($sum('fun_app$e'('of_nat$',X6),1)))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'('fun_app$q'(X2,X0),X1) ),
    inference(rectify,[],[f383]) ).

tff(f383,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_nat_bool_fun_fun$'] :
      ( ( ! [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
            ( ( 'fun_app$j'('fun_app$q'(X2,X4),X5)
              & 'fun_app$j'('fun_app$q'(X2,X3),X4)
              & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
           => 'fun_app$j'('fun_app$q'(X2,X3),X5) )
        & ! [X3: 'Nat$'] : 'fun_app$j'('fun_app$q'(X2,X3),'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'('fun_app$q'(X2,X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom381) ).

tff(f11798,plain,
    spl310_652,
    inference(avatar_split_clause,[],[f4189,f11796]) ).

tff(f11796,plain,
    ( spl310_652
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $sum('fun_app$e'('of_nat$',X0),1) = $sum('fun_app$e'('of_nat$',X1),$sum(1,'fun_app$e'('of_nat$',X2))) )
        | ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_652])]) ).

tff(f4189,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X0),1) = $sum('fun_app$e'('of_nat$',X1),$sum(1,'fun_app$e'('of_nat$',X2))) )
      | ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    inference(forward_demodulation,[],[f3118,f864]) ).

tff(f3118,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X0),1) = $sum('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),1)) )
      | ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    inference(cnf_transformation,[],[f1211]) ).

tff(f1211,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X0),1) = $sum('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),1)) )
      | ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    inference(ennf_transformation,[],[f527]) ).

tff(f527,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
     => ( $sum('fun_app$e'('of_nat$',X0),1) = $sum('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),1)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom525) ).

tff(f11794,plain,
    spl310_651,
    inference(avatar_split_clause,[],[f3890,f11792]) ).

tff(f11792,plain,
    ( spl310_651
  <=> ! [X4: 'Nat$',X3: 'Nat$',X2: $int,X1: $int] :
        ( sP79($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X1,X2,X3,X4)
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_651])]) ).

tff(f3890,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP79($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X1,X2,X3,X4)
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(equality_resolution,[],[f3667]) ).

tff(f3667,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP79(X0,X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != X0 )
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(cnf_transformation,[],[f2336]) ).

tff(f11790,plain,
    spl310_650,
    inference(avatar_split_clause,[],[f3887,f11788]) ).

tff(f11788,plain,
    ( spl310_650
  <=> ! [X4: 'Nat$',X3: 'Nat$',X2: $int,X1: $int] :
        ( sP77($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X1,X2,X3,X4)
        | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_650])]) ).

tff(f3887,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP77($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1),X1,X2,X3,X4)
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(equality_resolution,[],[f3658]) ).

tff(f3658,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP77(X0,X1,X2,X3,X4)
      | ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X4),$uminus('fun_app$e'('of_nat$',X3))))),X2),X1) != X0 )
      | $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(cnf_transformation,[],[f2331]) ).

tff(f11786,plain,
    spl310_649,
    inference(avatar_split_clause,[],[f3646,f11784]) ).

tff(f11784,plain,
    ( spl310_649
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less$',sK308(X1)),sK309(X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_649])]) ).

tff(f3646,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK308(X1)),sK309(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2326]) ).

tff(f11782,plain,
    spl310_648,
    inference(avatar_split_clause,[],[f3644,f11780]) ).

tff(f11780,plain,
    ( spl310_648
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK306(X1)),sK307(X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_648])]) ).

tff(f3644,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK306(X1)),sK307(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2324]) ).

tff(f11776,plain,
    spl310_647,
    inference(avatar_split_clause,[],[f3642,f11774]) ).

tff(f11774,plain,
    ( spl310_647
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK304(X1)),sK305(X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_647])]) ).

tff(f3642,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK304(X1)),sK305(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2322]) ).

tff(f11772,plain,
    spl310_646,
    inference(avatar_split_clause,[],[f3640,f11770]) ).

tff(f11770,plain,
    ( spl310_646
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N$',X1: 'N_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less$',sK302(X1)),sK303(X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_646])]) ).

tff(f3640,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK302(X1)),sK303(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X2)) ),
    inference(cnf_transformation,[],[f2320]) ).

tff(f11768,plain,
    spl310_645,
    inference(avatar_split_clause,[],[f3610,f11766]) ).

tff(f11766,plain,
    ( spl310_645
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
        | 'fun_app$d'('fun_app$g'('less_eq$',sK276(X2)),sK277(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_645])]) ).

tff(f3610,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | 'fun_app$d'('fun_app$g'('less_eq$',sK276(X2)),sK277(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2294]) ).

tff(f11764,plain,
    spl310_644,
    inference(avatar_split_clause,[],[f3608,f11762]) ).

tff(f11762,plain,
    ( spl310_644
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
        | 'fun_app$d'('fun_app$g'('less$',sK274(X2)),sK275(X2))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_644])]) ).

tff(f3608,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | 'fun_app$d'('fun_app$g'('less$',sK274(X2)),sK275(X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2292]) ).

tff(f11760,plain,
    spl310_643,
    inference(avatar_split_clause,[],[f3606,f11758]) ).

tff(f11758,plain,
    ( spl310_643
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
        | 'fun_app$d'('fun_app$g'('less$',sK272(X2)),sK273(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_643])]) ).

tff(f3606,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | 'fun_app$d'('fun_app$g'('less$',sK272(X2)),sK273(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2290]) ).

tff(f11756,plain,
    spl310_642,
    inference(avatar_split_clause,[],[f3604,f11754]) ).

tff(f11754,plain,
    ( spl310_642
  <=> ! [X0: 'N$',X3: 'N$',X2: 'N_n_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
        | 'fun_app$d'('fun_app$g'('less_eq$',sK270(X2)),sK271(X2))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_642])]) ).

tff(f3604,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | 'fun_app$d'('fun_app$g'('less_eq$',sK270(X2)),sK271(X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X1)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2288]) ).

tff(f11752,plain,
    spl310_641,
    inference(avatar_split_clause,[],[f3557,f11750]) ).

tff(f11750,plain,
    ( spl310_641
  <=> ! [X0: 'Int_set$',X3: 'Int_set$',X2: 'Int_set_int_set_fun$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
        | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK222(X2))),'fun_app$h'(X2,sK223(X2)))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_641])]) ).

tff(f3557,plain,
    ! [X2: 'Int_set_int_set_fun$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK222(X2))),'fun_app$h'(X2,sK223(X2)))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2240]) ).

tff(f2240,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK222(X2))),'fun_app$h'(X2,sK223(X2)))
        & 'fun_app$c'('less_eq$a'(sK222(X2)),sK223(X2)) )
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK222,sK223])],[f1478,f2239]) ).

tff(f2239,plain,
    ! [X2: 'Int_set_int_set_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK222(X2))),'fun_app$h'(X2,sK223(X2)))
        & 'fun_app$c'('less_eq$a'(sK222(X2)),sK223(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1478,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1477]) ).

tff(f1477,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f192]) ).

tff(f192,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5)) )
        & 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom190) ).

tff(f11748,plain,
    spl310_640,
    inference(avatar_split_clause,[],[f3401,f11746]) ).

tff(f11746,plain,
    ( spl310_640
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK168(X0)),sK169(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK171(X0)),sK172(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_640])]) ).

tff(f3401,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK168(X0)),sK169(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK171(X0)),sK172(X0)) ),
    inference(cnf_transformation,[],[f2149]) ).

tff(f2149,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK168(X0)),sK169(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK169(X0)),sK168(X0)) )
      | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK171(X0)),sK172(X0))
        & 'fun_app$d'('fun_app$g'('less$',sK171(X0)),sK172(X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK168,sK169,sK170,sK171,sK172])],[f1373,f2148,f2147,f2146]) ).

tff(f2146,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK168(X0)),sK169(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK169(X0)),sK168(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2147,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X5: 'N$'] : ~ 'fun_app$d'('fun_app$g'(X0,X5),X5)
     => ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0)) ),
    introduced(choice_axiom,[]) ).

tff(f2148,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X6: 'N$',X7: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X6),X7)
          & 'fun_app$d'('fun_app$g'('less$',X6),X7) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK171(X0)),sK172(X0))
        & 'fun_app$d'('fun_app$g'('less$',sK171(X0)),sK172(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1373,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$'] : ~ 'fun_app$d'('fun_app$g'(X0,X5),X5)
      | ? [X6: 'N$',X7: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X6),X7)
          & 'fun_app$d'('fun_app$g'('less$',X6),X7) ) ),
    inference(flattening,[],[f1372]) ).

tff(f1372,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$'] : ~ 'fun_app$d'('fun_app$g'(X0,X5),X5)
      | ? [X6: 'N$',X7: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X6),X7)
          & 'fun_app$d'('fun_app$g'('less$',X6),X7) ) ),
    inference(ennf_transformation,[],[f979]) ).

tff(f979,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X5: 'N$'] : 'fun_app$d'('fun_app$g'(X0,X5),X5)
        & ! [X6: 'N$',X7: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X6),X7)
           => 'fun_app$d'('fun_app$g'(X0,X6),X7) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    inference(rectify,[],[f329]) ).

tff(f329,axiom,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X3: 'N$'] : 'fun_app$d'('fun_app$g'(X0,X3),X3)
        & ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X3),X4)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom327) ).

tff(f11744,plain,
    spl310_639,
    inference(avatar_split_clause,[],[f3400,f11742]) ).

tff(f11742,plain,
    ( spl310_639
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK168(X0)),sK169(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
        | 'fun_app$d'('fun_app$g'('less$',sK171(X0)),sK172(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_639])]) ).

tff(f3400,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK168(X0)),sK169(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
      | 'fun_app$d'('fun_app$g'('less$',sK171(X0)),sK172(X0)) ),
    inference(cnf_transformation,[],[f2149]) ).

tff(f11740,plain,
    spl310_638,
    inference(avatar_split_clause,[],[f3399,f11738]) ).

tff(f11738,plain,
    ( spl310_638
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK169(X0)),sK168(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK171(X0)),sK172(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_638])]) ).

tff(f3399,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK169(X0)),sK168(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK171(X0)),sK172(X0)) ),
    inference(cnf_transformation,[],[f2149]) ).

tff(f11736,plain,
    spl310_637,
    inference(avatar_split_clause,[],[f3398,f11734]) ).

tff(f11734,plain,
    ( spl310_637
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK169(X0)),sK168(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
        | 'fun_app$d'('fun_app$g'('less$',sK171(X0)),sK172(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_637])]) ).

tff(f3398,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK169(X0)),sK168(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK170(X0)),sK170(X0))
      | 'fun_app$d'('fun_app$g'('less$',sK171(X0)),sK172(X0)) ),
    inference(cnf_transformation,[],[f2149]) ).

tff(f11732,plain,
    spl310_636,
    inference(avatar_split_clause,[],[f3397,f11730]) ).

tff(f11730,plain,
    ( spl310_636
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK163(X0)),sK164(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK166(X0)),sK167(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_636])]) ).

tff(f3397,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK163(X0)),sK164(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK166(X0)),sK167(X0)) ),
    inference(cnf_transformation,[],[f2145]) ).

tff(f2145,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK163(X0)),sK164(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK164(X0)),sK163(X0)) )
      | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK166(X0)),sK167(X0))
        & 'fun_app$d'('fun_app$g'('less$',sK167(X0)),sK166(X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK163,sK164,sK165,sK166,sK167])],[f1371,f2144,f2143,f2142]) ).

tff(f2142,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK163(X0)),sK164(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK164(X0)),sK163(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2143,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X5: 'N$'] : ~ 'fun_app$d'('fun_app$g'(X0,X5),X5)
     => ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0)) ),
    introduced(choice_axiom,[]) ).

tff(f2144,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X6: 'N$',X7: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X6),X7)
          & 'fun_app$d'('fun_app$g'('less$',X7),X6) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK166(X0)),sK167(X0))
        & 'fun_app$d'('fun_app$g'('less$',sK167(X0)),sK166(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1371,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$'] : ~ 'fun_app$d'('fun_app$g'(X0,X5),X5)
      | ? [X6: 'N$',X7: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X6),X7)
          & 'fun_app$d'('fun_app$g'('less$',X7),X6) ) ),
    inference(flattening,[],[f1370]) ).

tff(f1370,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$'] : ~ 'fun_app$d'('fun_app$g'(X0,X5),X5)
      | ? [X6: 'N$',X7: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X6),X7)
          & 'fun_app$d'('fun_app$g'('less$',X7),X6) ) ),
    inference(ennf_transformation,[],[f978]) ).

tff(f978,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X5: 'N$'] : 'fun_app$d'('fun_app$g'(X0,X5),X5)
        & ! [X6: 'N$',X7: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X7),X6)
           => 'fun_app$d'('fun_app$g'(X0,X6),X7) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    inference(rectify,[],[f324]) ).

tff(f324,axiom,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X3: 'N$'] : 'fun_app$d'('fun_app$g'(X0,X3),X3)
        & ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom322) ).

tff(f11728,plain,
    spl310_635,
    inference(avatar_split_clause,[],[f3396,f11726]) ).

tff(f11726,plain,
    ( spl310_635
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK163(X0)),sK164(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
        | 'fun_app$d'('fun_app$g'('less$',sK167(X0)),sK166(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_635])]) ).

tff(f3396,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK163(X0)),sK164(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
      | 'fun_app$d'('fun_app$g'('less$',sK167(X0)),sK166(X0)) ),
    inference(cnf_transformation,[],[f2145]) ).

tff(f11724,plain,
    spl310_634,
    inference(avatar_split_clause,[],[f3395,f11722]) ).

tff(f11722,plain,
    ( spl310_634
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK164(X0)),sK163(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK166(X0)),sK167(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_634])]) ).

tff(f3395,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK164(X0)),sK163(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK166(X0)),sK167(X0)) ),
    inference(cnf_transformation,[],[f2145]) ).

tff(f11720,plain,
    spl310_633,
    inference(avatar_split_clause,[],[f3394,f11718]) ).

tff(f11718,plain,
    ( spl310_633
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK164(X0)),sK163(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
        | 'fun_app$d'('fun_app$g'('less$',sK167(X0)),sK166(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_633])]) ).

tff(f3394,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK164(X0)),sK163(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK165(X0)),sK165(X0))
      | 'fun_app$d'('fun_app$g'('less$',sK167(X0)),sK166(X0)) ),
    inference(cnf_transformation,[],[f2145]) ).

tff(f11716,plain,
    spl310_632,
    inference(avatar_split_clause,[],[f3393,f11714]) ).

tff(f11714,plain,
    ( spl310_632
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | ~ 'fun_app$'('fun_app$a'(X0,sK158(X0)),sK159(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK161(X0)),sK162(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_632])]) ).

tff(f3393,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ~ 'fun_app$'('fun_app$a'(X0,sK158(X0)),sK159(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK161(X0)),sK162(X0)) ),
    inference(cnf_transformation,[],[f2141]) ).

tff(f2141,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ( ~ 'fun_app$'('fun_app$a'(X0,sK158(X0)),sK159(X0))
        & 'fun_app$'('fun_app$a'(X0,sK159(X0)),sK158(X0)) )
      | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
      | ( ~ 'fun_app$'('fun_app$a'(X0,sK161(X0)),sK162(X0))
        & $less(sK161(X0),sK162(X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK158,sK159,sK160,sK161,sK162])],[f1369,f2140,f2139,f2138]) ).

tff(f2138,plain,
    ! [X0: 'Int_int_bool_fun_fun$'] :
      ( ? [X3: $int,X4: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X3),X4)
          & 'fun_app$'('fun_app$a'(X0,X4),X3) )
     => ( ~ 'fun_app$'('fun_app$a'(X0,sK158(X0)),sK159(X0))
        & 'fun_app$'('fun_app$a'(X0,sK159(X0)),sK158(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2139,plain,
    ! [X0: 'Int_int_bool_fun_fun$'] :
      ( ? [X5: $int] : ~ 'fun_app$'('fun_app$a'(X0,X5),X5)
     => ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0)) ),
    introduced(choice_axiom,[]) ).

tff(f2140,plain,
    ! [X0: 'Int_int_bool_fun_fun$'] :
      ( ? [X6: $int,X7: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X6),X7)
          & $less(X6,X7) )
     => ( ~ 'fun_app$'('fun_app$a'(X0,sK161(X0)),sK162(X0))
        & $less(sK161(X0),sK162(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1369,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ? [X3: $int,X4: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X3),X4)
          & 'fun_app$'('fun_app$a'(X0,X4),X3) )
      | ? [X5: $int] : ~ 'fun_app$'('fun_app$a'(X0,X5),X5)
      | ? [X6: $int,X7: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X6),X7)
          & $less(X6,X7) ) ),
    inference(flattening,[],[f1368]) ).

tff(f1368,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ? [X3: $int,X4: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X3),X4)
          & 'fun_app$'('fun_app$a'(X0,X4),X3) )
      | ? [X5: $int] : ~ 'fun_app$'('fun_app$a'(X0,X5),X5)
      | ? [X6: $int,X7: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X6),X7)
          & $less(X6,X7) ) ),
    inference(ennf_transformation,[],[f977]) ).

tff(f977,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( ( ! [X3: $int,X4: $int] :
            ( 'fun_app$'('fun_app$a'(X0,X4),X3)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) )
        & ! [X5: $int] : 'fun_app$'('fun_app$a'(X0,X5),X5)
        & ! [X6: $int,X7: $int] :
            ( $less(X6,X7)
           => 'fun_app$'('fun_app$a'(X0,X6),X7) ) )
     => 'fun_app$'('fun_app$a'(X0,X1),X2) ),
    inference(rectify,[],[f328]) ).

tff(f328,axiom,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( ( ! [X3: $int,X4: $int] :
            ( 'fun_app$'('fun_app$a'(X0,X4),X3)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) )
        & ! [X3: $int] : 'fun_app$'('fun_app$a'(X0,X3),X3)
        & ! [X3: $int,X4: $int] :
            ( $less(X3,X4)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) ) )
     => 'fun_app$'('fun_app$a'(X0,X1),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom326) ).

tff(f11712,plain,
    spl310_631,
    inference(avatar_split_clause,[],[f3391,f11710]) ).

tff(f11710,plain,
    ( spl310_631
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | 'fun_app$'('fun_app$a'(X0,sK159(X0)),sK158(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK161(X0)),sK162(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_631])]) ).

tff(f3391,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | 'fun_app$'('fun_app$a'(X0,sK159(X0)),sK158(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK161(X0)),sK162(X0)) ),
    inference(cnf_transformation,[],[f2141]) ).

tff(f11708,plain,
    spl310_630,
    inference(avatar_split_clause,[],[f3370,f11706]) ).

tff(f11706,plain,
    ( spl310_630
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        | 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        | sP73(X0,X1,X2)
        | sP72(X1,X2,X0)
        | sP71(X2,X0,X1)
        | sP70(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_630])]) ).

tff(f3370,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      | 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | sP73(X0,X1,X2)
      | sP72(X1,X2,X0)
      | sP71(X2,X0,X1)
      | sP70(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1682]) ).

tff(f1682,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
      | sP73(X0,X1,X2)
      | sP72(X1,X2,X0)
      | sP71(X2,X0,X1)
      | sP70(X2,X1,X0) ),
    inference(definition_folding,[],[f1345,f1681,f1680,f1679,f1678]) ).

tff(f1678,plain,
    ! [X2: 'N$',X1: 'N$',X0: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
      | ~ sP70(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP70])]) ).

tff(f1679,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
      | ~ sP71(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP71])]) ).

tff(f1680,plain,
    ! [X1: 'N$',X2: 'N$',X0: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ~ sP72(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP72])]) ).

tff(f1681,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ~ sP73(X0,X1,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP73])]) ).

tff(f1345,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    inference(ennf_transformation,[],[f974]) ).

tff(f974,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ~ ( ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    inference(true_and_false_elimination,[],[f233]) ).

tff(f233,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
         => $false ) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom231) ).

tff(f11704,plain,
    spl310_629,
    inference(avatar_split_clause,[],[f3358,f11702]) ).

tff(f11702,plain,
    ( spl310_629
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        | 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        | sP69(X0,X1,X2)
        | sP68(X2,X0,X1)
        | sP67(X1,X2,X0)
        | sP66(X0,X2,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_629])]) ).

tff(f3358,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      | 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
      | sP69(X0,X1,X2)
      | sP68(X2,X0,X1)
      | sP67(X1,X2,X0)
      | sP66(X0,X2,X1) ),
    inference(cnf_transformation,[],[f1677]) ).

tff(f1677,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
      | sP69(X0,X1,X2)
      | sP68(X2,X0,X1)
      | sP67(X1,X2,X0)
      | sP66(X0,X2,X1) ),
    inference(definition_folding,[],[f1344,f1676,f1675,f1674,f1673]) ).

tff(f1673,plain,
    ! [X0: 'N$',X2: 'N$',X1: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
      | ~ sP66(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP66])]) ).

tff(f1674,plain,
    ! [X1: 'N$',X2: 'N$',X0: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
      | ~ sP67(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP67])]) ).

tff(f1675,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ~ sP68(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP68])]) ).

tff(f1676,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ~ sP69(X0,X1,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP69])]) ).

tff(f1344,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
      | ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    inference(ennf_transformation,[],[f973]) ).

tff(f973,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ~ ( ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
        & ~ ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    inference(true_and_false_elimination,[],[f230]) ).

tff(f230,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X2) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X0) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
            & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
            & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
         => $false )
        & ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
            & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
         => $false ) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom228) ).

tff(f11700,plain,
    spl310_628,
    inference(avatar_split_clause,[],[f3110,f11698]) ).

tff(f11698,plain,
    ( spl310_628
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_628])]) ).

tff(f3110,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1203]) ).

tff(f1203,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( $less(0,'fun_app$e'('of_nat$',X1))
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f759]) ).

tff(f759,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
         => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
        & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
         => $less(0,'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(theory_normalization,[],[f545]) ).

tff(f545,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
         => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
        & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
         => $less(0,'fun_app$e'('of_nat$',X1)) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom543) ).

tff(f11261,plain,
    spl310_627,
    inference(avatar_split_clause,[],[f4187,f11259]) ).

tff(f11259,plain,
    ( spl310_627
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),$sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))
        | sP29(X0,X1,X2)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_627])]) ).

tff(f4187,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),$sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))))
      | sP29(X0,X1,X2)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3914,f864]) ).

tff(f3914,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP29(X0,X1,X2)
      | $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(duplicate_literal_removal,[],[f3078]) ).

tff(f3078,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP29(X0,X1,X2)
      | $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1626]) ).

tff(f1626,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP29(X0,X1,X2)
          | ( ( $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X2),$sum(0,1))
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & sP30(X1,X0,X2) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1196,f1625,f1624,f1623]) ).

tff(f1623,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),$sum(0,1))
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP28(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP28])]) ).

tff(f1196,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X2),$sum(0,1))
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
            & ( ( 0 = 0 )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),$sum(0,1))
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1195]) ).

tff(f1195,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X2),$sum(0,1))
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) )
            & ( ( 0 = 0 )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),$sum(0,1))
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f886]) ).

tff(f886,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1)) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum(0,1)) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1))) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1)) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum(0,1)) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
             => ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f755]) ).

tff(f755,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1)) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum(0,1)) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1))) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum(0,1))) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1)) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum(0,1)) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
             => ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus($sum('fun_app$e'('of_nat$',X1),1))) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f245]) ).

tff(f245,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1)) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum(0,1)) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( $difference($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) = $difference('fun_app$e'('of_nat$',X2),$sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1)) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( 0 = $difference('fun_app$e'('of_nat$',X2),$sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1)) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( $difference($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) = $difference('fun_app$e'('of_nat$',X2),$sum(0,1)) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
                 => ( 0 = $difference('fun_app$e'('of_nat$',X2),$sum(0,1)) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1)) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum(0,1)) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
             => ( 0 = $difference($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$sum('fun_app$e'('of_nat$',X1),1))
             => ( 0 = 0 ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom243) ).

tff(f11257,plain,
    spl310_626,
    inference(avatar_split_clause,[],[f4167,f11255]) ).

tff(f11255,plain,
    ( spl310_626
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))),'fun_app$e'('of_nat$',X2))
        | sP26(X2,X0,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_626])]) ).

tff(f4167,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))),'fun_app$e'('of_nat$',X2))
      | sP26(X2,X0,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3918,f864]) ).

tff(f3918,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP26(X2,X0,X1)
      | $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(duplicate_literal_removal,[],[f3065]) ).

tff(f3065,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP26(X2,X0,X1)
      | $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1622]) ).

tff(f1622,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP26(X2,X0,X1)
          | ( ( $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less($sum(0,1),'fun_app$e'('of_nat$',X2))
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & sP27(X2,X0,X1) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1194,f1621,f1620,f1619]) ).

tff(f1619,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum(0,1),'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP25(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP25])]) ).

tff(f1194,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
                & ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
                & ( ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less($sum(0,1),'fun_app$e'('of_nat$',X2))
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) )
              | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = 0 )
              | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
          | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less($sum(0,1),'fun_app$e'('of_nat$',X2))
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1193]) ).

tff(f1193,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
                & ( ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
                & ( ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less($sum(0,1),'fun_app$e'('of_nat$',X2))
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) )
              | $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
            & ( ( 0 = 0 )
              | ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
          | ( ~ $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less($sum(0,1),'fun_app$e'('of_nat$',X2))
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f885]) ).

tff(f885,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2)) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less($sum(0,1),'fun_app$e'('of_nat$',X2)) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( 0 = $sum($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less($sum(0,1),'fun_app$e'('of_nat$',X2)) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
             => ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f754]) ).

tff(f754,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2)) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less($sum(0,1),'fun_app$e'('of_nat$',X2)) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1),$uminus('fun_app$e'('of_nat$',X2))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( 0 = $sum($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( 0 = $sum($sum(0,1),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less($sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),1),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less($sum(0,1),'fun_app$e'('of_nat$',X2)) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
             => ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),1),$uminus($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f246]) ).

tff(f246,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1),'fun_app$e'('of_nat$',X2)) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less($sum(0,1),'fun_app$e'('of_nat$',X2)) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( $difference($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) = $difference($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1),'fun_app$e'('of_nat$',X2)) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( 0 = $difference($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1),'fun_app$e'('of_nat$',X2)) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( $difference($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) = $difference($sum(0,1),'fun_app$e'('of_nat$',X2)) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
                 => ( 0 = $difference($sum(0,1),'fun_app$e'('of_nat$',X2)) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less($sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less($sum(0,1),'fun_app$e'('of_nat$',X2)) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
             => ( 0 = $difference($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X1),1),$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)))
             => ( 0 = 0 ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom244) ).

tff(f11253,plain,
    spl310_625,
    inference(avatar_split_clause,[],[f4043,f11251]) ).

tff(f11251,plain,
    ( spl310_625
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) != $sum(1,'fun_app$e'('of_nat$',X2)) )
        | $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_625])]) ).

tff(f4043,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) != $sum(1,'fun_app$e'('of_nat$',X2)) )
      | $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ),
    inference(forward_demodulation,[],[f4042,f864]) ).

tff(f4042,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X2),1) ) ),
    inference(forward_demodulation,[],[f2860,f864]) ).

tff(f2860,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X2),1) ) ),
    inference(cnf_transformation,[],[f1908]) ).

tff(f1908,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
        | ! [X2: 'Nat$'] :
            ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
            | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X2),1) ) ) )
      & ( ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK115(X0,X1)))
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK115(X0,X1)),1) ) )
        | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK115])],[f1906,f1907]) ).

tff(f1907,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X3: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3))
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X3),1) ) )
     => ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK115(X0,X1)))
        & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK115(X0,X1)),1) ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1906,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
        | ! [X2: 'Nat$'] :
            ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
            | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X2),1) ) ) )
      & ( ? [X3: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3))
            & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X3),1) ) )
        | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ),
    inference(rectify,[],[f1905]) ).

tff(f1905,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
        | ! [X2: 'Nat$'] :
            ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
            | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X2),1) ) ) )
      & ( ? [X2: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
            & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) ) )
        | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) ),
    inference(nnf_transformation,[],[f390]) ).

tff(f390,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
    <=> ? [X2: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom388) ).

tff(f11249,plain,
    spl310_624,
    inference(avatar_split_clause,[],[f4021,f11247]) ).

tff(f11247,plain,
    ( spl310_624
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',X0)) )
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK101(X0,X1)))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_624])]) ).

tff(f4021,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',X0)) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK101(X0,X1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2631,f864]) ).

tff(f2631,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK101(X0,X1)))
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1752]) ).

tff(f11245,plain,
    spl310_623,
    inference(avatar_split_clause,[],[f4018,f11243]) ).

tff(f11243,plain,
    ( spl310_623
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',X0)) )
        | $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_623])]) ).

tff(f4018,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',X0)) )
      | $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f4017,f864]) ).

tff(f4017,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2630,f864]) ).

tff(f2630,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1133]) ).

tff(f1133,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(flattening,[],[f1132]) ).

tff(f1132,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f400]) ).

tff(f400,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X0),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom398) ).

tff(f11241,plain,
    spl310_622,
    inference(avatar_split_clause,[],[f3953,f11239]) ).

tff(f11239,plain,
    ( spl310_622
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( $uminus('fun_app$e'('of_nat$',X0)) = $uminus('fun_app$e'('of_nat$',X1)) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP10(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_622])]) ).

tff(f3953,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $uminus('fun_app$e'('of_nat$',X0)) = $uminus('fun_app$e'('of_nat$',X1)) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP10(X0,X1,X2) ),
    inference(evaluation,[],[f3952]) ).

tff(f3952,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP10(X0,X1,X2) ),
    inference(duplicate_literal_removal,[],[f2938]) ).

tff(f2938,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X0))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP10(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1942]) ).

tff(f11237,plain,
    spl310_621,
    inference(avatar_split_clause,[],[f3884,f11235]) ).

tff(f11235,plain,
    ( spl310_621
  <=> ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK260(X2))),'fun_app$f'(X2,sK261(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_621])]) ).

tff(f3884,plain,
    ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK260(X2))),'fun_app$f'(X2,sK261(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(equality_resolution,[],[f3595]) ).

tff(f3595,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK260(X2))),'fun_app$f'(X2,sK261(X2)))
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2278]) ).

tff(f2278,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK260(X2))),'fun_app$f'(X2,sK261(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK260(X2)),sK261(X2)) )
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK260,sK261])],[f1516,f2277]) ).

tff(f2277,plain,
    ! [X2: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,sK260(X2))),'fun_app$f'(X2,sK261(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK260(X2)),sK261(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1516,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1515]) ).

tff(f1515,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f172]) ).

tff(f172,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5)) )
        & ( 'fun_app$f'(X2,X1) = X3 )
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom170) ).

tff(f11231,plain,
    spl310_620,
    inference(avatar_split_clause,[],[f3882,f11229]) ).

tff(f11229,plain,
    ( spl310_620
  <=> ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK258(X2))),'fun_app$f'(X2,sK259(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_620])]) ).

tff(f3882,plain,
    ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK258(X2))),'fun_app$f'(X2,sK259(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(equality_resolution,[],[f3593]) ).

tff(f3593,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK258(X2))),'fun_app$f'(X2,sK259(X2)))
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2276]) ).

tff(f2276,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK258(X2))),'fun_app$f'(X2,sK259(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK258(X2)),sK259(X2)) )
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK258,sK259])],[f1514,f2275]) ).

tff(f2275,plain,
    ! [X2: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,sK258(X2))),'fun_app$f'(X2,sK259(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK258(X2)),sK259(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1514,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1513]) ).

tff(f1513,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f298]) ).

tff(f298,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X4)),'fun_app$f'(X2,X5)) )
        & ( 'fun_app$f'(X2,X1) = X3 )
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom296) ).

tff(f11227,plain,
    spl310_619,
    inference(avatar_split_clause,[],[f3874,f11225]) ).

tff(f11225,plain,
    ( spl310_619
  <=> ! [X2: 'N$',X1: 'N_n_fun$',X3: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK250(X1))),'fun_app$f'(X1,sK251(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_619])]) ).

tff(f3874,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK250(X1))),'fun_app$f'(X1,sK251(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ),
    inference(equality_resolution,[],[f3585]) ).

tff(f3585,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK250(X1))),'fun_app$f'(X1,sK251(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2268]) ).

tff(f2268,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK250(X1))),'fun_app$f'(X1,sK251(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK250(X1)),sK251(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK250,sK251])],[f1506,f2267]) ).

tff(f2267,plain,
    ! [X1: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,sK250(X1))),'fun_app$f'(X1,sK251(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK250(X1)),sK251(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1506,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1505]) ).

tff(f1505,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f302]) ).

tff(f302,axiom,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less$',X2),X3)
        & ( 'fun_app$f'(X1,X2) = X0 ) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom300) ).

tff(f11223,plain,
    spl310_618,
    inference(avatar_split_clause,[],[f3872,f11221]) ).

tff(f11221,plain,
    ( spl310_618
  <=> ! [X2: 'N$',X1: 'N_n_fun$',X3: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK248(X1))),'fun_app$f'(X1,sK249(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_618])]) ).

tff(f3872,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK248(X1))),'fun_app$f'(X1,sK249(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ),
    inference(equality_resolution,[],[f3583]) ).

tff(f3583,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK248(X1))),'fun_app$f'(X1,sK249(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2266]) ).

tff(f2266,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK248(X1))),'fun_app$f'(X1,sK249(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK248(X1)),sK249(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK248,sK249])],[f1504,f2265]) ).

tff(f2265,plain,
    ! [X1: 'N_n_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,sK248(X1))),'fun_app$f'(X1,sK249(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK248(X1)),sK249(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1504,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1503]) ).

tff(f1503,plain,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f181]) ).

tff(f181,axiom,
    ! [X0: 'N$',X1: 'N_n_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X4)),'fun_app$f'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & ( 'fun_app$f'(X1,X2) = X0 ) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom179) ).

tff(f11219,plain,
    spl310_617,
    inference(avatar_split_clause,[],[f3698,f11217]) ).

tff(f11217,plain,
    ( spl310_617
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
        | ~ sP87(X4,X3,X2,X1,X0)
        | ~ sP88(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_617])]) ).

tff(f3698,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      | ~ sP87(X4,X3,X2,X1,X0)
      | ~ sP88(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2353]) ).

tff(f2353,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
          | ~ sP87(X4,X3,X2,X1,X0) )
        & ( sP87(X4,X3,X2,X1,X0)
          | ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4)) ) )
      | ~ sP88(X0,X1,X2,X3,X4) ),
    inference(rectify,[],[f2352]) ).

tff(f2352,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( $less($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4))
          | ~ sP87(X4,X3,X2,X0,X1) )
        & ( sP87(X4,X3,X2,X0,X1)
          | ~ $less($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4)) ) )
      | ~ sP88(X1,X0,X2,X3,X4) ),
    inference(nnf_transformation,[],[f1705]) ).

tff(f1705,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( $less($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4))
      <=> sP87(X4,X3,X2,X0,X1) )
      | ~ sP88(X1,X0,X2,X3,X4) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP88])]) ).

tff(f11215,plain,
    spl310_616,
    inference(avatar_split_clause,[],[f3697,f11213]) ).

tff(f11213,plain,
    ( spl310_616
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP87(X4,X3,X2,X1,X0)
        | ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
        | ~ sP88(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_616])]) ).

tff(f3697,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP87(X4,X3,X2,X1,X0)
      | ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      | ~ sP88(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2353]) ).

tff(f11211,plain,
    spl310_615,
    inference(avatar_split_clause,[],[f3689,f11209]) ).

tff(f11209,plain,
    ( spl310_615
  <=> ! [X2: 'Nat$',X4: $int,X0: $int,X3: $int,X1: 'Nat$'] :
        ( $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
        | ~ sP85(X4,X3,X2,X1,X0)
        | ~ sP86(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_615])]) ).

tff(f3689,plain,
    ! [X2: 'Nat$',X3: $int,X0: $int,X1: 'Nat$',X4: $int] :
      ( $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
      | ~ sP85(X4,X3,X2,X1,X0)
      | ~ sP86(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2348]) ).

tff(f2348,plain,
    ! [X0: $int,X1: 'Nat$',X2: 'Nat$',X3: $int,X4: $int] :
      ( ( ( $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
          | ~ sP85(X4,X3,X2,X1,X0) )
        & ( sP85(X4,X3,X2,X1,X0)
          | ~ $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4)) ) )
      | ~ sP86(X0,X1,X2,X3,X4) ),
    inference(rectify,[],[f2347]) ).

tff(f2347,plain,
    ! [X3: $int,X1: 'Nat$',X0: 'Nat$',X2: $int,X4: $int] :
      ( ( ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
          | ~ sP85(X4,X2,X0,X1,X3) )
        & ( sP85(X4,X2,X0,X1,X3)
          | ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4)) ) )
      | ~ sP86(X3,X1,X0,X2,X4) ),
    inference(nnf_transformation,[],[f1702]) ).

tff(f1702,plain,
    ! [X3: $int,X1: 'Nat$',X0: 'Nat$',X2: $int,X4: $int] :
      ( ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      <=> sP85(X4,X2,X0,X1,X3) )
      | ~ sP86(X3,X1,X0,X2,X4) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP86])]) ).

tff(f11207,plain,
    spl310_614,
    inference(avatar_split_clause,[],[f3688,f11205]) ).

tff(f11205,plain,
    ( spl310_614
  <=> ! [X4: $int,X0: $int,X3: $int,X2: 'Nat$',X1: 'Nat$'] :
        ( sP85(X4,X3,X2,X1,X0)
        | ~ $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
        | ~ sP86(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_614])]) ).

tff(f3688,plain,
    ! [X2: 'Nat$',X3: $int,X0: $int,X1: 'Nat$',X4: $int] :
      ( sP85(X4,X3,X2,X1,X0)
      | ~ $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
      | ~ sP86(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2348]) ).

tff(f11203,plain,
    spl310_613,
    inference(avatar_split_clause,[],[f3680,f11201]) ).

tff(f11201,plain,
    ( spl310_613
  <=> ! [X2: 'Nat$',X4: $int,X0: $int,X3: $int,X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
        | ~ sP83(X4,X3,X2,X1,X0)
        | ~ sP84(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_613])]) ).

tff(f3680,plain,
    ! [X2: 'Nat$',X3: $int,X0: $int,X1: 'Nat$',X4: $int] :
      ( ~ $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
      | ~ sP83(X4,X3,X2,X1,X0)
      | ~ sP84(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2343]) ).

tff(f2343,plain,
    ! [X0: $int,X1: 'Nat$',X2: 'Nat$',X3: $int,X4: $int] :
      ( ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
          | ~ sP83(X4,X3,X2,X1,X0) )
        & ( sP83(X4,X3,X2,X1,X0)
          | $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4)) ) )
      | ~ sP84(X0,X1,X2,X3,X4) ),
    inference(rectify,[],[f2342]) ).

tff(f2342,plain,
    ! [X4: $int,X1: 'Nat$',X0: 'Nat$',X2: $int,X3: $int] :
      ( ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X4),$sum('fun_app$b'('iterate_add$'(X1),X2),X3))
          | ~ sP83(X3,X2,X0,X1,X4) )
        & ( sP83(X3,X2,X0,X1,X4)
          | $less($sum('fun_app$b'('iterate_add$'(X0),X2),X4),$sum('fun_app$b'('iterate_add$'(X1),X2),X3)) ) )
      | ~ sP84(X4,X1,X0,X2,X3) ),
    inference(nnf_transformation,[],[f1699]) ).

tff(f1699,plain,
    ! [X4: $int,X1: 'Nat$',X0: 'Nat$',X2: $int,X3: $int] :
      ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X4),$sum('fun_app$b'('iterate_add$'(X1),X2),X3))
      <=> sP83(X3,X2,X0,X1,X4) )
      | ~ sP84(X4,X1,X0,X2,X3) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP84])]) ).

tff(f11199,plain,
    spl310_612,
    inference(avatar_split_clause,[],[f3679,f11197]) ).

tff(f11197,plain,
    ( spl310_612
  <=> ! [X4: $int,X0: $int,X3: $int,X2: 'Nat$',X1: 'Nat$'] :
        ( sP83(X4,X3,X2,X1,X0)
        | $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
        | ~ sP84(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_612])]) ).

tff(f3679,plain,
    ! [X2: 'Nat$',X3: $int,X0: $int,X1: 'Nat$',X4: $int] :
      ( sP83(X4,X3,X2,X1,X0)
      | $less($sum('fun_app$b'('iterate_add$'(X2),X3),X0),$sum('fun_app$b'('iterate_add$'(X1),X3),X4))
      | ~ sP84(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2343]) ).

tff(f11195,plain,
    spl310_611,
    inference(avatar_split_clause,[],[f3671,f11193]) ).

tff(f11193,plain,
    ( spl310_611
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
        | ~ sP81(X4,X3,X2,X1,X0)
        | ~ sP82(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_611])]) ).

tff(f3671,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      | ~ sP81(X4,X3,X2,X1,X0)
      | ~ sP82(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2338]) ).

tff(f2338,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
          | ~ sP81(X4,X3,X2,X1,X0) )
        & ( sP81(X4,X3,X2,X1,X0)
          | $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4)) ) )
      | ~ sP82(X0,X1,X2,X3,X4) ),
    inference(rectify,[],[f2337]) ).

tff(f2337,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X4: $int,X3: $int] :
      ( ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X1),X2),X4),$sum('fun_app$b'('iterate_add$'(X0),X2),X3))
          | ~ sP81(X3,X4,X2,X0,X1) )
        & ( sP81(X3,X4,X2,X0,X1)
          | $less($sum('fun_app$b'('iterate_add$'(X1),X2),X4),$sum('fun_app$b'('iterate_add$'(X0),X2),X3)) ) )
      | ~ sP82(X1,X0,X2,X4,X3) ),
    inference(nnf_transformation,[],[f1696]) ).

tff(f1696,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X4: $int,X3: $int] :
      ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X1),X2),X4),$sum('fun_app$b'('iterate_add$'(X0),X2),X3))
      <=> sP81(X3,X4,X2,X0,X1) )
      | ~ sP82(X1,X0,X2,X4,X3) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP82])]) ).

tff(f11191,plain,
    spl310_610,
    inference(avatar_split_clause,[],[f3670,f11189]) ).

tff(f11189,plain,
    ( spl310_610
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP81(X4,X3,X2,X1,X0)
        | $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
        | ~ sP82(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_610])]) ).

tff(f3670,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP81(X4,X3,X2,X1,X0)
      | $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      | ~ sP82(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2338]) ).

tff(f11187,plain,
    spl310_609,
    inference(avatar_split_clause,[],[f3662,f11185]) ).

tff(f11185,plain,
    ( spl310_609
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
        | ~ sP79(X4,X3,X2,X1,X0)
        | ~ sP80(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_609])]) ).

tff(f3662,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      | ~ sP79(X4,X3,X2,X1,X0)
      | ~ sP80(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2333]) ).

tff(f2333,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
          | ~ sP79(X4,X3,X2,X1,X0) )
        & ( sP79(X4,X3,X2,X1,X0)
          | ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) != $sum('fun_app$b'('iterate_add$'(X1),X2),X4) ) ) )
      | ~ sP80(X0,X1,X2,X3,X4) ),
    inference(rectify,[],[f2332]) ).

tff(f2332,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X4: $int,X3: $int] :
      ( ( ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
          | ~ sP79(X3,X4,X2,X0,X1) )
        & ( sP79(X3,X4,X2,X0,X1)
          | ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) != $sum('fun_app$b'('iterate_add$'(X1),X2),X4) ) ) )
      | ~ sP80(X1,X0,X2,X4,X3) ),
    inference(nnf_transformation,[],[f1693]) ).

tff(f1693,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X4: $int,X3: $int] :
      ( ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      <=> sP79(X3,X4,X2,X0,X1) )
      | ~ sP80(X1,X0,X2,X4,X3) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP80])]) ).

tff(f11183,plain,
    spl310_608,
    inference(avatar_split_clause,[],[f3661,f11181]) ).

tff(f11181,plain,
    ( spl310_608
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP79(X4,X3,X2,X1,X0)
        | ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) != $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
        | ~ sP80(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_608])]) ).

tff(f3661,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP79(X4,X3,X2,X1,X0)
      | ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) != $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      | ~ sP80(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2333]) ).

tff(f11179,plain,
    spl310_607,
    inference(avatar_split_clause,[],[f3653,f11177]) ).

tff(f11177,plain,
    ( spl310_607
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
        | ~ sP77(X4,X3,X2,X1,X0)
        | ~ sP78(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_607])]) ).

tff(f3653,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      | ~ sP77(X4,X3,X2,X1,X0)
      | ~ sP78(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2328]) ).

tff(f2328,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
          | ~ sP77(X4,X3,X2,X1,X0) )
        & ( sP77(X4,X3,X2,X1,X0)
          | ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) != $sum('fun_app$b'('iterate_add$'(X1),X2),X4) ) ) )
      | ~ sP78(X0,X1,X2,X3,X4) ),
    inference(rectify,[],[f2327]) ).

tff(f2327,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( ( $sum('fun_app$b'('iterate_add$'(X1),X2),X3) = $sum('fun_app$b'('iterate_add$'(X0),X2),X4) )
          | ~ sP77(X4,X3,X2,X0,X1) )
        & ( sP77(X4,X3,X2,X0,X1)
          | ( $sum('fun_app$b'('iterate_add$'(X1),X2),X3) != $sum('fun_app$b'('iterate_add$'(X0),X2),X4) ) ) )
      | ~ sP78(X1,X0,X2,X3,X4) ),
    inference(nnf_transformation,[],[f1690]) ).

tff(f1690,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( $sum('fun_app$b'('iterate_add$'(X1),X2),X3) = $sum('fun_app$b'('iterate_add$'(X0),X2),X4) )
      <=> sP77(X4,X3,X2,X0,X1) )
      | ~ sP78(X1,X0,X2,X3,X4) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP78])]) ).

tff(f11175,plain,
    spl310_606,
    inference(avatar_split_clause,[],[f3652,f11173]) ).

tff(f11173,plain,
    ( spl310_606
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP77(X4,X3,X2,X1,X0)
        | ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) != $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
        | ~ sP78(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_606])]) ).

tff(f3652,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP77(X4,X3,X2,X1,X0)
      | ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) != $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      | ~ sP78(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2328]) ).

tff(f11171,plain,
    spl310_605,
    inference(avatar_split_clause,[],[f3135,f11169]) ).

tff(f11169,plain,
    ( spl310_605
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ 'fun_app$j'(X0,sK122(X0))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_605])]) ).

tff(f3135,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$j'(X0,sK122(X0))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f2001]) ).

tff(f11149,plain,
    ( spl310_604
    | ~ spl310_117
    | ~ spl310_343 ),
    inference(avatar_split_clause,[],[f9433,f7842,f4838,f11147]) ).

tff(f11147,plain,
    ( spl310_604
  <=> ! [X0: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X0),0)
        | ~ sP76('to_nat$'('x$'),'to_nat$'('i$'),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_604])]) ).

tff(f4838,plain,
    ( spl310_117
  <=> $less('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','to_nat$'('x$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_117])]) ).

tff(f7842,plain,
    ( spl310_343
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),0)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP76(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_343])]) ).

tff(f9433,plain,
    ( ! [X0: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X0),0)
        | ~ sP76('to_nat$'('x$'),'to_nat$'('i$'),X0) )
    | ~ spl310_117
    | ~ spl310_343 ),
    inference(resolution,[],[f4840,f7843]) ).

tff(f7843,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),0)
        | ~ sP76(X0,X1,X2) )
    | ~ spl310_343 ),
    inference(avatar_component_clause,[],[f7842]) ).

tff(f4840,plain,
    ( $less('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','to_nat$'('x$')))
    | ~ spl310_117 ),
    inference(avatar_component_clause,[],[f4838]) ).

tff(f11084,plain,
    ( spl310_603
    | ~ spl310_117
    | ~ spl310_342 ),
    inference(avatar_split_clause,[],[f9432,f7838,f4838,f11082]) ).

tff(f11082,plain,
    ( spl310_603
  <=> ! [X0: 'Nat$'] :
        ( sP75('to_nat$'('x$'),'to_nat$'('i$'),X0)
        | $less('fun_app$e'('of_nat$',X0),0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_603])]) ).

tff(f7838,plain,
    ( spl310_342
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP75(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_342])]) ).

tff(f9432,plain,
    ( ! [X0: 'Nat$'] :
        ( sP75('to_nat$'('x$'),'to_nat$'('i$'),X0)
        | $less('fun_app$e'('of_nat$',X0),0) )
    | ~ spl310_117
    | ~ spl310_342 ),
    inference(resolution,[],[f4840,f7839]) ).

tff(f7839,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | sP75(X0,X1,X2)
        | $less('fun_app$e'('of_nat$',X2),0) )
    | ~ spl310_342 ),
    inference(avatar_component_clause,[],[f7838]) ).

tff(f11052,plain,
    spl310_602,
    inference(avatar_split_clause,[],[f4206,f11050]) ).

tff(f11050,plain,
    ( spl310_602
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK140(X1,X2))) )
        | 'fun_app$j'(X2,X0)
        | sP61(X2,X1)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_602])]) ).

tff(f4206,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK140(X1,X2))) )
      | 'fun_app$j'(X2,X0)
      | sP61(X2,X1)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3326,f864]) ).

tff(f3326,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X0)
      | sP61(X2,X1)
      | ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK140(X1,X2)),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f2085]) ).

tff(f2085,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | sP61(X2,X1)
      | ( ~ 'fun_app$j'(X2,sK140(X1,X2))
        & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK140(X1,X2)),1) ) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK140])],[f2083,f2084]) ).

tff(f2084,plain,
    ! [X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X3),1) ) )
     => ( ~ 'fun_app$j'(X2,sK140(X1,X2))
        & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK140(X1,X2)),1) ) ) ),
    introduced(choice_axiom,[]) ).

tff(f2083,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | sP61(X2,X1)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X3),1) ) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(rectify,[],[f1667]) ).

tff(f1667,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | sP61(X2,X1)
      | ? [X4: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X4)
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X4),1) ) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(definition_folding,[],[f1338,f1666]) ).

tff(f1666,plain,
    ! [X2: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1)) )
      | ~ sP61(X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP61])]) ).

tff(f1338,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1)) )
      | ? [X4: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X4)
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X4),1) ) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(flattening,[],[f1337]) ).

tff(f1337,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1)) )
      | ? [X4: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X4)
          & ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X4),1) ) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f967]) ).

tff(f967,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1)) )
           => 'fun_app$j'(X2,X3) )
        & ! [X4: 'Nat$'] :
            ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X4),1) )
           => 'fun_app$j'(X2,X4) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'(X2,X0) ),
    inference(rectify,[],[f378]) ).

tff(f378,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1)) )
           => 'fun_app$j'(X2,X3) )
        & ! [X3: 'Nat$'] :
            ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X3),1) )
           => 'fun_app$j'(X2,X3) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'(X2,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom376) ).

tff(f11048,plain,
    spl310_601,
    inference(avatar_split_clause,[],[f4199,f11046]) ).

tff(f11046,plain,
    ( spl310_601
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP45(X0,X1,X2)
        | ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_601])]) ).

tff(f4199,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP45(X0,X1,X2)
      | ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) ) ),
    inference(inner_rewriting,[],[f3234]) ).

tff(f3234,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP45(X0,X1,X2)
      | ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
      | ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) ) ),
    inference(cnf_transformation,[],[f2033]) ).

tff(f11044,plain,
    spl310_600,
    inference(avatar_split_clause,[],[f4194,f11042]) ).

tff(f11042,plain,
    ( spl310_600
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,'nat$'($sum(1,'fun_app$e'('of_nat$',sK129(X0,X1,X2)))))
        | 'fun_app$j'(X2,X0)
        | ~ 'fun_app$j'(X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_600])]) ).

tff(f4194,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,'nat$'($sum(1,'fun_app$e'('of_nat$',sK129(X0,X1,X2)))))
      | 'fun_app$j'(X2,X0)
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3189,f864]) ).

tff(f3189,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X0)
      | 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK129(X0,X1,X2)),1)))
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2011]) ).

tff(f2011,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | ( ~ 'fun_app$j'(X2,sK129(X0,X1,X2))
        & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK129(X0,X1,X2)),1)))
        & $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK129])],[f1314,f2010]) ).

tff(f2010,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
     => ( ~ 'fun_app$j'(X2,sK129(X0,X1,X2))
        & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK129(X0,X1,X2)),1)))
        & $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1314,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1313]) ).

tff(f1313,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X0)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f779]) ).

tff(f779,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
              & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
           => 'fun_app$j'(X2,X3) )
        & 'fun_app$j'(X2,X1)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => 'fun_app$j'(X2,X0) ),
    inference(theory_normalization,[],[f267]) ).

tff(f267,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
              & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
           => 'fun_app$j'(X2,X3) )
        & 'fun_app$j'(X2,X1)
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'(X2,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom265) ).

tff(f11040,plain,
    spl310_599,
    inference(avatar_split_clause,[],[f4193,f11038]) ).

tff(f11038,plain,
    ( spl310_599
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'(X2,'nat$'($sum(1,'fun_app$e'('of_nat$',sK128(X0,X1,X2)))))
        | 'fun_app$j'(X2,X1)
        | ~ 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_599])]) ).

tff(f4193,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'(X2,'nat$'($sum(1,'fun_app$e'('of_nat$',sK128(X0,X1,X2)))))
      | 'fun_app$j'(X2,X1)
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3186,f864]) ).

tff(f3186,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X1)
      | ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK128(X0,X1,X2)),1)))
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2009]) ).

tff(f2009,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X1)
      | ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK128(X0,X1,X2)),1)))
        & 'fun_app$j'(X2,sK128(X0,X1,X2))
        & $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK128])],[f1312,f2008]) ).

tff(f2008,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & 'fun_app$j'(X2,X3)
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
     => ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK128(X0,X1,X2)),1)))
        & 'fun_app$j'(X2,sK128(X0,X1,X2))
        & $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1312,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X1)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & 'fun_app$j'(X2,X3)
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1311]) ).

tff(f1311,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X1)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & 'fun_app$j'(X2,X3)
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f778]) ).

tff(f778,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,X3)
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
              & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
           => 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1))) )
        & 'fun_app$j'(X2,X0)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => 'fun_app$j'(X2,X1) ),
    inference(theory_normalization,[],[f266]) ).

tff(f266,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,X3)
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1))
              & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
           => 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1))) )
        & 'fun_app$j'(X2,X0)
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'(X2,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom264) ).

tff(f11036,plain,
    ( spl310_598
    | ~ spl310_117
    | ~ spl310_340 ),
    inference(avatar_split_clause,[],[f9431,f7830,f4838,f11034]) ).

tff(f11034,plain,
    ( spl310_598
  <=> ! [X0: 'Nat$'] :
        ( sP35(X0,'to_nat$'('x$'),'to_nat$'('i$'))
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_598])]) ).

tff(f7830,plain,
    ( spl310_340
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP35(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_340])]) ).

tff(f9431,plain,
    ( ! [X0: 'Nat$'] :
        ( sP35(X0,'to_nat$'('x$'),'to_nat$'('i$'))
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) )
    | ~ spl310_117
    | ~ spl310_340 ),
    inference(resolution,[],[f4840,f7831]) ).

tff(f7831,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | sP35(X0,X1,X2)
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) )
    | ~ spl310_340 ),
    inference(avatar_component_clause,[],[f7830]) ).

tff(f11032,plain,
    spl310_597,
    inference(avatar_split_clause,[],[f4135,f11030]) ).

tff(f11030,plain,
    ( spl310_597
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
        | sP14(X2,X1,X0)
        | sP16(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_597])]) ).

tff(f4135,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP14(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f4134,f2892]) ).

tff(f4134,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP14(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f4133,f865]) ).

tff(f4133,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP14(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f2993,f864]) ).

tff(f2993,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP14(X2,X1,X0)
      | ~ $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
      | sP16(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1614]) ).

tff(f11028,plain,
    spl310_596,
    inference(avatar_split_clause,[],[f4131,f11026]) ).

tff(f11026,plain,
    ( spl310_596
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
        | sP15(X2,X1,X0)
        | sP16(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_596])]) ).

tff(f4131,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP15(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f4130,f2892]) ).

tff(f4130,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X0),$sum(1,$uminus('fun_app$e'('of_nat$',X1)))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP15(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f4129,f865]) ).

tff(f4129,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP15(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f2995,f864]) ).

tff(f2995,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP15(X2,X1,X0)
      | ~ $less($sum($sum('fun_app$e'('of_nat$',X0),1),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X2),1))
      | sP16(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1614]) ).

tff(f11024,plain,
    spl310_595,
    inference(avatar_split_clause,[],[f4118,f11022]) ).

tff(f11022,plain,
    ( spl310_595
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP17(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_595])]) ).

tff(f4118,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus($sum(1,'fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP17(X0,X1,X2) ),
    inference(forward_demodulation,[],[f3936,f864]) ).

tff(f3936,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus($sum('fun_app$e'('of_nat$',X0),1)) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP17(X0,X1,X2) ),
    inference(evaluation,[],[f2979]) ).

tff(f2979,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X0),1))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP17(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1959]) ).

tff(f11020,plain,
    spl310_594,
    inference(avatar_split_clause,[],[f3856,f11018]) ).

tff(f11018,plain,
    ( spl310_594
  <=> ! [X2: 'Int_set$',X1: 'Int_set_n_fun$',X3: 'Int_set$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X2)),'fun_app$m'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,sK232(X1))),'fun_app$m'(X1,sK233(X1)))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_594])]) ).

tff(f3856,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X1: 'Int_set_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X2)),'fun_app$m'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,sK232(X1))),'fun_app$m'(X1,sK233(X1)))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3) ),
    inference(equality_resolution,[],[f3567]) ).

tff(f3567,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: 'N$',X1: 'Int_set_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$m'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,sK232(X1))),'fun_app$m'(X1,sK233(X1)))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$m'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2250]) ).

tff(f2250,plain,
    ! [X0: 'N$',X1: 'Int_set_n_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$m'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,sK232(X1))),'fun_app$m'(X1,sK233(X1)))
        & 'fun_app$c'('less_eq$a'(sK232(X1)),sK233(X1)) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$m'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK232,sK233])],[f1488,f2249]) ).

tff(f2249,plain,
    ! [X1: 'Int_set_n_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X4)),'fun_app$m'(X1,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,sK232(X1))),'fun_app$m'(X1,sK233(X1)))
        & 'fun_app$c'('less_eq$a'(sK232(X1)),sK233(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1488,plain,
    ! [X0: 'N$',X1: 'Int_set_n_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$m'(X1,X3))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X4)),'fun_app$m'(X1,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$m'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1487]) ).

tff(f1487,plain,
    ! [X0: 'N$',X1: 'Int_set_n_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$m'(X1,X3))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X4)),'fun_app$m'(X1,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$m'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f178]) ).

tff(f178,axiom,
    ! [X0: 'N$',X1: 'Int_set_n_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X4)),'fun_app$m'(X1,X5)) )
        & 'fun_app$c'('less_eq$a'(X2),X3)
        & ( 'fun_app$m'(X1,X2) = X0 ) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$m'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom176) ).

tff(f11016,plain,
    spl310_593,
    inference(avatar_split_clause,[],[f3846,f11014]) ).

tff(f11014,plain,
    ( spl310_593
  <=> ! [X2: 'Int_set_n_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),'fun_app$m'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK216(X2))),'fun_app$m'(X2,sK217(X2)))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_593])]) ).

tff(f3846,plain,
    ! [X2: 'Int_set_n_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),'fun_app$m'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK216(X2))),'fun_app$m'(X2,sK217(X2)))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(equality_resolution,[],[f3551]) ).

tff(f3551,plain,
    ! [X2: 'Int_set_n_fun$',X3: 'N$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK216(X2))),'fun_app$m'(X2,sK217(X2)))
      | ( 'fun_app$m'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2234]) ).

tff(f2234,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK216(X2))),'fun_app$m'(X2,sK217(X2)))
        & 'fun_app$c'('less_eq$a'(sK216(X2)),sK217(X2)) )
      | ( 'fun_app$m'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK216,sK217])],[f1472,f2233]) ).

tff(f2233,plain,
    ! [X2: 'Int_set_n_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,sK216(X2))),'fun_app$m'(X2,sK217(X2)))
        & 'fun_app$c'('less_eq$a'(sK216(X2)),sK217(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1472,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ( 'fun_app$m'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1471]) ).

tff(f1471,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ( 'fun_app$m'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f169]) ).

tff(f169,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_n_fun$',X3: 'N$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X4)),'fun_app$m'(X2,X5)) )
        & ( 'fun_app$m'(X2,X1) = X3 )
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom167) ).

tff(f11012,plain,
    spl310_592,
    inference(avatar_split_clause,[],[f3624,f11010]) ).

tff(f11010,plain,
    ( spl310_592
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( $less(X0,'fun_app$o'(X1,X3))
        | ~ $less('fun_app$o'(X1,sK288(X1)),'fun_app$o'(X1,sK289(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | ~ $less(X0,'fun_app$o'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_592])]) ).

tff(f3624,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ~ $less('fun_app$o'(X1,sK288(X1)),'fun_app$o'(X1,sK289(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(cnf_transformation,[],[f2306]) ).

tff(f2306,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ( ~ $less('fun_app$o'(X1,sK288(X1)),'fun_app$o'(X1,sK289(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK288(X1)),sK289(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK288,sK289])],[f1546,f2305]) ).

tff(f2305,plain,
    ! [X1: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ $less('fun_app$o'(X1,sK288(X1)),'fun_app$o'(X1,sK289(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK288(X1)),sK289(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1546,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(flattening,[],[f1545]) ).

tff(f1545,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(ennf_transformation,[],[f290]) ).

tff(f290,axiom,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less$',X2),X3)
        & $less(X0,'fun_app$o'(X1,X2)) )
     => $less(X0,'fun_app$o'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom288) ).

tff(f11008,plain,
    spl310_591,
    inference(avatar_split_clause,[],[f3622,f11006]) ).

tff(f11006,plain,
    ( spl310_591
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( $less(X0,'fun_app$o'(X1,X3))
        | $less('fun_app$o'(X1,sK287(X1)),'fun_app$o'(X1,sK286(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | ~ $less(X0,'fun_app$o'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_591])]) ).

tff(f3622,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | $less('fun_app$o'(X1,sK287(X1)),'fun_app$o'(X1,sK286(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(cnf_transformation,[],[f2304]) ).

tff(f2304,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ( $less('fun_app$o'(X1,sK287(X1)),'fun_app$o'(X1,sK286(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK286(X1)),sK287(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK286,sK287])],[f1544,f2303]) ).

tff(f2303,plain,
    ! [X1: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( $less('fun_app$o'(X1,sK287(X1)),'fun_app$o'(X1,sK286(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK286(X1)),sK287(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1544,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(flattening,[],[f1543]) ).

tff(f1543,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(ennf_transformation,[],[f851]) ).

tff(f851,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => ~ $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & $less(X0,'fun_app$o'(X1,X2)) )
     => $less(X0,'fun_app$o'(X1,X3)) ),
    inference(theory_normalization,[],[f492]) ).

tff(f492,axiom,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => $lesseq('fun_app$o'(X1,X4),'fun_app$o'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & $less(X0,'fun_app$o'(X1,X2)) )
     => $less(X0,'fun_app$o'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom490) ).

tff(f11004,plain,
    spl310_590,
    inference(avatar_split_clause,[],[f3615,f11002]) ).

tff(f11002,plain,
    ( spl310_590
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( $less(X0,'fun_app$o'(X1,X3))
        | ~ $less('fun_app$o'(X1,sK280(X1)),'fun_app$o'(X1,sK281(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | $less('fun_app$o'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_590])]) ).

tff(f3615,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ~ $less('fun_app$o'(X1,sK280(X1)),'fun_app$o'(X1,sK281(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2298]) ).

tff(f2298,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ( ~ $less('fun_app$o'(X1,sK280(X1)),'fun_app$o'(X1,sK281(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK280(X1)),sK281(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK280,sK281])],[f1536,f2297]) ).

tff(f2297,plain,
    ! [X1: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ $less('fun_app$o'(X1,sK280(X1)),'fun_app$o'(X1,sK281(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK280(X1)),sK281(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1536,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(flattening,[],[f1535]) ).

tff(f1535,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(ennf_transformation,[],[f847]) ).

tff(f847,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less$',X2),X3)
        & ~ $less('fun_app$o'(X1,X2),X0) )
     => $less(X0,'fun_app$o'(X1,X3)) ),
    inference(theory_normalization,[],[f486]) ).

tff(f486,axiom,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less$',X2),X3)
        & $lesseq(X0,'fun_app$o'(X1,X2)) )
     => $less(X0,'fun_app$o'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom484) ).

tff(f11000,plain,
    spl310_589,
    inference(avatar_split_clause,[],[f3613,f10998]) ).

tff(f10998,plain,
    ( spl310_589
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( ~ $less('fun_app$o'(X1,X3),X0)
        | $less('fun_app$o'(X1,sK279(X1)),'fun_app$o'(X1,sK278(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | $less('fun_app$o'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_589])]) ).

tff(f3613,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | $less('fun_app$o'(X1,sK279(X1)),'fun_app$o'(X1,sK278(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2296]) ).

tff(f2296,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | ( $less('fun_app$o'(X1,sK279(X1)),'fun_app$o'(X1,sK278(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK278(X1)),sK279(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK278,sK279])],[f1534,f2295]) ).

tff(f2295,plain,
    ! [X1: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( $less('fun_app$o'(X1,sK279(X1)),'fun_app$o'(X1,sK278(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK278(X1)),sK279(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1534,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(flattening,[],[f1533]) ).

tff(f1533,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(ennf_transformation,[],[f846]) ).

tff(f846,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => ~ $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & ~ $less('fun_app$o'(X1,X2),X0) )
     => ~ $less('fun_app$o'(X1,X3),X0) ),
    inference(theory_normalization,[],[f199]) ).

tff(f199,axiom,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => $lesseq('fun_app$o'(X1,X4),'fun_app$o'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & $lesseq(X0,'fun_app$o'(X1,X2)) )
     => $lesseq(X0,'fun_app$o'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom197) ).

tff(f10996,plain,
    spl310_588,
    inference(avatar_split_clause,[],[f3603,f10994]) ).

tff(f10994,plain,
    ( spl310_588
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),X3)
        | ~ $less('fun_app$o'(X2,sK268(X2)),'fun_app$o'(X2,sK269(X2)))
        | ~ $less('fun_app$o'(X2,X1),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_588])]) ).

tff(f3603,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ~ $less('fun_app$o'(X2,sK268(X2)),'fun_app$o'(X2,sK269(X2)))
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2286]) ).

tff(f2286,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ( ~ $less('fun_app$o'(X2,sK268(X2)),'fun_app$o'(X2,sK269(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK268(X2)),sK269(X2)) )
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK268,sK269])],[f1524,f2285]) ).

tff(f2285,plain,
    ! [X2: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ $less('fun_app$o'(X2,sK268(X2)),'fun_app$o'(X2,sK269(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK268(X2)),sK269(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1524,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1523]) ).

tff(f1523,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f287]) ).

tff(f287,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5)) )
        & $less('fun_app$o'(X2,X1),X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => $less('fun_app$o'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom285) ).

tff(f10992,plain,
    ( spl310_587
    | ~ spl310_117
    | ~ spl310_339 ),
    inference(avatar_split_clause,[],[f9430,f7826,f4838,f10990]) ).

tff(f10990,plain,
    ( spl310_587
  <=> ! [X0: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP35(X0,'to_nat$'('x$'),'to_nat$'('i$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_587])]) ).

tff(f7826,plain,
    ( spl310_339
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP35(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_339])]) ).

tff(f9430,plain,
    ( ! [X0: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP35(X0,'to_nat$'('x$'),'to_nat$'('i$')) )
    | ~ spl310_117
    | ~ spl310_339 ),
    inference(resolution,[],[f4840,f7827]) ).

tff(f7827,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP35(X0,X1,X2) )
    | ~ spl310_339 ),
    inference(avatar_component_clause,[],[f7826]) ).

tff(f10988,plain,
    spl310_586,
    inference(avatar_split_clause,[],[f3601,f10986]) ).

tff(f10986,plain,
    ( spl310_586
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),X3)
        | $less('fun_app$o'(X2,sK267(X2)),'fun_app$o'(X2,sK266(X2)))
        | ~ $less('fun_app$o'(X2,X1),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_586])]) ).

tff(f3601,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | $less('fun_app$o'(X2,sK267(X2)),'fun_app$o'(X2,sK266(X2)))
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2284]) ).

tff(f2284,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ( $less('fun_app$o'(X2,sK267(X2)),'fun_app$o'(X2,sK266(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK266(X2)),sK267(X2)) )
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK266,sK267])],[f1522,f2283]) ).

tff(f2283,plain,
    ! [X2: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( $less('fun_app$o'(X2,sK267(X2)),'fun_app$o'(X2,sK266(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK266(X2)),sK267(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1522,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1521]) ).

tff(f1521,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f845]) ).

tff(f845,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => ~ $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4)) )
        & $less('fun_app$o'(X2,X1),X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => $less('fun_app$o'(X2,X0),X3) ),
    inference(theory_normalization,[],[f488]) ).

tff(f488,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => $lesseq('fun_app$o'(X2,X4),'fun_app$o'(X2,X5)) )
        & $less('fun_app$o'(X2,X1),X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => $less('fun_app$o'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom486) ).

tff(f10984,plain,
    spl310_585,
    inference(avatar_split_clause,[],[f3599,f10982]) ).

tff(f10982,plain,
    ( spl310_585
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),X3)
        | ~ $less('fun_app$o'(X2,sK264(X2)),'fun_app$o'(X2,sK265(X2)))
        | $less(X3,'fun_app$o'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_585])]) ).

tff(f3599,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ~ $less('fun_app$o'(X2,sK264(X2)),'fun_app$o'(X2,sK265(X2)))
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2282]) ).

tff(f2282,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ( ~ $less('fun_app$o'(X2,sK264(X2)),'fun_app$o'(X2,sK265(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK264(X2)),sK265(X2)) )
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK264,sK265])],[f1520,f2281]) ).

tff(f2281,plain,
    ! [X2: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ $less('fun_app$o'(X2,sK264(X2)),'fun_app$o'(X2,sK265(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK264(X2)),sK265(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1520,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1519]) ).

tff(f1519,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f844]) ).

tff(f844,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5)) )
        & ~ $less(X3,'fun_app$o'(X2,X1))
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => $less('fun_app$o'(X2,X0),X3) ),
    inference(theory_normalization,[],[f498]) ).

tff(f498,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5)) )
        & $lesseq('fun_app$o'(X2,X1),X3)
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => $less('fun_app$o'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom496) ).

tff(f10980,plain,
    spl310_584,
    inference(avatar_split_clause,[],[f3597,f10978]) ).

tff(f10978,plain,
    ( spl310_584
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( ~ $less(X3,'fun_app$o'(X2,X0))
        | $less('fun_app$o'(X2,sK263(X2)),'fun_app$o'(X2,sK262(X2)))
        | $less(X3,'fun_app$o'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_584])]) ).

tff(f3597,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | $less('fun_app$o'(X2,sK263(X2)),'fun_app$o'(X2,sK262(X2)))
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2280]) ).

tff(f2280,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | ( $less('fun_app$o'(X2,sK263(X2)),'fun_app$o'(X2,sK262(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK262(X2)),sK263(X2)) )
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK262,sK263])],[f1518,f2279]) ).

tff(f2279,plain,
    ! [X2: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( $less('fun_app$o'(X2,sK263(X2)),'fun_app$o'(X2,sK262(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK262(X2)),sK263(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1518,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1517]) ).

tff(f1517,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f843]) ).

tff(f843,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => ~ $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4)) )
        & ~ $less(X3,'fun_app$o'(X2,X1))
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => ~ $less(X3,'fun_app$o'(X2,X0)) ),
    inference(theory_normalization,[],[f194]) ).

tff(f194,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => $lesseq('fun_app$o'(X2,X4),'fun_app$o'(X2,X5)) )
        & $lesseq('fun_app$o'(X2,X1),X3)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => $lesseq('fun_app$o'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom192) ).

tff(f10976,plain,
    spl310_583,
    inference(avatar_split_clause,[],[f3558,f10974]) ).

tff(f10974,plain,
    ( spl310_583
  <=> ! [X0: 'Int_set$',X3: 'N$',X2: 'Int_set_n_fun$',X1: 'Int_set$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
        | 'fun_app$c'('less_eq$a'(sK224(X2)),sK225(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_583])]) ).

tff(f3558,plain,
    ! [X2: 'Int_set_n_fun$',X3: 'N$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | 'fun_app$c'('less_eq$a'(sK224(X2)),sK225(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2242]) ).

tff(f10972,plain,
    spl310_582,
    inference(avatar_split_clause,[],[f3392,f10970]) ).

tff(f10970,plain,
    ( spl310_582
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | ~ 'fun_app$'('fun_app$a'(X0,sK158(X0)),sK159(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
        | $less(sK161(X0),sK162(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_582])]) ).

tff(f3392,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ~ 'fun_app$'('fun_app$a'(X0,sK158(X0)),sK159(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
      | $less(sK161(X0),sK162(X0)) ),
    inference(cnf_transformation,[],[f2141]) ).

tff(f10968,plain,
    spl310_581,
    inference(avatar_split_clause,[],[f3390,f10966]) ).

tff(f10966,plain,
    ( spl310_581
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | 'fun_app$'('fun_app$a'(X0,sK159(X0)),sK158(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
        | $less(sK161(X0),sK162(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_581])]) ).

tff(f3390,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | 'fun_app$'('fun_app$a'(X0,sK159(X0)),sK158(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK160(X0)),sK160(X0))
      | $less(sK161(X0),sK162(X0)) ),
    inference(cnf_transformation,[],[f2141]) ).

tff(f10949,plain,
    ( spl310_580
    | ~ spl310_117
    | ~ spl310_338 ),
    inference(avatar_split_clause,[],[f9429,f7822,f4838,f10947]) ).

tff(f10947,plain,
    ( spl310_580
  <=> ! [X0: 'Nat$'] :
        ( sP33(X0,'to_nat$'('x$'),'to_nat$'('i$'))
        | $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_580])]) ).

tff(f7822,plain,
    ( spl310_338
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP33(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_338])]) ).

tff(f9429,plain,
    ( ! [X0: 'Nat$'] :
        ( sP33(X0,'to_nat$'('x$'),'to_nat$'('i$'))
        | $less(0,'fun_app$e'('of_nat$',X0)) )
    | ~ spl310_117
    | ~ spl310_338 ),
    inference(resolution,[],[f4840,f7823]) ).

tff(f7823,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | sP33(X0,X1,X2)
        | $less(0,'fun_app$e'('of_nat$',X0)) )
    | ~ spl310_338 ),
    inference(avatar_component_clause,[],[f7822]) ).

tff(f10906,plain,
    ( spl310_579
    | ~ spl310_117
    | ~ spl310_337 ),
    inference(avatar_split_clause,[],[f9428,f7818,f4838,f10904]) ).

tff(f10904,plain,
    ( spl310_579
  <=> ! [X0: 'Nat$'] :
        ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP33(X0,'to_nat$'('x$'),'to_nat$'('i$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_579])]) ).

tff(f7818,plain,
    ( spl310_337
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP33(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_337])]) ).

tff(f9428,plain,
    ( ! [X0: 'Nat$'] :
        ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP33(X0,'to_nat$'('x$'),'to_nat$'('i$')) )
    | ~ spl310_117
    | ~ spl310_337 ),
    inference(resolution,[],[f4840,f7819]) ).

tff(f7819,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ $less(0,'fun_app$e'('of_nat$',X0))
        | ~ sP33(X0,X1,X2) )
    | ~ spl310_337 ),
    inference(avatar_component_clause,[],[f7818]) ).

tff(f10862,plain,
    ( spl310_578
    | ~ spl310_117
    | ~ spl310_335 ),
    inference(avatar_split_clause,[],[f9427,f7810,f4838,f10860]) ).

tff(f10860,plain,
    ( spl310_578
  <=> ! [X0: 'Nat$'] :
        ( ( 0 = 'fun_app$e'('of_nat$',X0) )
        | ~ sP31('to_nat$'('x$'),'to_nat$'('i$'),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_578])]) ).

tff(f7810,plain,
    ( spl310_335
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = 'fun_app$e'('of_nat$',X2) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP31(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_335])]) ).

tff(f9427,plain,
    ( ! [X0: 'Nat$'] :
        ( ( 0 = 'fun_app$e'('of_nat$',X0) )
        | ~ sP31('to_nat$'('x$'),'to_nat$'('i$'),X0) )
    | ~ spl310_117
    | ~ spl310_335 ),
    inference(resolution,[],[f4840,f7811]) ).

tff(f7811,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( 0 = 'fun_app$e'('of_nat$',X2) )
        | ~ sP31(X0,X1,X2) )
    | ~ spl310_335 ),
    inference(avatar_component_clause,[],[f7810]) ).

tff(f10843,plain,
    spl310_577,
    inference(avatar_split_clause,[],[f4203,f10841]) ).

tff(f10841,plain,
    ( spl310_577
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X2))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_577])]) ).

tff(f4203,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f3312,f864]) ).

tff(f3312,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1332]) ).

tff(f1332,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(flattening,[],[f1331]) ).

tff(f1331,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f384]) ).

tff(f384,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom382) ).

tff(f10839,plain,
    spl310_576,
    inference(avatar_split_clause,[],[f4192,f10837]) ).

tff(f10837,plain,
    ( spl310_576
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'(X2,'nat$'($sum(1,'fun_app$e'('of_nat$',sK127(X0,X2)))))
        | 'fun_app$j'(X2,X1)
        | ~ 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_576])]) ).

tff(f4192,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'(X2,'nat$'($sum(1,'fun_app$e'('of_nat$',sK127(X0,X2)))))
      | 'fun_app$j'(X2,X1)
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f3182,f864]) ).

tff(f3182,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X1)
      | ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK127(X0,X2)),1)))
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2007]) ).

tff(f2007,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X1)
      | ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK127(X0,X2)),1)))
        & 'fun_app$j'(X2,sK127(X0,X2))
        & ~ $less('fun_app$e'('of_nat$',sK127(X0,X2)),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK127])],[f1310,f2006]) ).

tff(f2006,plain,
    ! [X0: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & 'fun_app$j'(X2,X3)
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
     => ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',sK127(X0,X2)),1)))
        & 'fun_app$j'(X2,sK127(X0,X2))
        & ~ $less('fun_app$e'('of_nat$',sK127(X0,X2)),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1310,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X1)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & 'fun_app$j'(X2,X3)
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1309]) ).

tff(f1309,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X2,X1)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & 'fun_app$j'(X2,X3)
          & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f777]) ).

tff(f777,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,X3)
              & ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
           => 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1))) )
        & 'fun_app$j'(X2,X0)
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => 'fun_app$j'(X2,X1) ),
    inference(theory_normalization,[],[f516]) ).

tff(f516,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat_bool_fun$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( 'fun_app$j'(X2,X3)
              & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
           => 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1))) )
        & 'fun_app$j'(X2,X0)
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => 'fun_app$j'(X2,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom514) ).

tff(f10835,plain,
    spl310_575,
    inference(avatar_split_clause,[],[f4191,f10833]) ).

tff(f10833,plain,
    ( spl310_575
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK122(X0)))))
        | 'fun_app$j'(X0,'nat$'(0))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_575])]) ).

tff(f4191,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK122(X0)))))
      | 'fun_app$j'(X0,'nat$'(0))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(forward_demodulation,[],[f3132,f864]) ).

tff(f3132,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'(0))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK122(X0)),1)))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f2001]) ).

tff(f10831,plain,
    spl310_574,
    inference(avatar_split_clause,[],[f3996,f10829]) ).

tff(f10829,plain,
    ( spl310_574
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_574])]) ).

tff(f3996,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2558,f864]) ).

tff(f2558,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
      | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1741]) ).

tff(f1741,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
          | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) )
        & ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
          | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1)) ) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(nnf_transformation,[],[f1060]) ).

tff(f1060,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
      <=> ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) ) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f377]) ).

tff(f377,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
      <=> ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom375) ).

tff(f10827,plain,
    spl310_573,
    inference(avatar_split_clause,[],[f3922,f10825]) ).

tff(f10825,plain,
    ( spl310_573
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP23(X0,X1,X2)
        | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_573])]) ).

tff(f3922,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP23(X0,X1,X2)
      | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(duplicate_literal_removal,[],[f3052]) ).

tff(f3052,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP23(X0,X1,X2)
      | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1618]) ).

tff(f1618,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP23(X0,X1,X2)
          | ( ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X2),0)
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & sP24(X1,X0,X2) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1192,f1617,f1616,f1615]) ).

tff(f1615,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),0)
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP22(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP22])]) ).

tff(f1192,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X2),0)
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = 0 )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),0)
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1191]) ).

tff(f1191,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
                  | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
                & ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) )
                  | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X2),0)
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) )
              | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) )
            & ( ( 0 = 0 )
              | ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less('fun_app$e'('of_nat$',X2),0)
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f884]) ).

tff(f884,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),0) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),0) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
             => ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f753]) ).

tff(f753,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),0) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus(0)) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),0) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
             => ( 0 = $sum($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),$uminus('fun_app$e'('of_nat$',X1))) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f127]) ).

tff(f127,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X2),0) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( $difference($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) = $difference('fun_app$e'('of_nat$',X2),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( 0 = $difference('fun_app$e'('of_nat$',X2),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( $difference($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) = $difference('fun_app$e'('of_nat$',X2),0) ) )
                & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
                 => ( 0 = $difference('fun_app$e'('of_nat$',X2),0) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X2),0) ) )
         => ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
             => ( 0 = $difference($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) )
            & ( $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
             => ( 0 = 0 ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom125) ).

tff(f10823,plain,
    ( spl310_572
    | ~ spl310_117
    | ~ spl310_334 ),
    inference(avatar_split_clause,[],[f9426,f7806,f4838,f10821]) ).

tff(f10821,plain,
    ( spl310_572
  <=> ! [X0: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | sP13('to_nat$'('i$'),X0,'to_nat$'('x$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_572])]) ).

tff(f7806,plain,
    ( spl310_334
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP13(X0,X2,X1)
        | $less(0,'fun_app$e'('of_nat$',X2))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_334])]) ).

tff(f9426,plain,
    ( ! [X0: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | sP13('to_nat$'('i$'),X0,'to_nat$'('x$')) )
    | ~ spl310_117
    | ~ spl310_334 ),
    inference(resolution,[],[f4840,f7807]) ).

tff(f7807,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X2))
        | sP13(X0,X2,X1) )
    | ~ spl310_334 ),
    inference(avatar_component_clause,[],[f7806]) ).

tff(f10819,plain,
    spl310_571,
    inference(avatar_split_clause,[],[f3876,f10817]) ).

tff(f10817,plain,
    ( spl310_571
  <=> ! [X2: 'N_int_set_fun$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),'fun_app$k'(X2,X1))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,sK252(X2))),'fun_app$k'(X2,sK253(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_571])]) ).

tff(f3876,plain,
    ! [X2: 'N_int_set_fun$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),'fun_app$k'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,sK252(X2))),'fun_app$k'(X2,sK253(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(equality_resolution,[],[f3587]) ).

tff(f3587,plain,
    ! [X2: 'N_int_set_fun$',X3: 'Int_set$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),X3)
      | ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,sK252(X2))),'fun_app$k'(X2,sK253(X2)))
      | ( 'fun_app$k'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2270]) ).

tff(f2270,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),X3)
      | ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,sK252(X2))),'fun_app$k'(X2,sK253(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK252(X2)),sK253(X2)) )
      | ( 'fun_app$k'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK252,sK253])],[f1508,f2269]) ).

tff(f2269,plain,
    ! [X2: 'N_int_set_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X4)),'fun_app$k'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,sK252(X2))),'fun_app$k'(X2,sK253(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK252(X2)),sK253(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1508,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X4)),'fun_app$k'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ( 'fun_app$k'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1507]) ).

tff(f1507,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X4)),'fun_app$k'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ( 'fun_app$k'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f167]) ).

tff(f167,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_set_fun$',X3: 'Int_set$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X4)),'fun_app$k'(X2,X5)) )
        & ( 'fun_app$k'(X2,X1) = X3 )
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom165) ).

tff(f10815,plain,
    spl310_570,
    inference(avatar_split_clause,[],[f3860,f10813]) ).

tff(f10813,plain,
    ( spl310_570
  <=> ! [X2: $int,X1: 'Int_n_fun$',X3: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK236(X1))),'fun_app$p'(X1,sK237(X1)))
        | ~ $less(X2,X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_570])]) ).

tff(f3860,plain,
    ! [X2: $int,X3: $int,X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK236(X1))),'fun_app$p'(X1,sK237(X1)))
      | ~ $less(X2,X3) ),
    inference(equality_resolution,[],[f3571]) ).

tff(f3571,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK236(X1))),'fun_app$p'(X1,sK237(X1)))
      | ~ $less(X2,X3)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2254]) ).

tff(f2254,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK236(X1))),'fun_app$p'(X1,sK237(X1)))
        & $less(sK236(X1),sK237(X1)) )
      | ~ $less(X2,X3)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK236,sK237])],[f1492,f2253]) ).

tff(f2253,plain,
    ! [X1: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,sK236(X1))),'fun_app$p'(X1,sK237(X1)))
        & $less(sK236(X1),sK237(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1492,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1491]) ).

tff(f1491,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f300]) ).

tff(f300,axiom,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & $less(X2,X3)
        & ( 'fun_app$p'(X1,X2) = X0 ) )
     => 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom298) ).

tff(f10811,plain,
    spl310_569,
    inference(avatar_split_clause,[],[f3858,f10809]) ).

tff(f10809,plain,
    ( spl310_569
  <=> ! [X2: $int,X1: 'Int_n_fun$',X3: $int] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK234(X1))),'fun_app$p'(X1,sK235(X1)))
        | $less(X3,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_569])]) ).

tff(f3858,plain,
    ! [X2: $int,X3: $int,X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK234(X1))),'fun_app$p'(X1,sK235(X1)))
      | $less(X3,X2) ),
    inference(equality_resolution,[],[f3569]) ).

tff(f3569,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK234(X1))),'fun_app$p'(X1,sK235(X1)))
      | $less(X3,X2)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2252]) ).

tff(f2252,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK234(X1))),'fun_app$p'(X1,sK235(X1)))
        & ~ $less(sK235(X1),sK234(X1)) )
      | $less(X3,X2)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK234,sK235])],[f1490,f2251]) ).

tff(f2251,plain,
    ! [X1: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,sK234(X1))),'fun_app$p'(X1,sK235(X1)))
        & ~ $less(sK235(X1),sK234(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1490,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1489]) ).

tff(f1489,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f839]) ).

tff(f839,plain,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & ~ $less(X3,X2)
        & ( 'fun_app$p'(X1,X2) = X0 ) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3)) ),
    inference(theory_normalization,[],[f183]) ).

tff(f183,axiom,
    ! [X0: 'N$',X1: 'Int_n_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X4)),'fun_app$p'(X1,X5)) )
        & $lesseq(X2,X3)
        & ( 'fun_app$p'(X1,X2) = X0 ) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom181) ).

tff(f10807,plain,
    spl310_568,
    inference(avatar_split_clause,[],[f3852,f10805]) ).

tff(f10805,plain,
    ( spl310_568
  <=> ! [X2: 'N$',X1: 'N_int_set_fun$',X3: 'N$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X2)),'fun_app$k'(X1,X3))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,sK228(X1))),'fun_app$k'(X1,sK229(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_568])]) ).

tff(f3852,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X2)),'fun_app$k'(X1,X3))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,sK228(X1))),'fun_app$k'(X1,sK229(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ),
    inference(equality_resolution,[],[f3563]) ).

tff(f3563,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'Int_set$',X1: 'N_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$k'(X1,X3))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,sK228(X1))),'fun_app$k'(X1,sK229(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$k'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2246]) ).

tff(f2246,plain,
    ! [X0: 'Int_set$',X1: 'N_int_set_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$k'(X1,X3))
      | ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,sK228(X1))),'fun_app$k'(X1,sK229(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK228(X1)),sK229(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$k'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK228,sK229])],[f1484,f2245]) ).

tff(f2245,plain,
    ! [X1: 'N_int_set_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X4)),'fun_app$k'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,sK228(X1))),'fun_app$k'(X1,sK229(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK228(X1)),sK229(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1484,plain,
    ! [X0: 'Int_set$',X1: 'N_int_set_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$k'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X4)),'fun_app$k'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$k'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1483]) ).

tff(f1483,plain,
    ! [X0: 'Int_set$',X1: 'N_int_set_fun$',X2: 'N$',X3: 'N$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$k'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X4)),'fun_app$k'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$k'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f176]) ).

tff(f176,axiom,
    ! [X0: 'Int_set$',X1: 'N_int_set_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X4)),'fun_app$k'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & ( 'fun_app$k'(X1,X2) = X0 ) )
     => 'fun_app$c'('less_eq$a'(X0),'fun_app$k'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom174) ).

tff(f10803,plain,
    spl310_567,
    inference(avatar_split_clause,[],[f3840,f10801]) ).

tff(f10801,plain,
    ( spl310_567
  <=> ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK202(X2))),'fun_app$p'(X2,sK203(X2)))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_567])]) ).

tff(f3840,plain,
    ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK202(X2))),'fun_app$p'(X2,sK203(X2)))
      | ~ $less(X0,X1) ),
    inference(equality_resolution,[],[f3535]) ).

tff(f3535,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK202(X2))),'fun_app$p'(X2,sK203(X2)))
      | ( 'fun_app$p'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2220]) ).

tff(f2220,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK202(X2))),'fun_app$p'(X2,sK203(X2)))
        & $less(sK202(X2),sK203(X2)) )
      | ( 'fun_app$p'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK202,sK203])],[f1454,f2219]) ).

tff(f2219,plain,
    ! [X2: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
     => ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,sK202(X2))),'fun_app$p'(X2,sK203(X2)))
        & $less(sK202(X2),sK203(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1454,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
      | ( 'fun_app$p'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1453]) ).

tff(f1453,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & $less(X4,X5) )
      | ( 'fun_app$p'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f296]) ).

tff(f296,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & ( 'fun_app$p'(X2,X1) = X3 )
        & $less(X0,X1) )
     => 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom294) ).

tff(f10799,plain,
    spl310_566,
    inference(avatar_split_clause,[],[f3835,f10797]) ).

tff(f10797,plain,
    ( spl310_566
  <=> ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK190(X2))),'fun_app$p'(X2,sK191(X2)))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_566])]) ).

tff(f3835,plain,
    ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK190(X2))),'fun_app$p'(X2,sK191(X2)))
      | $less(X1,X0) ),
    inference(equality_resolution,[],[f3517]) ).

tff(f3517,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK190(X2))),'fun_app$p'(X2,sK191(X2)))
      | ( 'fun_app$p'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2208]) ).

tff(f2208,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK190(X2))),'fun_app$p'(X2,sK191(X2)))
        & ~ $less(sK191(X2),sK190(X2)) )
      | ( 'fun_app$p'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK190,sK191])],[f1430,f2207]) ).

tff(f2207,plain,
    ! [X2: 'Int_n_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,sK190(X2))),'fun_app$p'(X2,sK191(X2)))
        & ~ $less(sK191(X2),sK190(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1430,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
      | ( 'fun_app$p'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(flattening,[],[f1429]) ).

tff(f1429,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5))
          & ~ $less(X5,X4) )
      | ( 'fun_app$p'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f825]) ).

tff(f825,plain,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & ( 'fun_app$p'(X2,X1) = X3 )
        & ~ $less(X1,X0) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3) ),
    inference(theory_normalization,[],[f174]) ).

tff(f174,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_n_fun$',X3: 'N$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X4)),'fun_app$p'(X2,X5)) )
        & ( 'fun_app$p'(X2,X1) = X3 )
        & $lesseq(X0,X1) )
     => 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom172) ).

tff(f10795,plain,
    spl310_565,
    inference(avatar_split_clause,[],[f3555,f10793]) ).

tff(f10793,plain,
    ( spl310_565
  <=> ! [X0: 'Int_set$',X3: $int,X2: 'Int_set_int_fun$',X1: 'Int_set$'] :
        ( ~ $less(X3,'fun_app$n'(X2,X0))
        | $less('fun_app$n'(X2,sK221(X2)),'fun_app$n'(X2,sK220(X2)))
        | $less(X3,'fun_app$n'(X2,X1))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_565])]) ).

tff(f3555,plain,
    ! [X2: 'Int_set_int_fun$',X3: $int,X0: 'Int_set$',X1: 'Int_set$'] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | $less('fun_app$n'(X2,sK221(X2)),'fun_app$n'(X2,sK220(X2)))
      | $less(X3,'fun_app$n'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2238]) ).

tff(f2238,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | ( $less('fun_app$n'(X2,sK221(X2)),'fun_app$n'(X2,sK220(X2)))
        & 'fun_app$c'('less_eq$a'(sK220(X2)),sK221(X2)) )
      | $less(X3,'fun_app$n'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK220,sK221])],[f1476,f2237]) ).

tff(f2237,plain,
    ! [X2: 'Int_set_int_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( $less('fun_app$n'(X2,sK221(X2)),'fun_app$n'(X2,sK220(X2)))
        & 'fun_app$c'('less_eq$a'(sK220(X2)),sK221(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1476,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | $less(X3,'fun_app$n'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1475]) ).

tff(f1475,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | $less(X3,'fun_app$n'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f836]) ).

tff(f836,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => ~ $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4)) )
        & ~ $less(X3,'fun_app$n'(X2,X1))
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => ~ $less(X3,'fun_app$n'(X2,X0)) ),
    inference(theory_normalization,[],[f191]) ).

tff(f191,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => $lesseq('fun_app$n'(X2,X4),'fun_app$n'(X2,X5)) )
        & $lesseq('fun_app$n'(X2,X1),X3)
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => $lesseq('fun_app$n'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom189) ).

tff(f10791,plain,
    spl310_564,
    inference(avatar_split_clause,[],[f3479,f10789]) ).

tff(f10789,plain,
    ( spl310_564
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP76(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_564])]) ).

tff(f3479,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP76(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2199]) ).

tff(f2199,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP76(X0,X1,X2)
        | ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less('fun_app$e'('of_nat$',X2),0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less('fun_app$e'('of_nat$',X2),0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP76(X0,X1,X2) ) ),
    inference(rectify,[],[f2198]) ).

tff(f2198,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP76(X2,X1,X0)
        | ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( ~ $less('fun_app$e'('of_nat$',X0),0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP76(X2,X1,X0) ) ),
    inference(flattening,[],[f2197]) ).

tff(f2197,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP76(X2,X1,X0)
        | ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( ~ $less('fun_app$e'('of_nat$',X0),0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP76(X2,X1,X0) ) ),
    inference(nnf_transformation,[],[f1687]) ).

tff(f1687,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( sP76(X2,X1,X0)
    <=> ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X0),0)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP76])]) ).

tff(f10787,plain,
    spl310_563,
    inference(avatar_split_clause,[],[f3476,f10785]) ).

tff(f10785,plain,
    ( spl310_563
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP76(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_563])]) ).

tff(f3476,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP76(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2199]) ).

tff(f10783,plain,
    spl310_562,
    inference(avatar_split_clause,[],[f3471,f10781]) ).

tff(f10781,plain,
    ( spl310_562
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP75(X0,X1,X2)
        | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_562])]) ).

tff(f3471,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP75(X0,X1,X2)
      | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2195]) ).

tff(f2195,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP75(X0,X1,X2)
        | ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less('fun_app$e'('of_nat$',X2),0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less('fun_app$e'('of_nat$',X2),0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP75(X0,X1,X2) ) ),
    inference(rectify,[],[f2194]) ).

tff(f2194,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP75(X1,X0,X2)
        | ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( $less('fun_app$e'('of_nat$',X2),0)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ~ $less('fun_app$e'('of_nat$',X2),0)
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | ~ sP75(X1,X0,X2) ) ),
    inference(flattening,[],[f2193]) ).

tff(f2193,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP75(X1,X0,X2)
        | ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( $less('fun_app$e'('of_nat$',X2),0)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ~ $less('fun_app$e'('of_nat$',X2),0)
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | ~ sP75(X1,X0,X2) ) ),
    inference(nnf_transformation,[],[f1685]) ).

tff(f1685,plain,
    ! [X1: 'Nat$',X0: 'Nat$',X2: 'Nat$'] :
      ( sP75(X1,X0,X2)
    <=> ( ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ~ $less('fun_app$e'('of_nat$',X2),0)
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP75])]) ).

tff(f10779,plain,
    ( spl310_561
    | ~ spl310_117
    | ~ spl310_333 ),
    inference(avatar_split_clause,[],[f9425,f7802,f4838,f10777]) ).

tff(f10777,plain,
    ( spl310_561
  <=> ! [X0: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | sP5(X0,'to_nat$'('x$'),'to_nat$'('i$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_561])]) ).

tff(f7802,plain,
    ( spl310_333
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP5(X2,X1,X0)
        | $less(0,'fun_app$e'('of_nat$',X2))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_333])]) ).

tff(f9425,plain,
    ( ! [X0: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | sP5(X0,'to_nat$'('x$'),'to_nat$'('i$')) )
    | ~ spl310_117
    | ~ spl310_333 ),
    inference(resolution,[],[f4840,f7803]) ).

tff(f7803,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | $less(0,'fun_app$e'('of_nat$',X2))
        | sP5(X2,X1,X0) )
    | ~ spl310_333 ),
    inference(avatar_component_clause,[],[f7802]) ).

tff(f10775,plain,
    spl310_560,
    inference(avatar_split_clause,[],[f3468,f10773]) ).

tff(f10773,plain,
    ( spl310_560
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP75(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_560])]) ).

tff(f3468,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP75(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2195]) ).

tff(f10771,plain,
    spl310_559,
    inference(avatar_split_clause,[],[f3188,f10769]) ).

tff(f10769,plain,
    ( spl310_559
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
        | ~ 'fun_app$j'(X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_559])]) ).

tff(f3188,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2011]) ).

tff(f10767,plain,
    spl310_558,
    inference(avatar_split_clause,[],[f3187,f10765]) ).

tff(f10765,plain,
    ( spl310_558
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X0)
        | ~ $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X0))
        | ~ 'fun_app$j'(X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_558])]) ).

tff(f3187,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X0)
      | ~ $less('fun_app$e'('of_nat$',sK129(X0,X1,X2)),'fun_app$e'('of_nat$',X0))
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2011]) ).

tff(f10763,plain,
    spl310_557,
    inference(avatar_split_clause,[],[f3184,f10761]) ).

tff(f10761,plain,
    ( spl310_557
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X1)
        | $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
        | ~ 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_557])]) ).

tff(f3184,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2009]) ).

tff(f10759,plain,
    spl310_556,
    inference(avatar_split_clause,[],[f3183,f10757]) ).

tff(f10757,plain,
    ( spl310_556
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X1)
        | ~ $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X0))
        | ~ 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_556])]) ).

tff(f3183,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X1)
      | ~ $less('fun_app$e'('of_nat$',sK128(X0,X1,X2)),'fun_app$e'('of_nat$',X0))
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2009]) ).

tff(f10755,plain,
    spl310_555,
    inference(avatar_split_clause,[],[f3131,f10753]) ).

tff(f10753,plain,
    ( spl310_555
  <=> ! [X4: 'Nat$',X0: 'Nat_bool_fun$',X2: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',sK120(X0)),'fun_app$e'('of_nat$',X4))
        | ~ 'fun_app$j'(X0,X4)
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',sK121(X0,X2)))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_555])]) ).

tff(f3131,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$',X4: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',sK120(X0)),'fun_app$e'('of_nat$',X4))
      | ~ 'fun_app$j'(X0,X4)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',sK121(X0,X2)))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f1999]) ).

tff(f1999,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ! [X4: 'Nat$'] :
            ( ~ $less('fun_app$e'('of_nat$',sK120(X0)),'fun_app$e'('of_nat$',X4))
            | ~ 'fun_app$j'(X0,X4) )
        & 'fun_app$j'(X0,sK120(X0)) )
      | ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',sK121(X0,X2)))
        & 'fun_app$j'(X0,sK121(X0,X2)) )
      | ~ 'fun_app$j'(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK120,sK121])],[f1996,f1998,f1997]) ).

tff(f1997,plain,
    ! [X0: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( ! [X4: 'Nat$'] :
              ( ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
              | ~ 'fun_app$j'(X0,X4) )
          & 'fun_app$j'(X0,X3) )
     => ( ! [X4: 'Nat$'] :
            ( ~ $less('fun_app$e'('of_nat$',sK120(X0)),'fun_app$e'('of_nat$',X4))
            | ~ 'fun_app$j'(X0,X4) )
        & 'fun_app$j'(X0,sK120(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1998,plain,
    ! [X0: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( ? [X5: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X5))
          & 'fun_app$j'(X0,X5) )
     => ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',sK121(X0,X2)))
        & 'fun_app$j'(X0,sK121(X0,X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1996,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ? [X3: 'Nat$'] :
          ( ! [X4: 'Nat$'] :
              ( ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
              | ~ 'fun_app$j'(X0,X4) )
          & 'fun_app$j'(X0,X3) )
      | ? [X5: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X5))
          & 'fun_app$j'(X0,X5) )
      | ~ 'fun_app$j'(X0,X1) ),
    inference(rectify,[],[f1228]) ).

tff(f1228,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ? [X4: 'Nat$'] :
          ( ! [X5: 'Nat$'] :
              ( ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
              | ~ 'fun_app$j'(X0,X5) )
          & 'fun_app$j'(X0,X4) )
      | ? [X3: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
          & 'fun_app$j'(X0,X3) )
      | ~ 'fun_app$j'(X0,X1) ),
    inference(flattening,[],[f1227]) ).

tff(f1227,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ? [X4: 'Nat$'] :
          ( ! [X5: 'Nat$'] :
              ( ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
              | ~ 'fun_app$j'(X0,X5) )
          & 'fun_app$j'(X0,X4) )
      | ? [X3: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
          & 'fun_app$j'(X0,X3) )
      | ~ 'fun_app$j'(X0,X1) ),
    inference(ennf_transformation,[],[f962]) ).

tff(f962,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
           => ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3)) )
        & 'fun_app$j'(X0,X1) )
     => ? [X4: 'Nat$'] :
          ( ! [X5: 'Nat$'] :
              ( 'fun_app$j'(X0,X5)
             => ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5)) )
          & 'fun_app$j'(X0,X4) ) ),
    inference(rectify,[],[f765]) ).

tff(f765,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
           => ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3)) )
        & 'fun_app$j'(X0,X1) )
     => ? [X3: 'Nat$'] :
          ( ! [X4: 'Nat$'] :
              ( 'fun_app$j'(X0,X4)
             => ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
          & 'fun_app$j'(X0,X3) ) ),
    inference(theory_normalization,[],[f133]) ).

tff(f133,axiom,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
           => $lesseq('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) )
        & 'fun_app$j'(X0,X1) )
     => ? [X3: 'Nat$'] :
          ( ! [X4: 'Nat$'] :
              ( 'fun_app$j'(X0,X4)
             => $lesseq('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) )
          & 'fun_app$j'(X0,X3) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom131) ).

tff(f10751,plain,
    spl310_554,
    inference(avatar_split_clause,[],[f3103,f10749]) ).

tff(f10749,plain,
    ( spl310_554
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP35(X0,X1,X2)
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_554])]) ).

tff(f3103,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP35(X0,X1,X2)
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1995]) ).

tff(f1995,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP35(X0,X1,X2)
        | ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        | ( ~ $less(0,'fun_app$e'('of_nat$',X0))
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      & ( ( ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,'fun_app$e'('of_nat$',X0))
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
        | ~ sP35(X0,X1,X2) ) ),
    inference(rectify,[],[f1994]) ).

tff(f1994,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP35(X2,X0,X1)
        | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP35(X2,X0,X1) ) ),
    inference(flattening,[],[f1993]) ).

tff(f1993,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP35(X2,X0,X1)
        | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP35(X2,X0,X1) ) ),
    inference(nnf_transformation,[],[f1633]) ).

tff(f1633,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP35(X2,X0,X1)
    <=> ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP35])]) ).

tff(f10747,plain,
    spl310_553,
    inference(avatar_split_clause,[],[f3100,f10745]) ).

tff(f10745,plain,
    ( spl310_553
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP35(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_553])]) ).

tff(f3100,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP35(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1995]) ).

tff(f10743,plain,
    spl310_552,
    inference(avatar_split_clause,[],[f3094,f10741]) ).

tff(f10741,plain,
    ( spl310_552
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP33(X0,X1,X2)
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_552])]) ).

tff(f3094,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP33(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1990]) ).

tff(f1990,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP33(X0,X1,X2)
        | ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        | ( $less(0,'fun_app$e'('of_nat$',X0))
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
          & ( ~ $less(0,'fun_app$e'('of_nat$',X0))
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) )
        | ~ sP33(X0,X1,X2) ) ),
    inference(rectify,[],[f1989]) ).

tff(f1989,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP33(X2,X0,X1)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP33(X2,X0,X1) ) ),
    inference(flattening,[],[f1988]) ).

tff(f1988,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP33(X2,X0,X1)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP33(X2,X0,X1) ) ),
    inference(nnf_transformation,[],[f1630]) ).

tff(f1630,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP33(X2,X0,X1)
    <=> ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ~ $less(0,'fun_app$e'('of_nat$',X2))
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP33])]) ).

tff(f10739,plain,
    spl310_551,
    inference(avatar_split_clause,[],[f3091,f10737]) ).

tff(f10737,plain,
    ( spl310_551
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP33(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_551])]) ).

tff(f3091,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP33(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1990]) ).

tff(f10733,plain,
    spl310_550,
    inference(avatar_split_clause,[],[f3085,f10731]) ).

tff(f10731,plain,
    ( spl310_550
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP31(X0,X1,X2)
        | ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_550])]) ).

tff(f3085,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP31(X0,X1,X2)
      | ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1985]) ).

tff(f1985,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP31(X0,X1,X2)
        | ( ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != 'fun_app$e'('of_nat$',X2) )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = 'fun_app$e'('of_nat$',X2) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP31(X0,X1,X2) ) ),
    inference(flattening,[],[f1984]) ).

tff(f1984,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP31(X0,X1,X2)
        | ( ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != 'fun_app$e'('of_nat$',X2) )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = 'fun_app$e'('of_nat$',X2) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP31(X0,X1,X2) ) ),
    inference(nnf_transformation,[],[f1627]) ).

tff(f1627,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP31(X0,X1,X2)
    <=> ( ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( 0 = 'fun_app$e'('of_nat$',X2) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP31])]) ).

tff(f10729,plain,
    spl310_549,
    inference(avatar_split_clause,[],[f3082,f10727]) ).

tff(f10727,plain,
    ( spl310_549
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP31(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_549])]) ).

tff(f3082,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP31(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1985]) ).

tff(f10725,plain,
    spl310_548,
    inference(avatar_split_clause,[],[f2961,f10723]) ).

tff(f10723,plain,
    ( spl310_548
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP13(X0,X2,X1)
        | $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_548])]) ).

tff(f2961,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP13(X0,X2,X1)
      | $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1605]) ).

tff(f1605,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP13(X0,X2,X1)
        | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( sP7(X1,X2,X0)
          & sP6(X1,X2,X0) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | sP8(X2,X1,X0) ) ),
    inference(definition_folding,[],[f1161,f1604,f1603,f1602,f1601,f1600,f1599,f1598,f1597]) ).

tff(f1597,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( 0 = 0 )
      | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
        & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ~ sP6(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP6])]) ).

tff(f1599,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP8(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP8])]) ).

tff(f1604,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( sP12(X1,X2,X0)
            & sP11(X2,X1,X0) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( sP10(X1,X2,X0)
            & sP9(X2,X1,X0) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP13(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP13])]) ).

tff(f1161,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X1))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
              & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ( ( ( ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X1))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
            | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
              & ( $less(0,'fun_app$e'('of_nat$',X1))
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
          & ( ( 0 = 0 )
            | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
              & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
            | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(flattening,[],[f1160]) ).

tff(f1160,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X1))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
              & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ( ( ( ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
                | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                    | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                  & ( $less(0,'fun_app$e'('of_nat$',X1))
                    | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                  & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
                | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
                  & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( ( ( ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) )
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) )
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
            | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
                | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
              & ( $less(0,'fun_app$e'('of_nat$',X1))
                | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
          & ( ( 0 = 0 )
            | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1))
              & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
            | ( ~ $less(0,'fun_app$e'('of_nat$',X1))
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(ennf_transformation,[],[f882]) ).

tff(f882,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X1))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) ) ) ) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                 => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                 => $less(0,'fun_app$e'('of_nat$',X1)) ) )
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),$uminus('fun_app$e'('of_nat$',X1))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) ) ) ) )
          & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => $less(0,'fun_app$e'('of_nat$',X1)) ) )
           => ( 0 = 0 ) ) ) ) ),
    inference(theory_normalization,[],[f734]) ).

tff(f734,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) = $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),$uminus('fun_app$e'('of_nat$',X1))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( 0 = $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),$uminus('fun_app$e'('of_nat$',X1))) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $sum(0,$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) ) ) ) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                 => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
                & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                 => $less(0,'fun_app$e'('of_nat$',X1)) ) )
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => ( 0 = $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),$uminus('fun_app$e'('of_nat$',X1))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X1))) ) ) ) )
          & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X2))),'fun_app$e'('of_nat$',X1)) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => $less(0,'fun_app$e'('of_nat$',X1)) ) )
           => ( 0 = 0 ) ) ) ) ),
    inference(theory_normalization,[],[f80]) ).

tff(f80,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) = $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) = $difference(0,'fun_app$e'('of_nat$',X1)) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( 0 = $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
                    & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                     => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $difference(0,'fun_app$e'('of_nat$',X2)) = $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => ( $difference(0,'fun_app$e'('of_nat$',X2)) = $difference(0,'fun_app$e'('of_nat$',X1)) ) ) ) )
              & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
                  & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                   => $less(0,'fun_app$e'('of_nat$',X1)) ) )
               => ( 0 = $difference(0,'fun_app$e'('of_nat$',X2)) ) ) ) ) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                 => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
                & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
                 => $less(0,'fun_app$e'('of_nat$',X1)) ) )
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => ( 0 = $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => ( 0 = $difference(0,'fun_app$e'('of_nat$',X1)) ) ) ) )
          & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
              & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
               => $less(0,'fun_app$e'('of_nat$',X1)) ) )
           => ( 0 = 0 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom78) ).

tff(f10721,plain,
    spl310_547,
    inference(avatar_split_clause,[],[f2925,f10719]) ).

tff(f10719,plain,
    ( spl310_547
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP5(X2,X1,X0)
        | $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_547])]) ).

tff(f2925,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP5(X2,X1,X0)
      | $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1596]) ).

tff(f1596,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP5(X2,X1,X0)
        | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) )
            | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
          & ( ( 0 = 0 )
            | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | sP4(X2,X1,X0) ) ),
    inference(definition_folding,[],[f1159,f1595,f1594]) ).

tff(f1594,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP4(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP4])]) ).

tff(f1159,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
              & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) )
            | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
          & ( ( 0 = 0 )
            | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(flattening,[],[f1158]) ).

tff(f1158,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
              & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) )
                | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( ( ( ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
              & ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) )
                | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
        | ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) )
      & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) )
            | $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
          & ( ( 0 = 0 )
            | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
        | ( ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( ~ $less(0,'fun_app$e'('of_nat$',X2))
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(ennf_transformation,[],[f881]) ).

tff(f881,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( 0 = $sum($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) ) ) ) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
           => ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) ) )
          & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
           => ( 0 = 0 ) ) ) ) ),
    inference(theory_normalization,[],[f733]) ).

tff(f733,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( 0 = $sum($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$uminus('fun_app$e'('of_nat$',X2))) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) )
              & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X2))) ) ) ) ) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
           => ( 0 = $sum('fun_app$e'('of_nat$',X0),$uminus($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))) ) )
          & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
           => ( 0 = 0 ) ) ) ) ),
    inference(theory_normalization,[],[f9]) ).

tff(f9,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
            & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
             => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( $difference('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) = $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) ) )
              & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( 0 = $difference($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( $difference('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) = $difference(0,'fun_app$e'('of_nat$',X2)) ) )
              & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
               => ( 0 = $difference(0,'fun_app$e'('of_nat$',X2)) ) ) ) ) ) )
      & ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
       => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
           => ( 0 = $difference('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ) )
          & ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
           => ( 0 = 0 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom7) ).

tff(f10642,plain,
    ( spl310_58
    | spl310_546 ),
    inference(avatar_split_clause,[],[f2368,f10639,f4490]) ).

tff(f2368,plain,
    ( ( 'fun_app$e'('of_nat$','ja$') = $sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$')))) )
    | $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    inference(cnf_transformation,[],[f983]) ).

tff(f983,plain,
    ( ( ( 'fun_app$e'('of_nat$','ja$') = $sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$')))) )
      | $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    & ( ( 0 = 'fun_app$e'('of_nat$','ja$') )
      | ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) ) ),
    inference(ennf_transformation,[],[f624]) ).

tff(f624,plain,
    ( ( ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
     => ( 'fun_app$e'('of_nat$','ja$') = $sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$')))) ) )
    & ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
     => ( 0 = 'fun_app$e'('of_nat$','ja$') ) ) ),
    inference(theory_normalization,[],[f8]) ).

tff(f8,axiom,
    ( ( ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
     => ( $difference('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) = 'fun_app$e'('of_nat$','ja$') ) )
    & ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
     => ( 0 = 'fun_app$e'('of_nat$','ja$') ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom6) ).

tff(f10112,plain,
    spl310_545,
    inference(avatar_split_clause,[],[f4031,f10110]) ).

tff(f10110,plain,
    ( spl310_545
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK114(X0,X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_545])]) ).

tff(f4031,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK114(X0,X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2841,f864]) ).

tff(f2841,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK114(X0,X1))),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1895]) ).

tff(f1895,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | ! [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) != $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) ) )
      & ( ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK114(X0,X1))),1) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK114])],[f1893,f1894]) ).

tff(f1894,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X3: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)),1) )
     => ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK114(X0,X1))),1) ) ),
    introduced(choice_axiom,[]) ).

tff(f1893,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | ! [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) != $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) ) )
      & ( ? [X3: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)),1) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    inference(rectify,[],[f1892]) ).

tff(f1892,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | ! [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) != $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) ) )
      & ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    inference(nnf_transformation,[],[f370]) ).

tff(f370,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
    <=> ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom368) ).

tff(f10108,plain,
    spl310_544,
    inference(avatar_split_clause,[],[f4022,f10106]) ).

tff(f10106,plain,
    ( spl310_544
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK102(X0,X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_544])]) ).

tff(f4022,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK102(X0,X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2633,f864]) ).

tff(f2633,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK102(X0,X1))),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1754]) ).

tff(f1754,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK102(X0,X1))),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK102])],[f1135,f1753]) ).

tff(f1753,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
     => ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK102(X0,X1))),1) ) ),
    introduced(choice_axiom,[]) ).

tff(f1135,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f939]) ).

tff(f939,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ~ ( ! [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) != $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(flattening,[],[f938]) ).

tff(f938,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ~ ( ! [X2: 'Nat$'] :
            ( ~ 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(true_and_false_elimination,[],[f373]) ).

tff(f373,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ! [X2: 'Nat$'] :
            ( ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
           => $false )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom371) ).

tff(f10104,plain,
    spl310_543,
    inference(avatar_split_clause,[],[f4009,f10102]) ).

tff(f10102,plain,
    ( spl310_543
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK99(X0,X1)))) )
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_543])]) ).

tff(f4009,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK99(X0,X1)))) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2591,f864]) ).

tff(f2591,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK99(X0,X1))),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1748]) ).

tff(f1748,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK99(X0,X1))),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK99])],[f1077,f1747]) ).

tff(f1747,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
     => ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK99(X0,X1))),1) ) ),
    introduced(choice_axiom,[]) ).

tff(f1077,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) )
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f369]) ).

tff(f369,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom367) ).

tff(f10100,plain,
    spl310_542,
    inference(avatar_split_clause,[],[f3978,f10098]) ).

tff(f10098,plain,
    ( spl310_542
  <=> ! [X0: 'Nat$'] : ( 'fun_app$e'('of_nat$','fun_app$r'('triangle$','nat$'($sum(1,'fun_app$e'('of_nat$',X0))))) = $sum('fun_app$e'('of_nat$','fun_app$r'('triangle$',X0)),$sum(1,'fun_app$e'('of_nat$',X0))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_542])]) ).

tff(f3978,plain,
    ! [X0: 'Nat$'] : ( 'fun_app$e'('of_nat$','fun_app$r'('triangle$','nat$'($sum(1,'fun_app$e'('of_nat$',X0))))) = $sum('fun_app$e'('of_nat$','fun_app$r'('triangle$',X0)),$sum(1,'fun_app$e'('of_nat$',X0))) ),
    inference(forward_demodulation,[],[f2407,f864]) ).

tff(f2407,plain,
    ! [X0: 'Nat$'] : ( 'fun_app$e'('of_nat$','fun_app$r'('triangle$','nat$'($sum('fun_app$e'('of_nat$',X0),1)))) = $sum('fun_app$e'('of_nat$','fun_app$r'('triangle$',X0)),$sum('fun_app$e'('of_nat$',X0),1)) ),
    inference(cnf_transformation,[],[f584]) ).

tff(f584,axiom,
    ! [X0: 'Nat$'] : ( 'fun_app$e'('of_nat$','fun_app$r'('triangle$','nat$'($sum('fun_app$e'('of_nat$',X0),1)))) = $sum('fun_app$e'('of_nat$','fun_app$r'('triangle$',X0)),$sum('fun_app$e'('of_nat$',X0),1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom582) ).

tff(f10096,plain,
    spl310_541,
    inference(avatar_split_clause,[],[f3964,f10094]) ).

tff(f10094,plain,
    ( spl310_541
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),0)
        | ~ sP1(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_541])]) ).

tff(f3964,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),0)
      | ~ sP1(X0,X1) ),
    inference(evaluation,[],[f3963]) ).

tff(f3963,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),0)
      | ~ sP1(X0,X1) ),
    inference(duplicate_literal_removal,[],[f2567]) ).

tff(f2567,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),0)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP1(X0,X1) ),
    inference(cnf_transformation,[],[f1742]) ).

tff(f1742,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ( ( $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( $less('fun_app$e'('of_nat$',X1),0)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP1(X0,X1) ),
    inference(nnf_transformation,[],[f1588]) ).

tff(f1588,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) )
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ( ( $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( $less('fun_app$e'('of_nat$',X1),0)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP1(X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP1])]) ).

tff(f10092,plain,
    spl310_540,
    inference(avatar_split_clause,[],[f3949,f10090]) ).

tff(f10090,plain,
    ( spl310_540
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X1))
        | ~ sP9(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_540])]) ).

tff(f3949,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP9(X0,X1,X2) ),
    inference(evaluation,[],[f2943]) ).

tff(f2943,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X1))
      | ~ sP9(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1944]) ).

tff(f10086,plain,
    spl310_539,
    inference(avatar_split_clause,[],[f3866,f10084]) ).

tff(f10084,plain,
    ( spl310_539
  <=> ! [X2: 'Int_set$',X1: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X2)),'fun_app$h'(X1,X3))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,sK242(X1))),'fun_app$h'(X1,sK243(X1)))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_539])]) ).

tff(f3866,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X1: 'Int_set_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X2)),'fun_app$h'(X1,X3))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,sK242(X1))),'fun_app$h'(X1,sK243(X1)))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3) ),
    inference(equality_resolution,[],[f3577]) ).

tff(f3577,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$h'(X1,X3))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,sK242(X1))),'fun_app$h'(X1,sK243(X1)))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$h'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2260]) ).

tff(f2260,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_int_set_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$h'(X1,X3))
      | ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,sK242(X1))),'fun_app$h'(X1,sK243(X1)))
        & 'fun_app$c'('less_eq$a'(sK242(X1)),sK243(X1)) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$h'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK242,sK243])],[f1498,f2259]) ).

tff(f2259,plain,
    ! [X1: 'Int_set_int_set_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X4)),'fun_app$h'(X1,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,sK242(X1))),'fun_app$h'(X1,sK243(X1)))
        & 'fun_app$c'('less_eq$a'(sK242(X1)),sK243(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1498,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_int_set_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$h'(X1,X3))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X4)),'fun_app$h'(X1,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$h'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1497]) ).

tff(f1497,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_int_set_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$h'(X1,X3))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X4)),'fun_app$h'(X1,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$h'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f180]) ).

tff(f180,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set_int_set_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X4)),'fun_app$h'(X1,X5)) )
        & 'fun_app$c'('less_eq$a'(X2),X3)
        & ( 'fun_app$h'(X1,X2) = X0 ) )
     => 'fun_app$c'('less_eq$a'(X0),'fun_app$h'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom178) ).

tff(f10082,plain,
    spl310_538,
    inference(avatar_split_clause,[],[f3848,f10080]) ).

tff(f10080,plain,
    ( spl310_538
  <=> ! [X2: 'Int_set_int_set_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),'fun_app$h'(X2,X1))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK218(X2))),'fun_app$h'(X2,sK219(X2)))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_538])]) ).

tff(f3848,plain,
    ! [X2: 'Int_set_int_set_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),'fun_app$h'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK218(X2))),'fun_app$h'(X2,sK219(X2)))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(equality_resolution,[],[f3553]) ).

tff(f3553,plain,
    ! [X2: 'Int_set_int_set_fun$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK218(X2))),'fun_app$h'(X2,sK219(X2)))
      | ( 'fun_app$h'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2236]) ).

tff(f2236,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK218(X2))),'fun_app$h'(X2,sK219(X2)))
        & 'fun_app$c'('less_eq$a'(sK218(X2)),sK219(X2)) )
      | ( 'fun_app$h'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK218,sK219])],[f1474,f2235]) ).

tff(f2235,plain,
    ! [X2: 'Int_set_int_set_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,sK218(X2))),'fun_app$h'(X2,sK219(X2)))
        & 'fun_app$c'('less_eq$a'(sK218(X2)),sK219(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1474,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ( 'fun_app$h'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1473]) ).

tff(f1473,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ( 'fun_app$h'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f171]) ).

tff(f171,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X4)),'fun_app$h'(X2,X5)) )
        & ( 'fun_app$h'(X2,X1) = X3 )
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom169) ).

tff(f10078,plain,
    spl310_537,
    inference(avatar_split_clause,[],[f3708,f10076]) ).

tff(f10076,plain,
    ( spl310_537
  <=> ! [X4: $int,X0: 'Int_set$',X3: 'Int_set$',X2: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X1),X3))
        | ~ 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X0),X2))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_537])]) ).

tff(f3708,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$',X4: $int] :
      ( 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X0),X2))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1586]) ).

tff(f1586,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$',X3: 'Int_set$',X4: $int] :
      ( 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X0),X2))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1585]) ).

tff(f1585,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$',X3: 'Int_set$',X4: $int] :
      ( 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X0),X2))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f136]) ).

tff(f136,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$',X3: 'Int_set$',X4: $int] :
      ( ( 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X0),X2))
        & 'fun_app$c'('less_eq$a'(X2),X3)
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => 'fun_app$c'('member$a'(X4),'fun_app$h'('plus$b'(X1),X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom134) ).

tff(f10074,plain,
    spl310_536,
    inference(avatar_split_clause,[],[f3702,f10072]) ).

tff(f10072,plain,
    ( spl310_536
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( sP87(X0,X1,X2,X3,X4)
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_536])]) ).

tff(f3702,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP87(X0,X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ),
    inference(cnf_transformation,[],[f2356]) ).

tff(f10070,plain,
    spl310_535,
    inference(avatar_split_clause,[],[f3699,f10068]) ).

tff(f10068,plain,
    ( spl310_535
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP87(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_535])]) ).

tff(f3699,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP87(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2356]) ).

tff(f10066,plain,
    spl310_534,
    inference(avatar_split_clause,[],[f3693,f10064]) ).

tff(f10064,plain,
    ( spl310_534
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( sP85(X0,X1,X2,X3,X4)
        | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
        | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_534])]) ).

tff(f3693,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( sP85(X0,X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
      | ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ),
    inference(cnf_transformation,[],[f2351]) ).

tff(f10062,plain,
    spl310_533,
    inference(avatar_split_clause,[],[f3690,f10060]) ).

tff(f10060,plain,
    ( spl310_533
  <=> ! [X2: 'Nat$',X4: $int,X0: $int,X3: 'Nat$',X1: $int] :
        ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
        | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
        | ~ sP85(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_533])]) ).

tff(f3690,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
      | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
      | ~ sP85(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2351]) ).

tff(f10058,plain,
    spl310_532,
    inference(avatar_split_clause,[],[f3684,f10056]) ).

tff(f10056,plain,
    ( spl310_532
  <=> ! [X4: $int,X0: $int,X3: 'Nat$',X2: 'Nat$',X1: $int] :
        ( sP83(X0,X1,X2,X3,X4)
        | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
        | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_532])]) ).

tff(f3684,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( sP83(X0,X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
      | $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0)) ),
    inference(cnf_transformation,[],[f2346]) ).

tff(f10054,plain,
    spl310_531,
    inference(avatar_split_clause,[],[f3681,f10052]) ).

tff(f10052,plain,
    ( spl310_531
  <=> ! [X2: 'Nat$',X4: $int,X0: $int,X3: 'Nat$',X1: $int] :
        ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
        | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
        | ~ sP83(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_531])]) ).

tff(f3681,plain,
    ! [X2: 'Nat$',X3: 'Nat$',X0: $int,X1: $int,X4: $int] :
      ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X1),X0))
      | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
      | ~ sP83(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2346]) ).

tff(f10050,plain,
    spl310_530,
    inference(avatar_split_clause,[],[f3675,f10048]) ).

tff(f10048,plain,
    ( spl310_530
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( sP81(X0,X1,X2,X3,X4)
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_530])]) ).

tff(f3675,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP81(X0,X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0) ),
    inference(cnf_transformation,[],[f2341]) ).

tff(f10046,plain,
    spl310_529,
    inference(avatar_split_clause,[],[f3672,f10044]) ).

tff(f10044,plain,
    ( spl310_529
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP81(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_529])]) ).

tff(f3672,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X0)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP81(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2341]) ).

tff(f10042,plain,
    spl310_528,
    inference(avatar_split_clause,[],[f3663,f10040]) ).

tff(f10040,plain,
    ( spl310_528
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) = X0 )
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP79(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_528])]) ).

tff(f3663,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) = X0 )
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP79(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2336]) ).

tff(f10038,plain,
    spl310_527,
    inference(avatar_split_clause,[],[f3654,f10036]) ).

tff(f10036,plain,
    ( spl310_527
  <=> ! [X4: 'Nat$',X0: $int,X3: 'Nat$',X2: $int,X1: $int] :
        ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) = X0 )
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
        | ~ sP77(X0,X1,X2,X3,X4) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_527])]) ).

tff(f3654,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) = X0 )
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ~ sP77(X0,X1,X2,X3,X4) ),
    inference(cnf_transformation,[],[f2331]) ).

tff(f10034,plain,
    spl310_526,
    inference(avatar_split_clause,[],[f3638,f10032]) ).

tff(f10032,plain,
    ( spl310_526
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
        | $less(sK300(X1),sK301(X1))
        | ~ $less(X2,X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_526])]) ).

tff(f3638,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | $less(sK300(X1),sK301(X1))
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2318]) ).

tff(f10030,plain,
    spl310_525,
    inference(avatar_split_clause,[],[f3636,f10028]) ).

tff(f10028,plain,
    ( spl310_525
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
        | $less(sK298(X1),sK299(X1))
        | ~ $less(X2,X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_525])]) ).

tff(f3636,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | $less(sK298(X1),sK299(X1))
      | ~ $less(X2,X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2316]) ).

tff(f10026,plain,
    spl310_524,
    inference(avatar_split_clause,[],[f3634,f10024]) ).

tff(f10024,plain,
    ( spl310_524
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
        | ~ $less(sK297(X1),sK296(X1))
        | $less(X3,X2)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_524])]) ).

tff(f3634,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ~ $less(sK297(X1),sK296(X1))
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2314]) ).

tff(f10022,plain,
    spl310_523,
    inference(avatar_split_clause,[],[f3632,f10020]) ).

tff(f10020,plain,
    ( spl310_523
  <=> ! [X0: 'N$',X3: $int,X2: $int,X1: 'Int_n_fun$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
        | ~ $less(sK295(X1),sK294(X1))
        | $less(X3,X2)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_523])]) ).

tff(f3632,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | ~ $less(sK295(X1),sK294(X1))
      | $less(X3,X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X2)) ),
    inference(cnf_transformation,[],[f2312]) ).

tff(f10018,plain,
    spl310_522,
    inference(avatar_split_clause,[],[f3628,f10016]) ).

tff(f10016,plain,
    ( spl310_522
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( $less(X0,'fun_app$b'(X1,X3))
        | ~ $less('fun_app$b'(X1,sK292(X1)),'fun_app$b'(X1,sK293(X1)))
        | ~ $less(X2,X3)
        | ~ $less(X0,'fun_app$b'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_522])]) ).

tff(f3628,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ~ $less('fun_app$b'(X1,sK292(X1)),'fun_app$b'(X1,sK293(X1)))
      | ~ $less(X2,X3)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(cnf_transformation,[],[f2310]) ).

tff(f2310,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ( ~ $less('fun_app$b'(X1,sK292(X1)),'fun_app$b'(X1,sK293(X1)))
        & $less(sK292(X1),sK293(X1)) )
      | ~ $less(X2,X3)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK292,sK293])],[f1550,f2309]) ).

tff(f2309,plain,
    ! [X1: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
     => ( ~ $less('fun_app$b'(X1,sK292(X1)),'fun_app$b'(X1,sK293(X1)))
        & $less(sK292(X1),sK293(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1550,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(flattening,[],[f1549]) ).

tff(f1549,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(ennf_transformation,[],[f289]) ).

tff(f289,axiom,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5)) )
        & $less(X2,X3)
        & $less(X0,'fun_app$b'(X1,X2)) )
     => $less(X0,'fun_app$b'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom287) ).

tff(f10014,plain,
    spl310_521,
    inference(avatar_split_clause,[],[f3626,f10012]) ).

tff(f10012,plain,
    ( spl310_521
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( $less(X0,'fun_app$b'(X1,X3))
        | $less('fun_app$b'(X1,sK291(X1)),'fun_app$b'(X1,sK290(X1)))
        | $less(X3,X2)
        | ~ $less(X0,'fun_app$b'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_521])]) ).

tff(f3626,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | $less('fun_app$b'(X1,sK291(X1)),'fun_app$b'(X1,sK290(X1)))
      | $less(X3,X2)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(cnf_transformation,[],[f2308]) ).

tff(f2308,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ( $less('fun_app$b'(X1,sK291(X1)),'fun_app$b'(X1,sK290(X1)))
        & ~ $less(sK291(X1),sK290(X1)) )
      | $less(X3,X2)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK290,sK291])],[f1548,f2307]) ).

tff(f2307,plain,
    ! [X1: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
     => ( $less('fun_app$b'(X1,sK291(X1)),'fun_app$b'(X1,sK290(X1)))
        & ~ $less(sK291(X1),sK290(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1548,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(flattening,[],[f1547]) ).

tff(f1547,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(ennf_transformation,[],[f852]) ).

tff(f852,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => ~ $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4)) )
        & ~ $less(X3,X2)
        & $less(X0,'fun_app$b'(X1,X2)) )
     => $less(X0,'fun_app$b'(X1,X3)) ),
    inference(theory_normalization,[],[f494]) ).

tff(f494,axiom,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => $lesseq('fun_app$b'(X1,X4),'fun_app$b'(X1,X5)) )
        & $lesseq(X2,X3)
        & $less(X0,'fun_app$b'(X1,X2)) )
     => $less(X0,'fun_app$b'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom492) ).

tff(f10010,plain,
    spl310_520,
    inference(avatar_split_clause,[],[f3623,f10008]) ).

tff(f10008,plain,
    ( spl310_520
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( $less(X0,'fun_app$o'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less$',sK288(X1)),sK289(X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | ~ $less(X0,'fun_app$o'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_520])]) ).

tff(f3623,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK288(X1)),sK289(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(cnf_transformation,[],[f2306]) ).

tff(f10006,plain,
    ( spl310_519
    | ~ spl310_117
    | ~ spl310_261 ),
    inference(avatar_split_clause,[],[f7009,f6563,f4838,f10004]) ).

tff(f10004,plain,
    ( spl310_519
  <=> ! [X0: 'Nat$'] :
        ( sP58('to_nat$'('i$'),X0,'to_nat$'('x$'))
        | ~ sP56('to_nat$'('x$'),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_519])]) ).

tff(f6563,plain,
    ( spl310_261
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP58(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP56(X2,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_261])]) ).

tff(f7009,plain,
    ( ! [X0: 'Nat$'] :
        ( sP58('to_nat$'('i$'),X0,'to_nat$'('x$'))
        | ~ sP56('to_nat$'('x$'),X0) )
    | ~ spl310_117
    | ~ spl310_261 ),
    inference(resolution,[],[f4840,f6564]) ).

tff(f6564,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | sP58(X0,X1,X2)
        | ~ sP56(X2,X1) )
    | ~ spl310_261 ),
    inference(avatar_component_clause,[],[f6563]) ).

tff(f10002,plain,
    spl310_518,
    inference(avatar_split_clause,[],[f3621,f10000]) ).

tff(f10000,plain,
    ( spl310_518
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( $less(X0,'fun_app$o'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK286(X1)),sK287(X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | ~ $less(X0,'fun_app$o'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_518])]) ).

tff(f3621,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK286(X1)),sK287(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ~ $less(X0,'fun_app$o'(X1,X2)) ),
    inference(cnf_transformation,[],[f2304]) ).

tff(f9998,plain,
    spl310_517,
    inference(avatar_split_clause,[],[f3619,f9996]) ).

tff(f9996,plain,
    ( spl310_517
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( $less(X0,'fun_app$b'(X1,X3))
        | ~ $less('fun_app$b'(X1,sK284(X1)),'fun_app$b'(X1,sK285(X1)))
        | ~ $less(X2,X3)
        | $less('fun_app$b'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_517])]) ).

tff(f3619,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ~ $less('fun_app$b'(X1,sK284(X1)),'fun_app$b'(X1,sK285(X1)))
      | ~ $less(X2,X3)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2302]) ).

tff(f2302,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ( ~ $less('fun_app$b'(X1,sK284(X1)),'fun_app$b'(X1,sK285(X1)))
        & $less(sK284(X1),sK285(X1)) )
      | ~ $less(X2,X3)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK284,sK285])],[f1540,f2301]) ).

tff(f2301,plain,
    ! [X1: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
     => ( ~ $less('fun_app$b'(X1,sK284(X1)),'fun_app$b'(X1,sK285(X1)))
        & $less(sK284(X1),sK285(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1540,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(flattening,[],[f1539]) ).

tff(f1539,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(ennf_transformation,[],[f849]) ).

tff(f849,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5)) )
        & $less(X2,X3)
        & ~ $less('fun_app$b'(X1,X2),X0) )
     => $less(X0,'fun_app$b'(X1,X3)) ),
    inference(theory_normalization,[],[f485]) ).

tff(f485,axiom,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5)) )
        & $less(X2,X3)
        & $lesseq(X0,'fun_app$b'(X1,X2)) )
     => $less(X0,'fun_app$b'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom483) ).

tff(f9994,plain,
    spl310_516,
    inference(avatar_split_clause,[],[f3617,f9992]) ).

tff(f9992,plain,
    ( spl310_516
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( ~ $less('fun_app$b'(X1,X3),X0)
        | $less('fun_app$b'(X1,sK283(X1)),'fun_app$b'(X1,sK282(X1)))
        | $less(X3,X2)
        | $less('fun_app$b'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_516])]) ).

tff(f3617,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | $less('fun_app$b'(X1,sK283(X1)),'fun_app$b'(X1,sK282(X1)))
      | $less(X3,X2)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2300]) ).

tff(f2300,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ( $less('fun_app$b'(X1,sK283(X1)),'fun_app$b'(X1,sK282(X1)))
        & ~ $less(sK283(X1),sK282(X1)) )
      | $less(X3,X2)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK282,sK283])],[f1538,f2299]) ).

tff(f2299,plain,
    ! [X1: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
     => ( $less('fun_app$b'(X1,sK283(X1)),'fun_app$b'(X1,sK282(X1)))
        & ~ $less(sK283(X1),sK282(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1538,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(flattening,[],[f1537]) ).

tff(f1537,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(ennf_transformation,[],[f848]) ).

tff(f848,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => ~ $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4)) )
        & ~ $less(X3,X2)
        & ~ $less('fun_app$b'(X1,X2),X0) )
     => ~ $less('fun_app$b'(X1,X3),X0) ),
    inference(theory_normalization,[],[f200]) ).

tff(f200,axiom,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => $lesseq('fun_app$b'(X1,X4),'fun_app$b'(X1,X5)) )
        & $lesseq(X2,X3)
        & $lesseq(X0,'fun_app$b'(X1,X2)) )
     => $lesseq(X0,'fun_app$b'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom198) ).

tff(f9990,plain,
    spl310_515,
    inference(avatar_split_clause,[],[f3614,f9988]) ).

tff(f9988,plain,
    ( spl310_515
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( $less(X0,'fun_app$o'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less$',sK280(X1)),sK281(X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
        | $less('fun_app$o'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_515])]) ).

tff(f3614,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK280(X1)),sK281(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2298]) ).

tff(f9986,plain,
    spl310_514,
    inference(avatar_split_clause,[],[f3612,f9984]) ).

tff(f9984,plain,
    ( spl310_514
  <=> ! [X0: $int,X3: 'N$',X2: 'N$',X1: 'N_int_fun$'] :
        ( ~ $less('fun_app$o'(X1,X3),X0)
        | 'fun_app$d'('fun_app$g'('less_eq$',sK278(X1)),sK279(X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        | $less('fun_app$o'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_514])]) ).

tff(f3612,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | 'fun_app$d'('fun_app$g'('less_eq$',sK278(X1)),sK279(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | $less('fun_app$o'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2296]) ).

tff(f9982,plain,
    spl310_513,
    inference(avatar_split_clause,[],[f3602,f9980]) ).

tff(f9980,plain,
    ( spl310_513
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),X3)
        | 'fun_app$d'('fun_app$g'('less$',sK268(X2)),sK269(X2))
        | ~ $less('fun_app$o'(X2,X1),X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_513])]) ).

tff(f3602,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | 'fun_app$d'('fun_app$g'('less$',sK268(X2)),sK269(X2))
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2286]) ).

tff(f9978,plain,
    spl310_512,
    inference(avatar_split_clause,[],[f3600,f9976]) ).

tff(f9976,plain,
    ( spl310_512
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),X3)
        | 'fun_app$d'('fun_app$g'('less_eq$',sK266(X2)),sK267(X2))
        | ~ $less('fun_app$o'(X2,X1),X3)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_512])]) ).

tff(f3600,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | 'fun_app$d'('fun_app$g'('less_eq$',sK266(X2)),sK267(X2))
      | ~ $less('fun_app$o'(X2,X1),X3)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2284]) ).

tff(f9974,plain,
    spl310_511,
    inference(avatar_split_clause,[],[f3598,f9972]) ).

tff(f9972,plain,
    ( spl310_511
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),X3)
        | 'fun_app$d'('fun_app$g'('less$',sK264(X2)),sK265(X2))
        | $less(X3,'fun_app$o'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_511])]) ).

tff(f3598,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | 'fun_app$d'('fun_app$g'('less$',sK264(X2)),sK265(X2))
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2282]) ).

tff(f9970,plain,
    spl310_510,
    inference(avatar_split_clause,[],[f3596,f9968]) ).

tff(f9968,plain,
    ( spl310_510
  <=> ! [X0: 'N$',X3: $int,X2: 'N_int_fun$',X1: 'N$'] :
        ( ~ $less(X3,'fun_app$o'(X2,X0))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK262(X2)),sK263(X2))
        | $less(X3,'fun_app$o'(X2,X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_510])]) ).

tff(f3596,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK262(X2)),sK263(X2))
      | $less(X3,'fun_app$o'(X2,X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2280]) ).

tff(f9966,plain,
    spl310_509,
    inference(avatar_split_clause,[],[f3556,f9964]) ).

tff(f9964,plain,
    ( spl310_509
  <=> ! [X0: 'Int_set$',X3: 'Int_set$',X2: 'Int_set_int_set_fun$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
        | 'fun_app$c'('less_eq$a'(sK222(X2)),sK223(X2))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_509])]) ).

tff(f3556,plain,
    ! [X2: 'Int_set_int_set_fun$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | 'fun_app$c'('less_eq$a'(sK222(X2)),sK223(X2))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X1)),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2240]) ).

tff(f9962,plain,
    ( spl310_508
    | ~ spl310_117
    | ~ spl310_260 ),
    inference(avatar_split_clause,[],[f7008,f6559,f4838,f9960]) ).

tff(f9960,plain,
    ( spl310_508
  <=> ! [X0: 'Nat$'] :
        ( sP56('to_nat$'('x$'),X0)
        | ~ sP58('to_nat$'('i$'),X0,'to_nat$'('x$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_508])]) ).

tff(f6559,plain,
    ( spl310_260
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP56(X2,X1)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP58(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_260])]) ).

tff(f7008,plain,
    ( ! [X0: 'Nat$'] :
        ( sP56('to_nat$'('x$'),X0)
        | ~ sP58('to_nat$'('i$'),X0,'to_nat$'('x$')) )
    | ~ spl310_117
    | ~ spl310_260 ),
    inference(resolution,[],[f4840,f6560]) ).

tff(f6560,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | sP56(X2,X1)
        | ~ sP58(X0,X1,X2) )
    | ~ spl310_260 ),
    inference(avatar_component_clause,[],[f6559]) ).

tff(f9958,plain,
    spl310_507,
    inference(avatar_split_clause,[],[f3544,f9956]) ).

tff(f9956,plain,
    ( spl310_507
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
        | $less(sK212(X2),sK213(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_507])]) ).

tff(f3544,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | $less(sK212(X2),sK213(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2230]) ).

tff(f9954,plain,
    spl310_506,
    inference(avatar_split_clause,[],[f3542,f9952]) ).

tff(f9952,plain,
    ( spl310_506
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
        | $less(sK210(X2),sK211(X2))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_506])]) ).

tff(f3542,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | $less(sK210(X2),sK211(X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2228]) ).

tff(f9950,plain,
    spl310_505,
    inference(avatar_split_clause,[],[f3541,f9948]) ).

tff(f9948,plain,
    ( spl310_505
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( $less('fun_app$b'(X2,X0),X3)
        | ~ $less('fun_app$b'(X2,sK208(X2)),'fun_app$b'(X2,sK209(X2)))
        | ~ $less('fun_app$b'(X2,X1),X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_505])]) ).

tff(f3541,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ~ $less('fun_app$b'(X2,sK208(X2)),'fun_app$b'(X2,sK209(X2)))
      | ~ $less('fun_app$b'(X2,X1),X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2226]) ).

tff(f2226,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ( ~ $less('fun_app$b'(X2,sK208(X2)),'fun_app$b'(X2,sK209(X2)))
        & $less(sK208(X2),sK209(X2)) )
      | ~ $less('fun_app$b'(X2,X1),X3)
      | ~ $less(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK208,sK209])],[f1460,f2225]) ).

tff(f2225,plain,
    ! [X2: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
     => ( ~ $less('fun_app$b'(X2,sK208(X2)),'fun_app$b'(X2,sK209(X2)))
        & $less(sK208(X2),sK209(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1460,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
      | ~ $less('fun_app$b'(X2,X1),X3)
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1459]) ).

tff(f1459,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
      | ~ $less('fun_app$b'(X2,X1),X3)
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f285]) ).

tff(f285,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5)) )
        & $less('fun_app$b'(X2,X1),X3)
        & $less(X0,X1) )
     => $less('fun_app$b'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom283) ).

tff(f9946,plain,
    spl310_504,
    inference(avatar_split_clause,[],[f3539,f9944]) ).

tff(f9944,plain,
    ( spl310_504
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( $less('fun_app$b'(X2,X0),X3)
        | ~ $less('fun_app$b'(X2,sK206(X2)),'fun_app$b'(X2,sK207(X2)))
        | $less(X3,'fun_app$b'(X2,X1))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_504])]) ).

tff(f3539,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ~ $less('fun_app$b'(X2,sK206(X2)),'fun_app$b'(X2,sK207(X2)))
      | $less(X3,'fun_app$b'(X2,X1))
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2224]) ).

tff(f2224,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ( ~ $less('fun_app$b'(X2,sK206(X2)),'fun_app$b'(X2,sK207(X2)))
        & $less(sK206(X2),sK207(X2)) )
      | $less(X3,'fun_app$b'(X2,X1))
      | ~ $less(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK206,sK207])],[f1458,f2223]) ).

tff(f2223,plain,
    ! [X2: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
     => ( ~ $less('fun_app$b'(X2,sK206(X2)),'fun_app$b'(X2,sK207(X2)))
        & $less(sK206(X2),sK207(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1458,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
      | $less(X3,'fun_app$b'(X2,X1))
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1457]) ).

tff(f1457,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
      | $less(X3,'fun_app$b'(X2,X1))
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f834]) ).

tff(f834,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5)) )
        & ~ $less(X3,'fun_app$b'(X2,X1))
        & $less(X0,X1) )
     => $less('fun_app$b'(X2,X0),X3) ),
    inference(theory_normalization,[],[f497]) ).

tff(f497,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5)) )
        & $lesseq('fun_app$b'(X2,X1),X3)
        & $less(X0,X1) )
     => $less('fun_app$b'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom495) ).

tff(f9942,plain,
    spl310_503,
    inference(avatar_split_clause,[],[f3526,f9940]) ).

tff(f9940,plain,
    ( spl310_503
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
        | ~ $less(sK201(X2),sK200(X2))
        | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_503])]) ).

tff(f3526,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | ~ $less(sK201(X2),sK200(X2))
      | ~ 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2218]) ).

tff(f9938,plain,
    spl310_502,
    inference(avatar_split_clause,[],[f3524,f9936]) ).

tff(f9936,plain,
    ( spl310_502
  <=> ! [X0: $int,X3: 'N$',X2: 'Int_n_fun$',X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
        | ~ $less(sK199(X2),sK198(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_502])]) ).

tff(f3524,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ~ $less(sK199(X2),sK198(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X1)),X3)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2216]) ).

tff(f9934,plain,
    spl310_501,
    inference(avatar_split_clause,[],[f3523,f9932]) ).

tff(f9932,plain,
    ( spl310_501
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( $less('fun_app$b'(X2,X0),X3)
        | $less('fun_app$b'(X2,sK197(X2)),'fun_app$b'(X2,sK196(X2)))
        | ~ $less('fun_app$b'(X2,X1),X3)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_501])]) ).

tff(f3523,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | $less('fun_app$b'(X2,sK197(X2)),'fun_app$b'(X2,sK196(X2)))
      | ~ $less('fun_app$b'(X2,X1),X3)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2214]) ).

tff(f2214,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ( $less('fun_app$b'(X2,sK197(X2)),'fun_app$b'(X2,sK196(X2)))
        & ~ $less(sK197(X2),sK196(X2)) )
      | ~ $less('fun_app$b'(X2,X1),X3)
      | $less(X1,X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK196,sK197])],[f1436,f2213]) ).

tff(f2213,plain,
    ! [X2: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
     => ( $less('fun_app$b'(X2,sK197(X2)),'fun_app$b'(X2,sK196(X2)))
        & ~ $less(sK197(X2),sK196(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1436,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
      | ~ $less('fun_app$b'(X2,X1),X3)
      | $less(X1,X0) ),
    inference(flattening,[],[f1435]) ).

tff(f1435,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
      | ~ $less('fun_app$b'(X2,X1),X3)
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f828]) ).

tff(f828,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => ~ $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4)) )
        & $less('fun_app$b'(X2,X1),X3)
        & ~ $less(X1,X0) )
     => $less('fun_app$b'(X2,X0),X3) ),
    inference(theory_normalization,[],[f490]) ).

tff(f490,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => $lesseq('fun_app$b'(X2,X4),'fun_app$b'(X2,X5)) )
        & $less('fun_app$b'(X2,X1),X3)
        & $lesseq(X0,X1) )
     => $less('fun_app$b'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom488) ).

tff(f9930,plain,
    spl310_500,
    inference(avatar_split_clause,[],[f3521,f9928]) ).

tff(f9928,plain,
    ( spl310_500
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( ~ $less(X3,'fun_app$b'(X2,X0))
        | $less('fun_app$b'(X2,sK195(X2)),'fun_app$b'(X2,sK194(X2)))
        | $less(X3,'fun_app$b'(X2,X1))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_500])]) ).

tff(f3521,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | $less('fun_app$b'(X2,sK195(X2)),'fun_app$b'(X2,sK194(X2)))
      | $less(X3,'fun_app$b'(X2,X1))
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2212]) ).

tff(f2212,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ( $less('fun_app$b'(X2,sK195(X2)),'fun_app$b'(X2,sK194(X2)))
        & ~ $less(sK195(X2),sK194(X2)) )
      | $less(X3,'fun_app$b'(X2,X1))
      | $less(X1,X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK194,sK195])],[f1434,f2211]) ).

tff(f2211,plain,
    ! [X2: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
     => ( $less('fun_app$b'(X2,sK195(X2)),'fun_app$b'(X2,sK194(X2)))
        & ~ $less(sK195(X2),sK194(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1434,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
      | $less(X3,'fun_app$b'(X2,X1))
      | $less(X1,X0) ),
    inference(flattening,[],[f1433]) ).

tff(f1433,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
      | $less(X3,'fun_app$b'(X2,X1))
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f827]) ).

tff(f827,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => ~ $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4)) )
        & ~ $less(X3,'fun_app$b'(X2,X1))
        & ~ $less(X1,X0) )
     => ~ $less(X3,'fun_app$b'(X2,X0)) ),
    inference(theory_normalization,[],[f196]) ).

tff(f196,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => $lesseq('fun_app$b'(X2,X4),'fun_app$b'(X2,X5)) )
        & $lesseq('fun_app$b'(X2,X1),X3)
        & $lesseq(X0,X1) )
     => $lesseq('fun_app$b'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom194) ).

tff(f9926,plain,
    spl310_499,
    inference(avatar_split_clause,[],[f3346,f9924]) ).

tff(f9924,plain,
    ( spl310_499
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ $less(X2,X1)
        | sP65(X1,X0,X2)
        | sP64(X2,X1,X0)
        | sP63(X0,X2,X1)
        | sP62(X1,X2,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_499])]) ).

tff(f3346,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X1,X0)
      | ~ $less(X2,X1)
      | sP65(X1,X0,X2)
      | sP64(X2,X1,X0)
      | sP63(X0,X2,X1)
      | sP62(X1,X2,X0) ),
    inference(cnf_transformation,[],[f1672]) ).

tff(f1672,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X1,X0)
        & ~ $less(X0,X2) )
      | ( ~ $less(X0,X2)
        & ~ $less(X2,X1) )
      | sP65(X1,X0,X2)
      | sP64(X2,X1,X0)
      | sP63(X0,X2,X1)
      | sP62(X1,X2,X0) ),
    inference(definition_folding,[],[f1343,f1671,f1670,f1669,f1668]) ).

tff(f1668,plain,
    ! [X1: $int,X2: $int,X0: $int] :
      ( ( ~ $less(X2,X1)
        & ~ $less(X1,X0) )
      | ~ sP62(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP62])]) ).

tff(f1669,plain,
    ! [X0: $int,X2: $int,X1: $int] :
      ( ( ~ $less(X2,X0)
        & ~ $less(X0,X1) )
      | ~ sP63(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP63])]) ).

tff(f1670,plain,
    ! [X2: $int,X1: $int,X0: $int] :
      ( ( ~ $less(X1,X2)
        & ~ $less(X2,X0) )
      | ~ sP64(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP64])]) ).

tff(f1671,plain,
    ! [X1: $int,X0: $int,X2: $int] :
      ( ( ~ $less(X0,X1)
        & ~ $less(X1,X2) )
      | ~ sP65(X1,X0,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP65])]) ).

tff(f1343,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X1,X0)
        & ~ $less(X0,X2) )
      | ( ~ $less(X0,X2)
        & ~ $less(X2,X1) )
      | ( ~ $less(X0,X1)
        & ~ $less(X1,X2) )
      | ( ~ $less(X1,X2)
        & ~ $less(X2,X0) )
      | ( ~ $less(X2,X0)
        & ~ $less(X0,X1) )
      | ( ~ $less(X2,X1)
        & ~ $less(X1,X0) ) ),
    inference(ennf_transformation,[],[f972]) ).

tff(f972,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ~ ( ~ ( ~ $less(X1,X0)
            & ~ $less(X0,X2) )
        & ~ ( ~ $less(X0,X2)
            & ~ $less(X2,X1) )
        & ~ ( ~ $less(X0,X1)
            & ~ $less(X1,X2) )
        & ~ ( ~ $less(X1,X2)
            & ~ $less(X2,X0) )
        & ~ ( ~ $less(X2,X0)
            & ~ $less(X0,X1) )
        & ~ ( ~ $less(X2,X1)
            & ~ $less(X1,X0) ) ),
    inference(true_and_false_elimination,[],[f790]) ).

tff(f790,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ( ( ~ $less(X1,X0)
            & ~ $less(X0,X2) )
         => $false )
        & ( ( ~ $less(X0,X2)
            & ~ $less(X2,X1) )
         => $false )
        & ( ( ~ $less(X0,X1)
            & ~ $less(X1,X2) )
         => $false )
        & ( ( ~ $less(X1,X2)
            & ~ $less(X2,X0) )
         => $false )
        & ( ( ~ $less(X2,X0)
            & ~ $less(X0,X1) )
         => $false )
        & ( ( ~ $less(X2,X1)
            & ~ $less(X1,X0) )
         => $false ) )
     => $false ),
    inference(theory_normalization,[],[f234]) ).

tff(f234,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ( ( $lesseq(X0,X1)
            & $lesseq(X2,X0) )
         => $false )
        & ( ( $lesseq(X2,X0)
            & $lesseq(X1,X2) )
         => $false )
        & ( ( $lesseq(X1,X0)
            & $lesseq(X2,X1) )
         => $false )
        & ( ( $lesseq(X2,X1)
            & $lesseq(X0,X2) )
         => $false )
        & ( ( $lesseq(X0,X2)
            & $lesseq(X1,X0) )
         => $false )
        & ( ( $lesseq(X1,X2)
            & $lesseq(X0,X1) )
         => $false ) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom232) ).

tff(f9922,plain,
    spl310_498,
    inference(avatar_split_clause,[],[f3220,f9920]) ).

tff(f9920,plain,
    ( spl310_498
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP41(X0,X2,X1)
        | sP42(X0,X2,X1)
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_498])]) ).

tff(f3220,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP41(X0,X2,X1)
      | sP42(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1643]) ).

tff(f1643,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( sP41(X0,X2,X1)
            & sP40(X0,X2,X1) )
          | sP42(X0,X2,X1) )
        & sP43(X2,X1,X0) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1318,f1642,f1641,f1640,f1639,f1638,f1637,f1636]) ).

tff(f1636,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less(0,0)
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ~ sP37(X0,X2) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP37])]) ).

tff(f1638,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP37(X0,X2)
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP39(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP39])]) ).

tff(f1318,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus(0)) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum(0,$uminus(0)) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( $less(0,0)
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = 0 )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
                & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              | ( ~ $less(0,0)
                & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1317]) ).

tff(f1317,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus(0)) )
                      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
                    & ( ( 0 = $sum(0,$uminus(0)) )
                      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                  | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
                & ( $less(0,0)
                  | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
            & ( ( 0 = 0 )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
          | ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              | ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
                & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              | ( ~ $less(0,0)
                & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f887]) ).

tff(f887,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                  & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ) ) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                  & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less(0,0) ) ) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) ) ) ) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$uminus(0)) ) ) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum(0,$uminus($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))) ) ) ) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus(0)) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum(0,$uminus(0)) ) ) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less(0,0) ) ) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
             => ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f781]) ).

tff(f781,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                  & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ) ) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                  & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less(0,0) ) ) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$uminus($difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)))) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$uminus($difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)))) ) ) ) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$uminus(0)) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$uminus(0)) ) ) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus($difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)))) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum(0,$uminus($difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)))) ) ) ) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) = $sum(0,$uminus(0)) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $sum(0,$uminus(0)) ) ) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less(0,0) ) ) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
             => ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
             => ( 0 = 0 ) ) ) ) ) ),
    inference(theory_normalization,[],[f151]) ).

tff(f151,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
                  & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),0) ) ) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less(0,$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
                  & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                   => $less(0,0) ) ) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) = $difference($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $difference($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) ) ) ) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) = $difference($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),0) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $difference($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),0) ) ) ) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) = $difference(0,$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $difference(0,$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) ) ) ) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) = $difference(0,0) ) )
                    & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
                     => ( 0 = $difference(0,0) ) ) ) ) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),0) ) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less(0,$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
                & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
                 => $less(0,0) ) ) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
             => ( 0 = $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
             => ( 0 = 0 ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom149) ).

tff(f9918,plain,
    ( spl310_497
    | ~ spl310_117
    | ~ spl310_259 ),
    inference(avatar_split_clause,[],[f7007,f6555,f4838,f9916]) ).

tff(f9916,plain,
    ( spl310_497
  <=> ! [X0: 'Nat$'] :
        ( sP54(X0,'to_nat$'('i$'),'to_nat$'('x$'))
        | ~ sP52('to_nat$'('x$'),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_497])]) ).

tff(f6555,plain,
    ( spl310_259
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP54(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP52(X2,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_259])]) ).

tff(f7007,plain,
    ( ! [X0: 'Nat$'] :
        ( sP54(X0,'to_nat$'('i$'),'to_nat$'('x$'))
        | ~ sP52('to_nat$'('x$'),X0) )
    | ~ spl310_117
    | ~ spl310_259 ),
    inference(resolution,[],[f4840,f6556]) ).

tff(f6556,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | sP54(X0,X1,X2)
        | ~ sP52(X2,X0) )
    | ~ spl310_259 ),
    inference(avatar_component_clause,[],[f6555]) ).

tff(f9914,plain,
    spl310_496,
    inference(avatar_split_clause,[],[f3219,f9912]) ).

tff(f9912,plain,
    ( spl310_496
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP40(X0,X2,X1)
        | sP42(X0,X2,X1)
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_496])]) ).

tff(f3219,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP40(X0,X2,X1)
      | sP42(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1643]) ).

tff(f9910,plain,
    spl310_495,
    inference(avatar_split_clause,[],[f3180,f9908]) ).

tff(f9908,plain,
    ( spl310_495
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X1)
        | ~ $less('fun_app$e'('of_nat$',sK127(X0,X2)),'fun_app$e'('of_nat$',X0))
        | ~ 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_495])]) ).

tff(f3180,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X1)
      | ~ $less('fun_app$e'('of_nat$',sK127(X0,X2)),'fun_app$e'('of_nat$',X0))
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2007]) ).

tff(f9747,plain,
    ( spl310_494
    | ~ spl310_117
    | ~ spl310_258 ),
    inference(avatar_split_clause,[],[f7006,f6551,f4838,f9745]) ).

tff(f9745,plain,
    ( spl310_494
  <=> ! [X0: 'Nat$'] :
        ( sP52('to_nat$'('x$'),X0)
        | ~ sP54(X0,'to_nat$'('i$'),'to_nat$'('x$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_494])]) ).

tff(f6551,plain,
    ( spl310_258
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP52(X2,X0)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP54(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_258])]) ).

tff(f7006,plain,
    ( ! [X0: 'Nat$'] :
        ( sP52('to_nat$'('x$'),X0)
        | ~ sP54(X0,'to_nat$'('i$'),'to_nat$'('x$')) )
    | ~ spl310_117
    | ~ spl310_258 ),
    inference(resolution,[],[f4840,f6552]) ).

tff(f6552,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | sP52(X2,X0)
        | ~ sP54(X0,X1,X2) )
    | ~ spl310_258 ),
    inference(avatar_component_clause,[],[f6551]) ).

tff(f9676,plain,
    ( spl310_493
    | ~ spl310_117
    | ~ spl310_257 ),
    inference(avatar_split_clause,[],[f7005,f6547,f4838,f9674]) ).

tff(f9674,plain,
    ( spl310_493
  <=> ! [X0: 'Nat$'] :
        ( sP50(X0,'to_nat$'('i$'),'to_nat$'('x$'))
        | ~ sP48(X0,'to_nat$'('i$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_493])]) ).

tff(f6547,plain,
    ( spl310_257
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP50(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP48(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_257])]) ).

tff(f7005,plain,
    ( ! [X0: 'Nat$'] :
        ( sP50(X0,'to_nat$'('i$'),'to_nat$'('x$'))
        | ~ sP48(X0,'to_nat$'('i$')) )
    | ~ spl310_117
    | ~ spl310_257 ),
    inference(resolution,[],[f4840,f6548]) ).

tff(f6548,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | sP50(X0,X1,X2)
        | ~ sP48(X0,X1) )
    | ~ spl310_257 ),
    inference(avatar_component_clause,[],[f6547]) ).

tff(f9570,plain,
    spl310_492,
    inference(avatar_split_clause,[],[f3987,f9568]) ).

tff(f9568,plain,
    ( spl310_492
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$sum(1,'fun_app$e'('of_nat$',X0)))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_492])]) ).

tff(f3987,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$sum(1,'fun_app$e'('of_nat$',X0)))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2461,f864]) ).

tff(f2461,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X0),1))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f991]) ).

tff(f991,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X0),1))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      & ( $less(0,$sum('fun_app$e'('of_nat$',X0),1))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    inference(ennf_transformation,[],[f649]) ).

tff(f649,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),$sum('fun_app$e'('of_nat$',X0),1)) )
      & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => $less(0,$sum('fun_app$e'('of_nat$',X0),1)) ) ),
    inference(theory_normalization,[],[f374]) ).

tff(f374,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => $less($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),$sum('fun_app$e'('of_nat$',X0),1)) )
      & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => $less(0,$sum('fun_app$e'('of_nat$',X0),1)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom372) ).

tff(f9566,plain,
    spl310_491,
    inference(avatar_split_clause,[],[f3896,f9564]) ).

tff(f9564,plain,
    ( spl310_491
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        | sP69(X0,X1,X2)
        | sP68(X2,X0,X1)
        | sP67(X1,X2,X0)
        | sP66(X0,X2,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_491])]) ).

tff(f3896,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | sP69(X0,X1,X2)
      | sP68(X2,X0,X1)
      | sP67(X1,X2,X0)
      | sP66(X0,X2,X1) ),
    inference(duplicate_literal_removal,[],[f3357]) ).

tff(f3357,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | sP69(X0,X1,X2)
      | sP68(X2,X0,X1)
      | sP67(X1,X2,X0)
      | sP66(X0,X2,X1) ),
    inference(cnf_transformation,[],[f1677]) ).

tff(f9562,plain,
    ( spl310_490
    | ~ spl310_117
    | ~ spl310_256 ),
    inference(avatar_split_clause,[],[f7004,f6543,f4838,f9560]) ).

tff(f9560,plain,
    ( spl310_490
  <=> ! [X0: 'Nat$'] :
        ( sP48(X0,'to_nat$'('i$'))
        | ~ sP50(X0,'to_nat$'('i$'),'to_nat$'('x$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_490])]) ).

tff(f6543,plain,
    ( spl310_256
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP48(X0,X1)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP50(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_256])]) ).

tff(f7004,plain,
    ( ! [X0: 'Nat$'] :
        ( sP48(X0,'to_nat$'('i$'))
        | ~ sP50(X0,'to_nat$'('i$'),'to_nat$'('x$')) )
    | ~ spl310_117
    | ~ spl310_256 ),
    inference(resolution,[],[f4840,f6544]) ).

tff(f6544,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | sP48(X0,X1)
        | ~ sP50(X0,X1,X2) )
    | ~ spl310_256 ),
    inference(avatar_component_clause,[],[f6543]) ).

tff(f9558,plain,
    spl310_489,
    inference(avatar_split_clause,[],[f3895,f9556]) ).

tff(f9556,plain,
    ( spl310_489
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        | sP73(X0,X1,X2)
        | sP72(X1,X2,X0)
        | sP71(X2,X0,X1)
        | sP70(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_489])]) ).

tff(f3895,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
      | sP73(X0,X1,X2)
      | sP72(X1,X2,X0)
      | sP71(X2,X0,X1)
      | sP70(X2,X1,X0) ),
    inference(duplicate_literal_removal,[],[f3369]) ).

tff(f3369,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
      | 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
      | sP73(X0,X1,X2)
      | sP72(X1,X2,X0)
      | sP71(X2,X0,X1)
      | sP70(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1682]) ).

tff(f9554,plain,
    spl310_488,
    inference(avatar_split_clause,[],[f3885,f9552]) ).

tff(f9552,plain,
    ( spl310_488
  <=> ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK260(X2)),sK261(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_488])]) ).

tff(f3885,plain,
    ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK260(X2)),sK261(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(equality_resolution,[],[f3594]) ).

tff(f3594,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X2,X0)),X3)
      | 'fun_app$d'('fun_app$g'('less_eq$',sK260(X2)),sK261(X2))
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2278]) ).

tff(f9550,plain,
    spl310_487,
    inference(avatar_split_clause,[],[f3883,f9548]) ).

tff(f9548,plain,
    ( spl310_487
  <=> ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
        | 'fun_app$d'('fun_app$g'('less$',sK258(X2)),sK259(X2))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_487])]) ).

tff(f3883,plain,
    ! [X2: 'N_n_fun$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),'fun_app$f'(X2,X1))
      | 'fun_app$d'('fun_app$g'('less$',sK258(X2)),sK259(X2))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(equality_resolution,[],[f3592]) ).

tff(f3592,plain,
    ! [X2: 'N_n_fun$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X2,X0)),X3)
      | 'fun_app$d'('fun_app$g'('less$',sK258(X2)),sK259(X2))
      | ( 'fun_app$f'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2276]) ).

tff(f9546,plain,
    spl310_486,
    inference(avatar_split_clause,[],[f3880,f9544]) ).

tff(f9544,plain,
    ( spl310_486
  <=> ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),'fun_app$o'(X2,X1))
        | ~ $less('fun_app$o'(X2,sK256(X2)),'fun_app$o'(X2,sK257(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_486])]) ).

tff(f3880,plain,
    ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),'fun_app$o'(X2,X1))
      | ~ $less('fun_app$o'(X2,sK256(X2)),'fun_app$o'(X2,sK257(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(equality_resolution,[],[f3591]) ).

tff(f3591,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ~ $less('fun_app$o'(X2,sK256(X2)),'fun_app$o'(X2,sK257(X2)))
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2274]) ).

tff(f2274,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ( ~ $less('fun_app$o'(X2,sK256(X2)),'fun_app$o'(X2,sK257(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK256(X2)),sK257(X2)) )
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK256,sK257])],[f1512,f2273]) ).

tff(f2273,plain,
    ! [X2: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ $less('fun_app$o'(X2,sK256(X2)),'fun_app$o'(X2,sK257(X2)))
        & 'fun_app$d'('fun_app$g'('less$',sK256(X2)),sK257(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1512,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1511]) ).

tff(f1511,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( $less('fun_app$o'(X2,X0),X3)
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f297]) ).

tff(f297,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X2,X4),'fun_app$o'(X2,X5)) )
        & ( 'fun_app$o'(X2,X1) = X3 )
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => $less('fun_app$o'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom295) ).

tff(f9542,plain,
    spl310_485,
    inference(avatar_split_clause,[],[f3878,f9540]) ).

tff(f9540,plain,
    ( spl310_485
  <=> ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
        ( ~ $less('fun_app$o'(X2,X1),'fun_app$o'(X2,X0))
        | $less('fun_app$o'(X2,sK255(X2)),'fun_app$o'(X2,sK254(X2)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_485])]) ).

tff(f3878,plain,
    ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
      ( ~ $less('fun_app$o'(X2,X1),'fun_app$o'(X2,X0))
      | $less('fun_app$o'(X2,sK255(X2)),'fun_app$o'(X2,sK254(X2)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(equality_resolution,[],[f3589]) ).

tff(f3589,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | $less('fun_app$o'(X2,sK255(X2)),'fun_app$o'(X2,sK254(X2)))
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2272]) ).

tff(f2272,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | ( $less('fun_app$o'(X2,sK255(X2)),'fun_app$o'(X2,sK254(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK254(X2)),sK255(X2)) )
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK254,sK255])],[f1510,f2271]) ).

tff(f2271,plain,
    ! [X2: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( $less('fun_app$o'(X2,sK255(X2)),'fun_app$o'(X2,sK254(X2)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK254(X2)),sK255(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1510,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1509]) ).

tff(f1509,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f842]) ).

tff(f842,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => ~ $less('fun_app$o'(X2,X5),'fun_app$o'(X2,X4)) )
        & ( 'fun_app$o'(X2,X1) = X3 )
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => ~ $less(X3,'fun_app$o'(X2,X0)) ),
    inference(theory_normalization,[],[f173]) ).

tff(f173,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N_int_fun$',X3: $int] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => $lesseq('fun_app$o'(X2,X4),'fun_app$o'(X2,X5)) )
        & ( 'fun_app$o'(X2,X1) = X3 )
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => $lesseq('fun_app$o'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom171) ).

tff(f9538,plain,
    spl310_484,
    inference(avatar_split_clause,[],[f3875,f9536]) ).

tff(f9536,plain,
    ( spl310_484
  <=> ! [X2: 'N$',X1: 'N_n_fun$',X3: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less$',sK250(X1)),sK251(X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_484])]) ).

tff(f3875,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK250(X1)),sK251(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ),
    inference(equality_resolution,[],[f3584]) ).

tff(f3584,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK250(X1)),sK251(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2268]) ).

tff(f9534,plain,
    spl310_483,
    inference(avatar_split_clause,[],[f3873,f9532]) ).

tff(f9532,plain,
    ( spl310_483
  <=> ! [X2: 'N$',X1: 'N_n_fun$',X3: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK248(X1)),sK249(X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_483])]) ).

tff(f3873,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$f'(X1,X2)),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK248(X1)),sK249(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ),
    inference(equality_resolution,[],[f3582]) ).

tff(f3582,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$f'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK248(X1)),sK249(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$f'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2266]) ).

tff(f9530,plain,
    spl310_482,
    inference(avatar_split_clause,[],[f3864,f9528]) ).

tff(f9528,plain,
    ( spl310_482
  <=> ! [X2: 'N$',X1: 'N_int_fun$',X3: 'N$'] :
        ( $less('fun_app$o'(X1,X2),'fun_app$o'(X1,X3))
        | ~ $less('fun_app$o'(X1,sK240(X1)),'fun_app$o'(X1,sK241(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_482])]) ).

tff(f3864,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_int_fun$'] :
      ( $less('fun_app$o'(X1,X2),'fun_app$o'(X1,X3))
      | ~ $less('fun_app$o'(X1,sK240(X1)),'fun_app$o'(X1,sK241(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ),
    inference(equality_resolution,[],[f3575]) ).

tff(f3575,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ~ $less('fun_app$o'(X1,sK240(X1)),'fun_app$o'(X1,sK241(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2258]) ).

tff(f2258,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ( ~ $less('fun_app$o'(X1,sK240(X1)),'fun_app$o'(X1,sK241(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK240(X1)),sK241(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK240,sK241])],[f1496,f2257]) ).

tff(f2257,plain,
    ! [X1: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
     => ( ~ $less('fun_app$o'(X1,sK240(X1)),'fun_app$o'(X1,sK241(X1)))
        & 'fun_app$d'('fun_app$g'('less$',sK240(X1)),sK241(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1496,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1495]) ).

tff(f1495,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | ? [X4: 'N$',X5: 'N$'] :
          ( ~ $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5))
          & 'fun_app$d'('fun_app$g'('less$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f301]) ).

tff(f301,axiom,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less$',X4),X5)
           => $less('fun_app$o'(X1,X4),'fun_app$o'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less$',X2),X3)
        & ( 'fun_app$o'(X1,X2) = X0 ) )
     => $less(X0,'fun_app$o'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom299) ).

tff(f9526,plain,
    spl310_481,
    inference(avatar_split_clause,[],[f3862,f9524]) ).

tff(f9524,plain,
    ( spl310_481
  <=> ! [X2: 'N$',X1: 'N_int_fun$',X3: 'N$'] :
        ( ~ $less('fun_app$o'(X1,X3),'fun_app$o'(X1,X2))
        | $less('fun_app$o'(X1,sK239(X1)),'fun_app$o'(X1,sK238(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_481])]) ).

tff(f3862,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_int_fun$'] :
      ( ~ $less('fun_app$o'(X1,X3),'fun_app$o'(X1,X2))
      | $less('fun_app$o'(X1,sK239(X1)),'fun_app$o'(X1,sK238(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ),
    inference(equality_resolution,[],[f3573]) ).

tff(f3573,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | $less('fun_app$o'(X1,sK239(X1)),'fun_app$o'(X1,sK238(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2256]) ).

tff(f2256,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | ( $less('fun_app$o'(X1,sK239(X1)),'fun_app$o'(X1,sK238(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK238(X1)),sK239(X1)) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK238,sK239])],[f1494,f2255]) ).

tff(f2255,plain,
    ! [X1: 'N_int_fun$'] :
      ( ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
     => ( $less('fun_app$o'(X1,sK239(X1)),'fun_app$o'(X1,sK238(X1)))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK238(X1)),sK239(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1494,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1493]) ).

tff(f1493,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | ? [X4: 'N$',X5: 'N$'] :
          ( $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4))
          & 'fun_app$d'('fun_app$g'('less_eq$',X4),X5) )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f840]) ).

tff(f840,plain,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => ~ $less('fun_app$o'(X1,X5),'fun_app$o'(X1,X4)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & ( 'fun_app$o'(X1,X2) = X0 ) )
     => ~ $less('fun_app$o'(X1,X3),X0) ),
    inference(theory_normalization,[],[f182]) ).

tff(f182,axiom,
    ! [X0: $int,X1: 'N_int_fun$',X2: 'N$',X3: 'N$'] :
      ( ( ! [X4: 'N$',X5: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X5)
           => $lesseq('fun_app$o'(X1,X4),'fun_app$o'(X1,X5)) )
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
        & ( 'fun_app$o'(X1,X2) = X0 ) )
     => $lesseq(X0,'fun_app$o'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom180) ).

tff(f9522,plain,
    spl310_480,
    inference(avatar_split_clause,[],[f3850,f9520]) ).

tff(f9520,plain,
    ( spl310_480
  <=> ! [X2: $int,X1: 'Int_int_set_fun$',X3: $int] :
        ( 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X2)),'fun_app$l'(X1,X3))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,sK226(X1))),'fun_app$l'(X1,sK227(X1)))
        | $less(X3,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_480])]) ).

tff(f3850,plain,
    ! [X2: $int,X3: $int,X1: 'Int_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X2)),'fun_app$l'(X1,X3))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,sK226(X1))),'fun_app$l'(X1,sK227(X1)))
      | $less(X3,X2) ),
    inference(equality_resolution,[],[f3561]) ).

tff(f3561,plain,
    ! [X2: $int,X3: $int,X0: 'Int_set$',X1: 'Int_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$l'(X1,X3))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,sK226(X1))),'fun_app$l'(X1,sK227(X1)))
      | $less(X3,X2)
      | ( 'fun_app$l'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2244]) ).

tff(f2244,plain,
    ! [X0: 'Int_set$',X1: 'Int_int_set_fun$',X2: $int,X3: $int] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$l'(X1,X3))
      | ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,sK226(X1))),'fun_app$l'(X1,sK227(X1)))
        & ~ $less(sK227(X1),sK226(X1)) )
      | $less(X3,X2)
      | ( 'fun_app$l'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK226,sK227])],[f1482,f2243]) ).

tff(f2243,plain,
    ! [X1: 'Int_int_set_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X4)),'fun_app$l'(X1,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,sK226(X1))),'fun_app$l'(X1,sK227(X1)))
        & ~ $less(sK227(X1),sK226(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1482,plain,
    ! [X0: 'Int_set$',X1: 'Int_int_set_fun$',X2: $int,X3: $int] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$l'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X4)),'fun_app$l'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ( 'fun_app$l'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1481]) ).

tff(f1481,plain,
    ! [X0: 'Int_set$',X1: 'Int_int_set_fun$',X2: $int,X3: $int] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$l'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X4)),'fun_app$l'(X1,X5))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ( 'fun_app$l'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f837]) ).

tff(f837,plain,
    ! [X0: 'Int_set$',X1: 'Int_int_set_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X4)),'fun_app$l'(X1,X5)) )
        & ~ $less(X3,X2)
        & ( 'fun_app$l'(X1,X2) = X0 ) )
     => 'fun_app$c'('less_eq$a'(X0),'fun_app$l'(X1,X3)) ),
    inference(theory_normalization,[],[f177]) ).

tff(f177,axiom,
    ! [X0: 'Int_set$',X1: 'Int_int_set_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X4)),'fun_app$l'(X1,X5)) )
        & $lesseq(X2,X3)
        & ( 'fun_app$l'(X1,X2) = X0 ) )
     => 'fun_app$c'('less_eq$a'(X0),'fun_app$l'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom175) ).

tff(f9518,plain,
    ( spl310_479
    | ~ spl310_117
    | ~ spl310_255 ),
    inference(avatar_split_clause,[],[f7003,f6539,f4838,f9516]) ).

tff(f9516,plain,
    ( spl310_479
  <=> ! [X0: 'Nat$'] :
        ( sP46('to_nat$'('i$'),X0,'to_nat$'('x$'))
        | ~ sP44('to_nat$'('x$'),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_479])]) ).

tff(f6539,plain,
    ( spl310_255
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP46(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP44(X2,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_255])]) ).

tff(f7003,plain,
    ( ! [X0: 'Nat$'] :
        ( sP46('to_nat$'('i$'),X0,'to_nat$'('x$'))
        | ~ sP44('to_nat$'('x$'),X0) )
    | ~ spl310_117
    | ~ spl310_255 ),
    inference(resolution,[],[f4840,f6540]) ).

tff(f6540,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | sP46(X0,X1,X2)
        | ~ sP44(X2,X1) )
    | ~ spl310_255 ),
    inference(avatar_component_clause,[],[f6539]) ).

tff(f9514,plain,
    spl310_478,
    inference(avatar_split_clause,[],[f3833,f9512]) ).

tff(f9512,plain,
    ( spl310_478
  <=> ! [X2: 'Int_int_set_fun$',X0: $int,X1: $int] :
        ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),'fun_app$l'(X2,X1))
        | ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,sK188(X2))),'fun_app$l'(X2,sK189(X2)))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_478])]) ).

tff(f3833,plain,
    ! [X2: 'Int_int_set_fun$',X0: $int,X1: $int] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),'fun_app$l'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,sK188(X2))),'fun_app$l'(X2,sK189(X2)))
      | $less(X1,X0) ),
    inference(equality_resolution,[],[f3515]) ).

tff(f3515,plain,
    ! [X2: 'Int_int_set_fun$',X3: 'Int_set$',X0: $int,X1: $int] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),X3)
      | ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,sK188(X2))),'fun_app$l'(X2,sK189(X2)))
      | ( 'fun_app$l'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2206]) ).

tff(f2206,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),X3)
      | ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,sK188(X2))),'fun_app$l'(X2,sK189(X2)))
        & ~ $less(sK189(X2),sK188(X2)) )
      | ( 'fun_app$l'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK188,sK189])],[f1428,f2205]) ).

tff(f2205,plain,
    ! [X2: 'Int_int_set_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X4)),'fun_app$l'(X2,X5))
          & ~ $less(X5,X4) )
     => ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,sK188(X2))),'fun_app$l'(X2,sK189(X2)))
        & ~ $less(sK189(X2),sK188(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1428,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X4)),'fun_app$l'(X2,X5))
          & ~ $less(X5,X4) )
      | ( 'fun_app$l'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(flattening,[],[f1427]) ).

tff(f1427,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_set_fun$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X4)),'fun_app$l'(X2,X5))
          & ~ $less(X5,X4) )
      | ( 'fun_app$l'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f824]) ).

tff(f824,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_set_fun$',X3: 'Int_set$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X4)),'fun_app$l'(X2,X5)) )
        & ( 'fun_app$l'(X2,X1) = X3 )
        & ~ $less(X1,X0) )
     => 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),X3) ),
    inference(theory_normalization,[],[f168]) ).

tff(f168,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_int_set_fun$',X3: 'Int_set$'] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X4)),'fun_app$l'(X2,X5)) )
        & ( 'fun_app$l'(X2,X1) = X3 )
        & $lesseq(X0,X1) )
     => 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom166) ).

tff(f9510,plain,
    spl310_477,
    inference(avatar_split_clause,[],[f3472,f9508]) ).

tff(f9508,plain,
    ( spl310_477
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP75(X0,X1,X2)
        | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X2),0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_477])]) ).

tff(f3472,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP75(X0,X1,X2)
      | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X2),0) ),
    inference(cnf_transformation,[],[f2195]) ).

tff(f9506,plain,
    spl310_476,
    inference(avatar_split_clause,[],[f3295,f9504]) ).

tff(f9504,plain,
    ( spl310_476
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP57(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_476])]) ).

tff(f3295,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP57(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0) ),
    inference(cnf_transformation,[],[f2064]) ).

tff(f9502,plain,
    spl310_475,
    inference(avatar_split_clause,[],[f3292,f9500]) ).

tff(f9500,plain,
    ( spl310_475
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP57(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_475])]) ).

tff(f3292,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP57(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2064]) ).

tff(f9498,plain,
    spl310_474,
    inference(avatar_split_clause,[],[f3274,f9496]) ).

tff(f9496,plain,
    ( spl310_474
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP53(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_474])]) ).

tff(f3274,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP53(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ),
    inference(cnf_transformation,[],[f2053]) ).

tff(f9494,plain,
    spl310_473,
    inference(avatar_split_clause,[],[f3271,f9492]) ).

tff(f9492,plain,
    ( spl310_473
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP53(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_473])]) ).

tff(f3271,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP53(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2053]) ).

tff(f9490,plain,
    spl310_472,
    inference(avatar_split_clause,[],[f3253,f9488]) ).

tff(f9488,plain,
    ( spl310_472
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP49(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_472])]) ).

tff(f3253,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP49(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ),
    inference(cnf_transformation,[],[f2043]) ).

tff(f9486,plain,
    spl310_471,
    inference(avatar_split_clause,[],[f3250,f9484]) ).

tff(f9484,plain,
    ( spl310_471
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP49(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_471])]) ).

tff(f3250,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP49(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2043]) ).

tff(f9482,plain,
    spl310_470,
    inference(avatar_split_clause,[],[f3232,f9480]) ).

tff(f9480,plain,
    ( spl310_470
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP45(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_470])]) ).

tff(f3232,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP45(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) ) ),
    inference(cnf_transformation,[],[f2033]) ).

tff(f9478,plain,
    spl310_469,
    inference(avatar_split_clause,[],[f3229,f9476]) ).

tff(f9476,plain,
    ( spl310_469
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP45(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_469])]) ).

tff(f3229,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP45(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2033]) ).

tff(f9474,plain,
    ( spl310_468
    | ~ spl310_117
    | ~ spl310_254 ),
    inference(avatar_split_clause,[],[f7002,f6535,f4838,f9472]) ).

tff(f9472,plain,
    ( spl310_468
  <=> ! [X0: 'Nat$'] :
        ( sP44('to_nat$'('x$'),X0)
        | ~ sP46('to_nat$'('i$'),X0,'to_nat$'('x$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_468])]) ).

tff(f6535,plain,
    ( spl310_254
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP44(X2,X1)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP46(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_254])]) ).

tff(f7002,plain,
    ( ! [X0: 'Nat$'] :
        ( sP44('to_nat$'('x$'),X0)
        | ~ sP46('to_nat$'('i$'),X0,'to_nat$'('x$')) )
    | ~ spl310_117
    | ~ spl310_254 ),
    inference(resolution,[],[f4840,f6536]) ).

tff(f6536,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | sP44(X2,X1)
        | ~ sP46(X0,X1,X2) )
    | ~ spl310_254 ),
    inference(avatar_component_clause,[],[f6535]) ).

tff(f9470,plain,
    spl310_467,
    inference(avatar_split_clause,[],[f3211,f9468]) ).

tff(f9468,plain,
    ( spl310_467
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
        | ~ sP38(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_467])]) ).

tff(f3211,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
      | ~ sP38(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2023]) ).

tff(f9466,plain,
    spl310_466,
    inference(avatar_split_clause,[],[f3104,f9464]) ).

tff(f9464,plain,
    ( spl310_466
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP35(X0,X1,X2)
        | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_466])]) ).

tff(f3104,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP35(X0,X1,X2)
      | ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | ~ $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1995]) ).

tff(f9462,plain,
    spl310_465,
    inference(avatar_split_clause,[],[f3095,f9460]) ).

tff(f9460,plain,
    ( spl310_465
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP33(X0,X1,X2)
        | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
        | $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_465])]) ).

tff(f3095,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP33(X0,X1,X2)
      | $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X0))
      | $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1990]) ).

tff(f9458,plain,
    spl310_464,
    inference(avatar_split_clause,[],[f2905,f9456]) ).

tff(f9456,plain,
    ( spl310_464
  <=> ! [X2: $int,X0: 'Nat$',X1: 'Nat$'] : ( 'fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))),X2) = $sum('fun_app$b'('iterate_add$'(X0),X2),'fun_app$b'('iterate_add$'(X1),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_464])]) ).

tff(f2905,plain,
    ! [X2: $int,X0: 'Nat$',X1: 'Nat$'] : ( 'fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))),X2) = $sum('fun_app$b'('iterate_add$'(X0),X2),'fun_app$b'('iterate_add$'(X1),X2)) ),
    inference(cnf_transformation,[],[f263]) ).

tff(f263,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int] : ( 'fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))),X2) = $sum('fun_app$b'('iterate_add$'(X0),X2),'fun_app$b'('iterate_add$'(X1),X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom261) ).

tff(f9316,plain,
    ( spl310_117
    | ~ spl310_246
    | ~ spl310_308 ),
    inference(avatar_split_clause,[],[f7188,f7144,f6502,f4838]) ).

tff(f6502,plain,
    ( spl310_246
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1)))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_246])]) ).

tff(f7144,plain,
    ( spl310_308
  <=> 'fun_app$d'('fun_app$g'('less$','i$'),'x$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_308])]) ).

tff(f7188,plain,
    ( $less('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','to_nat$'('x$')))
    | ~ spl310_246
    | ~ spl310_308 ),
    inference(resolution,[],[f7146,f6503]) ).

tff(f6503,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$',X0),X1)
        | $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1))) )
    | ~ spl310_246 ),
    inference(avatar_component_clause,[],[f6502]) ).

tff(f7146,plain,
    ( 'fun_app$d'('fun_app$g'('less$','i$'),'x$')
    | ~ spl310_308 ),
    inference(avatar_component_clause,[],[f7144]) ).

tff(f9266,plain,
    spl310_463,
    inference(avatar_split_clause,[],[f4128,f9264]) ).

tff(f9264,plain,
    ( spl310_463
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | $less(0,$sum(1,'fun_app$e'('of_nat$',X2)))
        | sP21(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_463])]) ).

tff(f4128,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | $less(0,$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP21(X0,X1,X2) ),
    inference(forward_demodulation,[],[f4127,f864]) ).

tff(f4127,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,$sum(1,'fun_app$e'('of_nat$',X2)))
      | sP21(X0,X1,X2)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2996,f864]) ).

tff(f2996,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP21(X0,X1,X2)
      | $less(0,$sum('fun_app$e'('of_nat$',X2),1))
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1614]) ).

tff(f9262,plain,
    spl310_462,
    inference(avatar_split_clause,[],[f4046,f9260]) ).

tff(f9260,plain,
    ( spl310_462
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK115(X0,X1))) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_462])]) ).

tff(f4046,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK115(X0,X1))) ) ),
    inference(forward_demodulation,[],[f4045,f864]) ).

tff(f4045,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK115(X0,X1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2858,f864]) ).

tff(f2858,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK115(X0,X1)),1) )
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1908]) ).

tff(f9258,plain,
    spl310_461,
    inference(avatar_split_clause,[],[f4030,f9256]) ).

tff(f9256,plain,
    ( spl310_461
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X1) != $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_461])]) ).

tff(f4030,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) != $sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))) )
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2842,f864]) ).

tff(f2842,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | ( 'fun_app$e'('of_nat$',X1) != $sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),1) ) ),
    inference(cnf_transformation,[],[f1895]) ).

tff(f9254,plain,
    spl310_460,
    inference(avatar_split_clause,[],[f4027,f9252]) ).

tff(f9252,plain,
    ( spl310_460
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK103(X0,X1))) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_460])]) ).

tff(f4027,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK103(X0,X1))) ) ),
    inference(forward_demodulation,[],[f4026,f864]) ).

tff(f4026,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK103(X0,X1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2637,f864]) ).

tff(f2637,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK103(X0,X1)),1) )
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1756]) ).

tff(f1756,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK103(X0,X1)),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK103(X0,X1))) )
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK103])],[f1138,f1755]) ).

tff(f1755,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
     => ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK103(X0,X1)),1) )
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK103(X0,X1))) ) ),
    introduced(choice_axiom,[]) ).

tff(f1138,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f944]) ).

tff(f944,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ~ ( ! [X2: 'Nat$'] :
            ~ ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(true_and_false_elimination,[],[f401]) ).

tff(f401,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ! [X2: 'Nat$'] :
            ( ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
              & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
           => $false )
        & $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom399) ).

tff(f9250,plain,
    spl310_459,
    inference(avatar_split_clause,[],[f3877,f9248]) ).

tff(f9248,plain,
    ( spl310_459
  <=> ! [X2: 'N_int_set_fun$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),'fun_app$k'(X2,X1))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK252(X2)),sK253(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_459])]) ).

tff(f3877,plain,
    ! [X2: 'N_int_set_fun$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),'fun_app$k'(X2,X1))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK252(X2)),sK253(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(equality_resolution,[],[f3586]) ).

tff(f3586,plain,
    ! [X2: 'N_int_set_fun$',X3: 'Int_set$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X2,X0)),X3)
      | 'fun_app$d'('fun_app$g'('less_eq$',sK252(X2)),sK253(X2))
      | ( 'fun_app$k'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2270]) ).

tff(f9246,plain,
    spl310_458,
    inference(avatar_split_clause,[],[f3854,f9244]) ).

tff(f9244,plain,
    ( spl310_458
  <=> ! [X2: 'Int_set$',X1: 'Int_set_int_fun$',X3: 'Int_set$'] :
        ( ~ $less('fun_app$n'(X1,X3),'fun_app$n'(X1,X2))
        | $less('fun_app$n'(X1,sK231(X1)),'fun_app$n'(X1,sK230(X1)))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_458])]) ).

tff(f3854,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X1: 'Int_set_int_fun$'] :
      ( ~ $less('fun_app$n'(X1,X3),'fun_app$n'(X1,X2))
      | $less('fun_app$n'(X1,sK231(X1)),'fun_app$n'(X1,sK230(X1)))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3) ),
    inference(equality_resolution,[],[f3565]) ).

tff(f3565,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: $int,X1: 'Int_set_int_fun$'] :
      ( ~ $less('fun_app$n'(X1,X3),X0)
      | $less('fun_app$n'(X1,sK231(X1)),'fun_app$n'(X1,sK230(X1)))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$n'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2248]) ).

tff(f2248,plain,
    ! [X0: $int,X1: 'Int_set_int_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ~ $less('fun_app$n'(X1,X3),X0)
      | ( $less('fun_app$n'(X1,sK231(X1)),'fun_app$n'(X1,sK230(X1)))
        & 'fun_app$c'('less_eq$a'(sK230(X1)),sK231(X1)) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$n'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK230,sK231])],[f1486,f2247]) ).

tff(f2247,plain,
    ! [X1: 'Int_set_int_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X1,X5),'fun_app$n'(X1,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( $less('fun_app$n'(X1,sK231(X1)),'fun_app$n'(X1,sK230(X1)))
        & 'fun_app$c'('less_eq$a'(sK230(X1)),sK231(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1486,plain,
    ! [X0: $int,X1: 'Int_set_int_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ~ $less('fun_app$n'(X1,X3),X0)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X1,X5),'fun_app$n'(X1,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$n'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1485]) ).

tff(f1485,plain,
    ! [X0: $int,X1: 'Int_set_int_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ~ $less('fun_app$n'(X1,X3),X0)
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X1,X5),'fun_app$n'(X1,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$n'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f838]) ).

tff(f838,plain,
    ! [X0: $int,X1: 'Int_set_int_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => ~ $less('fun_app$n'(X1,X5),'fun_app$n'(X1,X4)) )
        & 'fun_app$c'('less_eq$a'(X2),X3)
        & ( 'fun_app$n'(X1,X2) = X0 ) )
     => ~ $less('fun_app$n'(X1,X3),X0) ),
    inference(theory_normalization,[],[f179]) ).

tff(f179,axiom,
    ! [X0: $int,X1: 'Int_set_int_fun$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => $lesseq('fun_app$n'(X1,X4),'fun_app$n'(X1,X5)) )
        & 'fun_app$c'('less_eq$a'(X2),X3)
        & ( 'fun_app$n'(X1,X2) = X0 ) )
     => $lesseq(X0,'fun_app$n'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom177) ).

tff(f9242,plain,
    spl310_457,
    inference(avatar_split_clause,[],[f3853,f9240]) ).

tff(f9240,plain,
    ( spl310_457
  <=> ! [X2: 'N$',X1: 'N_int_set_fun$',X3: 'N$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X2)),'fun_app$k'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK228(X1)),sK229(X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_457])]) ).

tff(f3853,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$k'(X1,X2)),'fun_app$k'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK228(X1)),sK229(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ),
    inference(equality_resolution,[],[f3562]) ).

tff(f3562,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'Int_set$',X1: 'N_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$k'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK228(X1)),sK229(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$k'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2246]) ).

tff(f9238,plain,
    spl310_456,
    inference(avatar_split_clause,[],[f3844,f9236]) ).

tff(f9236,plain,
    ( spl310_456
  <=> ! [X2: 'Int_set_int_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
        ( ~ $less('fun_app$n'(X2,X1),'fun_app$n'(X2,X0))
        | $less('fun_app$n'(X2,sK215(X2)),'fun_app$n'(X2,sK214(X2)))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_456])]) ).

tff(f3844,plain,
    ! [X2: 'Int_set_int_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( ~ $less('fun_app$n'(X2,X1),'fun_app$n'(X2,X0))
      | $less('fun_app$n'(X2,sK215(X2)),'fun_app$n'(X2,sK214(X2)))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(equality_resolution,[],[f3549]) ).

tff(f3549,plain,
    ! [X2: 'Int_set_int_fun$',X3: $int,X0: 'Int_set$',X1: 'Int_set$'] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | $less('fun_app$n'(X2,sK215(X2)),'fun_app$n'(X2,sK214(X2)))
      | ( 'fun_app$n'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2232]) ).

tff(f2232,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | ( $less('fun_app$n'(X2,sK215(X2)),'fun_app$n'(X2,sK214(X2)))
        & 'fun_app$c'('less_eq$a'(sK214(X2)),sK215(X2)) )
      | ( 'fun_app$n'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK214,sK215])],[f1470,f2231]) ).

tff(f2231,plain,
    ! [X2: 'Int_set_int_fun$'] :
      ( ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
     => ( $less('fun_app$n'(X2,sK215(X2)),'fun_app$n'(X2,sK214(X2)))
        & 'fun_app$c'('less_eq$a'(sK214(X2)),sK215(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1470,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ( 'fun_app$n'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1469]) ).

tff(f1469,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | ? [X4: 'Int_set$',X5: 'Int_set$'] :
          ( $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4))
          & 'fun_app$c'('less_eq$a'(X4),X5) )
      | ( 'fun_app$n'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f835]) ).

tff(f835,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => ~ $less('fun_app$n'(X2,X5),'fun_app$n'(X2,X4)) )
        & ( 'fun_app$n'(X2,X1) = X3 )
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => ~ $less(X3,'fun_app$n'(X2,X0)) ),
    inference(theory_normalization,[],[f170]) ).

tff(f170,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set_int_fun$',X3: $int] :
      ( ( ! [X4: 'Int_set$',X5: 'Int_set$'] :
            ( 'fun_app$c'('less_eq$a'(X4),X5)
           => $lesseq('fun_app$n'(X2,X4),'fun_app$n'(X2,X5)) )
        & ( 'fun_app$n'(X2,X1) = X3 )
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => $lesseq('fun_app$n'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom168) ).

tff(f9234,plain,
    spl310_455,
    inference(avatar_split_clause,[],[f3554,f9232]) ).

tff(f9232,plain,
    ( spl310_455
  <=> ! [X0: 'Int_set$',X3: $int,X2: 'Int_set_int_fun$',X1: 'Int_set$'] :
        ( ~ $less(X3,'fun_app$n'(X2,X0))
        | 'fun_app$c'('less_eq$a'(sK220(X2)),sK221(X2))
        | $less(X3,'fun_app$n'(X2,X1))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_455])]) ).

tff(f3554,plain,
    ! [X2: 'Int_set_int_fun$',X3: $int,X0: 'Int_set$',X1: 'Int_set$'] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | 'fun_app$c'('less_eq$a'(sK220(X2)),sK221(X2))
      | $less(X3,'fun_app$n'(X2,X1))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2238]) ).

tff(f9230,plain,
    ( spl310_454
    | ~ spl310_117
    | ~ spl310_118 ),
    inference(avatar_split_clause,[],[f6069,f4865,f4838,f9227]) ).

tff(f9227,plain,
    ( spl310_454
  <=> $less('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_454])]) ).

tff(f4865,plain,
    ( spl310_118
  <=> ! [X0: $int] :
        ( ~ $less(X0,'fun_app$e'('of_nat$','to_nat$'('x$')))
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_118])]) ).

tff(f6069,plain,
    ( $less('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
    | ~ spl310_117
    | ~ spl310_118 ),
    inference(resolution,[],[f4866,f4840]) ).

tff(f4866,plain,
    ( ! [X0: $int] :
        ( ~ $less(X0,'fun_app$e'('of_nat$','to_nat$'('x$')))
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    | ~ spl310_118 ),
    inference(avatar_component_clause,[],[f4865]) ).

tff(f9225,plain,
    spl310_453,
    inference(avatar_split_clause,[],[f3302,f9223]) ).

tff(f9223,plain,
    ( spl310_453
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP56(X0,X1)
        | ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_453])]) ).

tff(f3302,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP56(X0,X1)
      | ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2067]) ).

tff(f2067,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP56(X0,X1)
        | ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less(0,0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP56(X0,X1) ) ),
    inference(rectify,[],[f2066]) ).

tff(f2066,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP56(X0,X2)
        | ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less(0,0)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP56(X0,X2) ) ),
    inference(flattening,[],[f2065]) ).

tff(f2065,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP56(X0,X2)
        | ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ~ $less(0,0)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP56(X0,X2) ) ),
    inference(nnf_transformation,[],[f1659]) ).

tff(f1659,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( sP56(X0,X2)
    <=> ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( $less(0,0)
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP56])]) ).

tff(f9221,plain,
    spl310_452,
    inference(avatar_split_clause,[],[f3299,f9219]) ).

tff(f9219,plain,
    ( spl310_452
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP56(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_452])]) ).

tff(f3299,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP56(X0,X1) ),
    inference(cnf_transformation,[],[f2067]) ).

tff(f9217,plain,
    spl310_451,
    inference(avatar_split_clause,[],[f3236,f9215]) ).

tff(f9215,plain,
    ( spl310_451
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP44(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_451])]) ).

tff(f3236,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP44(X0,X1) ),
    inference(cnf_transformation,[],[f2036]) ).

tff(f2036,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP44(X0,X1)
        | ( ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != 0 )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = 0 )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP44(X0,X1) ) ),
    inference(rectify,[],[f2035]) ).

tff(f2035,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP44(X0,X2)
        | ( ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != 0 )
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = 0 )
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP44(X0,X2) ) ),
    inference(flattening,[],[f2034]) ).

tff(f2034,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP44(X0,X2)
        | ( ( 0 != $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( ( 0 != 0 )
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = 0 )
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP44(X0,X2) ) ),
    inference(nnf_transformation,[],[f1644]) ).

tff(f1644,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( sP44(X0,X2)
    <=> ( ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ( 0 = 0 )
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP44])]) ).

tff(f9213,plain,
    spl310_450,
    inference(avatar_split_clause,[],[f3216,f9211]) ).

tff(f9211,plain,
    ( spl310_450
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP37(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_450])]) ).

tff(f3216,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP37(X0,X1) ),
    inference(cnf_transformation,[],[f2025]) ).

tff(f2025,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less(0,0)
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP37(X0,X1) ),
    inference(rectify,[],[f2024]) ).

tff(f2024,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less(0,0)
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
      | ~ sP37(X0,X2) ),
    inference(nnf_transformation,[],[f1636]) ).

tff(f9209,plain,
    spl310_449,
    inference(avatar_split_clause,[],[f2959,f9207]) ).

tff(f9207,plain,
    ( spl310_449
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP7(X1,X2,X0)
        | ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        | sP8(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_449])]) ).

tff(f2959,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP7(X1,X2,X0)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
      | sP8(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1605]) ).

tff(f9205,plain,
    spl310_448,
    inference(avatar_split_clause,[],[f2957,f9203]) ).

tff(f9203,plain,
    ( spl310_448
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP6(X1,X2,X0)
        | ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
        | sP8(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_448])]) ).

tff(f2957,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP6(X1,X2,X0)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))),'fun_app$e'('of_nat$',X2))
      | sP8(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1605]) ).

tff(f9201,plain,
    spl310_447,
    inference(avatar_split_clause,[],[f2834,f9199]) ).

tff(f2834,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK113(X0,X1))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1888]) ).

tff(f1888,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ! [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK113(X0,X1))) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK113])],[f1886,f1887]) ).

tff(f1887,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X3: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
     => ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK113(X0,X1))) ) ),
    introduced(choice_axiom,[]) ).

tff(f1886,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ! [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      & ( ? [X3: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    inference(rectify,[],[f1885]) ).

tff(f1885,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ! [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      & ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    inference(nnf_transformation,[],[f716]) ).

tff(f716,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
    <=> ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    inference(theory_normalization,[],[f148]) ).

tff(f148,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
    <=> ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom146) ).

tff(f9197,plain,
    spl310_446,
    inference(avatar_split_clause,[],[f2579,f9195]) ).

tff(f2579,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK98(X0,X1))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1746]) ).

tff(f1746,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK98(X0,X1))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK98])],[f1068,f1745]) ).

tff(f1745,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
     => ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK98(X0,X1))) ) ),
    introduced(choice_axiom,[]) ).

tff(f1068,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f679]) ).

tff(f679,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    inference(theory_normalization,[],[f143]) ).

tff(f143,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom141) ).

tff(f9187,plain,
    ( ~ spl310_58
    | ~ spl310_24
    | ~ spl310_117 ),
    inference(avatar_split_clause,[],[f5065,f4838,f4326,f4490]) ).

tff(f5065,plain,
    ( ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
    | ~ spl310_24
    | ~ spl310_117 ),
    inference(resolution,[],[f4840,f4327]) ).

tff(f8885,plain,
    spl310_445,
    inference(avatar_split_clause,[],[f4188,f8883]) ).

tff(f8883,plain,
    ( spl310_445
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP31(X0,X1,X2)
        | ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
        | ( 0 != 'fun_app$e'('of_nat$',X2) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_445])]) ).

tff(f4188,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP31(X0,X1,X2)
      | ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | ( 0 != 'fun_app$e'('of_nat$',X2) ) ),
    inference(inner_rewriting,[],[f3086]) ).

tff(f3086,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP31(X0,X1,X2)
      | ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | ( 0 != 'fun_app$e'('of_nat$',X2) ) ),
    inference(cnf_transformation,[],[f1985]) ).

tff(f8881,plain,
    spl310_444,
    inference(avatar_split_clause,[],[f4012,f8879]) ).

tff(f8879,plain,
    ( spl310_444
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X1),$sum(1,'fun_app$e'('of_nat$',X0)))
        | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK100(X1))) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_444])]) ).

tff(f4012,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X1),$sum(1,'fun_app$e'('of_nat$',X0)))
      | ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK100(X1))) ) ),
    inference(forward_demodulation,[],[f4011,f864]) ).

tff(f4011,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum(1,'fun_app$e'('of_nat$',sK100(X1))) )
      | $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X0),1)) ),
    inference(forward_demodulation,[],[f2595,f864]) ).

tff(f2595,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK100(X1)),1) )
      | $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X0),1)) ),
    inference(cnf_transformation,[],[f1750]) ).

tff(f1750,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK100(X1)),1) )
      | $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X0),1)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK100])],[f1081,f1749]) ).

tff(f1749,plain,
    ! [X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
     => ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',sK100(X1)),1) ) ),
    introduced(choice_axiom,[]) ).

tff(f1081,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) )
      | $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X0),1)) ),
    inference(ennf_transformation,[],[f685]) ).

tff(f685,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X0),1))
     => ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) ) ),
    inference(theory_normalization,[],[f521]) ).

tff(f521,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $lesseq($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
     => ? [X2: 'Nat$'] : ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom519) ).

tff(f8877,plain,
    spl310_443,
    inference(avatar_split_clause,[],[f3897,f8875]) ).

tff(f8875,plain,
    ( spl310_443
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X0,X2)
        | sP65(X1,X0,X2)
        | sP64(X2,X1,X0)
        | sP63(X0,X2,X1)
        | sP62(X1,X2,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_443])]) ).

tff(f3897,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X2)
      | sP65(X1,X0,X2)
      | sP64(X2,X1,X0)
      | sP63(X0,X2,X1)
      | sP62(X1,X2,X0) ),
    inference(duplicate_literal_removal,[],[f3345]) ).

tff(f3345,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X2)
      | ~ $less(X0,X2)
      | sP65(X1,X0,X2)
      | sP64(X2,X1,X0)
      | sP63(X0,X2,X1)
      | sP62(X1,X2,X0) ),
    inference(cnf_transformation,[],[f1672]) ).

tff(f8873,plain,
    spl310_442,
    inference(avatar_split_clause,[],[f3891,f8871]) ).

tff(f8871,plain,
    ( spl310_442
  <=> ! [X4: 'Nat$',X3: 'Nat$',X2: $int,X1: $int] :
        ( sP79($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_442])]) ).

tff(f3891,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP79($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(equality_resolution,[],[f3666]) ).

tff(f3666,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP79(X0,X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) != X0 ) ),
    inference(cnf_transformation,[],[f2336]) ).

tff(f8869,plain,
    spl310_441,
    inference(avatar_split_clause,[],[f3888,f8867]) ).

tff(f8867,plain,
    ( spl310_441
  <=> ! [X4: 'Nat$',X3: 'Nat$',X2: $int,X1: $int] :
        ( sP77($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
        | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_441])]) ).

tff(f3888,plain,
    ! [X2: $int,X3: 'Nat$',X1: $int,X4: 'Nat$'] :
      ( sP77($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1),X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3)) ),
    inference(equality_resolution,[],[f3657]) ).

tff(f3657,plain,
    ! [X2: $int,X3: 'Nat$',X0: $int,X1: $int,X4: 'Nat$'] :
      ( sP77(X0,X1,X2,X3,X4)
      | ~ $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X3))
      | ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X1) != X0 ) ),
    inference(cnf_transformation,[],[f2331]) ).

tff(f8857,plain,
    spl310_440,
    inference(avatar_split_clause,[],[f3881,f8855]) ).

tff(f8855,plain,
    ( spl310_440
  <=> ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
        ( $less('fun_app$o'(X2,X0),'fun_app$o'(X2,X1))
        | 'fun_app$d'('fun_app$g'('less$',sK256(X2)),sK257(X2))
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_440])]) ).

tff(f3881,plain,
    ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),'fun_app$o'(X2,X1))
      | 'fun_app$d'('fun_app$g'('less$',sK256(X2)),sK257(X2))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(equality_resolution,[],[f3590]) ).

tff(f3590,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$o'(X2,X0),X3)
      | 'fun_app$d'('fun_app$g'('less$',sK256(X2)),sK257(X2))
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f2274]) ).

tff(f8853,plain,
    spl310_439,
    inference(avatar_split_clause,[],[f3879,f8851]) ).

tff(f8851,plain,
    ( spl310_439
  <=> ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
        ( ~ $less('fun_app$o'(X2,X1),'fun_app$o'(X2,X0))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK254(X2)),sK255(X2))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_439])]) ).

tff(f3879,plain,
    ! [X2: 'N_int_fun$',X0: 'N$',X1: 'N$'] :
      ( ~ $less('fun_app$o'(X2,X1),'fun_app$o'(X2,X0))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK254(X2)),sK255(X2))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(equality_resolution,[],[f3588]) ).

tff(f3588,plain,
    ! [X2: 'N_int_fun$',X3: $int,X0: 'N$',X1: 'N$'] :
      ( ~ $less(X3,'fun_app$o'(X2,X0))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK254(X2)),sK255(X2))
      | ( 'fun_app$o'(X2,X1) != X3 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f2272]) ).

tff(f8849,plain,
    spl310_438,
    inference(avatar_split_clause,[],[f3870,f8847]) ).

tff(f8847,plain,
    ( spl310_438
  <=> ! [X2: $int,X1: 'Int_int_fun$',X3: $int] :
        ( $less('fun_app$b'(X1,X2),'fun_app$b'(X1,X3))
        | ~ $less('fun_app$b'(X1,sK246(X1)),'fun_app$b'(X1,sK247(X1)))
        | ~ $less(X2,X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_438])]) ).

tff(f3870,plain,
    ! [X2: $int,X3: $int,X1: 'Int_int_fun$'] :
      ( $less('fun_app$b'(X1,X2),'fun_app$b'(X1,X3))
      | ~ $less('fun_app$b'(X1,sK246(X1)),'fun_app$b'(X1,sK247(X1)))
      | ~ $less(X2,X3) ),
    inference(equality_resolution,[],[f3581]) ).

tff(f3581,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ~ $less('fun_app$b'(X1,sK246(X1)),'fun_app$b'(X1,sK247(X1)))
      | ~ $less(X2,X3)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2264]) ).

tff(f2264,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ( ~ $less('fun_app$b'(X1,sK246(X1)),'fun_app$b'(X1,sK247(X1)))
        & $less(sK246(X1),sK247(X1)) )
      | ~ $less(X2,X3)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK246,sK247])],[f1502,f2263]) ).

tff(f2263,plain,
    ! [X1: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
     => ( ~ $less('fun_app$b'(X1,sK246(X1)),'fun_app$b'(X1,sK247(X1)))
        & $less(sK246(X1),sK247(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1502,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1501]) ).

tff(f1501,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5))
          & $less(X4,X5) )
      | ~ $less(X2,X3)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f299]) ).

tff(f299,axiom,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X1,X4),'fun_app$b'(X1,X5)) )
        & $less(X2,X3)
        & ( 'fun_app$b'(X1,X2) = X0 ) )
     => $less(X0,'fun_app$b'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom297) ).

tff(f8845,plain,
    spl310_437,
    inference(avatar_split_clause,[],[f3868,f8843]) ).

tff(f8843,plain,
    ( spl310_437
  <=> ! [X2: $int,X1: 'Int_int_fun$',X3: $int] :
        ( ~ $less('fun_app$b'(X1,X3),'fun_app$b'(X1,X2))
        | $less('fun_app$b'(X1,sK245(X1)),'fun_app$b'(X1,sK244(X1)))
        | $less(X3,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_437])]) ).

tff(f3868,plain,
    ! [X2: $int,X3: $int,X1: 'Int_int_fun$'] :
      ( ~ $less('fun_app$b'(X1,X3),'fun_app$b'(X1,X2))
      | $less('fun_app$b'(X1,sK245(X1)),'fun_app$b'(X1,sK244(X1)))
      | $less(X3,X2) ),
    inference(equality_resolution,[],[f3579]) ).

tff(f3579,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | $less('fun_app$b'(X1,sK245(X1)),'fun_app$b'(X1,sK244(X1)))
      | $less(X3,X2)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2262]) ).

tff(f2262,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ( $less('fun_app$b'(X1,sK245(X1)),'fun_app$b'(X1,sK244(X1)))
        & ~ $less(sK245(X1),sK244(X1)) )
      | $less(X3,X2)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK244,sK245])],[f1500,f2261]) ).

tff(f2261,plain,
    ! [X1: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
     => ( $less('fun_app$b'(X1,sK245(X1)),'fun_app$b'(X1,sK244(X1)))
        & ~ $less(sK245(X1),sK244(X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1500,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(flattening,[],[f1499]) ).

tff(f1499,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4))
          & ~ $less(X5,X4) )
      | $less(X3,X2)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(ennf_transformation,[],[f841]) ).

tff(f841,plain,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => ~ $less('fun_app$b'(X1,X5),'fun_app$b'(X1,X4)) )
        & ~ $less(X3,X2)
        & ( 'fun_app$b'(X1,X2) = X0 ) )
     => ~ $less('fun_app$b'(X1,X3),X0) ),
    inference(theory_normalization,[],[f184]) ).

tff(f184,axiom,
    ! [X0: $int,X1: 'Int_int_fun$',X2: $int,X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => $lesseq('fun_app$b'(X1,X4),'fun_app$b'(X1,X5)) )
        & $lesseq(X2,X3)
        & ( 'fun_app$b'(X1,X2) = X0 ) )
     => $lesseq(X0,'fun_app$b'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom182) ).

tff(f8841,plain,
    spl310_436,
    inference(avatar_split_clause,[],[f3865,f8839]) ).

tff(f8839,plain,
    ( spl310_436
  <=> ! [X2: 'N$',X1: 'N_int_fun$',X3: 'N$'] :
        ( $less('fun_app$o'(X1,X2),'fun_app$o'(X1,X3))
        | 'fun_app$d'('fun_app$g'('less$',sK240(X1)),sK241(X1))
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_436])]) ).

tff(f3865,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_int_fun$'] :
      ( $less('fun_app$o'(X1,X2),'fun_app$o'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK240(X1)),sK241(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ),
    inference(equality_resolution,[],[f3574]) ).

tff(f3574,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( $less(X0,'fun_app$o'(X1,X3))
      | 'fun_app$d'('fun_app$g'('less$',sK240(X1)),sK241(X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2258]) ).

tff(f8837,plain,
    spl310_435,
    inference(avatar_split_clause,[],[f3863,f8835]) ).

tff(f8835,plain,
    ( spl310_435
  <=> ! [X2: 'N$',X1: 'N_int_fun$',X3: 'N$'] :
        ( ~ $less('fun_app$o'(X1,X3),'fun_app$o'(X1,X2))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK238(X1)),sK239(X1))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_435])]) ).

tff(f3863,plain,
    ! [X2: 'N$',X3: 'N$',X1: 'N_int_fun$'] :
      ( ~ $less('fun_app$o'(X1,X3),'fun_app$o'(X1,X2))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK238(X1)),sK239(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3) ),
    inference(equality_resolution,[],[f3572]) ).

tff(f3572,plain,
    ! [X2: 'N$',X3: 'N$',X0: $int,X1: 'N_int_fun$'] :
      ( ~ $less('fun_app$o'(X1,X3),X0)
      | 'fun_app$d'('fun_app$g'('less_eq$',sK238(X1)),sK239(X1))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X3)
      | ( 'fun_app$o'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2256]) ).

tff(f8833,plain,
    spl310_434,
    inference(avatar_split_clause,[],[f3857,f8831]) ).

tff(f8831,plain,
    ( spl310_434
  <=> ! [X2: 'Int_set$',X1: 'Int_set_n_fun$',X3: 'Int_set$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X2)),'fun_app$m'(X1,X3))
        | 'fun_app$c'('less_eq$a'(sK232(X1)),sK233(X1))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_434])]) ).

tff(f3857,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X1: 'Int_set_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X1,X2)),'fun_app$m'(X1,X3))
      | 'fun_app$c'('less_eq$a'(sK232(X1)),sK233(X1))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3) ),
    inference(equality_resolution,[],[f3566]) ).

tff(f3566,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: 'N$',X1: 'Int_set_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$m'(X1,X3))
      | 'fun_app$c'('less_eq$a'(sK232(X1)),sK233(X1))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$m'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2250]) ).

tff(f8829,plain,
    spl310_433,
    inference(avatar_split_clause,[],[f3847,f8827]) ).

tff(f8827,plain,
    ( spl310_433
  <=> ! [X2: 'Int_set_n_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),'fun_app$m'(X2,X1))
        | 'fun_app$c'('less_eq$a'(sK216(X2)),sK217(X2))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_433])]) ).

tff(f3847,plain,
    ! [X2: 'Int_set_n_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),'fun_app$m'(X2,X1))
      | 'fun_app$c'('less_eq$a'(sK216(X2)),sK217(X2))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(equality_resolution,[],[f3550]) ).

tff(f3550,plain,
    ! [X2: 'Int_set_n_fun$',X3: 'N$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$m'(X2,X0)),X3)
      | 'fun_app$c'('less_eq$a'(sK216(X2)),sK217(X2))
      | ( 'fun_app$m'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2234]) ).

tff(f8825,plain,
    spl310_432,
    inference(avatar_split_clause,[],[f3842,f8823]) ).

tff(f8823,plain,
    ( spl310_432
  <=> ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
        ( $less('fun_app$b'(X2,X0),'fun_app$b'(X2,X1))
        | ~ $less('fun_app$b'(X2,sK204(X2)),'fun_app$b'(X2,sK205(X2)))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_432])]) ).

tff(f3842,plain,
    ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),'fun_app$b'(X2,X1))
      | ~ $less('fun_app$b'(X2,sK204(X2)),'fun_app$b'(X2,sK205(X2)))
      | ~ $less(X0,X1) ),
    inference(equality_resolution,[],[f3537]) ).

tff(f3537,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ~ $less('fun_app$b'(X2,sK204(X2)),'fun_app$b'(X2,sK205(X2)))
      | ( 'fun_app$b'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2222]) ).

tff(f2222,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ( ~ $less('fun_app$b'(X2,sK204(X2)),'fun_app$b'(X2,sK205(X2)))
        & $less(sK204(X2),sK205(X2)) )
      | ( 'fun_app$b'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK204,sK205])],[f1456,f2221]) ).

tff(f2221,plain,
    ! [X2: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
     => ( ~ $less('fun_app$b'(X2,sK204(X2)),'fun_app$b'(X2,sK205(X2)))
        & $less(sK204(X2),sK205(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1456,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
      | ( 'fun_app$b'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1455]) ).

tff(f1455,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ? [X4: $int,X5: $int] :
          ( ~ $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5))
          & $less(X4,X5) )
      | ( 'fun_app$b'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f295]) ).

tff(f295,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $less(X4,X5)
           => $less('fun_app$b'(X2,X4),'fun_app$b'(X2,X5)) )
        & ( 'fun_app$b'(X2,X1) = X3 )
        & $less(X0,X1) )
     => $less('fun_app$b'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom293) ).

tff(f8821,plain,
    spl310_431,
    inference(avatar_split_clause,[],[f3837,f8819]) ).

tff(f8819,plain,
    ( spl310_431
  <=> ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
        ( ~ $less('fun_app$b'(X2,X1),'fun_app$b'(X2,X0))
        | $less('fun_app$b'(X2,sK193(X2)),'fun_app$b'(X2,sK192(X2)))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_431])]) ).

tff(f3837,plain,
    ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
      ( ~ $less('fun_app$b'(X2,X1),'fun_app$b'(X2,X0))
      | $less('fun_app$b'(X2,sK193(X2)),'fun_app$b'(X2,sK192(X2)))
      | $less(X1,X0) ),
    inference(equality_resolution,[],[f3519]) ).

tff(f3519,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | $less('fun_app$b'(X2,sK193(X2)),'fun_app$b'(X2,sK192(X2)))
      | ( 'fun_app$b'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2210]) ).

tff(f2210,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ( $less('fun_app$b'(X2,sK193(X2)),'fun_app$b'(X2,sK192(X2)))
        & ~ $less(sK193(X2),sK192(X2)) )
      | ( 'fun_app$b'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK192,sK193])],[f1432,f2209]) ).

tff(f2209,plain,
    ! [X2: 'Int_int_fun$'] :
      ( ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
     => ( $less('fun_app$b'(X2,sK193(X2)),'fun_app$b'(X2,sK192(X2)))
        & ~ $less(sK193(X2),sK192(X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1432,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
      | ( 'fun_app$b'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(flattening,[],[f1431]) ).

tff(f1431,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ? [X4: $int,X5: $int] :
          ( $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4))
          & ~ $less(X5,X4) )
      | ( 'fun_app$b'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f826]) ).

tff(f826,plain,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( ~ $less(X5,X4)
           => ~ $less('fun_app$b'(X2,X5),'fun_app$b'(X2,X4)) )
        & ( 'fun_app$b'(X2,X1) = X3 )
        & ~ $less(X1,X0) )
     => ~ $less(X3,'fun_app$b'(X2,X0)) ),
    inference(theory_normalization,[],[f175]) ).

tff(f175,axiom,
    ! [X0: $int,X1: $int,X2: 'Int_int_fun$',X3: $int] :
      ( ( ! [X4: $int,X5: $int] :
            ( $lesseq(X4,X5)
           => $lesseq('fun_app$b'(X2,X4),'fun_app$b'(X2,X5)) )
        & ( 'fun_app$b'(X2,X1) = X3 )
        & $lesseq(X0,X1) )
     => $lesseq('fun_app$b'(X2,X0),X3) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom173) ).

tff(f8817,plain,
    ( ~ spl310_430
    | ~ spl310_126
    | ~ spl310_355 ),
    inference(avatar_split_clause,[],[f8149,f8052,f4908,f8814]) ).

tff(f8814,plain,
    ( spl310_430
  <=> 'fun_app$d'('fun_app$g'('less$',sK91('fun_app$g'('less$','i$'))),'i$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_430])]) ).

tff(f8052,plain,
    ( spl310_355
  <=> 'fun_app$d'('fun_app$g'('less$','i$'),sK91('fun_app$g'('less$','i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_355])]) ).

tff(f8149,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less$',sK91('fun_app$g'('less$','i$'))),'i$')
    | ~ spl310_126
    | ~ spl310_355 ),
    inference(resolution,[],[f8054,f4909]) ).

tff(f8054,plain,
    ( 'fun_app$d'('fun_app$g'('less$','i$'),sK91('fun_app$g'('less$','i$')))
    | ~ spl310_355 ),
    inference(avatar_component_clause,[],[f8052]) ).

tff(f8812,plain,
    spl310_429,
    inference(avatar_split_clause,[],[f3409,f8810]) ).

tff(f8810,plain,
    ( spl310_429
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK177(X0)),sK178(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK179(X0)),sK180(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_429])]) ).

tff(f3409,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK177(X0)),sK178(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK179(X0)),sK180(X0)) ),
    inference(cnf_transformation,[],[f2155]) ).

tff(f2155,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK177(X0)),sK178(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK178(X0)),sK177(X0)) )
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK179(X0)),sK180(X0))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK179(X0)),sK180(X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK177,sK178,sK179,sK180])],[f1377,f2154,f2153]) ).

tff(f2153,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK177(X0)),sK178(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK178(X0)),sK177(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2154,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X5: 'N$',X6: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X5),X6)
          & 'fun_app$d'('fun_app$g'('less_eq$',X5),X6) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK179(X0)),sK180(X0))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK179(X0)),sK180(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1377,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$',X6: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X5),X6)
          & 'fun_app$d'('fun_app$g'('less_eq$',X5),X6) ) ),
    inference(flattening,[],[f1376]) ).

tff(f1376,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$',X6: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X5),X6)
          & 'fun_app$d'('fun_app$g'('less_eq$',X5),X6) ) ),
    inference(ennf_transformation,[],[f981]) ).

tff(f981,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X5: 'N$',X6: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X5),X6)
           => 'fun_app$d'('fun_app$g'(X0,X5),X6) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    inference(rectify,[],[f215]) ).

tff(f215,axiom,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X3),X4)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom213) ).

tff(f8808,plain,
    spl310_428,
    inference(avatar_split_clause,[],[f3408,f8806]) ).

tff(f8806,plain,
    ( spl310_428
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK177(X0)),sK178(X0))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK179(X0)),sK180(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_428])]) ).

tff(f3408,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK177(X0)),sK178(X0))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK179(X0)),sK180(X0)) ),
    inference(cnf_transformation,[],[f2155]) ).

tff(f8804,plain,
    spl310_427,
    inference(avatar_split_clause,[],[f3407,f8802]) ).

tff(f8802,plain,
    ( spl310_427
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK178(X0)),sK177(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK179(X0)),sK180(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_427])]) ).

tff(f3407,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK178(X0)),sK177(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK179(X0)),sK180(X0)) ),
    inference(cnf_transformation,[],[f2155]) ).

tff(f8800,plain,
    spl310_426,
    inference(avatar_split_clause,[],[f3406,f8798]) ).

tff(f8798,plain,
    ( spl310_426
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK178(X0)),sK177(X0))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK179(X0)),sK180(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_426])]) ).

tff(f3406,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK178(X0)),sK177(X0))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK179(X0)),sK180(X0)) ),
    inference(cnf_transformation,[],[f2155]) ).

tff(f8796,plain,
    spl310_425,
    inference(avatar_split_clause,[],[f3405,f8794]) ).

tff(f8794,plain,
    ( spl310_425
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK173(X0)),sK174(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK175(X0)),sK176(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_425])]) ).

tff(f3405,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK173(X0)),sK174(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK175(X0)),sK176(X0)) ),
    inference(cnf_transformation,[],[f2152]) ).

tff(f2152,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK173(X0)),sK174(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK174(X0)),sK173(X0)) )
      | ( ~ 'fun_app$d'('fun_app$g'(X0,sK175(X0)),sK176(X0))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK176(X0)),sK175(X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK173,sK174,sK175,sK176])],[f1375,f2151,f2150]) ).

tff(f2150,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK173(X0)),sK174(X0))
        & 'fun_app$d'('fun_app$g'(X0,sK174(X0)),sK173(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2151,plain,
    ! [X0: 'N_n_bool_fun_fun$'] :
      ( ? [X5: 'N$',X6: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X5),X6)
          & 'fun_app$d'('fun_app$g'('less_eq$',X6),X5) )
     => ( ~ 'fun_app$d'('fun_app$g'(X0,sK175(X0)),sK176(X0))
        & 'fun_app$d'('fun_app$g'('less_eq$',sK176(X0)),sK175(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1375,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$',X6: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X5),X6)
          & 'fun_app$d'('fun_app$g'('less_eq$',X6),X5) ) ),
    inference(flattening,[],[f1374]) ).

tff(f1374,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ? [X3: 'N$',X4: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X3),X4)
          & 'fun_app$d'('fun_app$g'(X0,X4),X3) )
      | ? [X5: 'N$',X6: 'N$'] :
          ( ~ 'fun_app$d'('fun_app$g'(X0,X5),X6)
          & 'fun_app$d'('fun_app$g'('less_eq$',X6),X5) ) ),
    inference(ennf_transformation,[],[f980]) ).

tff(f980,plain,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X5: 'N$',X6: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X6),X5)
           => 'fun_app$d'('fun_app$g'(X0,X5),X6) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    inference(rectify,[],[f214]) ).

tff(f214,axiom,
    ! [X0: 'N_n_bool_fun_fun$',X1: 'N$',X2: 'N$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'(X0,X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) )
        & ! [X3: 'N$',X4: 'N$'] :
            ( 'fun_app$d'('fun_app$g'('less_eq$',X4),X3)
           => 'fun_app$d'('fun_app$g'(X0,X3),X4) ) )
     => 'fun_app$d'('fun_app$g'(X0,X1),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom212) ).

tff(f8792,plain,
    spl310_424,
    inference(avatar_split_clause,[],[f3404,f8790]) ).

tff(f8790,plain,
    ( spl310_424
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | ~ 'fun_app$d'('fun_app$g'(X0,sK173(X0)),sK174(X0))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK176(X0)),sK175(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_424])]) ).

tff(f3404,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | ~ 'fun_app$d'('fun_app$g'(X0,sK173(X0)),sK174(X0))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK176(X0)),sK175(X0)) ),
    inference(cnf_transformation,[],[f2152]) ).

tff(f8788,plain,
    spl310_423,
    inference(avatar_split_clause,[],[f3403,f8786]) ).

tff(f8786,plain,
    ( spl310_423
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK174(X0)),sK173(X0))
        | ~ 'fun_app$d'('fun_app$g'(X0,sK175(X0)),sK176(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_423])]) ).

tff(f3403,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK174(X0)),sK173(X0))
      | ~ 'fun_app$d'('fun_app$g'(X0,sK175(X0)),sK176(X0)) ),
    inference(cnf_transformation,[],[f2152]) ).

tff(f8784,plain,
    spl310_422,
    inference(avatar_split_clause,[],[f3402,f8782]) ).

tff(f8782,plain,
    ( spl310_422
  <=> ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
        | 'fun_app$d'('fun_app$g'(X0,sK174(X0)),sK173(X0))
        | 'fun_app$d'('fun_app$g'('less_eq$',sK176(X0)),sK175(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_422])]) ).

tff(f3402,plain,
    ! [X2: 'N$',X0: 'N_n_bool_fun_fun$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'(X0,X1),X2)
      | 'fun_app$d'('fun_app$g'(X0,sK174(X0)),sK173(X0))
      | 'fun_app$d'('fun_app$g'('less_eq$',sK176(X0)),sK175(X0)) ),
    inference(cnf_transformation,[],[f2152]) ).

tff(f8780,plain,
    spl310_421,
    inference(avatar_split_clause,[],[f3389,f8778]) ).

tff(f8778,plain,
    ( spl310_421
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | ~ 'fun_app$'('fun_app$a'(X0,sK154(X0)),sK155(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK156(X0)),sK157(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_421])]) ).

tff(f3389,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ~ 'fun_app$'('fun_app$a'(X0,sK154(X0)),sK155(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK156(X0)),sK157(X0)) ),
    inference(cnf_transformation,[],[f2137]) ).

tff(f2137,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ( ~ 'fun_app$'('fun_app$a'(X0,sK154(X0)),sK155(X0))
        & 'fun_app$'('fun_app$a'(X0,sK155(X0)),sK154(X0)) )
      | ( ~ 'fun_app$'('fun_app$a'(X0,sK156(X0)),sK157(X0))
        & ~ $less(sK157(X0),sK156(X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK154,sK155,sK156,sK157])],[f1367,f2136,f2135]) ).

tff(f2135,plain,
    ! [X0: 'Int_int_bool_fun_fun$'] :
      ( ? [X3: $int,X4: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X3),X4)
          & 'fun_app$'('fun_app$a'(X0,X4),X3) )
     => ( ~ 'fun_app$'('fun_app$a'(X0,sK154(X0)),sK155(X0))
        & 'fun_app$'('fun_app$a'(X0,sK155(X0)),sK154(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2136,plain,
    ! [X0: 'Int_int_bool_fun_fun$'] :
      ( ? [X5: $int,X6: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X5),X6)
          & ~ $less(X6,X5) )
     => ( ~ 'fun_app$'('fun_app$a'(X0,sK156(X0)),sK157(X0))
        & ~ $less(sK157(X0),sK156(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1367,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ? [X3: $int,X4: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X3),X4)
          & 'fun_app$'('fun_app$a'(X0,X4),X3) )
      | ? [X5: $int,X6: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X5),X6)
          & ~ $less(X6,X5) ) ),
    inference(flattening,[],[f1366]) ).

tff(f1366,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ? [X3: $int,X4: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X3),X4)
          & 'fun_app$'('fun_app$a'(X0,X4),X3) )
      | ? [X5: $int,X6: $int] :
          ( ~ 'fun_app$'('fun_app$a'(X0,X5),X6)
          & ~ $less(X6,X5) ) ),
    inference(ennf_transformation,[],[f976]) ).

tff(f976,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( ( ! [X3: $int,X4: $int] :
            ( 'fun_app$'('fun_app$a'(X0,X4),X3)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) )
        & ! [X5: $int,X6: $int] :
            ( ~ $less(X6,X5)
           => 'fun_app$'('fun_app$a'(X0,X5),X6) ) )
     => 'fun_app$'('fun_app$a'(X0,X1),X2) ),
    inference(rectify,[],[f798]) ).

tff(f798,plain,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( ( ! [X3: $int,X4: $int] :
            ( 'fun_app$'('fun_app$a'(X0,X4),X3)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) )
        & ! [X3: $int,X4: $int] :
            ( ~ $less(X4,X3)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) ) )
     => 'fun_app$'('fun_app$a'(X0,X1),X2) ),
    inference(theory_normalization,[],[f216]) ).

tff(f216,axiom,
    ! [X0: 'Int_int_bool_fun_fun$',X1: $int,X2: $int] :
      ( ( ! [X3: $int,X4: $int] :
            ( 'fun_app$'('fun_app$a'(X0,X4),X3)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) )
        & ! [X3: $int,X4: $int] :
            ( $lesseq(X3,X4)
           => 'fun_app$'('fun_app$a'(X0,X3),X4) ) )
     => 'fun_app$'('fun_app$a'(X0,X1),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom214) ).

tff(f8776,plain,
    spl310_420,
    inference(avatar_split_clause,[],[f3387,f8774]) ).

tff(f8774,plain,
    ( spl310_420
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | 'fun_app$'('fun_app$a'(X0,sK155(X0)),sK154(X0))
        | ~ 'fun_app$'('fun_app$a'(X0,sK156(X0)),sK157(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_420])]) ).

tff(f3387,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | 'fun_app$'('fun_app$a'(X0,sK155(X0)),sK154(X0))
      | ~ 'fun_app$'('fun_app$a'(X0,sK156(X0)),sK157(X0)) ),
    inference(cnf_transformation,[],[f2137]) ).

tff(f8772,plain,
    ( spl310_419
    | ~ spl310_127
    | ~ spl310_355 ),
    inference(avatar_split_clause,[],[f8148,f8052,f4912,f8769]) ).

tff(f8769,plain,
    ( spl310_419
  <=> 'fun_app$d'('fun_app$g'('less_eq$','i$'),sK91('fun_app$g'('less$','i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_419])]) ).

tff(f4912,plain,
    ( spl310_127
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_127])]) ).

tff(f8148,plain,
    ( 'fun_app$d'('fun_app$g'('less_eq$','i$'),sK91('fun_app$g'('less$','i$')))
    | ~ spl310_127
    | ~ spl310_355 ),
    inference(resolution,[],[f8054,f4913]) ).

tff(f4913,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less$',X0),X1)
        | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
    | ~ spl310_127 ),
    inference(avatar_component_clause,[],[f4912]) ).

tff(f8767,plain,
    spl310_418,
    inference(avatar_split_clause,[],[f3190,f8765]) ).

tff(f8765,plain,
    ( spl310_418
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X0)
        | ~ 'fun_app$j'(X2,sK129(X0,X1,X2))
        | ~ 'fun_app$j'(X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_418])]) ).

tff(f3190,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X0)
      | ~ 'fun_app$j'(X2,sK129(X0,X1,X2))
      | ~ 'fun_app$j'(X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2011]) ).

tff(f8763,plain,
    spl310_417,
    inference(avatar_split_clause,[],[f3185,f8761]) ).

tff(f8761,plain,
    ( spl310_417
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X1)
        | 'fun_app$j'(X2,sK128(X0,X1,X2))
        | ~ 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_417])]) ).

tff(f3185,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X1)
      | 'fun_app$j'(X2,sK128(X0,X1,X2))
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2009]) ).

tff(f8759,plain,
    spl310_416,
    inference(avatar_split_clause,[],[f3130,f8757]) ).

tff(f8757,plain,
    ( spl310_416
  <=> ! [X4: 'Nat$',X0: 'Nat_bool_fun$',X2: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',sK120(X0)),'fun_app$e'('of_nat$',X4))
        | ~ 'fun_app$j'(X0,X4)
        | 'fun_app$j'(X0,sK121(X0,X2))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_416])]) ).

tff(f3130,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$',X4: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',sK120(X0)),'fun_app$e'('of_nat$',X4))
      | ~ 'fun_app$j'(X0,X4)
      | 'fun_app$j'(X0,sK121(X0,X2))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f1999]) ).

tff(f8755,plain,
    spl310_415,
    inference(avatar_split_clause,[],[f3109,f8753]) ).

tff(f8753,plain,
    ( spl310_415
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_415])]) ).

tff(f3109,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,'fun_app$e'('of_nat$',X1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1203]) ).

tff(f8751,plain,
    spl310_414,
    inference(avatar_split_clause,[],[f3098,f8749]) ).

tff(f8749,plain,
    ( spl310_414
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP35(X1,X0,X2)
        | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ~ sP36(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_414])]) ).

tff(f3098,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP35(X1,X0,X2)
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ sP36(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1992]) ).

tff(f1992,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP35(X1,X0,X2)
          | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
        & ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
          | ~ sP35(X1,X0,X2) ) )
      | ~ sP36(X0,X1,X2) ),
    inference(rectify,[],[f1991]) ).

tff(f1991,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( sP35(X2,X0,X1)
          | ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
        & ( $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)))
          | ~ sP35(X2,X0,X1) ) )
      | ~ sP36(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1634]) ).

tff(f1634,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP35(X2,X0,X1)
      <=> $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
      | ~ sP36(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP36])]) ).

tff(f8747,plain,
    spl310_413,
    inference(avatar_split_clause,[],[f3097,f8745]) ).

tff(f8745,plain,
    ( spl310_413
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ~ sP35(X1,X0,X2)
        | ~ sP36(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_413])]) ).

tff(f3097,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ sP35(X1,X0,X2)
      | ~ sP36(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1992]) ).

tff(f8743,plain,
    spl310_412,
    inference(avatar_split_clause,[],[f3089,f8741]) ).

tff(f8741,plain,
    ( spl310_412
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP33(X1,X0,X2)
        | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ~ sP34(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_412])]) ).

tff(f3089,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP33(X1,X0,X2)
      | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ sP34(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1987]) ).

tff(f1987,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP33(X1,X0,X2)
          | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
        & ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
          | ~ sP33(X1,X0,X2) ) )
      | ~ sP34(X0,X1,X2) ),
    inference(rectify,[],[f1986]) ).

tff(f1986,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( sP33(X2,X0,X1)
          | $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
        & ( ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)))
          | ~ sP33(X2,X0,X1) ) )
      | ~ sP34(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1631]) ).

tff(f1631,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP33(X2,X0,X1)
      <=> ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
      | ~ sP34(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP34])]) ).

tff(f8739,plain,
    spl310_411,
    inference(avatar_split_clause,[],[f3088,f8737]) ).

tff(f8737,plain,
    ( spl310_411
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
        | ~ sP33(X1,X0,X2)
        | ~ sP34(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_411])]) ).

tff(f3088,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))
      | ~ sP33(X1,X0,X2)
      | ~ sP34(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1987]) ).

tff(f8735,plain,
    spl310_410,
    inference(avatar_split_clause,[],[f3080,f8733]) ).

tff(f8733,plain,
    ( spl310_410
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP31(X0,X2,X1)
        | ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ~ sP32(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_410])]) ).

tff(f3080,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP31(X0,X2,X1)
      | ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP32(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1983]) ).

tff(f1983,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP31(X0,X2,X1)
          | ( 'fun_app$e'('of_nat$',X2) != $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        & ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ~ sP31(X0,X2,X1) ) )
      | ~ sP32(X0,X1,X2) ),
    inference(rectify,[],[f1982]) ).

tff(f1982,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( sP31(X0,X1,X2)
          | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        & ( ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          | ~ sP31(X0,X1,X2) ) )
      | ~ sP32(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1628]) ).

tff(f1628,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP31(X0,X1,X2)
      <=> ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      | ~ sP32(X0,X2,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP32])]) ).

tff(f8731,plain,
    spl310_409,
    inference(avatar_split_clause,[],[f3079,f8729]) ).

tff(f8729,plain,
    ( spl310_409
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ~ sP31(X0,X2,X1)
        | ~ sP32(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_409])]) ).

tff(f3079,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP31(X0,X2,X1)
      | ~ sP32(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1983]) ).

tff(f8724,plain,
    spl310_408,
    inference(avatar_split_clause,[],[f2459,f8722]) ).

tff(f8722,plain,
    ( spl310_408
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_408])]) ).

tff(f2459,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f990]) ).

tff(f990,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      & ( ~ $less('fun_app$e'('of_nat$',X0),0)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    inference(ennf_transformation,[],[f648]) ).

tff(f648,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))) )
      & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => ~ $less('fun_app$e'('of_nat$',X0),0) ) ),
    inference(theory_normalization,[],[f153]) ).

tff(f153,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => $lesseq($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) )
      & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
       => $lesseq(0,'fun_app$e'('of_nat$',X0)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom151) ).

tff(f8521,plain,
    ( ~ spl310_407
    | ~ spl310_24
    | ~ spl310_329 ),
    inference(avatar_split_clause,[],[f8345,f7234,f4326,f8518]) ).

tff(f8518,plain,
    ( spl310_407
  <=> $less($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_407])]) ).

tff(f8345,plain,
    ( ~ $less($sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))),0)
    | ~ spl310_24
    | ~ spl310_329 ),
    inference(resolution,[],[f7236,f4327]) ).

tff(f8384,plain,
    ( spl310_406
    | ~ spl310_190
    | ~ spl310_329 ),
    inference(avatar_split_clause,[],[f8342,f7234,f5387,f8382]) ).

tff(f8382,plain,
    ( spl310_406
  <=> ! [X0: 'Nat$',X1: 'Nat$'] : ~ sP16('to_nat$'('i$'),X0,X1) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_406])]) ).

tff(f8342,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] : ~ sP16('to_nat$'('i$'),X0,X1)
    | ~ spl310_190
    | ~ spl310_329 ),
    inference(resolution,[],[f7236,f5388]) ).

tff(f8377,plain,
    spl310_405,
    inference(avatar_split_clause,[],[f4222,f8375]) ).

tff(f8375,plain,
    ( spl310_405
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] : ( $sum($sum(X0,$uminus(X2)),$sum(X1,$uminus(X3))) = $sum(X0,$sum(X1,$uminus($sum(X2,X3)))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_405])]) ).

tff(f4222,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] : ( $sum($sum(X0,$uminus(X2)),$sum(X1,$uminus(X3))) = $sum(X0,$sum(X1,$uminus($sum(X2,X3)))) ),
    inference(forward_demodulation,[],[f3483,f865]) ).

tff(f3483,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus($sum(X2,X3))) = $sum($sum(X0,$uminus(X2)),$sum(X1,$uminus(X3))) ),
    inference(cnf_transformation,[],[f812]) ).

tff(f812,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] : ( $sum($sum(X0,X1),$uminus($sum(X2,X3))) = $sum($sum(X0,$uminus(X2)),$sum(X1,$uminus(X3))) ),
    inference(theory_normalization,[],[f115]) ).

tff(f115,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] : ( $difference($sum(X0,X1),$sum(X2,X3)) = $sum($difference(X0,X2),$difference(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom113) ).

tff(f8373,plain,
    spl310_404,
    inference(avatar_split_clause,[],[f4044,f8371]) ).

tff(f8371,plain,
    ( spl310_404
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK115(X0,X1))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_404])]) ).

tff(f4044,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK115(X0,X1))) ),
    inference(forward_demodulation,[],[f2859,f864]) ).

tff(f2859,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK115(X0,X1)))
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1908]) ).

tff(f8369,plain,
    spl310_403,
    inference(avatar_split_clause,[],[f4035,f8367]) ).

tff(f8367,plain,
    ( spl310_403
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X0) != $sum(1,'fun_app$e'('of_nat$',X1)) )
        | ~ $less($sum(1,'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_403])]) ).

tff(f4035,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) != $sum(1,'fun_app$e'('of_nat$',X1)) )
      | ~ $less($sum(1,'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f4034,f864]) ).

tff(f4034,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
      | ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),1) ) ),
    inference(forward_demodulation,[],[f2850,f864]) ).

tff(f2850,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0))
      | ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),1) ) ),
    inference(cnf_transformation,[],[f1900]) ).

tff(f1900,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0))
        | ( ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),1) )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),1) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0)) ) ),
    inference(flattening,[],[f1899]) ).

tff(f1899,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0))
        | ( ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X1),1) )
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),1) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0)) ) ),
    inference(nnf_transformation,[],[f719]) ).

tff(f719,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0))
    <=> ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),1) )
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    inference(theory_normalization,[],[f520]) ).

tff(f520,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
    <=> ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),1) )
        | $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom518) ).

tff(f8365,plain,
    spl310_402,
    inference(avatar_split_clause,[],[f4028,f8363]) ).

tff(f8363,plain,
    ( spl310_402
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK103(X0,X1))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_402])]) ).

tff(f4028,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK103(X0,X1))) ),
    inference(forward_demodulation,[],[f2636,f864]) ).

tff(f2636,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK103(X0,X1)))
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1756]) ).

tff(f8361,plain,
    spl310_401,
    inference(avatar_split_clause,[],[f3913,f8359]) ).

tff(f8359,plain,
    ( spl310_401
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ sP42(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_401])]) ).

tff(f3913,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP42(X0,X1,X2) ),
    inference(evaluation,[],[f3196]) ).

tff(f3196,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,0)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | ~ sP42(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2015]) ).

tff(f8357,plain,
    spl310_400,
    inference(avatar_split_clause,[],[f3867,f8355]) ).

tff(f8355,plain,
    ( spl310_400
  <=> ! [X2: 'Int_set$',X1: 'Int_set_int_set_fun$',X3: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X2)),'fun_app$h'(X1,X3))
        | 'fun_app$c'('less_eq$a'(sK242(X1)),sK243(X1))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_400])]) ).

tff(f3867,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X1: 'Int_set_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X1,X2)),'fun_app$h'(X1,X3))
      | 'fun_app$c'('less_eq$a'(sK242(X1)),sK243(X1))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3) ),
    inference(equality_resolution,[],[f3576]) ).

tff(f3576,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$h'(X1,X3))
      | 'fun_app$c'('less_eq$a'(sK242(X1)),sK243(X1))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$h'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2260]) ).

tff(f8353,plain,
    spl310_399,
    inference(avatar_split_clause,[],[f3849,f8351]) ).

tff(f8351,plain,
    ( spl310_399
  <=> ! [X2: 'Int_set_int_set_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),'fun_app$h'(X2,X1))
        | 'fun_app$c'('less_eq$a'(sK218(X2)),sK219(X2))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_399])]) ).

tff(f3849,plain,
    ! [X2: 'Int_set_int_set_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),'fun_app$h'(X2,X1))
      | 'fun_app$c'('less_eq$a'(sK218(X2)),sK219(X2))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(equality_resolution,[],[f3552]) ).

tff(f3552,plain,
    ! [X2: 'Int_set_int_set_fun$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'(X2,X0)),X3)
      | 'fun_app$c'('less_eq$a'(sK218(X2)),sK219(X2))
      | ( 'fun_app$h'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2236]) ).

tff(f8349,plain,
    spl310_398,
    inference(avatar_split_clause,[],[f3627,f8347]) ).

tff(f8347,plain,
    ( spl310_398
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( $less(X0,'fun_app$b'(X1,X3))
        | $less(sK292(X1),sK293(X1))
        | ~ $less(X2,X3)
        | ~ $less(X0,'fun_app$b'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_398])]) ).

tff(f3627,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | $less(sK292(X1),sK293(X1))
      | ~ $less(X2,X3)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(cnf_transformation,[],[f2310]) ).

tff(f8341,plain,
    spl310_397,
    inference(avatar_split_clause,[],[f3625,f8339]) ).

tff(f8339,plain,
    ( spl310_397
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( $less(X0,'fun_app$b'(X1,X3))
        | ~ $less(sK291(X1),sK290(X1))
        | $less(X3,X2)
        | ~ $less(X0,'fun_app$b'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_397])]) ).

tff(f3625,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | ~ $less(sK291(X1),sK290(X1))
      | $less(X3,X2)
      | ~ $less(X0,'fun_app$b'(X1,X2)) ),
    inference(cnf_transformation,[],[f2308]) ).

tff(f8337,plain,
    spl310_396,
    inference(avatar_split_clause,[],[f3618,f8335]) ).

tff(f8335,plain,
    ( spl310_396
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( $less(X0,'fun_app$b'(X1,X3))
        | $less(sK284(X1),sK285(X1))
        | ~ $less(X2,X3)
        | $less('fun_app$b'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_396])]) ).

tff(f3618,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | $less(sK284(X1),sK285(X1))
      | ~ $less(X2,X3)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2302]) ).

tff(f8333,plain,
    spl310_395,
    inference(avatar_split_clause,[],[f3616,f8331]) ).

tff(f8331,plain,
    ( spl310_395
  <=> ! [X0: $int,X3: $int,X2: $int,X1: 'Int_int_fun$'] :
        ( ~ $less('fun_app$b'(X1,X3),X0)
        | ~ $less(sK283(X1),sK282(X1))
        | $less(X3,X2)
        | $less('fun_app$b'(X1,X2),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_395])]) ).

tff(f3616,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ~ $less(sK283(X1),sK282(X1))
      | $less(X3,X2)
      | $less('fun_app$b'(X1,X2),X0) ),
    inference(cnf_transformation,[],[f2300]) ).

tff(f8329,plain,
    spl310_394,
    inference(avatar_split_clause,[],[f3547,f8327]) ).

tff(f8327,plain,
    ( spl310_394
  <=> ! [X0: 'Int_set$',X3: 'Int_set$',X2: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'('fun_app$h'('plus$b'(X0),X2)),'fun_app$h'('plus$b'(X1),X3))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_394])]) ).

tff(f3547,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'('plus$b'(X0),X2)),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1468]) ).

tff(f1468,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'('plus$b'(X0),X2)),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(flattening,[],[f1467]) ).

tff(f1467,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$h'('plus$b'(X0),X2)),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f119]) ).

tff(f119,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$',X3: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X2),X3)
        & 'fun_app$c'('less_eq$a'(X0),X1) )
     => 'fun_app$c'('less_eq$a'('fun_app$h'('plus$b'(X0),X2)),'fun_app$h'('plus$b'(X1),X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom117) ).

tff(f8325,plain,
    spl310_393,
    inference(avatar_split_clause,[],[f3540,f8323]) ).

tff(f8323,plain,
    ( spl310_393
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( $less('fun_app$b'(X2,X0),X3)
        | $less(sK208(X2),sK209(X2))
        | ~ $less('fun_app$b'(X2,X1),X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_393])]) ).

tff(f3540,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | $less(sK208(X2),sK209(X2))
      | ~ $less('fun_app$b'(X2,X1),X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2226]) ).

tff(f8321,plain,
    spl310_392,
    inference(avatar_split_clause,[],[f3538,f8319]) ).

tff(f8319,plain,
    ( spl310_392
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( $less('fun_app$b'(X2,X0),X3)
        | $less(sK206(X2),sK207(X2))
        | $less(X3,'fun_app$b'(X2,X1))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_392])]) ).

tff(f3538,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | $less(sK206(X2),sK207(X2))
      | $less(X3,'fun_app$b'(X2,X1))
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2224]) ).

tff(f8317,plain,
    spl310_391,
    inference(avatar_split_clause,[],[f3522,f8315]) ).

tff(f8315,plain,
    ( spl310_391
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( $less('fun_app$b'(X2,X0),X3)
        | ~ $less(sK197(X2),sK196(X2))
        | ~ $less('fun_app$b'(X2,X1),X3)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_391])]) ).

tff(f3522,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | ~ $less(sK197(X2),sK196(X2))
      | ~ $less('fun_app$b'(X2,X1),X3)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2214]) ).

tff(f8313,plain,
    spl310_390,
    inference(avatar_split_clause,[],[f3520,f8311]) ).

tff(f8311,plain,
    ( spl310_390
  <=> ! [X0: $int,X3: $int,X2: 'Int_int_fun$',X1: $int] :
        ( ~ $less(X3,'fun_app$b'(X2,X0))
        | ~ $less(sK195(X2),sK194(X2))
        | $less(X3,'fun_app$b'(X2,X1))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_390])]) ).

tff(f3520,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ~ $less(sK195(X2),sK194(X2))
      | $less(X3,'fun_app$b'(X2,X1))
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2212]) ).

tff(f8309,plain,
    spl310_389,
    inference(avatar_split_clause,[],[f3331,f8307]) ).

tff(f8307,plain,
    ( spl310_389
  <=> ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
        ( ( $sum(sK141(X0,X1,X2),sK142(X0,X1,X2)) = X0 )
        | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_389])]) ).

tff(f3331,plain,
    ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
      ( ( $sum(sK141(X0,X1,X2),sK142(X0,X1,X2)) = X0 )
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ),
    inference(cnf_transformation,[],[f2087]) ).

tff(f2087,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( ( 'fun_app$c'('member$a'(sK142(X0,X1,X2)),X2)
        & 'fun_app$c'('member$a'(sK141(X0,X1,X2)),X1)
        & ( $sum(sK141(X0,X1,X2),sK142(X0,X1,X2)) = X0 ) )
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK141,sK142])],[f1341,f2086]) ).

tff(f2086,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( ? [X3: $int,X4: $int] :
          ( 'fun_app$c'('member$a'(X4),X2)
          & 'fun_app$c'('member$a'(X3),X1)
          & ( $sum(X3,X4) = X0 ) )
     => ( 'fun_app$c'('member$a'(sK142(X0,X1,X2)),X2)
        & 'fun_app$c'('member$a'(sK141(X0,X1,X2)),X1)
        & ( $sum(sK141(X0,X1,X2),sK142(X0,X1,X2)) = X0 ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1341,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( ? [X3: $int,X4: $int] :
          ( 'fun_app$c'('member$a'(X4),X2)
          & 'fun_app$c'('member$a'(X3),X1)
          & ( $sum(X3,X4) = X0 ) )
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ),
    inference(ennf_transformation,[],[f970]) ).

tff(f970,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ~ ( ! [X3: $int,X4: $int] :
            ~ ( 'fun_app$c'('member$a'(X4),X2)
              & 'fun_app$c'('member$a'(X3),X1)
              & ( $sum(X3,X4) = X0 ) )
        & 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ),
    inference(true_and_false_elimination,[],[f243]) ).

tff(f243,axiom,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( ( ! [X3: $int,X4: $int] :
            ( ( 'fun_app$c'('member$a'(X4),X2)
              & 'fun_app$c'('member$a'(X3),X1)
              & ( $sum(X3,X4) = X0 ) )
           => $false )
        & 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom241) ).

tff(f8305,plain,
    spl310_388,
    inference(avatar_split_clause,[],[f3327,f8303]) ).

tff(f8303,plain,
    ( spl310_388
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X0)
        | sP61(X2,X1)
        | ~ 'fun_app$j'(X2,sK140(X1,X2))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_388])]) ).

tff(f3327,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X0)
      | sP61(X2,X1)
      | ~ 'fun_app$j'(X2,sK140(X1,X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f2085]) ).

tff(f8301,plain,
    ( spl310_329
    | ~ spl310_117
    | ~ spl310_268 ),
    inference(avatar_split_clause,[],[f7010,f6601,f4838,f7234]) ).

tff(f7010,plain,
    ( $less(0,$sum(1,'fun_app$e'('of_nat$','to_nat$'('i$'))))
    | ~ spl310_117
    | ~ spl310_268 ),
    inference(resolution,[],[f4840,f6602]) ).

tff(f8300,plain,
    spl310_387,
    inference(avatar_split_clause,[],[f3304,f8298]) ).

tff(f8298,plain,
    ( spl310_387
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP59(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_387])]) ).

tff(f3304,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP59(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1663]) ).

tff(f1663,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP59(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1326,f1662,f1661,f1660,f1659]) ).

tff(f1661,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( sP58(X1,X2,X0)
    <=> ( ( sP57(X0,X2,X1)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( sP56(X0,X2)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP58])]) ).

tff(f1662,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP58(X1,X2,X0)
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | ~ sP59(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP59])]) ).

tff(f1326,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( $less(0,0)
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1325]) ).

tff(f1325,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( $less(0,0)
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f785]) ).

tff(f785,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less(0,$sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less(0,0) ) ) ) )
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    inference(theory_normalization,[],[f562]) ).

tff(f562,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),0) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less(0,$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $less(0,0) ) ) ) )
      <=> $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom560) ).

tff(f8296,plain,
    spl310_386,
    inference(avatar_split_clause,[],[f3283,f8294]) ).

tff(f8294,plain,
    ( spl310_386
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP55(X1,X2,X0)
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_386])]) ).

tff(f3283,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP55(X1,X2,X0)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1658]) ).

tff(f1658,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP55(X1,X2,X0)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1324,f1657,f1656,f1655,f1654]) ).

tff(f1654,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( sP52(X0,X2)
    <=> ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        & ( ~ $less(0,0)
          | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP52])]) ).

tff(f1656,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( sP54(X2,X1,X0)
    <=> ( ( sP53(X0,X1,X2)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( sP52(X0,X2)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP54])]) ).

tff(f1657,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( sP54(X2,X1,X0)
      <=> ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP55(X1,X2,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP55])]) ).

tff(f1324,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ~ $less(0,0)
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1323]) ).

tff(f1323,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ~ $less(0,0)
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f784]) ).

tff(f784,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ~ $less(0,0) ) ) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) ) ),
    inference(theory_normalization,[],[f150]) ).

tff(f150,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $lesseq($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $lesseq($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),0) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $lesseq(0,$difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => $lesseq(0,0) ) ) ) )
      <=> $lesseq('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom148) ).

tff(f8292,plain,
    spl310_385,
    inference(avatar_split_clause,[],[f3262,f8290]) ).

tff(f8290,plain,
    ( spl310_385
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP51(X2,X0,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_385])]) ).

tff(f3262,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP51(X2,X0,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1653]) ).

tff(f1653,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP51(X2,X0,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1322,f1652,f1651,f1650,f1649]) ).

tff(f1649,plain,
    ! [X2: 'Nat$',X1: 'Nat$'] :
      ( sP48(X2,X1)
    <=> ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( ~ $less(0,0)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP48])]) ).

tff(f1651,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( sP50(X2,X1,X0)
    <=> ( ( sP49(X0,X1,X2)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( sP48(X2,X1)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP50])]) ).

tff(f1652,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP50(X2,X1,X0)
      <=> ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | ~ sP51(X2,X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP51])]) ).

tff(f1322,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
              & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0)
                | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
              & ( ~ $less(0,0)
                | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1321]) ).

tff(f1321,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
              & ( ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
                | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0)
                | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
              & ( ~ $less(0,0)
                | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f783]) ).

tff(f783,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => ~ $less(0,$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => ~ $less(0,0) ) ) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    inference(theory_normalization,[],[f154]) ).

tff(f154,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => $lesseq($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => $lesseq($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),0) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => $lesseq(0,$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
               => $lesseq(0,0) ) ) ) )
      <=> $lesseq('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom152) ).

tff(f8288,plain,
    spl310_384,
    inference(avatar_split_clause,[],[f3241,f8286]) ).

tff(f8286,plain,
    ( spl310_384
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP47(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_384])]) ).

tff(f3241,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP47(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1648]) ).

tff(f1648,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP47(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1320,f1647,f1646,f1645,f1644]) ).

tff(f1646,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( sP46(X1,X2,X0)
    <=> ( ( sP45(X0,X2,X1)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( sP44(X0,X2)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP46])]) ).

tff(f1647,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP46(X1,X2,X0)
      <=> ( 'fun_app$e'('of_nat$',X1) = 'fun_app$e'('of_nat$',X2) ) )
      | ~ sP47(X2,X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP47])]) ).

tff(f1320,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ( 0 = 0 )
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ( 'fun_app$e'('of_nat$',X1) = 'fun_app$e'('of_nat$',X2) ) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1319]) ).

tff(f1319,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( ( ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( ( ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) )
                | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
              & ( ( 0 = 0 )
                | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ( 'fun_app$e'('of_nat$',X1) = 'fun_app$e'('of_nat$',X2) ) )
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f782]) ).

tff(f782,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) ) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( 0 = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( 0 = $sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))) ) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( 0 = 0 ) ) ) ) )
      <=> ( 'fun_app$e'('of_nat$',X1) = 'fun_app$e'('of_nat$',X2) ) ) ),
    inference(theory_normalization,[],[f149]) ).

tff(f149,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        & $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) = $difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( 0 = $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( 0 = $difference('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
              & ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
               => ( 0 = 0 ) ) ) ) )
      <=> ( 'fun_app$e'('of_nat$',X1) = 'fun_app$e'('of_nat$',X2) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom147) ).

tff(f8284,plain,
    spl310_383,
    inference(avatar_split_clause,[],[f3218,f8282]) ).

tff(f8282,plain,
    ( spl310_383
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP43(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_383])]) ).

tff(f3218,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP43(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1643]) ).

tff(f8280,plain,
    spl310_382,
    inference(avatar_split_clause,[],[f3181,f8278]) ).

tff(f8278,plain,
    ( spl310_382
  <=> ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
        ( 'fun_app$j'(X2,X1)
        | 'fun_app$j'(X2,sK127(X0,X2))
        | ~ 'fun_app$j'(X2,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_382])]) ).

tff(f3181,plain,
    ! [X2: 'Nat_bool_fun$',X0: 'Nat$',X1: 'Nat$'] :
      ( 'fun_app$j'(X2,X1)
      | 'fun_app$j'(X2,sK127(X0,X2))
      | ~ 'fun_app$j'(X2,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2007]) ).

tff(f8274,plain,
    ( spl310_381
    | ~ spl310_17
    | ~ spl310_265
    | ~ spl310_380 ),
    inference(avatar_split_clause,[],[f8270,f8267,f6588,f4298,f8272]) ).

tff(f8272,plain,
    ( spl310_381
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,'ja$')
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ 'fun_app$j'(X0,sK122(X0))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_381])]) ).

tff(f6588,plain,
    ( spl310_265
  <=> ( 0 = 'fun_app$e'('of_nat$','ja$') ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_265])]) ).

tff(f8267,plain,
    ( spl310_380
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,'nat$'(0))
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ 'fun_app$j'(X0,sK122(X0))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_380])]) ).

tff(f8270,plain,
    ( ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,'ja$')
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ 'fun_app$j'(X0,sK122(X0))
        | ~ 'fun_app$j'(X0,X1) )
    | ~ spl310_17
    | ~ spl310_265
    | ~ spl310_380 ),
    inference(forward_demodulation,[],[f8268,f7258]) ).

tff(f7258,plain,
    ( ( 'ja$' = 'nat$'(0) )
    | ~ spl310_17
    | ~ spl310_265 ),
    inference(superposition,[],[f4299,f6590]) ).

tff(f6590,plain,
    ( ( 0 = 'fun_app$e'('of_nat$','ja$') )
    | ~ spl310_265 ),
    inference(avatar_component_clause,[],[f6588]) ).

tff(f8268,plain,
    ( ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | 'fun_app$j'(X0,'nat$'(0))
        | ~ 'fun_app$j'(X0,sK122(X0))
        | ~ 'fun_app$j'(X0,X1) )
    | ~ spl310_380 ),
    inference(avatar_component_clause,[],[f8267]) ).

tff(f8269,plain,
    spl310_380,
    inference(avatar_split_clause,[],[f3133,f8267]) ).

tff(f3133,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'(0))
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ 'fun_app$j'(X0,sK122(X0))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f2001]) ).

tff(f8265,plain,
    ( ~ spl310_379
    | ~ spl310_126
    | ~ spl310_331 ),
    inference(avatar_split_clause,[],[f8099,f7525,f4908,f8262]) ).

tff(f8262,plain,
    ( spl310_379
  <=> 'fun_app$d'('fun_app$g'('less$',sK94('fun_app$g'('less$','i$'))),'i$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_379])]) ).

tff(f7525,plain,
    ( spl310_331
  <=> 'fun_app$d'('fun_app$g'('less$','i$'),sK94('fun_app$g'('less$','i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_331])]) ).

tff(f8099,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less$',sK94('fun_app$g'('less$','i$'))),'i$')
    | ~ spl310_126
    | ~ spl310_331 ),
    inference(resolution,[],[f7527,f4909]) ).

tff(f7527,plain,
    ( 'fun_app$d'('fun_app$g'('less$','i$'),sK94('fun_app$g'('less$','i$')))
    | ~ spl310_331 ),
    inference(avatar_component_clause,[],[f7525]) ).

tff(f8260,plain,
    spl310_378,
    inference(avatar_split_clause,[],[f3107,f8258]) ).

tff(f8258,plain,
    ( spl310_378
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_378])]) ).

tff(f3107,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1201]) ).

tff(f1201,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f539]) ).

tff(f539,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom537) ).

tff(f8256,plain,
    spl310_377,
    inference(avatar_split_clause,[],[f3106,f8254]) ).

tff(f8254,plain,
    ( spl310_377
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_377])]) ).

tff(f3106,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1200]) ).

tff(f1200,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f538]) ).

tff(f538,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom536) ).

tff(f8252,plain,
    spl310_376,
    inference(avatar_split_clause,[],[f3014,f8250]) ).

tff(f8250,plain,
    ( spl310_376
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_376])]) ).

tff(f3014,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1183]) ).

tff(f1183,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f745]) ).

tff(f745,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ~ $less($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X0)) ),
    inference(theory_normalization,[],[f146]) ).

tff(f146,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => $lesseq('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom144) ).

tff(f8248,plain,
    spl310_375,
    inference(avatar_split_clause,[],[f3013,f8246]) ).

tff(f8246,plain,
    ( spl310_375
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_375])]) ).

tff(f3013,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1182]) ).

tff(f1182,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f744]) ).

tff(f744,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ),
    inference(theory_normalization,[],[f147]) ).

tff(f147,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => $lesseq('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom145) ).

tff(f8244,plain,
    spl310_374,
    inference(avatar_split_clause,[],[f2862,f8242]) ).

tff(f8242,plain,
    ( spl310_374
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( $sum('fun_app$e'('of_nat$',X1),1) = $sum('fun_app$e'('of_nat$',X0),1) )
        | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_374])]) ).

tff(f2862,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X1),1) = $sum('fun_app$e'('of_nat$',X0),1) )
      | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ),
    inference(cnf_transformation,[],[f1909]) ).

tff(f1909,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( $sum('fun_app$e'('of_nat$',X1),1) = $sum('fun_app$e'('of_nat$',X0),1) )
        | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) )
      & ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
        | ( $sum('fun_app$e'('of_nat$',X1),1) != $sum('fun_app$e'('of_nat$',X0),1) ) ) ),
    inference(nnf_transformation,[],[f250]) ).

tff(f250,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $sum('fun_app$e'('of_nat$',X1),1) = $sum('fun_app$e'('of_nat$',X0),1) )
    <=> ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom248) ).

tff(f8240,plain,
    spl310_373,
    inference(avatar_split_clause,[],[f2835,f8238]) ).

tff(f8238,plain,
    ( spl310_373
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_373])]) ).

tff(f2835,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ( 'fun_app$e'('of_nat$',X1) != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    inference(cnf_transformation,[],[f1888]) ).

tff(f8216,plain,
    ( spl310_372
    | ~ spl310_127
    | ~ spl310_331 ),
    inference(avatar_split_clause,[],[f8098,f7525,f4912,f8213]) ).

tff(f8213,plain,
    ( spl310_372
  <=> 'fun_app$d'('fun_app$g'('less_eq$','i$'),sK94('fun_app$g'('less$','i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_372])]) ).

tff(f8098,plain,
    ( 'fun_app$d'('fun_app$g'('less_eq$','i$'),sK94('fun_app$g'('less$','i$')))
    | ~ spl310_127
    | ~ spl310_331 ),
    inference(resolution,[],[f7527,f4913]) ).

tff(f8125,plain,
    spl310_371,
    inference(avatar_split_clause,[],[f4136,f8123]) ).

tff(f8123,plain,
    ( spl310_371
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | sP14(X2,X1,X0)
        | sP16(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_371])]) ).

tff(f4136,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | sP14(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f2992,f864]) ).

tff(f2992,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP14(X2,X1,X0)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | sP16(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1614]) ).

tff(f8121,plain,
    spl310_370,
    inference(avatar_split_clause,[],[f4132,f8119]) ).

tff(f8119,plain,
    ( spl310_370
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | sP15(X2,X1,X0)
        | sP16(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_370])]) ).

tff(f4132,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | sP15(X2,X1,X0)
      | sP16(X2,X1,X0) ),
    inference(forward_demodulation,[],[f2994,f864]) ).

tff(f2994,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP15(X2,X1,X0)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | sP16(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1614]) ).

tff(f8117,plain,
    spl310_369,
    inference(avatar_split_clause,[],[f4079,f8115]) ).

tff(f8115,plain,
    ( spl310_369
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | sP17(X2,X1,X0)
        | ~ sP21(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_369])]) ).

tff(f4079,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | sP17(X2,X1,X0)
      | ~ sP21(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2962,f864]) ).

tff(f2962,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP17(X2,X1,X0)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ~ sP21(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1951]) ).

tff(f1951,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( sP20(X2,X1,X0)
            & sP19(X2,X1,X0) )
          | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
        & ( ( sP18(X2,X1,X0)
            & sP17(X2,X1,X0) )
          | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP21(X0,X1,X2) ),
    inference(nnf_transformation,[],[f1613]) ).

tff(f8113,plain,
    spl310_368,
    inference(avatar_split_clause,[],[f4078,f8111]) ).

tff(f8111,plain,
    ( spl310_368
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | sP18(X2,X1,X0)
        | ~ sP21(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_368])]) ).

tff(f4078,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | sP18(X2,X1,X0)
      | ~ sP21(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2963,f864]) ).

tff(f2963,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP18(X2,X1,X0)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ~ sP21(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1951]) ).

tff(f8109,plain,
    spl310_367,
    inference(avatar_split_clause,[],[f4077,f8107]) ).

tff(f8107,plain,
    ( spl310_367
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | sP19(X2,X1,X0)
        | ~ sP21(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_367])]) ).

tff(f4077,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | sP19(X2,X1,X0)
      | ~ sP21(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2964,f864]) ).

tff(f2964,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP19(X2,X1,X0)
      | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ~ sP21(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1951]) ).

tff(f8105,plain,
    spl310_366,
    inference(avatar_split_clause,[],[f4076,f8103]) ).

tff(f8103,plain,
    ( spl310_366
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
        | sP20(X2,X1,X0)
        | ~ sP21(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_366])]) ).

tff(f4076,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1))
      | sP20(X2,X1,X0)
      | ~ sP21(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2965,f864]) ).

tff(f2965,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP20(X2,X1,X0)
      | $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1))
      | ~ sP21(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1951]) ).

tff(f8095,plain,
    spl310_365,
    inference(avatar_split_clause,[],[f3861,f8093]) ).

tff(f8093,plain,
    ( spl310_365
  <=> ! [X2: $int,X1: 'Int_n_fun$',X3: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
        | $less(sK236(X1),sK237(X1))
        | ~ $less(X2,X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_365])]) ).

tff(f3861,plain,
    ! [X2: $int,X3: $int,X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
      | $less(sK236(X1),sK237(X1))
      | ~ $less(X2,X3) ),
    inference(equality_resolution,[],[f3570]) ).

tff(f3570,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$p'(X1,X3))
      | $less(sK236(X1),sK237(X1))
      | ~ $less(X2,X3)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2254]) ).

tff(f8091,plain,
    spl310_364,
    inference(avatar_split_clause,[],[f3859,f8089]) ).

tff(f8089,plain,
    ( spl310_364
  <=> ! [X2: $int,X1: 'Int_n_fun$',X3: $int] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
        | ~ $less(sK235(X1),sK234(X1))
        | $less(X3,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_364])]) ).

tff(f3859,plain,
    ! [X2: $int,X3: $int,X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X1,X2)),'fun_app$p'(X1,X3))
      | ~ $less(sK235(X1),sK234(X1))
      | $less(X3,X2) ),
    inference(equality_resolution,[],[f3568]) ).

tff(f3568,plain,
    ! [X2: $int,X3: $int,X0: 'N$',X1: 'Int_n_fun$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),'fun_app$p'(X1,X3))
      | ~ $less(sK235(X1),sK234(X1))
      | $less(X3,X2)
      | ( 'fun_app$p'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2252]) ).

tff(f8087,plain,
    spl310_363,
    inference(avatar_split_clause,[],[f3855,f8085]) ).

tff(f8085,plain,
    ( spl310_363
  <=> ! [X2: 'Int_set$',X1: 'Int_set_int_fun$',X3: 'Int_set$'] :
        ( ~ $less('fun_app$n'(X1,X3),'fun_app$n'(X1,X2))
        | 'fun_app$c'('less_eq$a'(sK230(X1)),sK231(X1))
        | ~ 'fun_app$c'('less_eq$a'(X2),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_363])]) ).

tff(f3855,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X1: 'Int_set_int_fun$'] :
      ( ~ $less('fun_app$n'(X1,X3),'fun_app$n'(X1,X2))
      | 'fun_app$c'('less_eq$a'(sK230(X1)),sK231(X1))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3) ),
    inference(equality_resolution,[],[f3564]) ).

tff(f3564,plain,
    ! [X2: 'Int_set$',X3: 'Int_set$',X0: $int,X1: 'Int_set_int_fun$'] :
      ( ~ $less('fun_app$n'(X1,X3),X0)
      | 'fun_app$c'('less_eq$a'(sK230(X1)),sK231(X1))
      | ~ 'fun_app$c'('less_eq$a'(X2),X3)
      | ( 'fun_app$n'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2248]) ).

tff(f8083,plain,
    spl310_362,
    inference(avatar_split_clause,[],[f3845,f8081]) ).

tff(f8081,plain,
    ( spl310_362
  <=> ! [X2: 'Int_set_int_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
        ( ~ $less('fun_app$n'(X2,X1),'fun_app$n'(X2,X0))
        | 'fun_app$c'('less_eq$a'(sK214(X2)),sK215(X2))
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_362])]) ).

tff(f3845,plain,
    ! [X2: 'Int_set_int_fun$',X0: 'Int_set$',X1: 'Int_set$'] :
      ( ~ $less('fun_app$n'(X2,X1),'fun_app$n'(X2,X0))
      | 'fun_app$c'('less_eq$a'(sK214(X2)),sK215(X2))
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(equality_resolution,[],[f3548]) ).

tff(f3548,plain,
    ! [X2: 'Int_set_int_fun$',X3: $int,X0: 'Int_set$',X1: 'Int_set$'] :
      ( ~ $less(X3,'fun_app$n'(X2,X0))
      | 'fun_app$c'('less_eq$a'(sK214(X2)),sK215(X2))
      | ( 'fun_app$n'(X2,X1) != X3 )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f2232]) ).

tff(f8079,plain,
    spl310_361,
    inference(avatar_split_clause,[],[f3841,f8077]) ).

tff(f8077,plain,
    ( spl310_361
  <=> ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
        | $less(sK202(X2),sK203(X2))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_361])]) ).

tff(f3841,plain,
    ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
      | $less(sK202(X2),sK203(X2))
      | ~ $less(X0,X1) ),
    inference(equality_resolution,[],[f3534]) ).

tff(f3534,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less$','fun_app$p'(X2,X0)),X3)
      | $less(sK202(X2),sK203(X2))
      | ( 'fun_app$p'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2220]) ).

tff(f8075,plain,
    spl310_360,
    inference(avatar_split_clause,[],[f3836,f8073]) ).

tff(f8073,plain,
    ( spl310_360
  <=> ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
        ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
        | ~ $less(sK191(X2),sK190(X2))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_360])]) ).

tff(f3836,plain,
    ! [X2: 'Int_n_fun$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),'fun_app$p'(X2,X1))
      | ~ $less(sK191(X2),sK190(X2))
      | $less(X1,X0) ),
    inference(equality_resolution,[],[f3516]) ).

tff(f3516,plain,
    ! [X2: 'Int_n_fun$',X3: 'N$',X0: $int,X1: $int] :
      ( 'fun_app$d'('fun_app$g'('less_eq$','fun_app$p'(X2,X0)),X3)
      | ~ $less(sK191(X2),sK190(X2))
      | ( 'fun_app$p'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2208]) ).

tff(f8071,plain,
    spl310_359,
    inference(avatar_split_clause,[],[f3546,f8069]) ).

tff(f8069,plain,
    ( spl310_359
  <=> ! [X0: $int,X3: 'Int_set$',X2: $int,X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'($sum(X0,X2)),'fun_app$h'('plus$b'(X1),X3))
        | ~ 'fun_app$c'('member$a'(X2),X3)
        | ~ 'fun_app$c'('member$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_359])]) ).

tff(f3546,plain,
    ! [X2: $int,X3: 'Int_set$',X0: $int,X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'($sum(X0,X2)),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('member$a'(X2),X3)
      | ~ 'fun_app$c'('member$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1466]) ).

tff(f1466,plain,
    ! [X0: $int,X1: 'Int_set$',X2: $int,X3: 'Int_set$'] :
      ( 'fun_app$c'('member$a'($sum(X0,X2)),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('member$a'(X2),X3)
      | ~ 'fun_app$c'('member$a'(X0),X1) ),
    inference(flattening,[],[f1465]) ).

tff(f1465,plain,
    ! [X0: $int,X1: 'Int_set$',X2: $int,X3: 'Int_set$'] :
      ( 'fun_app$c'('member$a'($sum(X0,X2)),'fun_app$h'('plus$b'(X1),X3))
      | ~ 'fun_app$c'('member$a'(X2),X3)
      | ~ 'fun_app$c'('member$a'(X0),X1) ),
    inference(ennf_transformation,[],[f107]) ).

tff(f107,axiom,
    ! [X0: $int,X1: 'Int_set$',X2: $int,X3: 'Int_set$'] :
      ( ( 'fun_app$c'('member$a'(X2),X3)
        & 'fun_app$c'('member$a'(X0),X1) )
     => 'fun_app$c'('member$a'($sum(X0,X2)),'fun_app$h'('plus$b'(X1),X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom105) ).

tff(f8067,plain,
    spl310_358,
    inference(avatar_split_clause,[],[f3388,f8065]) ).

tff(f8065,plain,
    ( spl310_358
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | ~ 'fun_app$'('fun_app$a'(X0,sK154(X0)),sK155(X0))
        | ~ $less(sK157(X0),sK156(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_358])]) ).

tff(f3388,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | ~ 'fun_app$'('fun_app$a'(X0,sK154(X0)),sK155(X0))
      | ~ $less(sK157(X0),sK156(X0)) ),
    inference(cnf_transformation,[],[f2137]) ).

tff(f8063,plain,
    spl310_357,
    inference(avatar_split_clause,[],[f3386,f8061]) ).

tff(f8061,plain,
    ( spl310_357
  <=> ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
        ( 'fun_app$'('fun_app$a'(X0,X1),X2)
        | 'fun_app$'('fun_app$a'(X0,sK155(X0)),sK154(X0))
        | ~ $less(sK157(X0),sK156(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_357])]) ).

tff(f3386,plain,
    ! [X2: $int,X0: 'Int_int_bool_fun_fun$',X1: $int] :
      ( 'fun_app$'('fun_app$a'(X0,X1),X2)
      | 'fun_app$'('fun_app$a'(X0,sK155(X0)),sK154(X0))
      | ~ $less(sK157(X0),sK156(X0)) ),
    inference(cnf_transformation,[],[f2137]) ).

tff(f8059,plain,
    spl310_356,
    inference(avatar_split_clause,[],[f3177,f8057]) ).

tff(f8057,plain,
    ( spl310_356
  <=> ! [X2: 'Int_set_set$',X0: 'Int_set$',X1: 'Int_set_set$'] :
        ( ( 'fun_app$h'('plus$b'(sK125(X0,X1,X2)),sK126(X0,X1,X2)) = X0 )
        | ~ 'member$'(X0,'plus$a'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_356])]) ).

tff(f3177,plain,
    ! [X2: 'Int_set_set$',X0: 'Int_set$',X1: 'Int_set_set$'] :
      ( ( 'fun_app$h'('plus$b'(sK125(X0,X1,X2)),sK126(X0,X1,X2)) = X0 )
      | ~ 'member$'(X0,'plus$a'(X1,X2)) ),
    inference(cnf_transformation,[],[f2005]) ).

tff(f2005,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set_set$'] :
      ( ( 'member$'(sK126(X0,X1,X2),X2)
        & 'member$'(sK125(X0,X1,X2),X1)
        & ( 'fun_app$h'('plus$b'(sK125(X0,X1,X2)),sK126(X0,X1,X2)) = X0 ) )
      | ~ 'member$'(X0,'plus$a'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK125,sK126])],[f1308,f2004]) ).

tff(f2004,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set_set$'] :
      ( ? [X3: 'Int_set$',X4: 'Int_set$'] :
          ( 'member$'(X4,X2)
          & 'member$'(X3,X1)
          & ( 'fun_app$h'('plus$b'(X3),X4) = X0 ) )
     => ( 'member$'(sK126(X0,X1,X2),X2)
        & 'member$'(sK125(X0,X1,X2),X1)
        & ( 'fun_app$h'('plus$b'(sK125(X0,X1,X2)),sK126(X0,X1,X2)) = X0 ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1308,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set_set$'] :
      ( ? [X3: 'Int_set$',X4: 'Int_set$'] :
          ( 'member$'(X4,X2)
          & 'member$'(X3,X1)
          & ( 'fun_app$h'('plus$b'(X3),X4) = X0 ) )
      | ~ 'member$'(X0,'plus$a'(X1,X2)) ),
    inference(ennf_transformation,[],[f964]) ).

tff(f964,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set_set$'] :
      ~ ( ! [X3: 'Int_set$',X4: 'Int_set$'] :
            ~ ( 'member$'(X4,X2)
              & 'member$'(X3,X1)
              & ( 'fun_app$h'('plus$b'(X3),X4) = X0 ) )
        & 'member$'(X0,'plus$a'(X1,X2)) ),
    inference(true_and_false_elimination,[],[f241]) ).

tff(f241,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set_set$'] :
      ( ( ! [X3: 'Int_set$',X4: 'Int_set$'] :
            ( ( 'member$'(X4,X2)
              & 'member$'(X3,X1)
              & ( 'fun_app$h'('plus$b'(X3),X4) = X0 ) )
           => $false )
        & 'member$'(X0,'plus$a'(X1,X2)) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom239) ).

tff(f8055,plain,
    ( spl310_355
    | ~ spl310_31
    | ~ spl310_308 ),
    inference(avatar_split_clause,[],[f7192,f7144,f4362,f8052]) ).

tff(f7192,plain,
    ( 'fun_app$d'('fun_app$g'('less$','i$'),sK91('fun_app$g'('less$','i$')))
    | ~ spl310_31
    | ~ spl310_308 ),
    inference(resolution,[],[f7146,f4363]) ).

tff(f8050,plain,
    spl310_354,
    inference(avatar_split_clause,[],[f3174,f8048]) ).

tff(f8048,plain,
    ( spl310_354
  <=> ! [X2: 'N_set$',X0: 'N$',X1: 'N_set$'] :
        ( ( 'fun_app$f'('plus$'(sK123(X0,X1,X2)),sK124(X0,X1,X2)) = X0 )
        | ~ 'member$b'(X0,'plus$c'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_354])]) ).

tff(f3174,plain,
    ! [X2: 'N_set$',X0: 'N$',X1: 'N_set$'] :
      ( ( 'fun_app$f'('plus$'(sK123(X0,X1,X2)),sK124(X0,X1,X2)) = X0 )
      | ~ 'member$b'(X0,'plus$c'(X1,X2)) ),
    inference(cnf_transformation,[],[f2003]) ).

tff(f2003,plain,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N_set$'] :
      ( ( 'member$b'(sK124(X0,X1,X2),X2)
        & 'member$b'(sK123(X0,X1,X2),X1)
        & ( 'fun_app$f'('plus$'(sK123(X0,X1,X2)),sK124(X0,X1,X2)) = X0 ) )
      | ~ 'member$b'(X0,'plus$c'(X1,X2)) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK123,sK124])],[f1307,f2002]) ).

tff(f2002,plain,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N_set$'] :
      ( ? [X3: 'N$',X4: 'N$'] :
          ( 'member$b'(X4,X2)
          & 'member$b'(X3,X1)
          & ( 'fun_app$f'('plus$'(X3),X4) = X0 ) )
     => ( 'member$b'(sK124(X0,X1,X2),X2)
        & 'member$b'(sK123(X0,X1,X2),X1)
        & ( 'fun_app$f'('plus$'(sK123(X0,X1,X2)),sK124(X0,X1,X2)) = X0 ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1307,plain,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N_set$'] :
      ( ? [X3: 'N$',X4: 'N$'] :
          ( 'member$b'(X4,X2)
          & 'member$b'(X3,X1)
          & ( 'fun_app$f'('plus$'(X3),X4) = X0 ) )
      | ~ 'member$b'(X0,'plus$c'(X1,X2)) ),
    inference(ennf_transformation,[],[f963]) ).

tff(f963,plain,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N_set$'] :
      ~ ( ! [X3: 'N$',X4: 'N$'] :
            ~ ( 'member$b'(X4,X2)
              & 'member$b'(X3,X1)
              & ( 'fun_app$f'('plus$'(X3),X4) = X0 ) )
        & 'member$b'(X0,'plus$c'(X1,X2)) ),
    inference(true_and_false_elimination,[],[f242]) ).

tff(f242,axiom,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N_set$'] :
      ( ( ! [X3: 'N$',X4: 'N$'] :
            ( ( 'member$b'(X4,X2)
              & 'member$b'(X3,X1)
              & ( 'fun_app$f'('plus$'(X3),X4) = X0 ) )
           => $false )
        & 'member$b'(X0,'plus$c'(X1,X2)) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom240) ).

tff(f8046,plain,
    spl310_353,
    inference(avatar_split_clause,[],[f2454,f8044]) ).

tff(f8044,plain,
    ( spl310_353
  <=> ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'(X0),X1) = 'fun_app$i'('from_nat$','nat$'($sum('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1))))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_353])]) ).

tff(f2454,plain,
    ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'(X0),X1) = 'fun_app$i'('from_nat$','nat$'($sum('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1))))) ),
    inference(cnf_transformation,[],[f161]) ).

tff(f161,axiom,
    ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'(X0),X1) = 'fun_app$i'('from_nat$','nat$'($sum('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1))))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom159) ).

tff(f7880,plain,
    spl310_352,
    inference(avatar_split_clause,[],[f4039,f7878]) ).

tff(f7878,plain,
    ( spl310_352
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_352])]) ).

tff(f4039,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ),
    inference(forward_demodulation,[],[f2857,f864]) ).

tff(f2857,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
      | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ),
    inference(cnf_transformation,[],[f1904]) ).

tff(f1904,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
        | ( ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) )
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      & ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1)) ) ),
    inference(flattening,[],[f1903]) ).

tff(f1903,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
        | ( ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) )
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      & ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | ~ $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1)) ) ),
    inference(nnf_transformation,[],[f395]) ).

tff(f395,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
    <=> ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom393) ).

tff(f7876,plain,
    spl310_351,
    inference(avatar_split_clause,[],[f4029,f7874]) ).

tff(f7874,plain,
    ( spl310_351
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X3: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',sK109(X0)),$sum(1,'fun_app$e'('of_nat$',X3)))
        | 'fun_app$j'(X0,X1)
        | 'fun_app$j'(X0,X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_351])]) ).

tff(f4029,plain,
    ! [X3: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',sK109(X0)),$sum(1,'fun_app$e'('of_nat$',X3)))
      | 'fun_app$j'(X0,X1)
      | 'fun_app$j'(X0,X3) ),
    inference(forward_demodulation,[],[f2656,f864]) ).

tff(f2656,plain,
    ! [X3: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | 'fun_app$j'(X0,X3)
      | $less('fun_app$e'('of_nat$',sK109(X0)),$sum('fun_app$e'('of_nat$',X3),1)) ),
    inference(cnf_transformation,[],[f1768]) ).

tff(f1768,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ( ~ 'fun_app$j'(X0,sK109(X0))
        & ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | $less('fun_app$e'('of_nat$',sK109(X0)),$sum('fun_app$e'('of_nat$',X3),1)) ) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK109])],[f1151,f1767]) ).

tff(f1767,plain,
    ! [X0: 'Nat_bool_fun$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X3),1)) ) )
     => ( ~ 'fun_app$j'(X0,sK109(X0))
        & ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | $less('fun_app$e'('of_nat$',sK109(X0)),$sum('fun_app$e'('of_nat$',X3),1)) ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1151,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X3),1)) ) ) ),
    inference(ennf_transformation,[],[f696]) ).

tff(f696,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ! [X2: 'Nat$'] :
          ( ! [X3: 'Nat$'] :
              ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X3),1))
             => 'fun_app$j'(X0,X3) )
         => 'fun_app$j'(X0,X2) )
     => 'fun_app$j'(X0,X1) ),
    inference(theory_normalization,[],[f517]) ).

tff(f517,axiom,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ! [X2: 'Nat$'] :
          ( ! [X3: 'Nat$'] :
              ( $lesseq($sum('fun_app$e'('of_nat$',X3),1),'fun_app$e'('of_nat$',X2))
             => 'fun_app$j'(X0,X3) )
         => 'fun_app$j'(X0,X2) )
     => 'fun_app$j'(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom515) ).

tff(f7872,plain,
    spl310_350,
    inference(avatar_split_clause,[],[f4005,f7870]) ).

tff(f7870,plain,
    ( spl310_350
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_350])]) ).

tff(f4005,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2585,f864]) ).

tff(f2585,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1074]) ).

tff(f1074,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f398]) ).

tff(f398,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom396) ).

tff(f7868,plain,
    spl310_349,
    inference(avatar_split_clause,[],[f3997,f7866]) ).

tff(f7866,plain,
    ( spl310_349
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum(1,'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_349])]) ).

tff(f3997,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(1,'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(forward_demodulation,[],[f2561,f864]) ).

tff(f2561,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1062]) ).

tff(f1062,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f674]) ).

tff(f674,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ~ $less($sum('fun_app$e'('of_nat$',X1),1),'fun_app$e'('of_nat$',X0)) ),
    inference(theory_normalization,[],[f522]) ).

tff(f522,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => $lesseq('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom520) ).

tff(f7864,plain,
    spl310_348,
    inference(avatar_split_clause,[],[f3983,f7862]) ).

tff(f7862,plain,
    ( spl310_348
  <=> ! [X0: 'Nat$',X1: $int] : ( $sum(X1,'fun_app$b'('iterate_add$'(X0),X1)) = 'fun_app$b'('iterate_add$'('nat$'($sum(1,'fun_app$e'('of_nat$',X0)))),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_348])]) ).

tff(f3983,plain,
    ! [X0: 'Nat$',X1: $int] : ( $sum(X1,'fun_app$b'('iterate_add$'(X0),X1)) = 'fun_app$b'('iterate_add$'('nat$'($sum(1,'fun_app$e'('of_nat$',X0)))),X1) ),
    inference(forward_demodulation,[],[f2453,f864]) ).

tff(f2453,plain,
    ! [X0: 'Nat$',X1: $int] : ( 'fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X0),1))),X1) = $sum(X1,'fun_app$b'('iterate_add$'(X0),X1)) ),
    inference(cnf_transformation,[],[f262]) ).

tff(f262,axiom,
    ! [X0: 'Nat$',X1: $int] : ( 'fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X0),1))),X1) = $sum(X1,'fun_app$b'('iterate_add$'(X0),X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom260) ).

tff(f7860,plain,
    spl310_347,
    inference(avatar_split_clause,[],[f3851,f7858]) ).

tff(f7858,plain,
    ( spl310_347
  <=> ! [X2: $int,X1: 'Int_int_set_fun$',X3: $int] :
        ( 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X2)),'fun_app$l'(X1,X3))
        | ~ $less(sK227(X1),sK226(X1))
        | $less(X3,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_347])]) ).

tff(f3851,plain,
    ! [X2: $int,X3: $int,X1: 'Int_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X1,X2)),'fun_app$l'(X1,X3))
      | ~ $less(sK227(X1),sK226(X1))
      | $less(X3,X2) ),
    inference(equality_resolution,[],[f3560]) ).

tff(f3560,plain,
    ! [X2: $int,X3: $int,X0: 'Int_set$',X1: 'Int_int_set_fun$'] :
      ( 'fun_app$c'('less_eq$a'(X0),'fun_app$l'(X1,X3))
      | ~ $less(sK227(X1),sK226(X1))
      | $less(X3,X2)
      | ( 'fun_app$l'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2244]) ).

tff(f7856,plain,
    spl310_346,
    inference(avatar_split_clause,[],[f3834,f7854]) ).

tff(f7854,plain,
    ( spl310_346
  <=> ! [X2: 'Int_int_set_fun$',X0: $int,X1: $int] :
        ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),'fun_app$l'(X2,X1))
        | ~ $less(sK189(X2),sK188(X2))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_346])]) ).

tff(f3834,plain,
    ! [X2: 'Int_int_set_fun$',X0: $int,X1: $int] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),'fun_app$l'(X2,X1))
      | ~ $less(sK189(X2),sK188(X2))
      | $less(X1,X0) ),
    inference(equality_resolution,[],[f3514]) ).

tff(f3514,plain,
    ! [X2: 'Int_int_set_fun$',X3: 'Int_set$',X0: $int,X1: $int] :
      ( 'fun_app$c'('less_eq$a'('fun_app$l'(X2,X0)),X3)
      | ~ $less(sK189(X2),sK188(X2))
      | ( 'fun_app$l'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2206]) ).

tff(f7852,plain,
    spl310_345,
    inference(avatar_split_clause,[],[f3707,f7850]) ).

tff(f7850,plain,
    ( spl310_345
  <=> ! [X4: 'Int_set$',X0: 'Int_set_set$',X3: 'Int_set_set$',X2: 'Int_set_set$',X1: 'Int_set_set$'] :
        ( 'member$'(X4,'plus$a'(X1,X3))
        | ~ 'member$'(X4,'plus$a'(X0,X2))
        | ~ 'less_eq$c'(X2,X3)
        | ~ 'less_eq$c'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_345])]) ).

tff(f3707,plain,
    ! [X2: 'Int_set_set$',X3: 'Int_set_set$',X0: 'Int_set_set$',X1: 'Int_set_set$',X4: 'Int_set$'] :
      ( 'member$'(X4,'plus$a'(X1,X3))
      | ~ 'member$'(X4,'plus$a'(X0,X2))
      | ~ 'less_eq$c'(X2,X3)
      | ~ 'less_eq$c'(X0,X1) ),
    inference(cnf_transformation,[],[f1584]) ).

tff(f1584,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set$',X4: 'Int_set$'] :
      ( 'member$'(X4,'plus$a'(X1,X3))
      | ~ 'member$'(X4,'plus$a'(X0,X2))
      | ~ 'less_eq$c'(X2,X3)
      | ~ 'less_eq$c'(X0,X1) ),
    inference(flattening,[],[f1583]) ).

tff(f1583,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set$',X4: 'Int_set$'] :
      ( 'member$'(X4,'plus$a'(X1,X3))
      | ~ 'member$'(X4,'plus$a'(X0,X2))
      | ~ 'less_eq$c'(X2,X3)
      | ~ 'less_eq$c'(X0,X1) ),
    inference(ennf_transformation,[],[f135]) ).

tff(f135,axiom,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set$',X4: 'Int_set$'] :
      ( ( 'member$'(X4,'plus$a'(X0,X2))
        & 'less_eq$c'(X2,X3)
        & 'less_eq$c'(X0,X1) )
     => 'member$'(X4,'plus$a'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom133) ).

tff(f7848,plain,
    spl310_344,
    inference(avatar_split_clause,[],[f3706,f7846]) ).

tff(f7846,plain,
    ( spl310_344
  <=> ! [X4: 'N$',X0: 'N_set$',X3: 'N_set$',X2: 'N_set$',X1: 'N_set$'] :
        ( 'member$b'(X4,'plus$c'(X1,X3))
        | ~ 'member$b'(X4,'plus$c'(X0,X2))
        | ~ 'less_eq$b'(X2,X3)
        | ~ 'less_eq$b'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_344])]) ).

tff(f3706,plain,
    ! [X2: 'N_set$',X3: 'N_set$',X0: 'N_set$',X1: 'N_set$',X4: 'N$'] :
      ( 'member$b'(X4,'plus$c'(X1,X3))
      | ~ 'member$b'(X4,'plus$c'(X0,X2))
      | ~ 'less_eq$b'(X2,X3)
      | ~ 'less_eq$b'(X0,X1) ),
    inference(cnf_transformation,[],[f1582]) ).

tff(f1582,plain,
    ! [X0: 'N_set$',X1: 'N_set$',X2: 'N_set$',X3: 'N_set$',X4: 'N$'] :
      ( 'member$b'(X4,'plus$c'(X1,X3))
      | ~ 'member$b'(X4,'plus$c'(X0,X2))
      | ~ 'less_eq$b'(X2,X3)
      | ~ 'less_eq$b'(X0,X1) ),
    inference(flattening,[],[f1581]) ).

tff(f1581,plain,
    ! [X0: 'N_set$',X1: 'N_set$',X2: 'N_set$',X3: 'N_set$',X4: 'N$'] :
      ( 'member$b'(X4,'plus$c'(X1,X3))
      | ~ 'member$b'(X4,'plus$c'(X0,X2))
      | ~ 'less_eq$b'(X2,X3)
      | ~ 'less_eq$b'(X0,X1) ),
    inference(ennf_transformation,[],[f134]) ).

tff(f134,axiom,
    ! [X0: 'N_set$',X1: 'N_set$',X2: 'N_set$',X3: 'N_set$',X4: 'N$'] :
      ( ( 'member$b'(X4,'plus$c'(X0,X2))
        & 'less_eq$b'(X2,X3)
        & 'less_eq$b'(X0,X1) )
     => 'member$b'(X4,'plus$c'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom132) ).

tff(f7844,plain,
    spl310_343,
    inference(avatar_split_clause,[],[f3475,f7842]) ).

tff(f3475,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),0)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP76(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2199]) ).

tff(f7840,plain,
    spl310_342,
    inference(avatar_split_clause,[],[f3470,f7838]) ).

tff(f3470,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP75(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less('fun_app$e'('of_nat$',X2),0) ),
    inference(cnf_transformation,[],[f2195]) ).

tff(f7836,plain,
    spl310_341,
    inference(avatar_split_clause,[],[f3129,f7834]) ).

tff(f7834,plain,
    ( spl310_341
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,sK120(X0))
        | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',sK121(X0,X2)))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_341])]) ).

tff(f3129,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,sK120(X0))
      | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',sK121(X0,X2)))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f1999]) ).

tff(f7832,plain,
    spl310_340,
    inference(avatar_split_clause,[],[f3102,f7830]) ).

tff(f3102,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP35(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1995]) ).

tff(f7828,plain,
    spl310_339,
    inference(avatar_split_clause,[],[f3099,f7826]) ).

tff(f3099,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP35(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1995]) ).

tff(f7824,plain,
    spl310_338,
    inference(avatar_split_clause,[],[f3093,f7822]) ).

tff(f3093,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP33(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1990]) ).

tff(f7820,plain,
    spl310_337,
    inference(avatar_split_clause,[],[f3090,f7818]) ).

tff(f3090,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP33(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1990]) ).

tff(f7816,plain,
    spl310_336,
    inference(avatar_split_clause,[],[f3084,f7814]) ).

tff(f7814,plain,
    ( spl310_336
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP31(X0,X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( 0 != 'fun_app$e'('of_nat$',X2) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_336])]) ).

tff(f3084,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP31(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ( 0 != 'fun_app$e'('of_nat$',X2) ) ),
    inference(cnf_transformation,[],[f1985]) ).

tff(f7812,plain,
    spl310_335,
    inference(avatar_split_clause,[],[f3081,f7810]) ).

tff(f3081,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = 'fun_app$e'('of_nat$',X2) )
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP31(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1985]) ).

tff(f7808,plain,
    spl310_334,
    inference(avatar_split_clause,[],[f2960,f7806]) ).

tff(f2960,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP13(X0,X2,X1)
      | $less(0,'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1605]) ).

tff(f7804,plain,
    spl310_333,
    inference(avatar_split_clause,[],[f2924,f7802]) ).

tff(f2924,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP5(X2,X1,X0)
      | $less(0,'fun_app$e'('of_nat$',X2))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f1596]) ).

tff(f7732,plain,
    ( ~ spl310_9
    | ~ spl310_58
    | ~ spl310_245 ),
    inference(avatar_split_clause,[],[f7238,f6498,f4490,f4265]) ).

tff(f6498,plain,
    ( spl310_245
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( ~ $less('fun_app$e'('of_nat$','to_nat$'(X1)),'fun_app$e'('of_nat$','to_nat$'(X0)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_245])]) ).

tff(f7238,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less_eq$','i$'),'x$')
    | ~ spl310_58
    | ~ spl310_245 ),
    inference(resolution,[],[f4492,f6499]) ).

tff(f6499,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( ~ $less('fun_app$e'('of_nat$','to_nat$'(X1)),'fun_app$e'('of_nat$','to_nat$'(X0)))
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
    | ~ spl310_245 ),
    inference(avatar_component_clause,[],[f6498]) ).

tff(f4492,plain,
    ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
    | ~ spl310_58 ),
    inference(avatar_component_clause,[],[f4490]) ).

tff(f7566,plain,
    ( ~ spl310_332
    | ~ spl310_58
    | ~ spl310_141 ),
    inference(avatar_split_clause,[],[f5256,f4971,f4490,f7563]) ).

tff(f7563,plain,
    ( spl310_332
  <=> sP56('to_nat$'('i$'),'to_nat$'('x$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_332])]) ).

tff(f4971,plain,
    ( spl310_141
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP56(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_141])]) ).

tff(f5256,plain,
    ( ~ sP56('to_nat$'('i$'),'to_nat$'('x$'))
    | ~ spl310_58
    | ~ spl310_141 ),
    inference(resolution,[],[f4492,f4972]) ).

tff(f4972,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP56(X0,X1) )
    | ~ spl310_141 ),
    inference(avatar_component_clause,[],[f4971]) ).

tff(f7528,plain,
    ( spl310_331
    | ~ spl310_32
    | ~ spl310_308 ),
    inference(avatar_split_clause,[],[f7191,f7144,f4366,f7525]) ).

tff(f7191,plain,
    ( 'fun_app$d'('fun_app$g'('less$','i$'),sK94('fun_app$g'('less$','i$')))
    | ~ spl310_32
    | ~ spl310_308 ),
    inference(resolution,[],[f7146,f4367]) ).

tff(f7427,plain,
    ( spl310_330
    | ~ spl310_58
    | ~ spl310_143 ),
    inference(avatar_split_clause,[],[f5255,f4982,f4490,f7424]) ).

tff(f7424,plain,
    ( spl310_330
  <=> sP52('to_nat$'('i$'),'to_nat$'('x$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_330])]) ).

tff(f4982,plain,
    ( spl310_143
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP52(X0,X1)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_143])]) ).

tff(f5255,plain,
    ( sP52('to_nat$'('i$'),'to_nat$'('x$'))
    | ~ spl310_58
    | ~ spl310_143 ),
    inference(resolution,[],[f4492,f4983]) ).

tff(f4983,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | sP52(X0,X1) )
    | ~ spl310_143 ),
    inference(avatar_component_clause,[],[f4982]) ).

tff(f7237,plain,
    ( spl310_329
    | ~ spl310_117
    | ~ spl310_268 ),
    inference(avatar_split_clause,[],[f7010,f6601,f4838,f7234]) ).

tff(f7232,plain,
    spl310_328,
    inference(avatar_split_clause,[],[f4059,f7230]) ).

tff(f7230,plain,
    ( spl310_328
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'(X0),'fun_app$f'('minus$'(X1),X2)) = 'fun_app$f'('minus$'(X0),'fun_app$f'('minus$'(X2),X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_328])]) ).

tff(f4059,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'(X0),'fun_app$f'('minus$'(X1),X2)) = 'fun_app$f'('minus$'(X0),'fun_app$f'('minus$'(X2),X1)) ),
    inference(forward_demodulation,[],[f2903,f2901]) ).

tff(f2901,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'(X0),'fun_app$f'('minus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X2)),X1) ),
    inference(cnf_transformation,[],[f86]) ).

tff(f86,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] : ( 'fun_app$f'('minus$'(X0),'fun_app$f'('minus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X2)),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom84) ).

tff(f2903,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'(X0),'fun_app$f'('minus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X1)),X2) ),
    inference(cnf_transformation,[],[f88]) ).

tff(f88,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] : ( 'fun_app$f'('plus$'(X0),'fun_app$f'('minus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X1)),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom86) ).

tff(f7228,plain,
    spl310_327,
    inference(avatar_split_clause,[],[f4053,f7226]) ).

tff(f7226,plain,
    ( spl310_327
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),$sum(1,'fun_app$e'('of_nat$',X1)))
        | sP3(X0,X1)
        | ~ 'fun_app$j'(X0,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_327])]) ).

tff(f4053,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X2),$sum(1,'fun_app$e'('of_nat$',X1)))
      | sP3(X0,X1)
      | ~ 'fun_app$j'(X0,X2) ),
    inference(forward_demodulation,[],[f2878,f864]) ).

tff(f2878,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( sP3(X0,X1)
      | ~ 'fun_app$j'(X0,X2)
      | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),1)) ),
    inference(cnf_transformation,[],[f1925]) ).

tff(f1925,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ( sP3(X0,X1)
        | ! [X2: 'Nat$'] :
            ( ~ 'fun_app$j'(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),1)) ) )
      & ( ( 'fun_app$j'(X0,sK118(X0,X1))
          & $less('fun_app$e'('of_nat$',sK118(X0,X1)),$sum('fun_app$e'('of_nat$',X1),1)) )
        | ~ sP3(X0,X1) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK118])],[f1923,f1924]) ).

tff(f1924,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X3: 'Nat$'] :
          ( 'fun_app$j'(X0,X3)
          & $less('fun_app$e'('of_nat$',X3),$sum('fun_app$e'('of_nat$',X1),1)) )
     => ( 'fun_app$j'(X0,sK118(X0,X1))
        & $less('fun_app$e'('of_nat$',sK118(X0,X1)),$sum('fun_app$e'('of_nat$',X1),1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1923,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ( sP3(X0,X1)
        | ! [X2: 'Nat$'] :
            ( ~ 'fun_app$j'(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),1)) ) )
      & ( ? [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            & $less('fun_app$e'('of_nat$',X3),$sum('fun_app$e'('of_nat$',X1),1)) )
        | ~ sP3(X0,X1) ) ),
    inference(rectify,[],[f1922]) ).

tff(f1922,plain,
    ! [X1: 'Nat_bool_fun$',X0: 'Nat$'] :
      ( ( sP3(X1,X0)
        | ! [X2: 'Nat$'] :
            ( ~ 'fun_app$j'(X1,X2)
            | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) ) )
      & ( ? [X2: 'Nat$'] :
            ( 'fun_app$j'(X1,X2)
            & $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) )
        | ~ sP3(X1,X0) ) ),
    inference(nnf_transformation,[],[f1592]) ).

tff(f1592,plain,
    ! [X1: 'Nat_bool_fun$',X0: 'Nat$'] :
      ( sP3(X1,X0)
    <=> ? [X2: 'Nat$'] :
          ( 'fun_app$j'(X1,X2)
          & $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP3])]) ).

tff(f7224,plain,
    spl310_326,
    inference(avatar_split_clause,[],[f4052,f7222]) ).

tff(f7222,plain,
    ( spl310_326
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X3: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X3),$sum(1,'fun_app$e'('of_nat$',X1)))
        | 'fun_app$j'(X0,X3)
        | ~ sP2(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_326])]) ).

tff(f4052,plain,
    ! [X3: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X3),$sum(1,'fun_app$e'('of_nat$',X1)))
      | 'fun_app$j'(X0,X3)
      | ~ sP2(X0,X1) ),
    inference(forward_demodulation,[],[f2869,f864]) ).

tff(f2869,plain,
    ! [X3: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X3)
      | ~ $less('fun_app$e'('of_nat$',X3),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ sP2(X0,X1) ),
    inference(cnf_transformation,[],[f1916]) ).

tff(f1916,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ( sP2(X0,X1)
        | ( ~ 'fun_app$j'(X0,sK116(X0,X1))
          & $less('fun_app$e'('of_nat$',sK116(X0,X1)),$sum('fun_app$e'('of_nat$',X1),1)) ) )
      & ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),$sum('fun_app$e'('of_nat$',X1),1)) )
        | ~ sP2(X0,X1) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK116])],[f1914,f1915]) ).

tff(f1915,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),1)) )
     => ( ~ 'fun_app$j'(X0,sK116(X0,X1))
        & $less('fun_app$e'('of_nat$',sK116(X0,X1)),$sum('fun_app$e'('of_nat$',X1),1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1914,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ( sP2(X0,X1)
        | ? [X2: 'Nat$'] :
            ( ~ 'fun_app$j'(X0,X2)
            & $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X1),1)) ) )
      & ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),$sum('fun_app$e'('of_nat$',X1),1)) )
        | ~ sP2(X0,X1) ) ),
    inference(rectify,[],[f1913]) ).

tff(f1913,plain,
    ! [X1: 'Nat_bool_fun$',X0: 'Nat$'] :
      ( ( sP2(X1,X0)
        | ? [X2: 'Nat$'] :
            ( ~ 'fun_app$j'(X1,X2)
            & $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) ) )
      & ( ! [X2: 'Nat$'] :
            ( 'fun_app$j'(X1,X2)
            | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) )
        | ~ sP2(X1,X0) ) ),
    inference(nnf_transformation,[],[f1590]) ).

tff(f1590,plain,
    ! [X1: 'Nat_bool_fun$',X0: 'Nat$'] :
      ( sP2(X1,X0)
    <=> ! [X2: 'Nat$'] :
          ( 'fun_app$j'(X1,X2)
          | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) ) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP2])]) ).

tff(f7220,plain,
    spl310_325,
    inference(avatar_split_clause,[],[f3969,f7218]) ).

tff(f7218,plain,
    ( spl310_325
  <=> ! [X0: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X0) = 1 )
        | ~ $less('fun_app$e'('of_nat$',X0),1)
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_325])]) ).

tff(f3969,plain,
    ! [X0: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = 1 )
      | ~ $less('fun_app$e'('of_nat$',X0),1)
      | ~ $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(evaluation,[],[f2415]) ).

tff(f2415,plain,
    ! [X0: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = $sum(0,1) )
      | ~ $less('fun_app$e'('of_nat$',X0),$sum(0,1))
      | ~ $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f987]) ).

tff(f987,plain,
    ! [X0: 'Nat$'] :
      ( ( ( ( 'fun_app$e'('of_nat$',X0) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus($sum(0,1))),1) )
          | $less('fun_app$e'('of_nat$',X0),$sum(0,1)) )
        & ( ( 'fun_app$e'('of_nat$',X0) = $sum(0,1) )
          | ~ $less('fun_app$e'('of_nat$',X0),$sum(0,1)) ) )
      | ~ $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f640]) ).

tff(f640,plain,
    ! [X0: 'Nat$'] :
      ( $less(0,'fun_app$e'('of_nat$',X0))
     => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum(0,1))
         => ( 'fun_app$e'('of_nat$',X0) = $sum($sum('fun_app$e'('of_nat$',X0),$uminus($sum(0,1))),1) ) )
        & ( $less('fun_app$e'('of_nat$',X0),$sum(0,1))
         => ( 'fun_app$e'('of_nat$',X0) = $sum(0,1) ) ) ) ),
    inference(theory_normalization,[],[f592]) ).

tff(f592,axiom,
    ! [X0: 'Nat$'] :
      ( $less(0,'fun_app$e'('of_nat$',X0))
     => ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum(0,1))
         => ( 'fun_app$e'('of_nat$',X0) = $sum($difference('fun_app$e'('of_nat$',X0),$sum(0,1)),1) ) )
        & ( $less('fun_app$e'('of_nat$',X0),$sum(0,1))
         => ( 'fun_app$e'('of_nat$',X0) = $sum(0,1) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom590) ).

tff(f7216,plain,
    spl310_324,
    inference(avatar_split_clause,[],[f3871,f7214]) ).

tff(f7214,plain,
    ( spl310_324
  <=> ! [X2: $int,X1: 'Int_int_fun$',X3: $int] :
        ( $less('fun_app$b'(X1,X2),'fun_app$b'(X1,X3))
        | $less(sK246(X1),sK247(X1))
        | ~ $less(X2,X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_324])]) ).

tff(f3871,plain,
    ! [X2: $int,X3: $int,X1: 'Int_int_fun$'] :
      ( $less('fun_app$b'(X1,X2),'fun_app$b'(X1,X3))
      | $less(sK246(X1),sK247(X1))
      | ~ $less(X2,X3) ),
    inference(equality_resolution,[],[f3580]) ).

tff(f3580,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( $less(X0,'fun_app$b'(X1,X3))
      | $less(sK246(X1),sK247(X1))
      | ~ $less(X2,X3)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2264]) ).

tff(f7212,plain,
    spl310_323,
    inference(avatar_split_clause,[],[f3869,f7210]) ).

tff(f7210,plain,
    ( spl310_323
  <=> ! [X2: $int,X1: 'Int_int_fun$',X3: $int] :
        ( ~ $less('fun_app$b'(X1,X3),'fun_app$b'(X1,X2))
        | ~ $less(sK245(X1),sK244(X1))
        | $less(X3,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_323])]) ).

tff(f3869,plain,
    ! [X2: $int,X3: $int,X1: 'Int_int_fun$'] :
      ( ~ $less('fun_app$b'(X1,X3),'fun_app$b'(X1,X2))
      | ~ $less(sK245(X1),sK244(X1))
      | $less(X3,X2) ),
    inference(equality_resolution,[],[f3578]) ).

tff(f3578,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: 'Int_int_fun$'] :
      ( ~ $less('fun_app$b'(X1,X3),X0)
      | ~ $less(sK245(X1),sK244(X1))
      | $less(X3,X2)
      | ( 'fun_app$b'(X1,X2) != X0 ) ),
    inference(cnf_transformation,[],[f2262]) ).

tff(f7208,plain,
    spl310_322,
    inference(avatar_split_clause,[],[f3843,f7206]) ).

tff(f7206,plain,
    ( spl310_322
  <=> ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
        ( $less('fun_app$b'(X2,X0),'fun_app$b'(X2,X1))
        | $less(sK204(X2),sK205(X2))
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_322])]) ).

tff(f3843,plain,
    ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),'fun_app$b'(X2,X1))
      | $less(sK204(X2),sK205(X2))
      | ~ $less(X0,X1) ),
    inference(equality_resolution,[],[f3536]) ).

tff(f3536,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( $less('fun_app$b'(X2,X0),X3)
      | $less(sK204(X2),sK205(X2))
      | ( 'fun_app$b'(X2,X1) != X3 )
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f2222]) ).

tff(f7204,plain,
    spl310_321,
    inference(avatar_split_clause,[],[f3838,f7202]) ).

tff(f7202,plain,
    ( spl310_321
  <=> ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
        ( ~ $less('fun_app$b'(X2,X1),'fun_app$b'(X2,X0))
        | ~ $less(sK193(X2),sK192(X2))
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_321])]) ).

tff(f3838,plain,
    ! [X2: 'Int_int_fun$',X0: $int,X1: $int] :
      ( ~ $less('fun_app$b'(X2,X1),'fun_app$b'(X2,X0))
      | ~ $less(sK193(X2),sK192(X2))
      | $less(X1,X0) ),
    inference(equality_resolution,[],[f3518]) ).

tff(f3518,plain,
    ! [X2: 'Int_int_fun$',X3: $int,X0: $int,X1: $int] :
      ( ~ $less(X3,'fun_app$b'(X2,X0))
      | ~ $less(sK193(X2),sK192(X2))
      | ( 'fun_app$b'(X2,X1) != X3 )
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f2210]) ).

tff(f7200,plain,
    spl310_320,
    inference(avatar_split_clause,[],[f3531,f7198]) ).

tff(f7198,plain,
    ( spl310_320
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( $less($sum(X0,$uminus(X3)),$sum(X1,$uminus(X2)))
        | ~ $less(X2,X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_320])]) ).

tff(f3531,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( $less($sum(X0,$uminus(X3)),$sum(X1,$uminus(X2)))
      | ~ $less(X2,X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f1448]) ).

tff(f1448,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,$uminus(X3)),$sum(X1,$uminus(X2)))
      | ~ $less(X2,X3)
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1447]) ).

tff(f1447,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,$uminus(X3)),$sum(X1,$uminus(X2)))
      | ~ $less(X2,X3)
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f833]) ).

tff(f833,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $less(X2,X3)
        & $less(X0,X1) )
     => $less($sum(X0,$uminus(X3)),$sum(X1,$uminus(X2))) ),
    inference(theory_normalization,[],[f514]) ).

tff(f514,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $less(X2,X3)
        & $less(X0,X1) )
     => $less($difference(X0,X3),$difference(X1,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom512) ).

tff(f7196,plain,
    spl310_319,
    inference(avatar_split_clause,[],[f3509,f7194]) ).

tff(f7194,plain,
    ( spl310_319
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( ~ $less($sum(X1,$uminus(X2)),$sum(X0,$uminus(X3)))
        | $less(X3,X2)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_319])]) ).

tff(f3509,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( ~ $less($sum(X1,$uminus(X2)),$sum(X0,$uminus(X3)))
      | $less(X3,X2)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f1418]) ).

tff(f1418,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ~ $less($sum(X1,$uminus(X2)),$sum(X0,$uminus(X3)))
      | $less(X3,X2)
      | $less(X1,X0) ),
    inference(flattening,[],[f1417]) ).

tff(f1417,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ~ $less($sum(X1,$uminus(X2)),$sum(X0,$uminus(X3)))
      | $less(X3,X2)
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f819]) ).

tff(f819,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ~ $less(X3,X2)
        & ~ $less(X1,X0) )
     => ~ $less($sum(X1,$uminus(X2)),$sum(X0,$uminus(X3))) ),
    inference(theory_normalization,[],[f38]) ).

tff(f38,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $lesseq(X2,X3)
        & $lesseq(X0,X1) )
     => $lesseq($difference(X0,X3),$difference(X1,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom36) ).

tff(f7187,plain,
    spl310_318,
    inference(avatar_split_clause,[],[f3496,f7185]) ).

tff(f7185,plain,
    ( spl310_318
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( $less(X0,X1)
        | ~ $less(X2,X3)
        | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_318])]) ).

tff(f3496,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( $less(X0,X1)
      | ~ $less(X2,X3)
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(cnf_transformation,[],[f2203]) ).

tff(f2203,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ( $less(X0,X1)
          | ~ $less(X2,X3) )
        & ( $less(X2,X3)
          | ~ $less(X0,X1) ) )
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(nnf_transformation,[],[f1395]) ).

tff(f1395,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $less(X0,X1)
      <=> $less(X2,X3) )
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(ennf_transformation,[],[f816]) ).

tff(f816,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $sum(X0,$uminus(X1)) = $sum(X2,$uminus(X3)) )
     => ( $less(X0,X1)
      <=> $less(X2,X3) ) ),
    inference(theory_normalization,[],[f513]) ).

tff(f513,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $difference(X0,X1) = $difference(X2,X3) )
     => ( $less(X0,X1)
      <=> $less(X2,X3) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom511) ).

tff(f7183,plain,
    spl310_317,
    inference(avatar_split_clause,[],[f3494,f7181]) ).

tff(f7181,plain,
    ( spl310_317
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | $less(X3,X2)
        | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_317])]) ).

tff(f3494,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( ~ $less(X1,X0)
      | $less(X3,X2)
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(cnf_transformation,[],[f2202]) ).

tff(f2202,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ( ~ $less(X1,X0)
          | $less(X3,X2) )
        & ( ~ $less(X3,X2)
          | $less(X1,X0) ) )
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(nnf_transformation,[],[f1394]) ).

tff(f1394,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ~ $less(X1,X0)
      <=> ~ $less(X3,X2) )
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(ennf_transformation,[],[f815]) ).

tff(f815,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $sum(X0,$uminus(X1)) = $sum(X2,$uminus(X3)) )
     => ( ~ $less(X1,X0)
      <=> ~ $less(X3,X2) ) ),
    inference(theory_normalization,[],[f35]) ).

tff(f35,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $difference(X0,X1) = $difference(X2,X3) )
     => ( $lesseq(X0,X1)
      <=> $lesseq(X2,X3) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom33) ).

tff(f7179,plain,
    spl310_316,
    inference(avatar_split_clause,[],[f3482,f7177]) ).

tff(f7177,plain,
    ( spl310_316
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP76(X2,X1,X0)
        | ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_316])]) ).

tff(f3482,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP76(X2,X1,X0)
      | ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f2200]) ).

tff(f2200,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP76(X2,X1,X0)
        | ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) )
      & ( $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1))
        | ~ sP76(X2,X1,X0) ) ),
    inference(nnf_transformation,[],[f1688]) ).

tff(f1688,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP76(X2,X1,X0)
    <=> $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ),
    inference(definition_folding,[],[f1385,f1687]) ).

tff(f1385,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))))
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X0),0)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
    <=> $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ),
    inference(ennf_transformation,[],[f811]) ).

tff(f811,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => $less('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2)))) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => $less('fun_app$e'('of_nat$',X0),0) ) )
    <=> $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ),
    inference(theory_normalization,[],[f565]) ).

tff(f565,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => $less('fun_app$e'('of_nat$',X0),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
         => $less('fun_app$e'('of_nat$',X0),0) ) )
    <=> $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom563) ).

tff(f7175,plain,
    spl310_315,
    inference(avatar_split_clause,[],[f3481,f7173]) ).

tff(f7173,plain,
    ( spl310_315
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1))
        | ~ sP76(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_315])]) ).

tff(f3481,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1))
      | ~ sP76(X2,X1,X0) ),
    inference(cnf_transformation,[],[f2200]) ).

tff(f7171,plain,
    spl310_314,
    inference(avatar_split_clause,[],[f3474,f7169]) ).

tff(f7169,plain,
    ( spl310_314
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP75(X1,X0,X2)
        | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_314])]) ).

tff(f3474,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP75(X1,X0,X2)
      | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f2196]) ).

tff(f2196,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP75(X1,X0,X2)
        | $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) )
      & ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
        | ~ sP75(X1,X0,X2) ) ),
    inference(nnf_transformation,[],[f1686]) ).

tff(f1686,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP75(X1,X0,X2)
    <=> ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1384,f1685]) ).

tff(f1384,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1))))
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ~ $less('fun_app$e'('of_nat$',X2),0)
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
    <=> ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f810]) ).

tff(f810,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
         => ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),$uminus('fun_app$e'('of_nat$',X1)))) )
        & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
         => ~ $less('fun_app$e'('of_nat$',X2),0) ) )
    <=> ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ),
    inference(theory_normalization,[],[f156]) ).

tff(f156,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
         => $lesseq($difference('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
         => $lesseq(0,'fun_app$e'('of_nat$',X2)) ) )
    <=> $lesseq('fun_app$e'('of_nat$',X0),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom154) ).

tff(f7167,plain,
    spl310_313,
    inference(avatar_split_clause,[],[f3473,f7165]) ).

tff(f7165,plain,
    ( spl310_313
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
        | ~ sP75(X1,X0,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_313])]) ).

tff(f3473,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0))
      | ~ sP75(X1,X0,X2) ),
    inference(cnf_transformation,[],[f2196]) ).

tff(f7163,plain,
    spl310_312,
    inference(avatar_split_clause,[],[f3384,f7161]) ).

tff(f7161,plain,
    ( spl310_312
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( 'fun_app$j'(X1,X2)
        | sP74(X1,X0)
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK153(X0,X1))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_312])]) ).

tff(f3384,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X1,X2)
      | sP74(X1,X0)
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK153(X0,X1))) ),
    inference(cnf_transformation,[],[f2134]) ).

tff(f2134,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( 'fun_app$j'(X1,X2)
      | sP74(X1,X0)
      | ( ~ 'fun_app$j'(X1,sK153(X0,X1))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK153(X0,X1))) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK153])],[f2132,f2133]) ).

tff(f2133,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X3)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
     => ( ~ 'fun_app$j'(X1,sK153(X0,X1))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',sK153(X0,X1))) ) ),
    introduced(choice_axiom,[]) ).

tff(f2132,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( 'fun_app$j'(X1,X2)
      | sP74(X1,X0)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X3)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) ) ),
    inference(rectify,[],[f1684]) ).

tff(f1684,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( 'fun_app$j'(X1,X2)
      | sP74(X1,X0)
      | ? [X5: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X5)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X5)) ) ),
    inference(definition_folding,[],[f1365,f1683]) ).

tff(f1683,plain,
    ! [X1: 'Nat_bool_fun$',X0: 'Nat$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X3)
          & ! [X4: 'Nat$'] :
              ( 'fun_app$j'(X1,X4)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
      | ~ sP74(X1,X0) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP74])]) ).

tff(f1365,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( 'fun_app$j'(X1,X2)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X3)
          & ! [X4: 'Nat$'] :
              ( 'fun_app$j'(X1,X4)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
      | ? [X5: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X5)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X5)) ) ),
    inference(flattening,[],[f1364]) ).

tff(f1364,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( 'fun_app$j'(X1,X2)
      | ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X3)
          & ! [X4: 'Nat$'] :
              ( 'fun_app$j'(X1,X4)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
      | ? [X5: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X5)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X5)) ) ),
    inference(ennf_transformation,[],[f975]) ).

tff(f975,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( ! [X4: 'Nat$'] :
                  ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
                 => 'fun_app$j'(X1,X4) )
              & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
           => 'fun_app$j'(X1,X3) )
        & ! [X5: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X5))
           => 'fun_app$j'(X1,X5) ) )
     => 'fun_app$j'(X1,X2) ),
    inference(rectify,[],[f797]) ).

tff(f797,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( ! [X4: 'Nat$'] :
                  ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
                 => 'fun_app$j'(X1,X4) )
              & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
           => 'fun_app$j'(X1,X3) )
        & ! [X3: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3))
           => 'fun_app$j'(X1,X3) ) )
     => 'fun_app$j'(X1,X2) ),
    inference(theory_normalization,[],[f571]) ).

tff(f571,axiom,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X2: 'Nat$'] :
      ( ( ! [X3: 'Nat$'] :
            ( ( ! [X4: 'Nat$'] :
                  ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4))
                 => 'fun_app$j'(X1,X4) )
              & $lesseq('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
           => 'fun_app$j'(X1,X3) )
        & ! [X3: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3))
           => 'fun_app$j'(X1,X3) ) )
     => 'fun_app$j'(X1,X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom569) ).

tff(f7159,plain,
    spl310_311,
    inference(avatar_split_clause,[],[f3382,f7157]) ).

tff(f7157,plain,
    ( spl310_311
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X3: 'Nat$'] :
        ( 'fun_app$j'(X0,X3)
        | ~ $less('fun_app$e'('of_nat$',sK152(X0,X1)),'fun_app$e'('of_nat$',X3))
        | ~ sP74(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_311])]) ).

tff(f3382,plain,
    ! [X3: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X3)
      | ~ $less('fun_app$e'('of_nat$',sK152(X0,X1)),'fun_app$e'('of_nat$',X3))
      | ~ sP74(X0,X1) ),
    inference(cnf_transformation,[],[f2131]) ).

tff(f2131,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ( ~ 'fun_app$j'(X0,sK152(X0,X1))
        & ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',sK152(X0,X1)),'fun_app$e'('of_nat$',X3)) )
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',sK152(X0,X1))) )
      | ~ sP74(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK152])],[f2129,f2130]) ).

tff(f2130,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3)) )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
     => ( ~ 'fun_app$j'(X0,sK152(X0,X1))
        & ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',sK152(X0,X1)),'fun_app$e'('of_nat$',X3)) )
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',sK152(X0,X1))) ) ),
    introduced(choice_axiom,[]) ).

tff(f2129,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3)) )
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | ~ sP74(X0,X1) ),
    inference(rectify,[],[f2128]) ).

tff(f2128,plain,
    ! [X1: 'Nat_bool_fun$',X0: 'Nat$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X3)
          & ! [X4: 'Nat$'] :
              ( 'fun_app$j'(X1,X4)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X3)) )
      | ~ sP74(X1,X0) ),
    inference(nnf_transformation,[],[f1683]) ).

tff(f7155,plain,
    spl310_310,
    inference(avatar_split_clause,[],[f3290,f7153]) ).

tff(f7153,plain,
    ( spl310_310
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP58(X0,X1,X2)
        | ~ sP57(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_310])]) ).

tff(f3290,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP58(X0,X1,X2)
      | ~ sP57(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ),
    inference(cnf_transformation,[],[f2061]) ).

tff(f2061,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP58(X0,X1,X2)
        | ( ~ sP57(X2,X1,X0)
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        | ( ~ sP56(X2,X1)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( sP57(X2,X1,X0)
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
          & ( sP56(X2,X1)
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP58(X0,X1,X2) ) ),
    inference(rectify,[],[f2060]) ).

tff(f2060,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( sP58(X1,X2,X0)
        | ( ~ sP57(X0,X2,X1)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP56(X0,X2)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP57(X0,X2,X1)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP56(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP58(X1,X2,X0) ) ),
    inference(flattening,[],[f2059]) ).

tff(f2059,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( sP58(X1,X2,X0)
        | ( ~ sP57(X0,X2,X1)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP56(X0,X2)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP57(X0,X2,X1)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP56(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP58(X1,X2,X0) ) ),
    inference(nnf_transformation,[],[f1661]) ).

tff(f7151,plain,
    spl310_309,
    inference(avatar_split_clause,[],[f3287,f7149]) ).

tff(f7149,plain,
    ( spl310_309
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP57(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP58(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_309])]) ).

tff(f3287,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP57(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP58(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2061]) ).

tff(f7147,plain,
    ( spl310_308
    | ~ spl310_123
    | spl310_287 ),
    inference(avatar_split_clause,[],[f7102,f7056,f4896,f7144]) ).

tff(f4896,plain,
    ( spl310_123
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X1),X0)
        | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_123])]) ).

tff(f7056,plain,
    ( spl310_287
  <=> 'fun_app$d'('fun_app$g'('less_eq$','x$'),'i$') ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_287])]) ).

tff(f7102,plain,
    ( 'fun_app$d'('fun_app$g'('less$','i$'),'x$')
    | ~ spl310_123
    | spl310_287 ),
    inference(resolution,[],[f7058,f4897]) ).

tff(f4897,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X1),X0)
        | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
    | ~ spl310_123 ),
    inference(avatar_component_clause,[],[f4896]) ).

tff(f7058,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less_eq$','x$'),'i$')
    | spl310_287 ),
    inference(avatar_component_clause,[],[f7056]) ).

tff(f7142,plain,
    spl310_307,
    inference(avatar_split_clause,[],[f3285,f7140]) ).

tff(f7140,plain,
    ( spl310_307
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP58(X1,X0,X2)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP59(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_307])]) ).

tff(f3285,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP58(X1,X0,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP59(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2058]) ).

tff(f2058,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP58(X1,X0,X2)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
          | ~ sP58(X1,X0,X2) ) )
      | ~ sP59(X0,X1,X2) ),
    inference(rectify,[],[f2057]) ).

tff(f2057,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( sP58(X1,X2,X0)
          | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
          | ~ sP58(X1,X2,X0) ) )
      | ~ sP59(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1662]) ).

tff(f7138,plain,
    spl310_306,
    inference(avatar_split_clause,[],[f3284,f7136]) ).

tff(f7136,plain,
    ( spl310_306
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP58(X1,X0,X2)
        | ~ sP59(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_306])]) ).

tff(f3284,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP58(X1,X0,X2)
      | ~ sP59(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2058]) ).

tff(f7134,plain,
    spl310_305,
    inference(avatar_split_clause,[],[f3269,f7132]) ).

tff(f7132,plain,
    ( spl310_305
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP54(X0,X1,X2)
        | ~ sP53(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_305])]) ).

tff(f3269,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP54(X0,X1,X2)
      | ~ sP53(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(cnf_transformation,[],[f2051]) ).

tff(f2051,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP54(X0,X1,X2)
        | ( ~ sP53(X2,X1,X0)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( ~ sP52(X2,X0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( sP53(X2,X1,X0)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( sP52(X2,X0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP54(X0,X1,X2) ) ),
    inference(rectify,[],[f2050]) ).

tff(f2050,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP54(X2,X1,X0)
        | ( ~ sP53(X0,X1,X2)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP52(X0,X2)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP53(X0,X1,X2)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP52(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP54(X2,X1,X0) ) ),
    inference(flattening,[],[f2049]) ).

tff(f2049,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP54(X2,X1,X0)
        | ( ~ sP53(X0,X1,X2)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP52(X0,X2)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP53(X0,X1,X2)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP52(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP54(X2,X1,X0) ) ),
    inference(nnf_transformation,[],[f1656]) ).

tff(f7130,plain,
    spl310_304,
    inference(avatar_split_clause,[],[f3266,f7128]) ).

tff(f7128,plain,
    ( spl310_304
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP53(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP54(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_304])]) ).

tff(f3266,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP53(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP54(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2051]) ).

tff(f7126,plain,
    spl310_303,
    inference(avatar_split_clause,[],[f3264,f7124]) ).

tff(f7124,plain,
    ( spl310_303
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP54(X1,X0,X2)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP55(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_303])]) ).

tff(f3264,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP54(X1,X0,X2)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP55(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2048]) ).

tff(f2048,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP54(X1,X0,X2)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
          | ~ sP54(X1,X0,X2) ) )
      | ~ sP55(X0,X1,X2) ),
    inference(rectify,[],[f2047]) ).

tff(f2047,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( ( sP54(X2,X1,X0)
          | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
        & ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
          | ~ sP54(X2,X1,X0) ) )
      | ~ sP55(X1,X2,X0) ),
    inference(nnf_transformation,[],[f1657]) ).

tff(f7122,plain,
    spl310_302,
    inference(avatar_split_clause,[],[f3263,f7120]) ).

tff(f7120,plain,
    ( spl310_302
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP54(X1,X0,X2)
        | ~ sP55(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_302])]) ).

tff(f3263,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP54(X1,X0,X2)
      | ~ sP55(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2048]) ).

tff(f7118,plain,
    spl310_301,
    inference(avatar_split_clause,[],[f3248,f7116]) ).

tff(f7116,plain,
    ( spl310_301
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP50(X0,X1,X2)
        | ~ sP49(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_301])]) ).

tff(f3248,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP50(X0,X1,X2)
      | ~ sP49(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ),
    inference(cnf_transformation,[],[f2041]) ).

tff(f2041,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP50(X0,X1,X2)
        | ( ~ sP49(X2,X1,X0)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( ~ sP48(X0,X1)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( sP49(X2,X1,X0)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( sP48(X0,X1)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP50(X0,X1,X2) ) ),
    inference(rectify,[],[f2040]) ).

tff(f2040,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP50(X2,X1,X0)
        | ( ~ sP49(X0,X1,X2)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP48(X2,X1)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP49(X0,X1,X2)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP48(X2,X1)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP50(X2,X1,X0) ) ),
    inference(flattening,[],[f2039]) ).

tff(f2039,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( sP50(X2,X1,X0)
        | ( ~ sP49(X0,X1,X2)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP48(X2,X1)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP49(X0,X1,X2)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP48(X2,X1)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP50(X2,X1,X0) ) ),
    inference(nnf_transformation,[],[f1651]) ).

tff(f7114,plain,
    spl310_300,
    inference(avatar_split_clause,[],[f3245,f7112]) ).

tff(f7112,plain,
    ( spl310_300
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP49(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
        | ~ sP50(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_300])]) ).

tff(f3245,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP49(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP50(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2041]) ).

tff(f7110,plain,
    spl310_299,
    inference(avatar_split_clause,[],[f3243,f7108]) ).

tff(f7108,plain,
    ( spl310_299
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP50(X0,X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP51(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_299])]) ).

tff(f3243,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP50(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP51(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2038]) ).

tff(f2038,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP50(X0,X2,X1)
          | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        & ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
          | ~ sP50(X0,X2,X1) ) )
      | ~ sP51(X0,X1,X2) ),
    inference(rectify,[],[f2037]) ).

tff(f2037,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( sP50(X2,X1,X0)
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
          | ~ sP50(X2,X1,X0) ) )
      | ~ sP51(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1652]) ).

tff(f7106,plain,
    spl310_298,
    inference(avatar_split_clause,[],[f3242,f7104]) ).

tff(f7104,plain,
    ( spl310_298
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ~ sP50(X0,X2,X1)
        | ~ sP51(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_298])]) ).

tff(f3242,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP50(X0,X2,X1)
      | ~ sP51(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2038]) ).

tff(f7099,plain,
    spl310_297,
    inference(avatar_split_clause,[],[f3227,f7097]) ).

tff(f7097,plain,
    ( spl310_297
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP46(X0,X1,X2)
        | ~ sP45(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_297])]) ).

tff(f3227,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP46(X0,X1,X2)
      | ~ sP45(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ),
    inference(cnf_transformation,[],[f2030]) ).

tff(f2030,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP46(X0,X1,X2)
        | ( ~ sP45(X2,X1,X0)
          & ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        | ( ~ sP44(X2,X1)
          & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( sP45(X2,X1,X0)
            | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
          & ( sP44(X2,X1)
            | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP46(X0,X1,X2) ) ),
    inference(rectify,[],[f2029]) ).

tff(f2029,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( sP46(X1,X2,X0)
        | ( ~ sP45(X0,X2,X1)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP44(X0,X2)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP45(X0,X2,X1)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP44(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP46(X1,X2,X0) ) ),
    inference(flattening,[],[f2028]) ).

tff(f2028,plain,
    ! [X1: 'Nat$',X2: 'Nat$',X0: 'Nat$'] :
      ( ( sP46(X1,X2,X0)
        | ( ~ sP45(X0,X2,X1)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( ~ sP44(X0,X2)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( sP45(X0,X2,X1)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( sP44(X0,X2)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP46(X1,X2,X0) ) ),
    inference(nnf_transformation,[],[f1646]) ).

tff(f7095,plain,
    spl310_296,
    inference(avatar_split_clause,[],[f3224,f7093]) ).

tff(f7093,plain,
    ( spl310_296
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP45(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP46(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_296])]) ).

tff(f3224,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP45(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP46(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2030]) ).

tff(f7091,plain,
    spl310_295,
    inference(avatar_split_clause,[],[f3222,f7089]) ).

tff(f3222,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP46(X1,X0,X2)
      | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) )
      | ~ sP47(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2027]) ).

tff(f2027,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( sP46(X1,X0,X2)
          | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) )
        & ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
          | ~ sP46(X1,X0,X2) ) )
      | ~ sP47(X0,X1,X2) ),
    inference(rectify,[],[f2026]) ).

tff(f2026,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ( sP46(X1,X2,X0)
          | ( 'fun_app$e'('of_nat$',X1) != 'fun_app$e'('of_nat$',X2) ) )
        & ( ( 'fun_app$e'('of_nat$',X1) = 'fun_app$e'('of_nat$',X2) )
          | ~ sP46(X1,X2,X0) ) )
      | ~ sP47(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1647]) ).

tff(f7087,plain,
    spl310_294,
    inference(avatar_split_clause,[],[f3221,f7085]) ).

tff(f7085,plain,
    ( spl310_294
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
        | ~ sP46(X1,X0,X2)
        | ~ sP47(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_294])]) ).

tff(f3221,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
      | ~ sP46(X1,X0,X2)
      | ~ sP47(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2027]) ).

tff(f7083,plain,
    spl310_293,
    inference(avatar_split_clause,[],[f3165,f7081]) ).

tff(f7081,plain,
    ( spl310_293
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_293])]) ).

tff(f3165,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f1290]) ).

tff(f1290,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1289]) ).

tff(f1289,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f432]) ).

tff(f432,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less$',X2),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom430) ).

tff(f7079,plain,
    spl310_292,
    inference(avatar_split_clause,[],[f3164,f7077]) ).

tff(f7077,plain,
    ( spl310_292
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X0),X2)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_292])]) ).

tff(f3164,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f1288]) ).

tff(f1288,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1287]) ).

tff(f1287,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f481]) ).

tff(f481,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less$',X0),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom479) ).

tff(f7075,plain,
    spl310_291,
    inference(avatar_split_clause,[],[f3160,f7073]) ).

tff(f7073,plain,
    ( spl310_291
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_291])]) ).

tff(f3160,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f1280]) ).

tff(f1280,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(flattening,[],[f1279]) ).

tff(f1279,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f222]) ).

tff(f222,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom220) ).

tff(f7071,plain,
    spl310_290,
    inference(avatar_split_clause,[],[f3158,f7069]) ).

tff(f7069,plain,
    ( spl310_290
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
        | ~ 'fun_app$d'('fun_app$g'('less$',X2),X0)
        | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_290])]) ).

tff(f3158,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f1276]) ).

tff(f1276,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(flattening,[],[f1275]) ).

tff(f1275,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
      | ~ 'fun_app$d'('fun_app$g'('less$',X2),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f358]) ).

tff(f358,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less$',X2),X0)
        & 'fun_app$d'('fun_app$g'('less$',X0),X1) )
     => 'fun_app$d'('fun_app$g'('less$',X2),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom356) ).

tff(f7067,plain,
    spl310_289,
    inference(avatar_split_clause,[],[f3151,f7065]) ).

tff(f7065,plain,
    ( spl310_289
  <=> ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2))
        | 'fun_app$c'('member$a'(X0),X2)
        | ~ 'fun_app$c'('member$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_289])]) ).

tff(f3151,plain,
    ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2))
      | 'fun_app$c'('member$a'(X0),X2)
      | ~ 'fun_app$c'('member$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1262]) ).

tff(f1262,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2))
      | 'fun_app$c'('member$a'(X0),X2)
      | ~ 'fun_app$c'('member$a'(X0),X1) ),
    inference(flattening,[],[f1261]) ).

tff(f1261,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2))
      | 'fun_app$c'('member$a'(X0),X2)
      | ~ 'fun_app$c'('member$a'(X0),X1) ),
    inference(ennf_transformation,[],[f585]) ).

tff(f585,axiom,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( ( ~ 'fun_app$c'('member$a'(X0),X2)
        & 'fun_app$c'('member$a'(X0),X1) )
     => 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom583) ).

tff(f7063,plain,
    spl310_288,
    inference(avatar_split_clause,[],[f2958,f7061]) ).

tff(f7061,plain,
    ( spl310_288
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP7(X1,X2,X0)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | sP8(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_288])]) ).

tff(f2958,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP7(X1,X2,X0)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP8(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1605]) ).

tff(f7059,plain,
    ( ~ spl310_287
    | ~ spl310_117
    | ~ spl310_245 ),
    inference(avatar_split_clause,[],[f6995,f6498,f4838,f7056]) ).

tff(f6995,plain,
    ( ~ 'fun_app$d'('fun_app$g'('less_eq$','x$'),'i$')
    | ~ spl310_117
    | ~ spl310_245 ),
    inference(resolution,[],[f4840,f6499]) ).

tff(f7054,plain,
    spl310_286,
    inference(avatar_split_clause,[],[f2956,f7052]) ).

tff(f7052,plain,
    ( spl310_286
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP6(X1,X2,X0)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
        | sP8(X2,X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_286])]) ).

tff(f2956,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP6(X1,X2,X0)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
      | sP8(X2,X1,X0) ),
    inference(cnf_transformation,[],[f1605]) ).

tff(f7050,plain,
    spl310_285,
    inference(avatar_split_clause,[],[f2929,f7048]) ).

tff(f7048,plain,
    ( spl310_285
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP12(X2,X1,X0)
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP13(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_285])]) ).

tff(f2929,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP12(X2,X1,X0)
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP13(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1936]) ).

tff(f1936,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( sP12(X2,X1,X0)
            & sP11(X1,X2,X0) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) )
        & ( ( sP10(X2,X1,X0)
            & sP9(X1,X2,X0) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2)) ) )
      | ~ sP13(X0,X1,X2) ),
    inference(rectify,[],[f1935]) ).

tff(f1935,plain,
    ! [X0: 'Nat$',X2: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( sP12(X1,X2,X0)
            & sP11(X2,X1,X0) )
          | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        & ( ( sP10(X1,X2,X0)
            & sP9(X2,X1,X0) )
          | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) )
      | ~ sP13(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1604]) ).

tff(f7046,plain,
    spl310_284,
    inference(avatar_split_clause,[],[f2928,f7044]) ).

tff(f7044,plain,
    ( spl310_284
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP11(X1,X2,X0)
        | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP13(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_284])]) ).

tff(f2928,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP11(X1,X2,X0)
      | $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP13(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1936]) ).

tff(f7042,plain,
    spl310_283,
    inference(avatar_split_clause,[],[f2927,f7040]) ).

tff(f7040,plain,
    ( spl310_283
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP10(X2,X1,X0)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP13(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_283])]) ).

tff(f2927,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP10(X2,X1,X0)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP13(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1936]) ).

tff(f7038,plain,
    spl310_282,
    inference(avatar_split_clause,[],[f2926,f7036]) ).

tff(f7036,plain,
    ( spl310_282
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP9(X1,X2,X0)
        | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
        | ~ sP13(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_282])]) ).

tff(f2926,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP9(X1,X2,X0)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP13(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1936]) ).

tff(f7034,plain,
    spl310_281,
    inference(avatar_split_clause,[],[f2904,f7032]) ).

tff(f7032,plain,
    ( spl310_281
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'('fun_app$f'('plus$'(X0),X1)),X2) = 'fun_app$f'('plus$'(X0),'fun_app$f'('plus$'(X1),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_281])]) ).

tff(f2904,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'('fun_app$f'('plus$'(X0),X1)),X2) = 'fun_app$f'('plus$'(X0),'fun_app$f'('plus$'(X1),X2)) ),
    inference(cnf_transformation,[],[f65]) ).

tff(f65,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] : ( 'fun_app$f'('plus$'('fun_app$f'('plus$'(X0),X1)),X2) = 'fun_app$f'('plus$'(X0),'fun_app$f'('plus$'(X1),X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom63) ).

tff(f7030,plain,
    spl310_280,
    inference(avatar_split_clause,[],[f2902,f7028]) ).

tff(f7028,plain,
    ( spl310_280
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'(X0),'fun_app$f'('plus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('minus$'(X0),X2)),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_280])]) ).

tff(f2902,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'(X0),'fun_app$f'('plus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('minus$'(X0),X2)),X1) ),
    inference(cnf_transformation,[],[f83]) ).

tff(f83,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] : ( 'fun_app$f'('minus$'(X0),'fun_app$f'('plus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('minus$'(X0),X2)),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom81) ).

tff(f7026,plain,
    spl310_279,
    inference(avatar_split_clause,[],[f2901,f7024]) ).

tff(f7024,plain,
    ( spl310_279
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'(X0),'fun_app$f'('minus$'(X1),X2)) = 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X2)),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_279])]) ).

tff(f7022,plain,
    spl310_278,
    inference(avatar_split_clause,[],[f2899,f7020]) ).

tff(f7020,plain,
    ( spl310_278
  <=> ! [X2: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'fun_app$h'('plus$b'(X1),X2)) = 'fun_app$h'('plus$b'('fun_app$h'('plus$b'(X0),X1)),X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_278])]) ).

tff(f2899,plain,
    ! [X2: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'fun_app$h'('plus$b'(X1),X2)) = 'fun_app$h'('plus$b'('fun_app$h'('plus$b'(X0),X1)),X2) ),
    inference(cnf_transformation,[],[f64]) ).

tff(f64,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'fun_app$h'('plus$b'(X1),X2)) = 'fun_app$h'('plus$b'('fun_app$h'('plus$b'(X0),X1)),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom62) ).

tff(f7018,plain,
    spl310_277,
    inference(avatar_split_clause,[],[f2898,f7016]) ).

tff(f7016,plain,
    ( spl310_277
  <=> ! [X2: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'fun_app$h'('plus$b'(X1),X2)) = 'fun_app$h'('plus$b'(X1),'fun_app$h'('plus$b'(X0),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_277])]) ).

tff(f2898,plain,
    ! [X2: 'Int_set$',X0: 'Int_set$',X1: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'fun_app$h'('plus$b'(X1),X2)) = 'fun_app$h'('plus$b'(X1),'fun_app$h'('plus$b'(X0),X2)) ),
    inference(cnf_transformation,[],[f46]) ).

tff(f46,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'fun_app$h'('plus$b'(X1),X2)) = 'fun_app$h'('plus$b'(X1),'fun_app$h'('plus$b'(X0),X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom44) ).

tff(f6994,plain,
    spl310_276,
    inference(avatar_split_clause,[],[f2879,f6992]) ).

tff(f6992,plain,
    ( spl310_276
  <=> ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( $less('fun_app$e'('of_nat$',sK119(X0,X1)),'fun_app$e'('of_nat$',X0))
        | 'fun_app$j'(X1,X0)
        | ~ sP3(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_276])]) ).

tff(f2879,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( $less('fun_app$e'('of_nat$',sK119(X0,X1)),'fun_app$e'('of_nat$',X0))
      | 'fun_app$j'(X1,X0)
      | ~ sP3(X1,X0) ),
    inference(cnf_transformation,[],[f1930]) ).

tff(f1930,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP3(X1,X0)
        | ( ! [X2: 'Nat$'] :
              ( ~ 'fun_app$j'(X1,X2)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ~ 'fun_app$j'(X1,X0) ) )
      & ( ( 'fun_app$j'(X1,sK119(X0,X1))
          & $less('fun_app$e'('of_nat$',sK119(X0,X1)),'fun_app$e'('of_nat$',X0)) )
        | 'fun_app$j'(X1,X0)
        | ~ sP3(X1,X0) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK119])],[f1928,f1929]) ).

tff(f1929,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ? [X3: 'Nat$'] :
          ( 'fun_app$j'(X1,X3)
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
     => ( 'fun_app$j'(X1,sK119(X0,X1))
        & $less('fun_app$e'('of_nat$',sK119(X0,X1)),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1928,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP3(X1,X0)
        | ( ! [X2: 'Nat$'] :
              ( ~ 'fun_app$j'(X1,X2)
              | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ~ 'fun_app$j'(X1,X0) ) )
      & ( ? [X3: 'Nat$'] :
            ( 'fun_app$j'(X1,X3)
            & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        | 'fun_app$j'(X1,X0)
        | ~ sP3(X1,X0) ) ),
    inference(rectify,[],[f1927]) ).

tff(f1927,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP3(X1,X0)
        | ( ! [X3: 'Nat$'] :
              ( ~ 'fun_app$j'(X1,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
          & ~ 'fun_app$j'(X1,X0) ) )
      & ( ? [X3: 'Nat$'] :
            ( 'fun_app$j'(X1,X3)
            & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        | 'fun_app$j'(X1,X0)
        | ~ sP3(X1,X0) ) ),
    inference(flattening,[],[f1926]) ).

tff(f1926,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP3(X1,X0)
        | ( ! [X3: 'Nat$'] :
              ( ~ 'fun_app$j'(X1,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
          & ~ 'fun_app$j'(X1,X0) ) )
      & ( ? [X3: 'Nat$'] :
            ( 'fun_app$j'(X1,X3)
            & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        | 'fun_app$j'(X1,X0)
        | ~ sP3(X1,X0) ) ),
    inference(nnf_transformation,[],[f1593]) ).

tff(f1593,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( sP3(X1,X0)
    <=> ( ? [X3: 'Nat$'] :
            ( 'fun_app$j'(X1,X3)
            & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        | 'fun_app$j'(X1,X0) ) ),
    inference(definition_folding,[],[f958,f1592]) ).

tff(f958,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ? [X2: 'Nat$'] :
          ( 'fun_app$j'(X1,X2)
          & $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) )
    <=> ( ? [X3: 'Nat$'] :
            ( 'fun_app$j'(X1,X3)
            & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        | 'fun_app$j'(X1,X0) ) ),
    inference(rectify,[],[f396]) ).

tff(f396,axiom,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ? [X2: 'Nat$'] :
          ( 'fun_app$j'(X1,X2)
          & $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) )
    <=> ( ? [X2: 'Nat$'] :
            ( 'fun_app$j'(X1,X2)
            & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | 'fun_app$j'(X1,X0) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom394) ).

tff(f6990,plain,
    spl310_275,
    inference(avatar_split_clause,[],[f2874,f6988]) ).

tff(f6988,plain,
    ( spl310_275
  <=> ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( sP2(X1,X0)
        | $less('fun_app$e'('of_nat$',sK117(X0,X1)),'fun_app$e'('of_nat$',X0))
        | ~ 'fun_app$j'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_275])]) ).

tff(f2874,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( sP2(X1,X0)
      | $less('fun_app$e'('of_nat$',sK117(X0,X1)),'fun_app$e'('of_nat$',X0))
      | ~ 'fun_app$j'(X1,X0) ),
    inference(cnf_transformation,[],[f1921]) ).

tff(f1921,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP2(X1,X0)
        | ( ~ 'fun_app$j'(X1,sK117(X0,X1))
          & $less('fun_app$e'('of_nat$',sK117(X0,X1)),'fun_app$e'('of_nat$',X0)) )
        | ~ 'fun_app$j'(X1,X0) )
      & ( ( ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X1,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
          & 'fun_app$j'(X1,X0) )
        | ~ sP2(X1,X0) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK117])],[f1919,f1920]) ).

tff(f1920,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X1,X2)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
     => ( ~ 'fun_app$j'(X1,sK117(X0,X1))
        & $less('fun_app$e'('of_nat$',sK117(X0,X1)),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1919,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP2(X1,X0)
        | ? [X2: 'Nat$'] :
            ( ~ 'fun_app$j'(X1,X2)
            & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ~ 'fun_app$j'(X1,X0) )
      & ( ( ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X1,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
          & 'fun_app$j'(X1,X0) )
        | ~ sP2(X1,X0) ) ),
    inference(rectify,[],[f1918]) ).

tff(f1918,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP2(X1,X0)
        | ? [X3: 'Nat$'] :
            ( ~ 'fun_app$j'(X1,X3)
            & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        | ~ 'fun_app$j'(X1,X0) )
      & ( ( ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X1,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
          & 'fun_app$j'(X1,X0) )
        | ~ sP2(X1,X0) ) ),
    inference(flattening,[],[f1917]) ).

tff(f1917,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ( sP2(X1,X0)
        | ? [X3: 'Nat$'] :
            ( ~ 'fun_app$j'(X1,X3)
            & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        | ~ 'fun_app$j'(X1,X0) )
      & ( ( ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X1,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
          & 'fun_app$j'(X1,X0) )
        | ~ sP2(X1,X0) ) ),
    inference(nnf_transformation,[],[f1591]) ).

tff(f1591,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( sP2(X1,X0)
    <=> ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X1,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        & 'fun_app$j'(X1,X0) ) ),
    inference(definition_folding,[],[f1155,f1590]) ).

tff(f1155,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ! [X2: 'Nat$'] :
          ( 'fun_app$j'(X1,X2)
          | ~ $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1)) )
    <=> ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X1,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0)) )
        & 'fun_app$j'(X1,X0) ) ),
    inference(ennf_transformation,[],[f957]) ).

tff(f957,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ! [X2: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1))
         => 'fun_app$j'(X1,X2) )
    <=> ( ! [X3: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0))
           => 'fun_app$j'(X1,X3) )
        & 'fun_app$j'(X1,X0) ) ),
    inference(rectify,[],[f391]) ).

tff(f391,axiom,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( ! [X2: 'Nat$'] :
          ( $less('fun_app$e'('of_nat$',X2),$sum('fun_app$e'('of_nat$',X0),1))
         => 'fun_app$j'(X1,X2) )
    <=> ( ! [X2: 'Nat$'] :
            ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
           => 'fun_app$j'(X1,X2) )
        & 'fun_app$j'(X1,X0) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom389) ).

tff(f6986,plain,
    spl310_274,
    inference(avatar_split_clause,[],[f2659,f6984]) ).

tff(f6984,plain,
    ( spl310_274
  <=> ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
        ( ( 'collect$b'(X1) = 'collect$b'(X0) )
        | ~ 'fun_app$'(X1,sK110(X0,X1))
        | ~ 'fun_app$'(X0,sK110(X0,X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_274])]) ).

tff(f2659,plain,
    ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
      ( ( 'collect$b'(X1) = 'collect$b'(X0) )
      | ~ 'fun_app$'(X1,sK110(X0,X1))
      | ~ 'fun_app$'(X0,sK110(X0,X1)) ),
    inference(cnf_transformation,[],[f1771]) ).

tff(f1771,plain,
    ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
      ( ( 'collect$b'(X1) = 'collect$b'(X0) )
      | ( ( ~ 'fun_app$'(X1,sK110(X0,X1))
          | ~ 'fun_app$'(X0,sK110(X0,X1)) )
        & ( 'fun_app$'(X1,sK110(X0,X1))
          | 'fun_app$'(X0,sK110(X0,X1)) ) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK110])],[f1769,f1770]) ).

tff(f1770,plain,
    ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
      ( ? [X2: $int] :
          ( ( ~ 'fun_app$'(X1,X2)
            | ~ 'fun_app$'(X0,X2) )
          & ( 'fun_app$'(X1,X2)
            | 'fun_app$'(X0,X2) ) )
     => ( ( ~ 'fun_app$'(X1,sK110(X0,X1))
          | ~ 'fun_app$'(X0,sK110(X0,X1)) )
        & ( 'fun_app$'(X1,sK110(X0,X1))
          | 'fun_app$'(X0,sK110(X0,X1)) ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1769,plain,
    ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
      ( ( 'collect$b'(X1) = 'collect$b'(X0) )
      | ? [X2: $int] :
          ( ( ~ 'fun_app$'(X1,X2)
            | ~ 'fun_app$'(X0,X2) )
          & ( 'fun_app$'(X1,X2)
            | 'fun_app$'(X0,X2) ) ) ),
    inference(nnf_transformation,[],[f1152]) ).

tff(f1152,plain,
    ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
      ( ( 'collect$b'(X1) = 'collect$b'(X0) )
      | ? [X2: $int] :
          ( 'fun_app$'(X0,X2)
        <~> 'fun_app$'(X1,X2) ) ),
    inference(ennf_transformation,[],[f54]) ).

tff(f54,axiom,
    ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
      ( ! [X2: $int] :
          ( 'fun_app$'(X0,X2)
        <=> 'fun_app$'(X1,X2) )
     => ( 'collect$b'(X1) = 'collect$b'(X0) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom52) ).

tff(f6982,plain,
    spl310_273,
    inference(avatar_split_clause,[],[f2658,f6980]) ).

tff(f6980,plain,
    ( spl310_273
  <=> ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
        ( ( 'collect$b'(X1) = 'collect$b'(X0) )
        | 'fun_app$'(X1,sK110(X0,X1))
        | 'fun_app$'(X0,sK110(X0,X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_273])]) ).

tff(f2658,plain,
    ! [X0: 'Int_bool_fun$',X1: 'Int_bool_fun$'] :
      ( ( 'collect$b'(X1) = 'collect$b'(X0) )
      | 'fun_app$'(X1,sK110(X0,X1))
      | 'fun_app$'(X0,sK110(X0,X1)) ),
    inference(cnf_transformation,[],[f1771]) ).

tff(f6978,plain,
    spl310_272,
    inference(avatar_split_clause,[],[f2580,f6976]) ).

tff(f2580,plain,
    ! [X0: 'N$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$i'('from_nat$',X1)) ),
    inference(cnf_transformation,[],[f1069]) ).

tff(f1069,plain,
    ! [X0: 'N$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$',X1))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$i'('from_nat$',X1)) ),
    inference(ennf_transformation,[],[f408]) ).

tff(f408,axiom,
    ! [X0: 'N$',X1: 'Nat$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),'fun_app$i'('from_nat$',X1))
     => $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$',X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom406) ).

tff(f6615,plain,
    spl310_271,
    inference(avatar_split_clause,[],[f4057,f6613]) ).

tff(f6613,plain,
    ( spl310_271
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$uminus($sum(X1,$uminus(X2)))) = $sum(X0,$sum(X2,$uminus(X1))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_271])]) ).

tff(f6611,plain,
    spl310_270,
    inference(avatar_split_clause,[],[f4054,f6609]) ).

tff(f6609,plain,
    ( spl310_270
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',sK118(X0,X1)),$sum(1,'fun_app$e'('of_nat$',X1)))
        | ~ sP3(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_270])]) ).

tff(f4054,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',sK118(X0,X1)),$sum(1,'fun_app$e'('of_nat$',X1)))
      | ~ sP3(X0,X1) ),
    inference(forward_demodulation,[],[f2876,f864]) ).

tff(f2876,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',sK118(X0,X1)),$sum('fun_app$e'('of_nat$',X1),1))
      | ~ sP3(X0,X1) ),
    inference(cnf_transformation,[],[f1925]) ).

tff(f6607,plain,
    spl310_269,
    inference(avatar_split_clause,[],[f4051,f6605]) ).

tff(f6605,plain,
    ( spl310_269
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',sK116(X0,X1)),$sum(1,'fun_app$e'('of_nat$',X1)))
        | sP2(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_269])]) ).

tff(f4051,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',sK116(X0,X1)),$sum(1,'fun_app$e'('of_nat$',X1)))
      | sP2(X0,X1) ),
    inference(forward_demodulation,[],[f2870,f864]) ).

tff(f2870,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( sP2(X0,X1)
      | $less('fun_app$e'('of_nat$',sK116(X0,X1)),$sum('fun_app$e'('of_nat$',X1),1)) ),
    inference(cnf_transformation,[],[f1916]) ).

tff(f6603,plain,
    spl310_268,
    inference(avatar_split_clause,[],[f3988,f6601]) ).

tff(f3988,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(forward_demodulation,[],[f2460,f864]) ).

tff(f2460,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,$sum('fun_app$e'('of_nat$',X0),1))
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    inference(cnf_transformation,[],[f991]) ).

tff(f6599,plain,
    spl310_267,
    inference(avatar_split_clause,[],[f3507,f6597]) ).

tff(f6597,plain,
    ( spl310_267
  <=> ! [X0: 'N$',X3: 'N_set$',X2: 'N$',X1: 'N_set$'] :
        ( 'member$b'('fun_app$f'('plus$'(X0),X2),'plus$c'(X1,X3))
        | ~ 'member$b'(X2,X3)
        | ~ 'member$b'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_267])]) ).

tff(f3507,plain,
    ! [X2: 'N$',X3: 'N_set$',X0: 'N$',X1: 'N_set$'] :
      ( 'member$b'('fun_app$f'('plus$'(X0),X2),'plus$c'(X1,X3))
      | ~ 'member$b'(X2,X3)
      | ~ 'member$b'(X0,X1) ),
    inference(cnf_transformation,[],[f1414]) ).

tff(f1414,plain,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N$',X3: 'N_set$'] :
      ( 'member$b'('fun_app$f'('plus$'(X0),X2),'plus$c'(X1,X3))
      | ~ 'member$b'(X2,X3)
      | ~ 'member$b'(X0,X1) ),
    inference(flattening,[],[f1413]) ).

tff(f1413,plain,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N$',X3: 'N_set$'] :
      ( 'member$b'('fun_app$f'('plus$'(X0),X2),'plus$c'(X1,X3))
      | ~ 'member$b'(X2,X3)
      | ~ 'member$b'(X0,X1) ),
    inference(ennf_transformation,[],[f106]) ).

tff(f106,axiom,
    ! [X0: 'N$',X1: 'N_set$',X2: 'N$',X3: 'N_set$'] :
      ( ( 'member$b'(X2,X3)
        & 'member$b'(X0,X1) )
     => 'member$b'('fun_app$f'('plus$'(X0),X2),'plus$c'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom104) ).

tff(f6595,plain,
    spl310_266,
    inference(avatar_split_clause,[],[f3506,f6593]) ).

tff(f6593,plain,
    ( spl310_266
  <=> ! [X0: 'Int_set$',X3: 'Int_set_set$',X2: 'Int_set$',X1: 'Int_set_set$'] :
        ( 'member$'('fun_app$h'('plus$b'(X0),X2),'plus$a'(X1,X3))
        | ~ 'member$'(X2,X3)
        | ~ 'member$'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_266])]) ).

tff(f3506,plain,
    ! [X2: 'Int_set$',X3: 'Int_set_set$',X0: 'Int_set$',X1: 'Int_set_set$'] :
      ( 'member$'('fun_app$h'('plus$b'(X0),X2),'plus$a'(X1,X3))
      | ~ 'member$'(X2,X3)
      | ~ 'member$'(X0,X1) ),
    inference(cnf_transformation,[],[f1412]) ).

tff(f1412,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set$',X3: 'Int_set_set$'] :
      ( 'member$'('fun_app$h'('plus$b'(X0),X2),'plus$a'(X1,X3))
      | ~ 'member$'(X2,X3)
      | ~ 'member$'(X0,X1) ),
    inference(flattening,[],[f1411]) ).

tff(f1411,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set$',X3: 'Int_set_set$'] :
      ( 'member$'('fun_app$h'('plus$b'(X0),X2),'plus$a'(X1,X3))
      | ~ 'member$'(X2,X3)
      | ~ 'member$'(X0,X1) ),
    inference(ennf_transformation,[],[f105]) ).

tff(f105,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set_set$',X2: 'Int_set$',X3: 'Int_set_set$'] :
      ( ( 'member$'(X2,X3)
        & 'member$'(X0,X1) )
     => 'member$'('fun_app$h'('plus$b'(X0),X2),'plus$a'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom103) ).

tff(f6591,plain,
    ( ~ spl310_58
    | spl310_265 ),
    inference(avatar_split_clause,[],[f2367,f6588,f4490]) ).

tff(f2367,plain,
    ( ( 0 = 'fun_app$e'('of_nat$','ja$') )
    | ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    inference(cnf_transformation,[],[f983]) ).

tff(f6577,plain,
    ( spl310_264
    | ~ spl310_117
    | ~ spl310_216 ),
    inference(avatar_split_clause,[],[f6089,f5934,f4838,f6575]) ).

tff(f6575,plain,
    ( spl310_264
  <=> ! [X0: 'Nat_bool_fun$'] :
        ( ~ 'fun_app$j'(X0,'to_nat$'('i$'))
        | sP3(X0,'to_nat$'('x$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_264])]) ).

tff(f5934,plain,
    ( spl310_216
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( sP3(X1,X0)
        | ~ 'fun_app$j'(X1,X2)
        | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_216])]) ).

tff(f6089,plain,
    ( ! [X0: 'Nat_bool_fun$'] :
        ( ~ 'fun_app$j'(X0,'to_nat$'('i$'))
        | sP3(X0,'to_nat$'('x$')) )
    | ~ spl310_117
    | ~ spl310_216 ),
    inference(resolution,[],[f5935,f4840]) ).

tff(f5935,plain,
    ( ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))
        | ~ 'fun_app$j'(X1,X2)
        | sP3(X1,X0) )
    | ~ spl310_216 ),
    inference(avatar_component_clause,[],[f5934]) ).

tff(f6573,plain,
    spl310_263,
    inference(avatar_split_clause,[],[f3333,f6571]) ).

tff(f6571,plain,
    ( spl310_263
  <=> ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'(sK142(X0,X1,X2)),X2)
        | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_263])]) ).

tff(f3333,plain,
    ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(sK142(X0,X1,X2)),X2)
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ),
    inference(cnf_transformation,[],[f2087]) ).

tff(f6569,plain,
    spl310_262,
    inference(avatar_split_clause,[],[f3332,f6567]) ).

tff(f6567,plain,
    ( spl310_262
  <=> ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'(sK141(X0,X1,X2)),X1)
        | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_262])]) ).

tff(f3332,plain,
    ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(sK141(X0,X1,X2)),X1)
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('plus$b'(X1),X2)) ),
    inference(cnf_transformation,[],[f2087]) ).

tff(f6565,plain,
    spl310_261,
    inference(avatar_split_clause,[],[f3289,f6563]) ).

tff(f3289,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP58(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP56(X2,X1) ),
    inference(cnf_transformation,[],[f2061]) ).

tff(f6561,plain,
    spl310_260,
    inference(avatar_split_clause,[],[f3286,f6559]) ).

tff(f3286,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP56(X2,X1)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP58(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2061]) ).

tff(f6557,plain,
    spl310_259,
    inference(avatar_split_clause,[],[f3268,f6555]) ).

tff(f3268,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP54(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP52(X2,X0) ),
    inference(cnf_transformation,[],[f2051]) ).

tff(f6553,plain,
    spl310_258,
    inference(avatar_split_clause,[],[f3265,f6551]) ).

tff(f3265,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP52(X2,X0)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP54(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2051]) ).

tff(f6549,plain,
    spl310_257,
    inference(avatar_split_clause,[],[f3247,f6547]) ).

tff(f3247,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP50(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP48(X0,X1) ),
    inference(cnf_transformation,[],[f2041]) ).

tff(f6545,plain,
    spl310_256,
    inference(avatar_split_clause,[],[f3244,f6543]) ).

tff(f3244,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP48(X0,X1)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2))
      | ~ sP50(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2041]) ).

tff(f6541,plain,
    spl310_255,
    inference(avatar_split_clause,[],[f3226,f6539]) ).

tff(f3226,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP46(X0,X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP44(X2,X1) ),
    inference(cnf_transformation,[],[f2030]) ).

tff(f6537,plain,
    spl310_254,
    inference(avatar_split_clause,[],[f3223,f6535]) ).

tff(f3223,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP44(X2,X1)
      | ~ $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X2))
      | ~ sP46(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2030]) ).

tff(f6532,plain,
    spl310_253,
    inference(avatar_split_clause,[],[f2888,f6530]) ).

tff(f6530,plain,
    ( spl310_253
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$uminus($sum(X1,X2))) = $sum($sum(X0,$uminus(X2)),$uminus(X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_253])]) ).

tff(f2888,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$uminus($sum(X1,X2))) = $sum($sum(X0,$uminus(X2)),$uminus(X1)) ),
    inference(cnf_transformation,[],[f878]) ).

tff(f878,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum(X0,$uminus($sum(X1,X2))) = $sum($sum(X0,$uminus(X2)),$uminus(X1)) ),
    inference(theory_normalization,[],[f724]) ).

tff(f724,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum(X0,$uminus($sum(X1,X2))) = $sum($difference(X0,X2),$uminus(X1)) ),
    inference(theory_normalization,[],[f84]) ).

tff(f84,axiom,
    ! [X0: $int,X1: $int,X2: $int] : ( $difference($difference(X0,X2),X1) = $difference(X0,$sum(X1,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom82) ).

tff(f6528,plain,
    spl310_252,
    inference(avatar_split_clause,[],[f2887,f6526]) ).

tff(f6526,plain,
    ( spl310_252
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$uminus($sum(X1,X2))) = $sum($sum(X0,$uminus(X1)),$uminus(X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_252])]) ).

tff(f2887,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$uminus($sum(X1,X2))) = $sum($sum(X0,$uminus(X1)),$uminus(X2)) ),
    inference(cnf_transformation,[],[f877]) ).

tff(f877,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum(X0,$uminus($sum(X1,X2))) = $sum($sum(X0,$uminus(X1)),$uminus(X2)) ),
    inference(theory_normalization,[],[f723]) ).

tff(f723,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum($difference(X0,X1),$uminus(X2)) = $sum(X0,$uminus($sum(X1,X2))) ),
    inference(theory_normalization,[],[f81]) ).

tff(f81,axiom,
    ! [X0: $int,X1: $int,X2: $int] : ( $difference($difference(X0,X1),X2) = $difference(X0,$sum(X1,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom79) ).

tff(f6524,plain,
    spl310_251,
    inference(avatar_split_clause,[],[f2844,f6522]) ).

tff(f6522,plain,
    ( spl310_251
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = 'fun_app$e'('of_nat$',X1) )
        | ( 0 != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_251])]) ).

tff(f2844,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = 'fun_app$e'('of_nat$',X1) )
      | ( 0 != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    inference(cnf_transformation,[],[f1897]) ).

tff(f1897,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( 0 = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( 0 != 'fun_app$e'('of_nat$',X1) )
        | ( 0 != 'fun_app$e'('of_nat$',X0) ) )
      & ( ( ( 0 = 'fun_app$e'('of_nat$',X1) )
          & ( 0 = 'fun_app$e'('of_nat$',X0) ) )
        | ( 0 != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(flattening,[],[f1896]) ).

tff(f1896,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( 0 = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
        | ( 0 != 'fun_app$e'('of_nat$',X1) )
        | ( 0 != 'fun_app$e'('of_nat$',X0) ) )
      & ( ( ( 0 = 'fun_app$e'('of_nat$',X1) )
          & ( 0 = 'fun_app$e'('of_nat$',X0) ) )
        | ( 0 != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    inference(nnf_transformation,[],[f614]) ).

tff(f614,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
    <=> ( ( 0 = 'fun_app$e'('of_nat$',X1) )
        & ( 0 = 'fun_app$e'('of_nat$',X0) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom612) ).

tff(f6520,plain,
    spl310_250,
    inference(avatar_split_clause,[],[f2843,f6518]) ).

tff(f6518,plain,
    ( spl310_250
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ( 0 = 'fun_app$e'('of_nat$',X0) )
        | ( 0 != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_250])]) ).

tff(f2843,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = 'fun_app$e'('of_nat$',X0) )
      | ( 0 != $sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ) ),
    inference(cnf_transformation,[],[f1897]) ).

tff(f6516,plain,
    spl310_249,
    inference(avatar_split_clause,[],[f2654,f6514]) ).

tff(f6514,plain,
    ( spl310_249
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X3: 'Nat$'] :
        ( 'fun_app$j'(X0,X1)
        | 'fun_app$j'(X0,X3)
        | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK108(X0))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_249])]) ).

tff(f2654,plain,
    ! [X3: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | 'fun_app$j'(X0,X3)
      | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK108(X0))) ),
    inference(cnf_transformation,[],[f1766]) ).

tff(f1766,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ( ~ 'fun_app$j'(X0,sK108(X0))
        & ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK108(X0))) ) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK108])],[f1150,f1765]) ).

tff(f1765,plain,
    ! [X0: 'Nat_bool_fun$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) )
     => ( ~ 'fun_app$j'(X0,sK108(X0))
        & ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK108(X0))) ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1150,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) ) ),
    inference(ennf_transformation,[],[f381]) ).

tff(f381,axiom,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ! [X2: 'Nat$'] :
          ( ! [X3: 'Nat$'] :
              ( $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2))
             => 'fun_app$j'(X0,X3) )
         => 'fun_app$j'(X0,X2) )
     => 'fun_app$j'(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom379) ).

tff(f6512,plain,
    spl310_248,
    inference(avatar_split_clause,[],[f2647,f6510]) ).

tff(f6510,plain,
    ( spl310_248
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$',X3: 'Nat$'] :
        ( 'fun_app$j'(X0,X1)
        | 'fun_app$j'(X0,X3)
        | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK104(X0))) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_248])]) ).

tff(f2647,plain,
    ! [X3: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | 'fun_app$j'(X0,X3)
      | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK104(X0))) ),
    inference(cnf_transformation,[],[f1758]) ).

tff(f1758,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK104(X0))) )
        & ~ 'fun_app$j'(X0,sK104(X0)) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK104])],[f1146,f1757]) ).

tff(f1757,plain,
    ! [X0: 'Nat_bool_fun$'] :
      ( ? [X2: 'Nat$'] :
          ( ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) )
          & ~ 'fun_app$j'(X0,X2) )
     => ( ! [X3: 'Nat$'] :
            ( 'fun_app$j'(X0,X3)
            | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',sK104(X0))) )
        & ~ 'fun_app$j'(X0,sK104(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1146,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ? [X2: 'Nat$'] :
          ( ! [X3: 'Nat$'] :
              ( 'fun_app$j'(X0,X3)
              | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) )
          & ~ 'fun_app$j'(X0,X2) ) ),
    inference(ennf_transformation,[],[f380]) ).

tff(f380,axiom,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ! [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
         => ? [X3: 'Nat$'] :
              ( ~ 'fun_app$j'(X0,X3)
              & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X2)) ) )
     => 'fun_app$j'(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom378) ).

tff(f6508,plain,
    spl310_247,
    inference(avatar_split_clause,[],[f2560,f6506]) ).

tff(f6506,plain,
    ( spl310_247
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_247])]) ).

tff(f2560,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ),
    inference(cnf_transformation,[],[f1061]) ).

tff(f1061,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ( 'fun_app$e'('of_nat$',X0) != 'fun_app$e'('of_nat$',X1) ) ),
    inference(ennf_transformation,[],[f673]) ).

tff(f673,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
     => ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(theory_normalization,[],[f130]) ).

tff(f130,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X0) = 'fun_app$e'('of_nat$',X1) )
     => $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom128) ).

tff(f6504,plain,
    spl310_246,
    inference(avatar_split_clause,[],[f2539,f6502]) ).

tff(f2539,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f1044]) ).

tff(f1044,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1)))
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f376]) ).

tff(f376,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),X1)
     => $less('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom374) ).

tff(f6500,plain,
    spl310_245,
    inference(avatar_split_clause,[],[f2538,f6498]) ).

tff(f2538,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ $less('fun_app$e'('of_nat$','to_nat$'(X1)),'fun_app$e'('of_nat$','to_nat$'(X0)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f1043]) ).

tff(f1043,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ $less('fun_app$e'('of_nat$','to_nat$'(X1)),'fun_app$e'('of_nat$','to_nat$'(X0)))
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f671]) ).

tff(f671,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
     => ~ $less('fun_app$e'('of_nat$','to_nat$'(X1)),'fun_app$e'('of_nat$','to_nat$'(X0))) ),
    inference(theory_normalization,[],[f26]) ).

tff(f26,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
     => $lesseq('fun_app$e'('of_nat$','to_nat$'(X0)),'fun_app$e'('of_nat$','to_nat$'(X1))) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom24) ).

tff(f6496,plain,
    spl310_244,
    inference(avatar_split_clause,[],[f2433,f6494]) ).

tff(f6494,plain,
    ( spl310_244
  <=> ! [X2: 'N$',X0: 'N_bool_fun$'] :
        ( 'fun_app$d'(X0,sK92(X0))
        | 'fun_app$d'('fun_app$g'('less$',X2),sK93(X0,X2))
        | ~ 'fun_app$d'(X0,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_244])]) ).

tff(f2433,plain,
    ! [X2: 'N$',X0: 'N_bool_fun$'] :
      ( 'fun_app$d'(X0,sK92(X0))
      | 'fun_app$d'('fun_app$g'('less$',X2),sK93(X0,X2))
      | ~ 'fun_app$d'(X0,X2) ),
    inference(cnf_transformation,[],[f1721]) ).

tff(f1721,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ( 'fun_app$d'(X0,sK92(X0))
        | ! [X2: 'N$'] :
            ( ( 'fun_app$d'(X0,sK93(X0,X2))
              & 'fun_app$d'('fun_app$g'('less$',X2),sK93(X0,X2)) )
            | ~ 'fun_app$d'(X0,X2) ) )
      & ( ( ! [X5: 'N$'] :
              ( ~ 'fun_app$d'(X0,X5)
              | ~ 'fun_app$d'('fun_app$g'('less$',sK94(X0)),X5) )
          & 'fun_app$d'(X0,sK94(X0)) )
        | ! [X6: 'N$'] : ~ 'fun_app$d'(X0,X6) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK92,sK93,sK94])],[f1717,f1720,f1719,f1718]) ).

tff(f1718,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
     => 'fun_app$d'(X0,sK92(X0)) ),
    introduced(choice_axiom,[]) ).

tff(f1719,plain,
    ! [X0: 'N_bool_fun$',X2: 'N$'] :
      ( ? [X3: 'N$'] :
          ( 'fun_app$d'(X0,X3)
          & 'fun_app$d'('fun_app$g'('less$',X2),X3) )
     => ( 'fun_app$d'(X0,sK93(X0,X2))
        & 'fun_app$d'('fun_app$g'('less$',X2),sK93(X0,X2)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1720,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X4: 'N$'] :
          ( ! [X5: 'N$'] :
              ( ~ 'fun_app$d'(X0,X5)
              | ~ 'fun_app$d'('fun_app$g'('less$',X4),X5) )
          & 'fun_app$d'(X0,X4) )
     => ( ! [X5: 'N$'] :
            ( ~ 'fun_app$d'(X0,X5)
            | ~ 'fun_app$d'('fun_app$g'('less$',sK94(X0)),X5) )
        & 'fun_app$d'(X0,sK94(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1717,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
        | ! [X2: 'N$'] :
            ( ? [X3: 'N$'] :
                ( 'fun_app$d'(X0,X3)
                & 'fun_app$d'('fun_app$g'('less$',X2),X3) )
            | ~ 'fun_app$d'(X0,X2) ) )
      & ( ? [X4: 'N$'] :
            ( ! [X5: 'N$'] :
                ( ~ 'fun_app$d'(X0,X5)
                | ~ 'fun_app$d'('fun_app$g'('less$',X4),X5) )
            & 'fun_app$d'(X0,X4) )
        | ! [X6: 'N$'] : ~ 'fun_app$d'(X0,X6) ) ),
    inference(rectify,[],[f1716]) ).

tff(f1716,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
        | ! [X2: 'N$'] :
            ( ? [X3: 'N$'] :
                ( 'fun_app$d'(X0,X3)
                & 'fun_app$d'('fun_app$g'('less$',X2),X3) )
            | ~ 'fun_app$d'(X0,X2) ) )
      & ( ? [X2: 'N$'] :
            ( ! [X3: 'N$'] :
                ( ~ 'fun_app$d'(X0,X3)
                | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) )
            & 'fun_app$d'(X0,X2) )
        | ! [X1: 'N$'] : ~ 'fun_app$d'(X0,X1) ) ),
    inference(nnf_transformation,[],[f989]) ).

tff(f989,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
    <=> ? [X2: 'N$'] :
          ( ! [X3: 'N$'] :
              ( ~ 'fun_app$d'(X0,X3)
              | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) )
          & 'fun_app$d'(X0,X2) ) ),
    inference(ennf_transformation,[],[f900]) ).

tff(f900,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
    <=> ? [X2: 'N$'] :
          ( ! [X3: 'N$'] :
              ( 'fun_app$d'('fun_app$g'('less$',X2),X3)
             => ~ 'fun_app$d'(X0,X3) )
          & 'fun_app$d'(X0,X2) ) ),
    inference(rectify,[],[f337]) ).

tff(f337,axiom,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
    <=> ? [X1: 'N$'] :
          ( ! [X2: 'N$'] :
              ( 'fun_app$d'('fun_app$g'('less$',X1),X2)
             => ~ 'fun_app$d'(X0,X2) )
          & 'fun_app$d'(X0,X1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom335) ).

tff(f6492,plain,
    spl310_243,
    inference(avatar_split_clause,[],[f2429,f6490]) ).

tff(f6490,plain,
    ( spl310_243
  <=> ! [X2: 'N$',X0: 'N_bool_fun$'] :
        ( 'fun_app$d'(X0,sK89(X0))
        | 'fun_app$d'('fun_app$g'('less$',sK90(X0,X2)),X2)
        | ~ 'fun_app$d'(X0,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_243])]) ).

tff(f2429,plain,
    ! [X2: 'N$',X0: 'N_bool_fun$'] :
      ( 'fun_app$d'(X0,sK89(X0))
      | 'fun_app$d'('fun_app$g'('less$',sK90(X0,X2)),X2)
      | ~ 'fun_app$d'(X0,X2) ),
    inference(cnf_transformation,[],[f1715]) ).

tff(f1715,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ( 'fun_app$d'(X0,sK89(X0))
        | ! [X2: 'N$'] :
            ( ( 'fun_app$d'(X0,sK90(X0,X2))
              & 'fun_app$d'('fun_app$g'('less$',sK90(X0,X2)),X2) )
            | ~ 'fun_app$d'(X0,X2) ) )
      & ( ( ! [X5: 'N$'] :
              ( ~ 'fun_app$d'(X0,X5)
              | ~ 'fun_app$d'('fun_app$g'('less$',X5),sK91(X0)) )
          & 'fun_app$d'(X0,sK91(X0)) )
        | ! [X6: 'N$'] : ~ 'fun_app$d'(X0,X6) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK89,sK90,sK91])],[f1711,f1714,f1713,f1712]) ).

tff(f1712,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
     => 'fun_app$d'(X0,sK89(X0)) ),
    introduced(choice_axiom,[]) ).

tff(f1713,plain,
    ! [X0: 'N_bool_fun$',X2: 'N$'] :
      ( ? [X3: 'N$'] :
          ( 'fun_app$d'(X0,X3)
          & 'fun_app$d'('fun_app$g'('less$',X3),X2) )
     => ( 'fun_app$d'(X0,sK90(X0,X2))
        & 'fun_app$d'('fun_app$g'('less$',sK90(X0,X2)),X2) ) ),
    introduced(choice_axiom,[]) ).

tff(f1714,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X4: 'N$'] :
          ( ! [X5: 'N$'] :
              ( ~ 'fun_app$d'(X0,X5)
              | ~ 'fun_app$d'('fun_app$g'('less$',X5),X4) )
          & 'fun_app$d'(X0,X4) )
     => ( ! [X5: 'N$'] :
            ( ~ 'fun_app$d'(X0,X5)
            | ~ 'fun_app$d'('fun_app$g'('less$',X5),sK91(X0)) )
        & 'fun_app$d'(X0,sK91(X0)) ) ),
    introduced(choice_axiom,[]) ).

tff(f1711,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
        | ! [X2: 'N$'] :
            ( ? [X3: 'N$'] :
                ( 'fun_app$d'(X0,X3)
                & 'fun_app$d'('fun_app$g'('less$',X3),X2) )
            | ~ 'fun_app$d'(X0,X2) ) )
      & ( ? [X4: 'N$'] :
            ( ! [X5: 'N$'] :
                ( ~ 'fun_app$d'(X0,X5)
                | ~ 'fun_app$d'('fun_app$g'('less$',X5),X4) )
            & 'fun_app$d'(X0,X4) )
        | ! [X6: 'N$'] : ~ 'fun_app$d'(X0,X6) ) ),
    inference(rectify,[],[f1710]) ).

tff(f1710,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
        | ! [X2: 'N$'] :
            ( ? [X3: 'N$'] :
                ( 'fun_app$d'(X0,X3)
                & 'fun_app$d'('fun_app$g'('less$',X3),X2) )
            | ~ 'fun_app$d'(X0,X2) ) )
      & ( ? [X2: 'N$'] :
            ( ! [X3: 'N$'] :
                ( ~ 'fun_app$d'(X0,X3)
                | ~ 'fun_app$d'('fun_app$g'('less$',X3),X2) )
            & 'fun_app$d'(X0,X2) )
        | ! [X1: 'N$'] : ~ 'fun_app$d'(X0,X1) ) ),
    inference(nnf_transformation,[],[f988]) ).

tff(f988,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
    <=> ? [X2: 'N$'] :
          ( ! [X3: 'N$'] :
              ( ~ 'fun_app$d'(X0,X3)
              | ~ 'fun_app$d'('fun_app$g'('less$',X3),X2) )
          & 'fun_app$d'(X0,X2) ) ),
    inference(ennf_transformation,[],[f899]) ).

tff(f899,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
    <=> ? [X2: 'N$'] :
          ( ! [X3: 'N$'] :
              ( 'fun_app$d'('fun_app$g'('less$',X3),X2)
             => ~ 'fun_app$d'(X0,X3) )
          & 'fun_app$d'(X0,X2) ) ),
    inference(rectify,[],[f333]) ).

tff(f333,axiom,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X1: 'N$'] : 'fun_app$d'(X0,X1)
    <=> ? [X1: 'N$'] :
          ( ! [X2: 'N$'] :
              ( 'fun_app$d'('fun_app$g'('less$',X2),X1)
             => ~ 'fun_app$d'(X0,X2) )
          & 'fun_app$d'(X0,X1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom331) ).

tff(f6472,plain,
    ( spl310_242
    | ~ spl310_58
    | ~ spl310_144 ),
    inference(avatar_split_clause,[],[f5254,f4986,f4490,f6469]) ).

tff(f6469,plain,
    ( spl310_242
  <=> sP48('to_nat$'('i$'),'to_nat$'('x$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_242])]) ).

tff(f4986,plain,
    ( spl310_144
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP48(X0,X1)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_144])]) ).

tff(f5254,plain,
    ( sP48('to_nat$'('i$'),'to_nat$'('x$'))
    | ~ spl310_58
    | ~ spl310_144 ),
    inference(resolution,[],[f4492,f4987]) ).

tff(f4987,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | sP48(X0,X1) )
    | ~ spl310_144 ),
    inference(avatar_component_clause,[],[f4986]) ).

tff(f6302,plain,
    ( spl310_241
    | ~ spl310_117
    | ~ spl310_215 ),
    inference(avatar_split_clause,[],[f6086,f5930,f4838,f6300]) ).

tff(f6300,plain,
    ( spl310_241
  <=> ! [X0: 'Nat_bool_fun$'] :
        ( 'fun_app$j'(X0,'to_nat$'('i$'))
        | ~ sP2(X0,'to_nat$'('x$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_241])]) ).

tff(f5930,plain,
    ( spl310_215
  <=> ! [X0: 'Nat$',X1: 'Nat_bool_fun$',X3: 'Nat$'] :
        ( 'fun_app$j'(X1,X3)
        | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0))
        | ~ sP2(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_215])]) ).

tff(f6086,plain,
    ( ! [X0: 'Nat_bool_fun$'] :
        ( 'fun_app$j'(X0,'to_nat$'('i$'))
        | ~ sP2(X0,'to_nat$'('x$')) )
    | ~ spl310_117
    | ~ spl310_215 ),
    inference(resolution,[],[f5931,f4840]) ).

tff(f5931,plain,
    ( ! [X3: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0))
        | 'fun_app$j'(X1,X3)
        | ~ sP2(X1,X0) )
    | ~ spl310_215 ),
    inference(avatar_component_clause,[],[f5930]) ).

tff(f6035,plain,
    spl310_240,
    inference(avatar_split_clause,[],[f4205,f6033]) ).

tff(f6033,plain,
    ( spl310_240
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK139(X0,X1)))))
        | ~ sP61(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_240])]) ).

tff(f4205,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'($sum(1,'fun_app$e'('of_nat$',sK139(X0,X1)))))
      | ~ sP61(X0,X1) ),
    inference(forward_demodulation,[],[f3324,f864]) ).

tff(f3324,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK139(X0,X1)),1)))
      | ~ sP61(X0,X1) ),
    inference(cnf_transformation,[],[f2082]) ).

tff(f2082,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ( ~ 'fun_app$j'(X0,sK139(X0,X1))
        & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK139(X0,X1)),1)))
        & $less('fun_app$e'('of_nat$',sK139(X0,X1)),'fun_app$e'('of_nat$',X1)) )
      | ~ sP61(X0,X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK139])],[f2080,f2081]) ).

tff(f2081,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X2),1)))
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
     => ( ~ 'fun_app$j'(X0,sK139(X0,X1))
        & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',sK139(X0,X1)),1)))
        & $less('fun_app$e'('of_nat$',sK139(X0,X1)),'fun_app$e'('of_nat$',X1)) ) ),
    introduced(choice_axiom,[]) ).

tff(f2080,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X2: 'Nat$'] :
          ( ~ 'fun_app$j'(X0,X2)
          & 'fun_app$j'(X0,'nat$'($sum('fun_app$e'('of_nat$',X2),1)))
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP61(X0,X1) ),
    inference(rectify,[],[f2079]) ).

tff(f2079,plain,
    ! [X2: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ? [X3: 'Nat$'] :
          ( ~ 'fun_app$j'(X2,X3)
          & 'fun_app$j'(X2,'nat$'($sum('fun_app$e'('of_nat$',X3),1)))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X1)) )
      | ~ sP61(X2,X1) ),
    inference(nnf_transformation,[],[f1666]) ).

tff(f6031,plain,
    spl310_239,
    inference(avatar_split_clause,[],[f4120,f6029]) ).

tff(f6029,plain,
    ( spl310_239
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less($sum(1,'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1))
        | ~ sP16(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_239])]) ).

tff(f4120,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum(1,'fun_app$e'('of_nat$',X2)),'fun_app$e'('of_nat$',X1))
      | ~ sP16(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2982,f864]) ).

tff(f2982,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less($sum('fun_app$e'('of_nat$',X2),1),'fun_app$e'('of_nat$',X1))
      | ~ sP16(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1961]) ).

tff(f1961,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X0),1))
        & $less($sum('fun_app$e'('of_nat$',X2),1),'fun_app$e'('of_nat$',X1)) )
      | ~ sP16(X0,X1,X2) ),
    inference(rectify,[],[f1960]) ).

tff(f1960,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ~ $less(0,$sum('fun_app$e'('of_nat$',X2),1))
        & $less($sum('fun_app$e'('of_nat$',X0),1),'fun_app$e'('of_nat$',X1)) )
      | ~ sP16(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1608]) ).

tff(f6027,plain,
    spl310_238,
    inference(avatar_split_clause,[],[f3982,f6025]) ).

tff(f6025,plain,
    ( spl310_238
  <=> ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum(1,$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_238])]) ).

tff(f3982,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum(1,$sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),
    inference(forward_demodulation,[],[f2452,f864]) ).

tff(f2452,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1)),
    inference(cnf_transformation,[],[f371]) ).

tff(f371,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum($sum('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),1)),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom369) ).

tff(f6023,plain,
    spl310_237,
    inference(avatar_split_clause,[],[f3981,f6021]) ).

tff(f6021,plain,
    ( spl310_237
  <=> ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_237])]) ).

tff(f3981,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum(1,$sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)))),
    inference(forward_demodulation,[],[f2451,f864]) ).

tff(f2451,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),1)),
    inference(cnf_transformation,[],[f372]) ).

tff(f372,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] : $less('fun_app$e'('of_nat$',X0),$sum($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),1)),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom370) ).

tff(f6019,plain,
    spl310_236,
    inference(avatar_split_clause,[],[f3907,f6017]) ).

tff(f3907,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP44(X0,X1)
      | ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) ) ),
    inference(trivial_inequality_removal,[],[f3240]) ).

tff(f3240,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP44(X0,X1)
      | ( 0 != $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
      | ( 0 != 0 ) ),
    inference(cnf_transformation,[],[f2036]) ).

tff(f6015,plain,
    spl310_235,
    inference(avatar_split_clause,[],[f3905,f6013]) ).

tff(f6013,plain,
    ( spl310_235
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP48(X0,X1)
        | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_235])]) ).

tff(f3905,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP48(X0,X1)
      | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ),
    inference(evaluation,[],[f3261]) ).

tff(f3261,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP48(X0,X1)
      | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
      | $less(0,0) ),
    inference(cnf_transformation,[],[f2046]) ).

tff(f2046,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP48(X0,X1)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP48(X0,X1) ) ),
    inference(rectify,[],[f2045]) ).

tff(f2045,plain,
    ! [X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP48(X2,X1)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( $less(0,0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( ~ $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP48(X2,X1) ) ),
    inference(flattening,[],[f2044]) ).

tff(f2044,plain,
    ! [X2: 'Nat$',X1: 'Nat$'] :
      ( ( sP48(X2,X1)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
        | ( $less(0,0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X2))),0)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
          & ( ~ $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) ) )
        | ~ sP48(X2,X1) ) ),
    inference(nnf_transformation,[],[f1649]) ).

tff(f6011,plain,
    spl310_234,
    inference(avatar_split_clause,[],[f3903,f6009]) ).

tff(f6009,plain,
    ( spl310_234
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP52(X0,X1)
        | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_234])]) ).

tff(f3903,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP52(X0,X1)
      | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0) ),
    inference(evaluation,[],[f3282]) ).

tff(f3282,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP52(X0,X1)
      | $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
      | $less(0,0) ),
    inference(cnf_transformation,[],[f2056]) ).

tff(f2056,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP52(X0,X1)
        | ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
          & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,0)
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),0)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP52(X0,X1) ) ),
    inference(rectify,[],[f2055]) ).

tff(f2055,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP52(X0,X2)
        | ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,0)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP52(X0,X2) ) ),
    inference(flattening,[],[f2054]) ).

tff(f2054,plain,
    ! [X0: 'Nat$',X2: 'Nat$'] :
      ( ( sP52(X0,X2)
        | ( $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
          & ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
        | ( $less(0,0)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      & ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X2),$uminus('fun_app$e'('of_nat$',X0))),0)
            | $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,0)
            | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
        | ~ sP52(X0,X2) ) ),
    inference(nnf_transformation,[],[f1654]) ).

tff(f6007,plain,
    spl310_233,
    inference(avatar_split_clause,[],[f3705,f6005]) ).

tff(f6005,plain,
    ( spl310_233
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP88(X1,X0,X2,X3,X4)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_233])]) ).

tff(f3705,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP88(X1,X0,X2,X3,X4)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1706]) ).

tff(f1706,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( sP88(X1,X0,X2,X3,X4)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1580,f1705,f1704]) ).

tff(f1580,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( $less($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4))
      <=> ( ( $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),X4)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f863]) ).

tff(f863,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( $less($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3),X4) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4) ) ) ) ),
    inference(theory_normalization,[],[f529]) ).

tff(f529,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( $less($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less($sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X3),X4) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom527) ).

tff(f6001,plain,
    spl310_232,
    inference(avatar_split_clause,[],[f3696,f5999]) ).

tff(f5999,plain,
    ( spl310_232
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP86(X3,X1,X0,X2,X4)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_232])]) ).

tff(f3696,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP86(X3,X1,X0,X2,X4)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1703]) ).

tff(f1703,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( sP86(X3,X1,X0,X2,X4)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1579,f1702,f1701]) ).

tff(f1579,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      <=> ( ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f862]) ).

tff(f862,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4)) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4)) ) ) ) ),
    inference(theory_normalization,[],[f528]) ).

tff(f528,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( $less($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X4)) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4)) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom526) ).

tff(f5997,plain,
    spl310_231,
    inference(avatar_split_clause,[],[f3687,f5995]) ).

tff(f5995,plain,
    ( spl310_231
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP84(X4,X1,X0,X2,X3)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_231])]) ).

tff(f3687,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP84(X4,X1,X0,X2,X3)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1700]) ).

tff(f1700,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( sP84(X4,X1,X0,X2,X3)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1578,f1699,f1698]) ).

tff(f1578,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X4),$sum('fun_app$b'('iterate_add$'(X1),X2),X3))
      <=> ( ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f861]) ).

tff(f861,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ~ $less($sum('fun_app$b'('iterate_add$'(X0),X2),X4),$sum('fun_app$b'('iterate_add$'(X1),X2),X3))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3)) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ~ $less(X4,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3)) ) ) ) ),
    inference(theory_normalization,[],[f569]) ).

tff(f569,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( $lesseq($sum('fun_app$b'('iterate_add$'(X1),X2),X3),$sum('fun_app$b'('iterate_add$'(X0),X2),X4))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $lesseq($sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X3),X4) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $lesseq($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3),X4) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom567) ).

tff(f5993,plain,
    spl310_230,
    inference(avatar_split_clause,[],[f3678,f5991]) ).

tff(f5991,plain,
    ( spl310_230
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP82(X1,X0,X2,X4,X3)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_230])]) ).

tff(f3678,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP82(X1,X0,X2,X4,X3)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1697]) ).

tff(f1697,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( sP82(X1,X0,X2,X4,X3)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1577,f1696,f1695]) ).

tff(f1577,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ~ $less($sum('fun_app$b'('iterate_add$'(X1),X2),X4),$sum('fun_app$b'('iterate_add$'(X0),X2),X3))
      <=> ( ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4),X3)
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4),X3)
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f860]) ).

tff(f860,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ~ $less($sum('fun_app$b'('iterate_add$'(X1),X2),X4),$sum('fun_app$b'('iterate_add$'(X0),X2),X3))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ~ $less($sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4),X3) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ~ $less($sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4),X3) ) ) ) ),
    inference(theory_normalization,[],[f568]) ).

tff(f568,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( $lesseq($sum('fun_app$b'('iterate_add$'(X0),X2),X3),$sum('fun_app$b'('iterate_add$'(X1),X2),X4))
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $lesseq(X3,$sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X4)) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $lesseq(X3,$sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4)) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom566) ).

tff(f5989,plain,
    spl310_229,
    inference(avatar_split_clause,[],[f3669,f5987]) ).

tff(f5987,plain,
    ( spl310_229
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP80(X1,X0,X2,X4,X3)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_229])]) ).

tff(f3669,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP80(X1,X0,X2,X4,X3)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1694]) ).

tff(f1694,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( sP80(X1,X0,X2,X4,X3)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1576,f1693,f1692]) ).

tff(f1576,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      <=> ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4) = X3 )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) = X3 )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f859]) ).

tff(f859,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X4) = X3 ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) = X3 ) ) ) ) ),
    inference(theory_normalization,[],[f566]) ).

tff(f566,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( $sum('fun_app$b'('iterate_add$'(X0),X2),X3) = $sum('fun_app$b'('iterate_add$'(X1),X2),X4) )
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X4) = X3 ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X4) = X3 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom564) ).

tff(f5985,plain,
    spl310_228,
    inference(avatar_split_clause,[],[f3660,f5983]) ).

tff(f5983,plain,
    ( spl310_228
  <=> ! [X4: $int,X0: 'Nat$',X3: $int,X2: $int,X1: 'Nat$'] :
        ( sP78(X1,X0,X2,X3,X4)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_228])]) ).

tff(f3660,plain,
    ! [X2: $int,X3: $int,X0: 'Nat$',X1: 'Nat$',X4: $int] :
      ( sP78(X1,X0,X2,X3,X4)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1691]) ).

tff(f1691,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( sP78(X1,X0,X2,X3,X4)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1575,f1690,f1689]) ).

tff(f1575,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ( ( $sum('fun_app$b'('iterate_add$'(X1),X2),X3) = $sum('fun_app$b'('iterate_add$'(X0),X2),X4) )
      <=> ( ( ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3) = X4 )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) = X4 )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f858]) ).

tff(f858,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( $sum('fun_app$b'('iterate_add$'(X1),X2),X3) = $sum('fun_app$b'('iterate_add$'(X0),X2),X4) )
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))),X2),X3) = X4 ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) = X4 ) ) ) ) ),
    inference(theory_normalization,[],[f567]) ).

tff(f567,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: $int,X3: $int,X4: $int] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( $sum('fun_app$b'('iterate_add$'(X1),X2),X3) = $sum('fun_app$b'('iterate_add$'(X0),X2),X4) )
      <=> ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))),X2),X3) = X4 ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( $sum('fun_app$b'('iterate_add$'('nat$'(0)),X2),X3) = X4 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom565) ).

tff(f5981,plain,
    spl310_227,
    inference(avatar_split_clause,[],[f3532,f5979]) ).

tff(f5979,plain,
    ( spl310_227
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( $less($sum(X0,X2),$sum(X1,X3))
        | ~ $less(X2,X3)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_227])]) ).

tff(f3532,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | ~ $less(X2,X3)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f1450]) ).

tff(f1450,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | ~ $less(X2,X3)
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1449]) ).

tff(f1449,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | ~ $less(X2,X3)
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f507]) ).

tff(f507,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $less(X2,X3)
        & $less(X0,X1) )
     => $less($sum(X0,X2),$sum(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom505) ).

tff(f5977,plain,
    spl310_226,
    inference(avatar_split_clause,[],[f3529,f5975]) ).

tff(f5975,plain,
    ( spl310_226
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( $less($sum(X0,X2),$sum(X1,X3))
        | $less(X3,X2)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_226])]) ).

tff(f3529,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | $less(X3,X2)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f1444]) ).

tff(f1444,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | $less(X3,X2)
      | ~ $less(X0,X1) ),
    inference(flattening,[],[f1443]) ).

tff(f1443,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | $less(X3,X2)
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f831]) ).

tff(f831,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ~ $less(X3,X2)
        & $less(X0,X1) )
     => $less($sum(X0,X2),$sum(X1,X3)) ),
    inference(theory_normalization,[],[f548]) ).

tff(f548,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $lesseq(X2,X3)
        & $less(X0,X1) )
     => $less($sum(X0,X2),$sum(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom546) ).

tff(f5973,plain,
    spl310_225,
    inference(avatar_split_clause,[],[f3512,f5971]) ).

tff(f5971,plain,
    ( spl310_225
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( $less($sum(X0,X2),$sum(X1,X3))
        | ~ $less(X2,X3)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_225])]) ).

tff(f3512,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | ~ $less(X2,X3)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f1424]) ).

tff(f1424,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | ~ $less(X2,X3)
      | $less(X1,X0) ),
    inference(flattening,[],[f1423]) ).

tff(f1423,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( $less($sum(X0,X2),$sum(X1,X3))
      | ~ $less(X2,X3)
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f822]) ).

tff(f822,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $less(X2,X3)
        & ~ $less(X1,X0) )
     => $less($sum(X0,X2),$sum(X1,X3)) ),
    inference(theory_normalization,[],[f549]) ).

tff(f549,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $less(X2,X3)
        & $lesseq(X0,X1) )
     => $less($sum(X0,X2),$sum(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom547) ).

tff(f5969,plain,
    spl310_224,
    inference(avatar_split_clause,[],[f3510,f5967]) ).

tff(f5967,plain,
    ( spl310_224
  <=> ! [X0: $int,X3: $int,X2: $int,X1: $int] :
        ( ~ $less($sum(X1,X3),$sum(X0,X2))
        | $less(X3,X2)
        | $less(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_224])]) ).

tff(f3510,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( ~ $less($sum(X1,X3),$sum(X0,X2))
      | $less(X3,X2)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f1420]) ).

tff(f1420,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ~ $less($sum(X1,X3),$sum(X0,X2))
      | $less(X3,X2)
      | $less(X1,X0) ),
    inference(flattening,[],[f1419]) ).

tff(f1419,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ~ $less($sum(X1,X3),$sum(X0,X2))
      | $less(X3,X2)
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f820]) ).

tff(f820,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ~ $less(X3,X2)
        & ~ $less(X1,X0) )
     => ~ $less($sum(X1,X3),$sum(X0,X2)) ),
    inference(theory_normalization,[],[f31]) ).

tff(f31,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $lesseq(X2,X3)
        & $lesseq(X0,X1) )
     => $lesseq($sum(X0,X2),$sum(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom29) ).

tff(f5965,plain,
    spl310_223,
    inference(avatar_split_clause,[],[f3505,f5963]) ).

tff(f5963,plain,
    ( spl310_223
  <=> ! [X0: 'Int_set_set$',X3: 'Int_set_set$',X2: 'Int_set_set$',X1: 'Int_set_set$'] :
        ( 'less_eq$c'('plus$a'(X0,X2),'plus$a'(X1,X3))
        | ~ 'less_eq$c'(X2,X3)
        | ~ 'less_eq$c'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_223])]) ).

tff(f3505,plain,
    ! [X2: 'Int_set_set$',X3: 'Int_set_set$',X0: 'Int_set_set$',X1: 'Int_set_set$'] :
      ( 'less_eq$c'('plus$a'(X0,X2),'plus$a'(X1,X3))
      | ~ 'less_eq$c'(X2,X3)
      | ~ 'less_eq$c'(X0,X1) ),
    inference(cnf_transformation,[],[f1410]) ).

tff(f1410,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set$'] :
      ( 'less_eq$c'('plus$a'(X0,X2),'plus$a'(X1,X3))
      | ~ 'less_eq$c'(X2,X3)
      | ~ 'less_eq$c'(X0,X1) ),
    inference(flattening,[],[f1409]) ).

tff(f1409,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set$'] :
      ( 'less_eq$c'('plus$a'(X0,X2),'plus$a'(X1,X3))
      | ~ 'less_eq$c'(X2,X3)
      | ~ 'less_eq$c'(X0,X1) ),
    inference(ennf_transformation,[],[f118]) ).

tff(f118,axiom,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set$'] :
      ( ( 'less_eq$c'(X2,X3)
        & 'less_eq$c'(X0,X1) )
     => 'less_eq$c'('plus$a'(X0,X2),'plus$a'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom116) ).

tff(f5961,plain,
    ( ~ spl310_222
    | ~ spl310_24
    | ~ spl310_142 ),
    inference(avatar_split_clause,[],[f5916,f4975,f4326,f5958]) ).

tff(f5958,plain,
    ( spl310_222
  <=> $less('fun_app$e'('of_nat$','to_nat$'('i$')),sK95('fun_app$e'('of_nat$','to_nat$'('x$')))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_222])]) ).

tff(f4975,plain,
    ( spl310_142
  <=> $less(sK95('fun_app$e'('of_nat$','to_nat$'('x$'))),'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_142])]) ).

tff(f5916,plain,
    ( ~ $less('fun_app$e'('of_nat$','to_nat$'('i$')),sK95('fun_app$e'('of_nat$','to_nat$'('x$'))))
    | ~ spl310_24
    | ~ spl310_142 ),
    inference(resolution,[],[f4977,f4327]) ).

tff(f4977,plain,
    ( $less(sK95('fun_app$e'('of_nat$','to_nat$'('x$'))),'fun_app$e'('of_nat$','to_nat$'('i$')))
    | ~ spl310_142 ),
    inference(avatar_component_clause,[],[f4975]) ).

tff(f5956,plain,
    spl310_221,
    inference(avatar_split_clause,[],[f3504,f5954]) ).

tff(f5954,plain,
    ( spl310_221
  <=> ! [X0: 'N_set$',X3: 'N_set$',X2: 'N_set$',X1: 'N_set$'] :
        ( 'less_eq$b'('plus$c'(X0,X2),'plus$c'(X1,X3))
        | ~ 'less_eq$b'(X2,X3)
        | ~ 'less_eq$b'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_221])]) ).

tff(f3504,plain,
    ! [X2: 'N_set$',X3: 'N_set$',X0: 'N_set$',X1: 'N_set$'] :
      ( 'less_eq$b'('plus$c'(X0,X2),'plus$c'(X1,X3))
      | ~ 'less_eq$b'(X2,X3)
      | ~ 'less_eq$b'(X0,X1) ),
    inference(cnf_transformation,[],[f1408]) ).

tff(f1408,plain,
    ! [X0: 'N_set$',X1: 'N_set$',X2: 'N_set$',X3: 'N_set$'] :
      ( 'less_eq$b'('plus$c'(X0,X2),'plus$c'(X1,X3))
      | ~ 'less_eq$b'(X2,X3)
      | ~ 'less_eq$b'(X0,X1) ),
    inference(flattening,[],[f1407]) ).

tff(f1407,plain,
    ! [X0: 'N_set$',X1: 'N_set$',X2: 'N_set$',X3: 'N_set$'] :
      ( 'less_eq$b'('plus$c'(X0,X2),'plus$c'(X1,X3))
      | ~ 'less_eq$b'(X2,X3)
      | ~ 'less_eq$b'(X0,X1) ),
    inference(ennf_transformation,[],[f117]) ).

tff(f117,axiom,
    ! [X0: 'N_set$',X1: 'N_set$',X2: 'N_set$',X3: 'N_set$'] :
      ( ( 'less_eq$b'(X2,X3)
        & 'less_eq$b'(X0,X1) )
     => 'less_eq$b'('plus$c'(X0,X2),'plus$c'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom115) ).

tff(f5952,plain,
    spl310_220,
    inference(avatar_split_clause,[],[f3503,f5950]) ).

tff(f5950,plain,
    ( spl310_220
  <=> ! [X0: 'Int_set_set$',X3: 'Int_set_set_set$',X2: 'Int_set_set$',X1: 'Int_set_set_set$'] :
        ( 'member$d'('plus$a'(X0,X2),'plus$e'(X1,X3))
        | ~ 'member$d'(X2,X3)
        | ~ 'member$d'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_220])]) ).

tff(f3503,plain,
    ! [X2: 'Int_set_set$',X3: 'Int_set_set_set$',X0: 'Int_set_set$',X1: 'Int_set_set_set$'] :
      ( 'member$d'('plus$a'(X0,X2),'plus$e'(X1,X3))
      | ~ 'member$d'(X2,X3)
      | ~ 'member$d'(X0,X1) ),
    inference(cnf_transformation,[],[f1406]) ).

tff(f1406,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set_set$'] :
      ( 'member$d'('plus$a'(X0,X2),'plus$e'(X1,X3))
      | ~ 'member$d'(X2,X3)
      | ~ 'member$d'(X0,X1) ),
    inference(flattening,[],[f1405]) ).

tff(f1405,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set_set$'] :
      ( 'member$d'('plus$a'(X0,X2),'plus$e'(X1,X3))
      | ~ 'member$d'(X2,X3)
      | ~ 'member$d'(X0,X1) ),
    inference(ennf_transformation,[],[f104]) ).

tff(f104,axiom,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set_set$',X2: 'Int_set_set$',X3: 'Int_set_set_set$'] :
      ( ( 'member$d'(X2,X3)
        & 'member$d'(X0,X1) )
     => 'member$d'('plus$a'(X0,X2),'plus$e'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom102) ).

tff(f5948,plain,
    spl310_219,
    inference(avatar_split_clause,[],[f3502,f5946]) ).

tff(f5946,plain,
    ( spl310_219
  <=> ! [X0: 'N_set$',X3: 'N_set_set$',X2: 'N_set$',X1: 'N_set_set$'] :
        ( 'member$c'('plus$c'(X0,X2),'plus$d'(X1,X3))
        | ~ 'member$c'(X2,X3)
        | ~ 'member$c'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_219])]) ).

tff(f3502,plain,
    ! [X2: 'N_set$',X3: 'N_set_set$',X0: 'N_set$',X1: 'N_set_set$'] :
      ( 'member$c'('plus$c'(X0,X2),'plus$d'(X1,X3))
      | ~ 'member$c'(X2,X3)
      | ~ 'member$c'(X0,X1) ),
    inference(cnf_transformation,[],[f1404]) ).

tff(f1404,plain,
    ! [X0: 'N_set$',X1: 'N_set_set$',X2: 'N_set$',X3: 'N_set_set$'] :
      ( 'member$c'('plus$c'(X0,X2),'plus$d'(X1,X3))
      | ~ 'member$c'(X2,X3)
      | ~ 'member$c'(X0,X1) ),
    inference(flattening,[],[f1403]) ).

tff(f1403,plain,
    ! [X0: 'N_set$',X1: 'N_set_set$',X2: 'N_set$',X3: 'N_set_set$'] :
      ( 'member$c'('plus$c'(X0,X2),'plus$d'(X1,X3))
      | ~ 'member$c'(X2,X3)
      | ~ 'member$c'(X0,X1) ),
    inference(ennf_transformation,[],[f103]) ).

tff(f103,axiom,
    ! [X0: 'N_set$',X1: 'N_set_set$',X2: 'N_set$',X3: 'N_set_set$'] :
      ( ( 'member$c'(X2,X3)
        & 'member$c'(X0,X1) )
     => 'member$c'('plus$c'(X0,X2),'plus$d'(X1,X3)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom101) ).

tff(f5944,plain,
    spl310_218,
    inference(avatar_split_clause,[],[f2897,f5942]) ).

tff(f2897,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus($sum(X2,X1))) = $sum(X0,$uminus(X2)) ),
    inference(cnf_transformation,[],[f730]) ).

tff(f730,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum($sum(X0,X1),$uminus($sum(X2,X1))) = $sum(X0,$uminus(X2)) ),
    inference(theory_normalization,[],[f16]) ).

tff(f16,axiom,
    ! [X0: $int,X1: $int,X2: $int] : ( $difference($sum(X0,X1),$sum(X2,X1)) = $difference(X0,X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom14) ).

tff(f5940,plain,
    spl310_217,
    inference(avatar_split_clause,[],[f2896,f5938]) ).

tff(f2896,plain,
    ! [X2: $int,X0: $int,X1: $int] : ( $sum(X1,$uminus(X2)) = $sum($sum(X0,X1),$uminus($sum(X0,X2))) ),
    inference(cnf_transformation,[],[f729]) ).

tff(f729,plain,
    ! [X0: $int,X1: $int,X2: $int] : ( $sum(X1,$uminus(X2)) = $sum($sum(X0,X1),$uminus($sum(X0,X2))) ),
    inference(theory_normalization,[],[f14]) ).

tff(f14,axiom,
    ! [X0: $int,X1: $int,X2: $int] : ( $difference($sum(X0,X1),$sum(X0,X2)) = $difference(X1,X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom12) ).

tff(f5936,plain,
    spl310_216,
    inference(avatar_split_clause,[],[f2882,f5934]) ).

tff(f2882,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( sP3(X1,X0)
      | ~ 'fun_app$j'(X1,X2)
      | ~ $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1930]) ).

tff(f5932,plain,
    spl310_215,
    inference(avatar_split_clause,[],[f2873,f5930]) ).

tff(f2873,plain,
    ! [X3: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X1,X3)
      | ~ $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X0))
      | ~ sP2(X1,X0) ),
    inference(cnf_transformation,[],[f1921]) ).

tff(f5928,plain,
    spl310_214,
    inference(avatar_split_clause,[],[f2592,f5926]) ).

tff(f5926,plain,
    ( spl310_214
  <=> ! [X0: 'N$',X1: 'Nat$'] :
        ( ( 'fun_app$i'('from_nat$',X1) = X0 )
        | ( 'fun_app$e'('of_nat$',X1) != 'fun_app$e'('of_nat$','to_nat$'(X0)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_214])]) ).

tff(f2592,plain,
    ! [X0: 'N$',X1: 'Nat$'] :
      ( ( 'fun_app$i'('from_nat$',X1) = X0 )
      | ( 'fun_app$e'('of_nat$',X1) != 'fun_app$e'('of_nat$','to_nat$'(X0)) ) ),
    inference(cnf_transformation,[],[f1078]) ).

tff(f1078,plain,
    ! [X0: 'N$',X1: 'Nat$'] :
      ( ( 'fun_app$i'('from_nat$',X1) = X0 )
      | ( 'fun_app$e'('of_nat$',X1) != 'fun_app$e'('of_nat$','to_nat$'(X0)) ) ),
    inference(ennf_transformation,[],[f137]) ).

tff(f137,axiom,
    ! [X0: 'N$',X1: 'Nat$'] :
      ( ( 'fun_app$e'('of_nat$',X1) = 'fun_app$e'('of_nat$','to_nat$'(X0)) )
     => ( 'fun_app$i'('from_nat$',X1) = X0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom135) ).

tff(f5924,plain,
    spl310_213,
    inference(avatar_split_clause,[],[f2522,f5922]) ).

tff(f5922,plain,
    ( spl310_213
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( ( X0 = X1 )
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        | 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_213])]) ).

tff(f2522,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ( X0 = X1 )
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      | 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f1735]) ).

tff(f1735,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
          | ( X0 != X1 ) )
        & ( ( X0 = X1 )
          | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ) )
      | 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(nnf_transformation,[],[f1029]) ).

tff(f1029,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      <=> ( X0 = X1 ) )
      | 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f429]) ).

tff(f429,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$',X0),X1)
     => ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      <=> ( X0 = X1 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom427) ).

tff(f5920,plain,
    spl310_212,
    inference(avatar_split_clause,[],[f2511,f5918]) ).

tff(f5918,plain,
    ( spl310_212
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'(sK97(X0,X1)),'fun_app$h'('minus$a'(X1),X0))
        | ~ 'fun_app$c'('less$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_212])]) ).

tff(f2511,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(sK97(X0,X1)),'fun_app$h'('minus$a'(X1),X0))
      | ~ 'fun_app$c'('less$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1731]) ).

tff(f1731,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(sK97(X0,X1)),'fun_app$h'('minus$a'(X1),X0))
      | ~ 'fun_app$c'('less$a'(X0),X1) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK97])],[f1021,f1730]) ).

tff(f1730,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ? [X2: $int] : 'fun_app$c'('member$a'(X2),'fun_app$h'('minus$a'(X1),X0))
     => 'fun_app$c'('member$a'(sK97(X0,X1)),'fun_app$h'('minus$a'(X1),X0)) ),
    introduced(choice_axiom,[]) ).

tff(f1021,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ? [X2: $int] : 'fun_app$c'('member$a'(X2),'fun_app$h'('minus$a'(X1),X0))
      | ~ 'fun_app$c'('less$a'(X0),X1) ),
    inference(ennf_transformation,[],[f588]) ).

tff(f588,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less$a'(X0),X1)
     => ? [X2: $int] : 'fun_app$c'('member$a'(X2),'fun_app$h'('minus$a'(X1),X0)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom586) ).

tff(f5913,plain,
    spl310_211,
    inference(avatar_split_clause,[],[f2475,f5911]) ).

tff(f5911,plain,
    ( spl310_211
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less$',X1),X0)
        | ( X0 = X1 )
        | 'fun_app$d'('fun_app$g'('less$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_211])]) ).

tff(f2475,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | ( X0 = X1 )
      | 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f355]) ).

tff(f355,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | ( X0 = X1 )
      | 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom353) ).

tff(f5909,plain,
    spl310_210,
    inference(avatar_split_clause,[],[f2471,f5907]) ).

tff(f5907,plain,
    ( spl310_210
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
        | ( X0 = X1 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_210])]) ).

tff(f2471,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      | ( X0 = X1 ) ),
    inference(cnf_transformation,[],[f574]) ).

tff(f574,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      | ( X0 = X1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom572) ).

tff(f5844,plain,
    ( ~ spl310_58
    | ~ spl310_24
    | ~ spl310_117 ),
    inference(avatar_split_clause,[],[f5065,f4838,f4326,f4490]) ).

tff(f5777,plain,
    ( spl310_209
    | ~ spl310_21
    | ~ spl310_87
    | ~ spl310_208 ),
    inference(avatar_split_clause,[],[f5773,f5770,f4676,f4314,f5775]) ).

tff(f5775,plain,
    ( spl310_209
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( ( 'fun_app$f'('minus$'(X0),X1) != 'zero$a' )
        | ( X0 = X1 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_209])]) ).

tff(f5770,plain,
    ( spl310_208
  <=> ! [X0: 'N$',X1: 'N$',X3: 'N$'] :
        ( ( X0 = X1 )
        | ( 'fun_app$f'('minus$'(X0),X1) != 'fun_app$f'('minus$'(X3),X3) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_208])]) ).

tff(f5773,plain,
    ( ! [X0: 'N$',X1: 'N$'] :
        ( ( 'fun_app$f'('minus$'(X0),X1) != 'zero$a' )
        | ( X0 = X1 ) )
    | ~ spl310_21
    | ~ spl310_87
    | ~ spl310_208 ),
    inference(forward_demodulation,[],[f5771,f4844]) ).

tff(f5771,plain,
    ( ! [X3: 'N$',X0: 'N$',X1: 'N$'] :
        ( ( X0 = X1 )
        | ( 'fun_app$f'('minus$'(X0),X1) != 'fun_app$f'('minus$'(X3),X3) ) )
    | ~ spl310_208 ),
    inference(avatar_component_clause,[],[f5770]) ).

tff(f5772,plain,
    spl310_208,
    inference(avatar_split_clause,[],[f3826,f5770]) ).

tff(f3826,plain,
    ! [X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( X0 = X1 )
      | ( 'fun_app$f'('minus$'(X0),X1) != 'fun_app$f'('minus$'(X3),X3) ) ),
    inference(equality_resolution,[],[f3498]) ).

tff(f3498,plain,
    ! [X2: 'N$',X3: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( X0 = X1 )
      | ( X2 != X3 )
      | ( 'fun_app$f'('minus$'(X0),X1) != 'fun_app$f'('minus$'(X2),X3) ) ),
    inference(cnf_transformation,[],[f2204]) ).

tff(f2204,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$',X3: 'N$'] :
      ( ( ( ( X0 = X1 )
          | ( X2 != X3 ) )
        & ( ( X2 = X3 )
          | ( X0 != X1 ) ) )
      | ( 'fun_app$f'('minus$'(X0),X1) != 'fun_app$f'('minus$'(X2),X3) ) ),
    inference(nnf_transformation,[],[f1396]) ).

tff(f1396,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$',X3: 'N$'] :
      ( ( ( X0 = X1 )
      <=> ( X2 = X3 ) )
      | ( 'fun_app$f'('minus$'(X0),X1) != 'fun_app$f'('minus$'(X2),X3) ) ),
    inference(ennf_transformation,[],[f78]) ).

tff(f78,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$',X3: 'N$'] :
      ( ( 'fun_app$f'('minus$'(X0),X1) = 'fun_app$f'('minus$'(X2),X3) )
     => ( ( X0 = X1 )
      <=> ( X2 = X3 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom76) ).

tff(f5768,plain,
    spl310_207,
    inference(avatar_split_clause,[],[f3381,f5766]) ).

tff(f5766,plain,
    ( spl310_207
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',sK152(X0,X1)))
        | ~ sP74(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_207])]) ).

tff(f3381,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',sK152(X0,X1)))
      | ~ sP74(X0,X1) ),
    inference(cnf_transformation,[],[f2131]) ).

tff(f5764,plain,
    spl310_206,
    inference(avatar_split_clause,[],[f3323,f5762]) ).

tff(f5762,plain,
    ( spl310_206
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',sK139(X0,X1)),'fun_app$e'('of_nat$',X1))
        | ~ sP61(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_206])]) ).

tff(f3323,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',sK139(X0,X1)),'fun_app$e'('of_nat$',X1))
      | ~ sP61(X0,X1) ),
    inference(cnf_transformation,[],[f2082]) ).

tff(f5760,plain,
    spl310_205,
    inference(avatar_split_clause,[],[f3150,f5758]) ).

tff(f5758,plain,
    ( spl310_205
  <=> ! [X2: $int,X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'(X2),X1)
        | ~ 'fun_app$c'('member$a'(X2),X0)
        | ~ 'fun_app$c'('less$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_205])]) ).

tff(f3150,plain,
    ! [X2: $int,X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X2),X1)
      | ~ 'fun_app$c'('member$a'(X2),X0)
      | ~ 'fun_app$c'('less$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1260]) ).

tff(f1260,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: $int] :
      ( 'fun_app$c'('member$a'(X2),X1)
      | ~ 'fun_app$c'('member$a'(X2),X0)
      | ~ 'fun_app$c'('less$a'(X0),X1) ),
    inference(flattening,[],[f1259]) ).

tff(f1259,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: $int] :
      ( 'fun_app$c'('member$a'(X2),X1)
      | ~ 'fun_app$c'('member$a'(X2),X0)
      | ~ 'fun_app$c'('less$a'(X0),X1) ),
    inference(ennf_transformation,[],[f587]) ).

tff(f587,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$',X2: $int] :
      ( ( 'fun_app$c'('member$a'(X2),X0)
        & 'fun_app$c'('less$a'(X0),X1) )
     => 'fun_app$c'('member$a'(X2),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom585) ).

tff(f5756,plain,
    spl310_204,
    inference(avatar_split_clause,[],[f3128,f5754]) ).

tff(f5754,plain,
    ( spl310_204
  <=> ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,sK120(X0))
        | 'fun_app$j'(X0,sK121(X0,X2))
        | ~ 'fun_app$j'(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_204])]) ).

tff(f3128,plain,
    ! [X2: 'Nat$',X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,sK120(X0))
      | 'fun_app$j'(X0,sK121(X0,X2))
      | ~ 'fun_app$j'(X0,X1) ),
    inference(cnf_transformation,[],[f1999]) ).

tff(f5752,plain,
    spl310_203,
    inference(avatar_split_clause,[],[f3117,f5750]) ).

tff(f5750,plain,
    ( spl310_203
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( ( X0 = X2 )
        | ( 'fun_app$f'('plus$'(X0),X1) != 'fun_app$f'('plus$'(X2),X1) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_203])]) ).

tff(f3117,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( X0 = X2 )
      | ( 'fun_app$f'('plus$'(X0),X1) != 'fun_app$f'('plus$'(X2),X1) ) ),
    inference(cnf_transformation,[],[f1210]) ).

tff(f1210,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( X0 = X2 )
      | ( 'fun_app$f'('plus$'(X0),X1) != 'fun_app$f'('plus$'(X2),X1) ) ),
    inference(ennf_transformation,[],[f41]) ).

tff(f41,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$f'('plus$'(X0),X1) = 'fun_app$f'('plus$'(X2),X1) )
     => ( X0 = X2 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom39) ).

tff(f5748,plain,
    spl310_202,
    inference(avatar_split_clause,[],[f3116,f5746]) ).

tff(f5746,plain,
    ( spl310_202
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( ( X1 = X2 )
        | ( 'fun_app$f'('plus$'(X0),X1) != 'fun_app$f'('plus$'(X0),X2) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_202])]) ).

tff(f3116,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( X1 = X2 )
      | ( 'fun_app$f'('plus$'(X0),X1) != 'fun_app$f'('plus$'(X0),X2) ) ),
    inference(cnf_transformation,[],[f1209]) ).

tff(f1209,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( X1 = X2 )
      | ( 'fun_app$f'('plus$'(X0),X1) != 'fun_app$f'('plus$'(X0),X2) ) ),
    inference(ennf_transformation,[],[f43]) ).

tff(f43,axiom,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$f'('plus$'(X0),X1) = 'fun_app$f'('plus$'(X0),X2) )
     => ( X1 = X2 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom41) ).

tff(f5744,plain,
    spl310_201,
    inference(avatar_split_clause,[],[f3002,f5742]) ).

tff(f3002,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( $less($sum(X2,$uminus(X1)),$sum(X2,$uminus(X0)))
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f1168]) ).

tff(f1168,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $less($sum(X2,$uminus(X1)),$sum(X2,$uminus(X0)))
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f740]) ).

tff(f740,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $less(X0,X1)
     => $less($sum(X2,$uminus(X1)),$sum(X2,$uminus(X0))) ),
    inference(theory_normalization,[],[f512]) ).

tff(f512,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $less(X0,X1)
     => $less($difference(X2,X1),$difference(X2,X0)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom510) ).

tff(f5740,plain,
    spl310_200,
    inference(avatar_split_clause,[],[f2999,f5738]) ).

tff(f2999,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less($sum(X2,$uminus(X0)),$sum(X2,$uminus(X1)))
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f1165]) ).

tff(f1165,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less($sum(X2,$uminus(X0)),$sum(X2,$uminus(X1)))
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f737]) ).

tff(f737,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less(X1,X0)
     => ~ $less($sum(X2,$uminus(X0)),$sum(X2,$uminus(X1))) ),
    inference(theory_normalization,[],[f37]) ).

tff(f37,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $lesseq(X0,X1)
     => $lesseq($difference(X2,X1),$difference(X2,X0)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom35) ).

tff(f5736,plain,
    spl310_199,
    inference(avatar_split_clause,[],[f2846,f5734]) ).

tff(f5734,plain,
    ( spl310_199
  <=> ! [X0: 'N$',X1: 'N$'] :
        ( ( X0 = X1 )
        | ( 'fun_app$e'('of_nat$','to_nat$'(X0)) != 'fun_app$e'('of_nat$','to_nat$'(X1)) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_199])]) ).

tff(f2846,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ( X0 = X1 )
      | ( 'fun_app$e'('of_nat$','to_nat$'(X0)) != 'fun_app$e'('of_nat$','to_nat$'(X1)) ) ),
    inference(cnf_transformation,[],[f1898]) ).

tff(f1898,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ( ( 'fun_app$e'('of_nat$','to_nat$'(X0)) = 'fun_app$e'('of_nat$','to_nat$'(X1)) )
        | ( X0 != X1 ) )
      & ( ( X0 = X1 )
        | ( 'fun_app$e'('of_nat$','to_nat$'(X0)) != 'fun_app$e'('of_nat$','to_nat$'(X1)) ) ) ),
    inference(nnf_transformation,[],[f19]) ).

tff(f19,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( ( 'fun_app$e'('of_nat$','to_nat$'(X0)) = 'fun_app$e'('of_nat$','to_nat$'(X1)) )
    <=> ( X0 = X1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom17) ).

tff(f5732,plain,
    spl310_198,
    inference(avatar_split_clause,[],[f2740,f5730]) ).

tff(f5730,plain,
    ( spl310_198
  <=> ! [X0: 'Int_set$',X1: 'Int_set$',X3: $int] :
        ( 'fun_app$c'('member$a'(X3),X1)
        | ~ 'fun_app$c'('member$a'(X3),X0)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_198])]) ).

tff(f2740,plain,
    ! [X3: $int,X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X3),X1)
      | ~ 'fun_app$c'('member$a'(X3),X0)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1826]) ).

tff(f1826,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ( ~ 'fun_app$c'('member$a'(sK111(X0,X1)),X1)
          & 'fun_app$c'('member$a'(sK111(X0,X1)),X0) ) )
      & ( ! [X3: $int] :
            ( 'fun_app$c'('member$a'(X3),X1)
            | ~ 'fun_app$c'('member$a'(X3),X0) )
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK111])],[f1824,f1825]) ).

tff(f1825,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ? [X2: $int] :
          ( ~ 'fun_app$c'('member$a'(X2),X1)
          & 'fun_app$c'('member$a'(X2),X0) )
     => ( ~ 'fun_app$c'('member$a'(sK111(X0,X1)),X1)
        & 'fun_app$c'('member$a'(sK111(X0,X1)),X0) ) ),
    introduced(choice_axiom,[]) ).

tff(f1824,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ? [X2: $int] :
            ( ~ 'fun_app$c'('member$a'(X2),X1)
            & 'fun_app$c'('member$a'(X2),X0) ) )
      & ( ! [X3: $int] :
            ( 'fun_app$c'('member$a'(X3),X1)
            | ~ 'fun_app$c'('member$a'(X3),X0) )
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    inference(rectify,[],[f1823]) ).

tff(f1823,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ? [X2: $int] :
            ( ~ 'fun_app$c'('member$a'(X2),X1)
            & 'fun_app$c'('member$a'(X2),X0) ) )
      & ( ! [X2: $int] :
            ( 'fun_app$c'('member$a'(X2),X1)
            | ~ 'fun_app$c'('member$a'(X2),X0) )
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    inference(nnf_transformation,[],[f1153]) ).

tff(f1153,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
    <=> ! [X2: $int] :
          ( 'fun_app$c'('member$a'(X2),X1)
          | ~ 'fun_app$c'('member$a'(X2),X0) ) ),
    inference(ennf_transformation,[],[f581]) ).

tff(f581,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
    <=> ! [X2: $int] :
          ( 'fun_app$c'('member$a'(X2),X0)
         => 'fun_app$c'('member$a'(X2),X1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom579) ).

tff(f5728,plain,
    spl310_197,
    inference(avatar_split_clause,[],[f2652,f5726]) ).

tff(f5726,plain,
    ( spl310_197
  <=> ! [X0: 'N_bool_fun$',X1: 'N$',X3: 'N$'] :
        ( 'fun_app$d'(X0,X1)
        | 'fun_app$d'(X0,X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',sK107(X0)),X3) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_197])]) ).

tff(f2652,plain,
    ! [X3: 'N$',X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | 'fun_app$d'(X0,X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',sK107(X0)),X3) ),
    inference(cnf_transformation,[],[f1764]) ).

tff(f1764,plain,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | ( ~ 'fun_app$d'(X0,sK107(X0))
        & ! [X3: 'N$'] :
            ( 'fun_app$d'(X0,X3)
            | ~ 'fun_app$d'('fun_app$g'('less$',sK107(X0)),X3) ) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK107])],[f1149,f1763]) ).

tff(f1763,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X2: 'N$'] :
          ( ~ 'fun_app$d'(X0,X2)
          & ! [X3: 'N$'] :
              ( 'fun_app$d'(X0,X3)
              | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ) )
     => ( ~ 'fun_app$d'(X0,sK107(X0))
        & ! [X3: 'N$'] :
            ( 'fun_app$d'(X0,X3)
            | ~ 'fun_app$d'('fun_app$g'('less$',sK107(X0)),X3) ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1149,plain,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | ? [X2: 'N$'] :
          ( ~ 'fun_app$d'(X0,X2)
          & ! [X3: 'N$'] :
              ( 'fun_app$d'(X0,X3)
              | ~ 'fun_app$d'('fun_app$g'('less$',X2),X3) ) ) ),
    inference(ennf_transformation,[],[f357]) ).

tff(f357,axiom,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( ! [X2: 'N$'] :
          ( ! [X3: 'N$'] :
              ( 'fun_app$d'('fun_app$g'('less$',X2),X3)
             => 'fun_app$d'(X0,X3) )
         => 'fun_app$d'(X0,X2) )
     => 'fun_app$d'(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom355) ).

tff(f5724,plain,
    spl310_196,
    inference(avatar_split_clause,[],[f2650,f5722]) ).

tff(f5722,plain,
    ( spl310_196
  <=> ! [X0: 'N_bool_fun$',X1: 'N$',X3: 'N$'] :
        ( 'fun_app$d'(X0,X1)
        | 'fun_app$d'(X0,X3)
        | ~ 'fun_app$d'('fun_app$g'('less$',X3),sK106(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_196])]) ).

tff(f2650,plain,
    ! [X3: 'N$',X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | 'fun_app$d'(X0,X3)
      | ~ 'fun_app$d'('fun_app$g'('less$',X3),sK106(X0)) ),
    inference(cnf_transformation,[],[f1762]) ).

tff(f1762,plain,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | ( ~ 'fun_app$d'(X0,sK106(X0))
        & ! [X3: 'N$'] :
            ( 'fun_app$d'(X0,X3)
            | ~ 'fun_app$d'('fun_app$g'('less$',X3),sK106(X0)) ) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK106])],[f1148,f1761]) ).

tff(f1761,plain,
    ! [X0: 'N_bool_fun$'] :
      ( ? [X2: 'N$'] :
          ( ~ 'fun_app$d'(X0,X2)
          & ! [X3: 'N$'] :
              ( 'fun_app$d'(X0,X3)
              | ~ 'fun_app$d'('fun_app$g'('less$',X3),X2) ) )
     => ( ~ 'fun_app$d'(X0,sK106(X0))
        & ! [X3: 'N$'] :
            ( 'fun_app$d'(X0,X3)
            | ~ 'fun_app$d'('fun_app$g'('less$',X3),sK106(X0)) ) ) ),
    introduced(choice_axiom,[]) ).

tff(f1148,plain,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | ? [X2: 'N$'] :
          ( ~ 'fun_app$d'(X0,X2)
          & ! [X3: 'N$'] :
              ( 'fun_app$d'(X0,X3)
              | ~ 'fun_app$d'('fun_app$g'('less$',X3),X2) ) ) ),
    inference(ennf_transformation,[],[f350]) ).

tff(f350,axiom,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( ! [X2: 'N$'] :
          ( ! [X3: 'N$'] :
              ( 'fun_app$d'('fun_app$g'('less$',X3),X2)
             => 'fun_app$d'(X0,X3) )
         => 'fun_app$d'(X0,X2) )
     => 'fun_app$d'(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom348) ).

tff(f5720,plain,
    spl310_195,
    inference(avatar_split_clause,[],[f2434,f5718]) ).

tff(f5718,plain,
    ( spl310_195
  <=> ! [X2: 'N$',X0: 'N_bool_fun$'] :
        ( 'fun_app$d'(X0,sK92(X0))
        | 'fun_app$d'(X0,sK93(X0,X2))
        | ~ 'fun_app$d'(X0,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_195])]) ).

tff(f2434,plain,
    ! [X2: 'N$',X0: 'N_bool_fun$'] :
      ( 'fun_app$d'(X0,sK92(X0))
      | 'fun_app$d'(X0,sK93(X0,X2))
      | ~ 'fun_app$d'(X0,X2) ),
    inference(cnf_transformation,[],[f1721]) ).

tff(f5716,plain,
    spl310_194,
    inference(avatar_split_clause,[],[f2432,f5714]) ).

tff(f5714,plain,
    ( spl310_194
  <=> ! [X6: 'N$',X5: 'N$',X0: 'N_bool_fun$'] :
        ( ~ 'fun_app$d'(X0,X5)
        | ~ 'fun_app$d'('fun_app$g'('less$',sK94(X0)),X5)
        | ~ 'fun_app$d'(X0,X6) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_194])]) ).

tff(f2432,plain,
    ! [X0: 'N_bool_fun$',X6: 'N$',X5: 'N$'] :
      ( ~ 'fun_app$d'(X0,X5)
      | ~ 'fun_app$d'('fun_app$g'('less$',sK94(X0)),X5)
      | ~ 'fun_app$d'(X0,X6) ),
    inference(cnf_transformation,[],[f1721]) ).

tff(f5712,plain,
    spl310_193,
    inference(avatar_split_clause,[],[f2430,f5710]) ).

tff(f5710,plain,
    ( spl310_193
  <=> ! [X2: 'N$',X0: 'N_bool_fun$'] :
        ( 'fun_app$d'(X0,sK89(X0))
        | 'fun_app$d'(X0,sK90(X0,X2))
        | ~ 'fun_app$d'(X0,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_193])]) ).

tff(f2430,plain,
    ! [X2: 'N$',X0: 'N_bool_fun$'] :
      ( 'fun_app$d'(X0,sK89(X0))
      | 'fun_app$d'(X0,sK90(X0,X2))
      | ~ 'fun_app$d'(X0,X2) ),
    inference(cnf_transformation,[],[f1715]) ).

tff(f5708,plain,
    spl310_192,
    inference(avatar_split_clause,[],[f2428,f5706]) ).

tff(f5706,plain,
    ( spl310_192
  <=> ! [X6: 'N$',X5: 'N$',X0: 'N_bool_fun$'] :
        ( ~ 'fun_app$d'(X0,X5)
        | ~ 'fun_app$d'('fun_app$g'('less$',X5),sK91(X0))
        | ~ 'fun_app$d'(X0,X6) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_192])]) ).

tff(f2428,plain,
    ! [X0: 'N_bool_fun$',X6: 'N$',X5: 'N$'] :
      ( ~ 'fun_app$d'(X0,X5)
      | ~ 'fun_app$d'('fun_app$g'('less$',X5),sK91(X0))
      | ~ 'fun_app$d'(X0,X6) ),
    inference(cnf_transformation,[],[f1715]) ).

tff(f5502,plain,
    ( spl310_191
    | ~ spl310_117
    | ~ spl310_145 ),
    inference(avatar_split_clause,[],[f5196,f4990,f4838,f5499]) ).

tff(f5499,plain,
    ( spl310_191
  <=> sP44('to_nat$'('x$'),'to_nat$'('i$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_191])]) ).

tff(f4990,plain,
    ( spl310_145
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP44(X0,X1)
        | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_145])]) ).

tff(f5196,plain,
    ( sP44('to_nat$'('x$'),'to_nat$'('i$'))
    | ~ spl310_117
    | ~ spl310_145 ),
    inference(resolution,[],[f4991,f4840]) ).

tff(f4991,plain,
    ( ! [X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
        | sP44(X0,X1) )
    | ~ spl310_145 ),
    inference(avatar_component_clause,[],[f4990]) ).

tff(f5389,plain,
    spl310_190,
    inference(avatar_split_clause,[],[f4119,f5387]) ).

tff(f4119,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,$sum(1,'fun_app$e'('of_nat$',X0)))
      | ~ sP16(X0,X1,X2) ),
    inference(forward_demodulation,[],[f2983,f864]) ).

tff(f2983,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,$sum('fun_app$e'('of_nat$',X0),1))
      | ~ sP16(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1961]) ).

tff(f5385,plain,
    ( spl310_189
    | ~ spl310_117
    | ~ spl310_144 ),
    inference(avatar_split_clause,[],[f5193,f4986,f4838,f5382]) ).

tff(f5382,plain,
    ( spl310_189
  <=> sP48('to_nat$'('x$'),'to_nat$'('i$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_189])]) ).

tff(f5193,plain,
    ( sP48('to_nat$'('x$'),'to_nat$'('i$'))
    | ~ spl310_117
    | ~ spl310_144 ),
    inference(resolution,[],[f4987,f4840]) ).

tff(f5380,plain,
    spl310_188,
    inference(avatar_split_clause,[],[f3970,f5378]) ).

tff(f5378,plain,
    ( spl310_188
  <=> ! [X0: 'Nat$'] :
        ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
        | $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_188])]) ).

tff(f3970,plain,
    ! [X0: 'Nat$'] :
      ( ( 0 = $uminus('fun_app$e'('of_nat$',X0)) )
      | $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(evaluation,[],[f2411]) ).

tff(f2411,plain,
    ! [X0: 'Nat$'] :
      ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
      | $less(0,'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f985]) ).

tff(f985,plain,
    ! [X0: 'Nat$'] :
      ( ( ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) )
        | $less(0,'fun_app$e'('of_nat$',X0)) )
      & ( ( 0 = 0 )
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) ) ),
    inference(ennf_transformation,[],[f638]) ).

tff(f638,plain,
    ! [X0: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X0))
       => ( 0 = $sum(0,$uminus('fun_app$e'('of_nat$',X0))) ) )
      & ( $less(0,'fun_app$e'('of_nat$',X0))
       => ( 0 = 0 ) ) ),
    inference(theory_normalization,[],[f615]) ).

tff(f615,axiom,
    ! [X0: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X0))
       => ( 0 = $difference(0,'fun_app$e'('of_nat$',X0)) ) )
      & ( $less(0,'fun_app$e'('of_nat$',X0))
       => ( 0 = 0 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom613) ).

tff(f5376,plain,
    spl310_187,
    inference(avatar_split_clause,[],[f3427,f5374]) ).

tff(f3427,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X2,$sum(X0,$uminus(X1)))
      | $less($sum(X2,X1),X0) ),
    inference(cnf_transformation,[],[f2172]) ).

tff(f2172,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X2,$sum(X0,$uminus(X1)))
        | $less($sum(X2,X1),X0) )
      & ( ~ $less($sum(X2,X1),X0)
        | $less(X2,$sum(X0,$uminus(X1))) ) ),
    inference(nnf_transformation,[],[f804]) ).

tff(f804,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less(X2,$sum(X0,$uminus(X1)))
    <=> ~ $less($sum(X2,X1),X0) ),
    inference(theory_normalization,[],[f40]) ).

tff(f40,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $lesseq($difference(X0,X1),X2)
    <=> $lesseq(X0,$sum(X2,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom38) ).

tff(f5372,plain,
    spl310_186,
    inference(avatar_split_clause,[],[f3426,f5370]) ).

tff(f3426,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less($sum(X2,X1),X0)
      | $less(X2,$sum(X0,$uminus(X1))) ),
    inference(cnf_transformation,[],[f2172]) ).

tff(f5368,plain,
    spl310_185,
    inference(avatar_split_clause,[],[f3424,f5366]) ).

tff(f3424,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X1,$sum(X0,X2))
      | $less($sum(X1,$uminus(X2)),X0) ),
    inference(cnf_transformation,[],[f2171]) ).

tff(f2171,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less($sum(X1,$uminus(X2)),X0)
        | $less(X1,$sum(X0,X2)) )
      & ( ~ $less(X1,$sum(X0,X2))
        | $less($sum(X1,$uminus(X2)),X0) ) ),
    inference(nnf_transformation,[],[f803]) ).

tff(f803,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less($sum(X1,$uminus(X2)),X0)
    <=> ~ $less(X1,$sum(X0,X2)) ),
    inference(theory_normalization,[],[f39]) ).

tff(f39,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $lesseq(X0,$difference(X1,X2))
    <=> $lesseq($sum(X0,X2),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom37) ).

tff(f5364,plain,
    spl310_184,
    inference(avatar_split_clause,[],[f3385,f5362]) ).

tff(f5362,plain,
    ( spl310_184
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( 'fun_app$j'(X1,X2)
        | sP74(X1,X0)
        | ~ 'fun_app$j'(X1,sK153(X0,X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_184])]) ).

tff(f3385,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X1,X2)
      | sP74(X1,X0)
      | ~ 'fun_app$j'(X1,sK153(X0,X1)) ),
    inference(cnf_transformation,[],[f2134]) ).

tff(f5360,plain,
    spl310_183,
    inference(avatar_split_clause,[],[f3328,f5358]) ).

tff(f5358,plain,
    ( spl310_183
  <=> ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
        ( ~ 'fun_app$c'('member$a'(X0),X2)
        | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_183])]) ).

tff(f3328,plain,
    ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
      ( ~ 'fun_app$c'('member$a'(X0),X2)
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ),
    inference(cnf_transformation,[],[f1339]) ).

tff(f1339,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( ~ 'fun_app$c'('member$a'(X0),X2)
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ),
    inference(ennf_transformation,[],[f968]) ).

tff(f968,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ~ ( 'fun_app$c'('member$a'(X0),X2)
        & 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ),
    inference(true_and_false_elimination,[],[f591]) ).

tff(f591,axiom,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( ( 'fun_app$c'('member$a'(X0),X2)
        & 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) )
     => $false ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom589) ).

tff(f5356,plain,
    spl310_182,
    inference(avatar_split_clause,[],[f3318,f5354]) ).

tff(f5354,plain,
    ( spl310_182
  <=> ! [X0: 'Nat_nat_bool_fun_fun$'] :
        ( $less('fun_app$e'('of_nat$',sK136(X0)),'fun_app$e'('of_nat$',sK137(X0)))
        | ~ sP60(X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_182])]) ).

tff(f3318,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( $less('fun_app$e'('of_nat$',sK136(X0)),'fun_app$e'('of_nat$',sK137(X0)))
      | ~ sP60(X0) ),
    inference(cnf_transformation,[],[f2075]) ).

tff(f2075,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( ( ~ 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK137(X0))
        & 'fun_app$j'('fun_app$q'(X0,sK136(X0)),sK137(X0))
        & 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK136(X0))
        & $less('fun_app$e'('of_nat$',sK136(X0)),'fun_app$e'('of_nat$',sK137(X0)))
        & $less('fun_app$e'('of_nat$',sK135(X0)),'fun_app$e'('of_nat$',sK136(X0))) )
      | ~ sP60(X0) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK135,sK136,sK137])],[f2073,f2074]) ).

tff(f2074,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X1: 'Nat$',X2: 'Nat$',X3: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X0,X1),X3)
          & 'fun_app$j'('fun_app$q'(X0,X2),X3)
          & 'fun_app$j'('fun_app$q'(X0,X1),X2)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
     => ( ~ 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK137(X0))
        & 'fun_app$j'('fun_app$q'(X0,sK136(X0)),sK137(X0))
        & 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK136(X0))
        & $less('fun_app$e'('of_nat$',sK136(X0)),'fun_app$e'('of_nat$',sK137(X0)))
        & $less('fun_app$e'('of_nat$',sK135(X0)),'fun_app$e'('of_nat$',sK136(X0))) ) ),
    introduced(choice_axiom,[]) ).

tff(f2073,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X1: 'Nat$',X2: 'Nat$',X3: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X0,X1),X3)
          & 'fun_app$j'('fun_app$q'(X0,X2),X3)
          & 'fun_app$j'('fun_app$q'(X0,X1),X2)
          & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X3))
          & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X2)) )
      | ~ sP60(X0) ),
    inference(rectify,[],[f2072]) ).

tff(f2072,plain,
    ! [X2: 'Nat_nat_bool_fun_fun$'] :
      ( ? [X3: 'Nat$',X4: 'Nat$',X5: 'Nat$'] :
          ( ~ 'fun_app$j'('fun_app$q'(X2,X3),X5)
          & 'fun_app$j'('fun_app$q'(X2,X4),X5)
          & 'fun_app$j'('fun_app$q'(X2,X3),X4)
          & $less('fun_app$e'('of_nat$',X4),'fun_app$e'('of_nat$',X5))
          & $less('fun_app$e'('of_nat$',X3),'fun_app$e'('of_nat$',X4)) )
      | ~ sP60(X2) ),
    inference(nnf_transformation,[],[f1664]) ).

tff(f5352,plain,
    spl310_181,
    inference(avatar_split_clause,[],[f3317,f5350]) ).

tff(f5350,plain,
    ( spl310_181
  <=> ! [X0: 'Nat_nat_bool_fun_fun$'] :
        ( $less('fun_app$e'('of_nat$',sK135(X0)),'fun_app$e'('of_nat$',sK136(X0)))
        | ~ sP60(X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_181])]) ).

tff(f3317,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( $less('fun_app$e'('of_nat$',sK135(X0)),'fun_app$e'('of_nat$',sK136(X0)))
      | ~ sP60(X0) ),
    inference(cnf_transformation,[],[f2075]) ).

tff(f5348,plain,
    spl310_180,
    inference(avatar_split_clause,[],[f3291,f5346]) ).

tff(f5346,plain,
    ( spl310_180
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP58(X0,X1,X2)
        | ~ sP57(X2,X1,X0)
        | ~ sP56(X2,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_180])]) ).

tff(f3291,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP58(X0,X1,X2)
      | ~ sP57(X2,X1,X0)
      | ~ sP56(X2,X1) ),
    inference(cnf_transformation,[],[f2061]) ).

tff(f5344,plain,
    spl310_179,
    inference(avatar_split_clause,[],[f3270,f5342]) ).

tff(f5342,plain,
    ( spl310_179
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP54(X0,X1,X2)
        | ~ sP53(X2,X1,X0)
        | ~ sP52(X2,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_179])]) ).

tff(f3270,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP54(X0,X1,X2)
      | ~ sP53(X2,X1,X0)
      | ~ sP52(X2,X0) ),
    inference(cnf_transformation,[],[f2051]) ).

tff(f5340,plain,
    spl310_178,
    inference(avatar_split_clause,[],[f3249,f5338]) ).

tff(f5338,plain,
    ( spl310_178
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP50(X0,X1,X2)
        | ~ sP49(X2,X1,X0)
        | ~ sP48(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_178])]) ).

tff(f3249,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP50(X0,X1,X2)
      | ~ sP49(X2,X1,X0)
      | ~ sP48(X0,X1) ),
    inference(cnf_transformation,[],[f2041]) ).

tff(f5336,plain,
    spl310_177,
    inference(avatar_split_clause,[],[f3228,f5334]) ).

tff(f5334,plain,
    ( spl310_177
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP46(X0,X1,X2)
        | ~ sP45(X2,X1,X0)
        | ~ sP44(X2,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_177])]) ).

tff(f3228,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP46(X0,X1,X2)
      | ~ sP45(X2,X1,X0)
      | ~ sP44(X2,X1) ),
    inference(cnf_transformation,[],[f2030]) ).

tff(f5332,plain,
    spl310_176,
    inference(avatar_split_clause,[],[f3208,f5330]) ).

tff(f5330,plain,
    ( spl310_176
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP39(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_176])]) ).

tff(f3208,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP39(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2021]) ).

tff(f2021,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( sP37(X1,X0)
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP39(X0,X1,X2) ),
    inference(rectify,[],[f2020]) ).

tff(f2020,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP37(X0,X2)
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP39(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1638]) ).

tff(f5328,plain,
    spl310_175,
    inference(avatar_split_clause,[],[f3179,f5326]) ).

tff(f5326,plain,
    ( spl310_175
  <=> ! [X2: 'Int_set_set$',X0: 'Int_set$',X1: 'Int_set_set$'] :
        ( 'member$'(sK126(X0,X1,X2),X2)
        | ~ 'member$'(X0,'plus$a'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_175])]) ).

tff(f3179,plain,
    ! [X2: 'Int_set_set$',X0: 'Int_set$',X1: 'Int_set_set$'] :
      ( 'member$'(sK126(X0,X1,X2),X2)
      | ~ 'member$'(X0,'plus$a'(X1,X2)) ),
    inference(cnf_transformation,[],[f2005]) ).

tff(f5324,plain,
    spl310_174,
    inference(avatar_split_clause,[],[f3178,f5322]) ).

tff(f5322,plain,
    ( spl310_174
  <=> ! [X2: 'Int_set_set$',X0: 'Int_set$',X1: 'Int_set_set$'] :
        ( 'member$'(sK125(X0,X1,X2),X1)
        | ~ 'member$'(X0,'plus$a'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_174])]) ).

tff(f3178,plain,
    ! [X2: 'Int_set_set$',X0: 'Int_set$',X1: 'Int_set_set$'] :
      ( 'member$'(sK125(X0,X1,X2),X1)
      | ~ 'member$'(X0,'plus$a'(X1,X2)) ),
    inference(cnf_transformation,[],[f2005]) ).

tff(f5320,plain,
    spl310_173,
    inference(avatar_split_clause,[],[f3176,f5318]) ).

tff(f5318,plain,
    ( spl310_173
  <=> ! [X2: 'N_set$',X0: 'N$',X1: 'N_set$'] :
        ( 'member$b'(sK124(X0,X1,X2),X2)
        | ~ 'member$b'(X0,'plus$c'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_173])]) ).

tff(f3176,plain,
    ! [X2: 'N_set$',X0: 'N$',X1: 'N_set$'] :
      ( 'member$b'(sK124(X0,X1,X2),X2)
      | ~ 'member$b'(X0,'plus$c'(X1,X2)) ),
    inference(cnf_transformation,[],[f2003]) ).

tff(f5316,plain,
    spl310_172,
    inference(avatar_split_clause,[],[f3175,f5314]) ).

tff(f5314,plain,
    ( spl310_172
  <=> ! [X2: 'N_set$',X0: 'N$',X1: 'N_set$'] :
        ( 'member$b'(sK123(X0,X1,X2),X1)
        | ~ 'member$b'(X0,'plus$c'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_172])]) ).

tff(f3175,plain,
    ! [X2: 'N_set$',X0: 'N$',X1: 'N_set$'] :
      ( 'member$b'(sK123(X0,X1,X2),X1)
      | ~ 'member$b'(X0,'plus$c'(X1,X2)) ),
    inference(cnf_transformation,[],[f2003]) ).

tff(f5312,plain,
    spl310_171,
    inference(avatar_split_clause,[],[f3111,f5310]) ).

tff(f5310,plain,
    ( spl310_171
  <=> ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
        ( 'fun_app$c'('member$a'(X0),X1)
        | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_171])]) ).

tff(f3111,plain,
    ! [X2: 'Int_set$',X0: $int,X1: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X0),X1)
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ),
    inference(cnf_transformation,[],[f1204]) ).

tff(f1204,plain,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X0),X1)
      | ~ 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2)) ),
    inference(ennf_transformation,[],[f590]) ).

tff(f590,axiom,
    ! [X0: $int,X1: 'Int_set$',X2: 'Int_set$'] :
      ( 'fun_app$c'('member$a'(X0),'fun_app$h'('minus$a'(X1),X2))
     => 'fun_app$c'('member$a'(X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom588) ).

tff(f5308,plain,
    spl310_170,
    inference(avatar_split_clause,[],[f3105,f5306]) ).

tff(f5306,plain,
    ( spl310_170
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP36(X0,X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_170])]) ).

tff(f3105,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP36(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1635]) ).

tff(f1635,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP36(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1199,f1634,f1633]) ).

tff(f1199,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f758]) ).

tff(f758,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
      <=> $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) ) ),
    inference(theory_normalization,[],[f247]) ).

tff(f247,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $less(0,'fun_app$e'('of_nat$',X2)) ) )
      <=> $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom245) ).

tff(f5304,plain,
    spl310_169,
    inference(avatar_split_clause,[],[f3096,f5302]) ).

tff(f5302,plain,
    ( spl310_169
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP34(X0,X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_169])]) ).

tff(f3096,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP34(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1632]) ).

tff(f1632,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP34(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1198,f1631,f1630]) ).

tff(f1198,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2))
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ~ $less(0,'fun_app$e'('of_nat$',X2))
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f757]) ).

tff(f757,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ~ $less($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))),'fun_app$e'('of_nat$',X2)) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ~ $less(0,'fun_app$e'('of_nat$',X2)) ) )
      <=> ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0))) ) ),
    inference(theory_normalization,[],[f157]) ).

tff(f157,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $lesseq('fun_app$e'('of_nat$',X2),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => $lesseq('fun_app$e'('of_nat$',X2),0) ) )
      <=> $lesseq($sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)),'fun_app$e'('of_nat$',X1)) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom155) ).

tff(f5300,plain,
    ( spl310_168
    | ~ spl310_58
    | ~ spl310_145 ),
    inference(avatar_split_clause,[],[f5253,f4990,f4490,f5297]) ).

tff(f5297,plain,
    ( spl310_168
  <=> sP44('to_nat$'('i$'),'to_nat$'('x$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_168])]) ).

tff(f5253,plain,
    ( sP44('to_nat$'('i$'),'to_nat$'('x$'))
    | ~ spl310_58
    | ~ spl310_145 ),
    inference(resolution,[],[f4492,f4991]) ).

tff(f5295,plain,
    spl310_167,
    inference(avatar_split_clause,[],[f3087,f5293]) ).

tff(f5293,plain,
    ( spl310_167
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP32(X0,X2,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_167])]) ).

tff(f3087,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP32(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1629]) ).

tff(f1629,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( sP32(X0,X2,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1197,f1628,f1627]) ).

tff(f1197,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ( ( ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) )
            | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          & ( ( 0 = 'fun_app$e'('of_nat$',X2) )
            | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
      <=> ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f756]) ).

tff(f756,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( 'fun_app$e'('of_nat$',X2) = $sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( 0 = 'fun_app$e'('of_nat$',X2) ) ) )
      <=> ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    inference(theory_normalization,[],[f160]) ).

tff(f160,axiom,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( 'fun_app$e'('of_nat$',X2) = $difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
           => ( 0 = 'fun_app$e'('of_nat$',X2) ) ) )
      <=> ( 'fun_app$e'('of_nat$',X1) = $sum('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X0)) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom158) ).

tff(f5291,plain,
    spl310_166,
    inference(avatar_split_clause,[],[f3076,f5289]) ).

tff(f5289,plain,
    ( spl310_166
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP30(X1,X0,X2)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_166])]) ).

tff(f3076,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP30(X1,X0,X2)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1626]) ).

tff(f5287,plain,
    spl310_165,
    inference(avatar_split_clause,[],[f3074,f5285]) ).

tff(f5285,plain,
    ( spl310_165
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP28(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_165])]) ).

tff(f3074,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP28(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1981]) ).

tff(f1981,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X0),$sum(0,1))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP28(X0,X1,X2) ),
    inference(rectify,[],[f1980]) ).

tff(f1980,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),$sum(0,1))
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP28(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1623]) ).

tff(f5283,plain,
    spl310_164,
    inference(avatar_split_clause,[],[f3063,f5281]) ).

tff(f5281,plain,
    ( spl310_164
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP27(X2,X0,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_164])]) ).

tff(f3063,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP27(X2,X0,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1622]) ).

tff(f5279,plain,
    spl310_163,
    inference(avatar_split_clause,[],[f3061,f5277]) ).

tff(f5277,plain,
    ( spl310_163
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP25(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_163])]) ).

tff(f3061,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP25(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1976]) ).

tff(f1976,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less($sum(0,1),'fun_app$e'('of_nat$',X0))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP25(X0,X1,X2) ),
    inference(rectify,[],[f1975]) ).

tff(f1975,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less($sum(0,1),'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP25(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1619]) ).

tff(f5275,plain,
    spl310_162,
    inference(avatar_split_clause,[],[f3050,f5273]) ).

tff(f5273,plain,
    ( spl310_162
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP24(X1,X0,X2)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_162])]) ).

tff(f3050,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP24(X1,X0,X2)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1618]) ).

tff(f5271,plain,
    spl310_161,
    inference(avatar_split_clause,[],[f3048,f5269]) ).

tff(f5269,plain,
    ( spl310_161
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP22(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_161])]) ).

tff(f3048,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP22(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1970]) ).

tff(f1970,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X0),0)
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP22(X0,X1,X2) ),
    inference(rectify,[],[f1969]) ).

tff(f1969,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ( ~ $less('fun_app$e'('of_nat$',X2),0)
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP22(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1615]) ).

tff(f5267,plain,
    spl310_160,
    inference(avatar_split_clause,[],[f3010,f5265]) ).

tff(f3010,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less($sum(X2,$uminus(X1)),X0)
      | $less(X2,$sum(X0,X1)) ),
    inference(cnf_transformation,[],[f1179]) ).

tff(f1179,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less($sum(X2,$uminus(X1)),X0)
      | $less(X2,$sum(X0,X1)) ),
    inference(ennf_transformation,[],[f743]) ).

tff(f743,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less(X2,$sum(X0,X1))
     => ~ $less($sum(X2,$uminus(X1)),X0) ),
    inference(theory_normalization,[],[f102]) ).

tff(f102,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $lesseq($sum(X0,X1),X2)
     => $lesseq(X0,$difference(X2,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom100) ).

tff(f5263,plain,
    spl310_159,
    inference(avatar_split_clause,[],[f2946,f5261]) ).

tff(f5261,plain,
    ( spl310_159
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP8(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_159])]) ).

tff(f2946,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP8(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1946]) ).

tff(f1946,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP8(X0,X1,X2) ),
    inference(rectify,[],[f1945]) ).

tff(f1945,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP8(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1599]) ).

tff(f5252,plain,
    ( spl310_158
    | ~ spl310_117
    | ~ spl310_143 ),
    inference(avatar_split_clause,[],[f5190,f4982,f4838,f5249]) ).

tff(f5249,plain,
    ( spl310_158
  <=> sP52('to_nat$'('x$'),'to_nat$'('i$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_158])]) ).

tff(f5190,plain,
    ( sP52('to_nat$'('x$'),'to_nat$'('i$'))
    | ~ spl310_117
    | ~ spl310_143 ),
    inference(resolution,[],[f4983,f4840]) ).

tff(f5247,plain,
    spl310_157,
    inference(avatar_split_clause,[],[f2918,f5245]) ).

tff(f5245,plain,
    ( spl310_157
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
        | ~ sP4(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_157])]) ).

tff(f2918,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1))
      | ~ sP4(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1934]) ).

tff(f1934,plain,
    ! [X0: 'Nat$',X1: 'Nat$',X2: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X0))
        & $less('fun_app$e'('of_nat$',X2),'fun_app$e'('of_nat$',X1)) )
      | ~ sP4(X0,X1,X2) ),
    inference(rectify,[],[f1933]) ).

tff(f1933,plain,
    ! [X2: 'Nat$',X1: 'Nat$',X0: 'Nat$'] :
      ( ( ~ $less(0,'fun_app$e'('of_nat$',X2))
        & $less('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)) )
      | ~ sP4(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1594]) ).

tff(f5243,plain,
    spl310_156,
    inference(avatar_split_clause,[],[f2892,f5241]) ).

tff(f5241,plain,
    ( spl310_156
  <=> ! [X2: $int,X0: $int,X1: $int] : ( $sum(X0,$sum(X1,X2)) = $sum(X1,$sum(X0,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_156])]) ).

tff(f5239,plain,
    spl310_155,
    inference(avatar_split_clause,[],[f2885,f5237]) ).

tff(f5237,plain,
    ( spl310_155
  <=> ! [X2: 'Int_set_set$',X0: 'Int_set_set$',X1: 'Int_set_set$'] : ( 'plus$a'(X0,'plus$a'(X1,X2)) = 'plus$a'('plus$a'(X0,X1),X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_155])]) ).

tff(f2885,plain,
    ! [X2: 'Int_set_set$',X0: 'Int_set_set$',X1: 'Int_set_set$'] : ( 'plus$a'(X0,'plus$a'(X1,X2)) = 'plus$a'('plus$a'(X0,X1),X2) ),
    inference(cnf_transformation,[],[f63]) ).

tff(f63,axiom,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$'] : ( 'plus$a'(X0,'plus$a'(X1,X2)) = 'plus$a'('plus$a'(X0,X1),X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom61) ).

tff(f5235,plain,
    spl310_154,
    inference(avatar_split_clause,[],[f2884,f5233]) ).

tff(f5233,plain,
    ( spl310_154
  <=> ! [X2: 'Int_set_set$',X0: 'Int_set_set$',X1: 'Int_set_set$'] : ( 'plus$a'(X0,'plus$a'(X1,X2)) = 'plus$a'(X1,'plus$a'(X0,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_154])]) ).

tff(f2884,plain,
    ! [X2: 'Int_set_set$',X0: 'Int_set_set$',X1: 'Int_set_set$'] : ( 'plus$a'(X0,'plus$a'(X1,X2)) = 'plus$a'(X1,'plus$a'(X0,X2)) ),
    inference(cnf_transformation,[],[f45]) ).

tff(f45,axiom,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$',X2: 'Int_set_set$'] : ( 'plus$a'(X0,'plus$a'(X1,X2)) = 'plus$a'(X1,'plus$a'(X0,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom43) ).

tff(f5231,plain,
    spl310_153,
    inference(avatar_split_clause,[],[f2883,f5229]) ).

tff(f5229,plain,
    ( spl310_153
  <=> ! [X2: 'N_set$',X0: 'N_set$',X1: 'N_set$'] : ( 'plus$c'('plus$c'(X0,X1),X2) = 'plus$c'(X0,'plus$c'(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_153])]) ).

tff(f2883,plain,
    ! [X2: 'N_set$',X0: 'N_set$',X1: 'N_set$'] : ( 'plus$c'('plus$c'(X0,X1),X2) = 'plus$c'(X0,'plus$c'(X1,X2)) ),
    inference(cnf_transformation,[],[f62]) ).

tff(f62,axiom,
    ! [X0: 'N_set$',X1: 'N_set$',X2: 'N_set$'] : ( 'plus$c'('plus$c'(X0,X1),X2) = 'plus$c'(X0,'plus$c'(X1,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom60) ).

tff(f5227,plain,
    spl310_152,
    inference(avatar_split_clause,[],[f2880,f5225]) ).

tff(f5225,plain,
    ( spl310_152
  <=> ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( 'fun_app$j'(X1,sK119(X0,X1))
        | 'fun_app$j'(X1,X0)
        | ~ sP3(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_152])]) ).

tff(f2880,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X1,sK119(X0,X1))
      | 'fun_app$j'(X1,X0)
      | ~ sP3(X1,X0) ),
    inference(cnf_transformation,[],[f1930]) ).

tff(f5223,plain,
    spl310_151,
    inference(avatar_split_clause,[],[f2875,f5221]) ).

tff(f5221,plain,
    ( spl310_151
  <=> ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( sP2(X1,X0)
        | ~ 'fun_app$j'(X1,sK117(X0,X1))
        | ~ 'fun_app$j'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_151])]) ).

tff(f2875,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( sP2(X1,X0)
      | ~ 'fun_app$j'(X1,sK117(X0,X1))
      | ~ 'fun_app$j'(X1,X0) ),
    inference(cnf_transformation,[],[f1921]) ).

tff(f5219,plain,
    spl310_150,
    inference(avatar_split_clause,[],[f2512,f5217]) ).

tff(f5217,plain,
    ( spl310_150
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( ( X0 = X1 )
        | ~ 'fun_app$c'('less_eq$a'(X1),X0)
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_150])]) ).

tff(f2512,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( X0 = X1 )
      | ~ 'fun_app$c'('less_eq$a'(X1),X0)
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(cnf_transformation,[],[f1732]) ).

tff(f1732,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( ( 'fun_app$c'('less_eq$a'(X1),X0)
          | ( X0 != X1 ) )
        & ( ( X0 = X1 )
          | ~ 'fun_app$c'('less_eq$a'(X1),X0) ) )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(nnf_transformation,[],[f1022]) ).

tff(f1022,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X1),X0)
      <=> ( X0 = X1 ) )
      | ~ 'fun_app$c'('less_eq$a'(X0),X1) ),
    inference(ennf_transformation,[],[f162]) ).

tff(f162,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
     => ( 'fun_app$c'('less_eq$a'(X1),X0)
      <=> ( X0 = X1 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom160) ).

tff(f5215,plain,
    spl310_149,
    inference(avatar_split_clause,[],[f2448,f5213]) ).

tff(f5213,plain,
    ( spl310_149
  <=> ! [X0: 'Nat$',X1: 'Nat$'] : ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_149])]) ).

tff(f2448,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] : ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)),
    inference(cnf_transformation,[],[f542]) ).

tff(f542,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] : ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X0)),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom540) ).

tff(f5211,plain,
    spl310_148,
    inference(avatar_split_clause,[],[f2447,f5209]) ).

tff(f5209,plain,
    ( spl310_148
  <=> ! [X0: 'Nat$',X1: 'Nat$'] : ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X1)) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_148])]) ).

tff(f2447,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] : ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X1)),
    inference(cnf_transformation,[],[f541]) ).

tff(f541,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] : ~ $less($sum('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1)),'fun_app$e'('of_nat$',X1)),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom539) ).

tff(f5207,plain,
    ( ~ spl310_147
    | ~ spl310_117
    | ~ spl310_141 ),
    inference(avatar_split_clause,[],[f5187,f4971,f4838,f5204]) ).

tff(f5204,plain,
    ( spl310_147
  <=> sP56('to_nat$'('x$'),'to_nat$'('i$')) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_147])]) ).

tff(f5187,plain,
    ( ~ sP56('to_nat$'('x$'),'to_nat$'('i$'))
    | ~ spl310_117
    | ~ spl310_141 ),
    inference(resolution,[],[f4972,f4840]) ).

tff(f5202,plain,
    spl310_146,
    inference(avatar_split_clause,[],[f865,f5200]) ).

tff(f4992,plain,
    spl310_145,
    inference(avatar_split_clause,[],[f3906,f4990]) ).

tff(f3906,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP44(X0,X1)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(trivial_inequality_removal,[],[f3238]) ).

tff(f3238,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP44(X0,X1)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ( 0 != 0 ) ),
    inference(cnf_transformation,[],[f2036]) ).

tff(f4988,plain,
    spl310_144,
    inference(avatar_split_clause,[],[f3904,f4986]) ).

tff(f3904,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP48(X0,X1)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(evaluation,[],[f3259]) ).

tff(f3259,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP48(X0,X1)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less(0,0) ),
    inference(cnf_transformation,[],[f2046]) ).

tff(f4984,plain,
    spl310_143,
    inference(avatar_split_clause,[],[f3902,f4982]) ).

tff(f3902,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP52(X0,X1)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(evaluation,[],[f3280]) ).

tff(f3280,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP52(X0,X1)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | $less(0,0) ),
    inference(cnf_transformation,[],[f2056]) ).

tff(f4978,plain,
    ( spl310_142
    | ~ spl310_5
    | ~ spl310_118 ),
    inference(avatar_split_clause,[],[f4931,f4865,f4249,f4975]) ).

tff(f4931,plain,
    ( $less(sK95('fun_app$e'('of_nat$','to_nat$'('x$'))),'fun_app$e'('of_nat$','to_nat$'('i$')))
    | ~ spl310_5
    | ~ spl310_118 ),
    inference(resolution,[],[f4866,f4250]) ).

tff(f4973,plain,
    spl310_141,
    inference(avatar_split_clause,[],[f3901,f4971]) ).

tff(f3901,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP56(X0,X1) ),
    inference(evaluation,[],[f3298]) ).

tff(f3298,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $less(0,0)
      | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
      | ~ sP56(X0,X1) ),
    inference(cnf_transformation,[],[f2067]) ).

tff(f4969,plain,
    spl310_140,
    inference(avatar_split_clause,[],[f3012,f4967]) ).

tff(f4967,plain,
    ( spl310_140
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ( X0 = X2 )
        | ( $sum(X0,X1) != $sum(X2,X1) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_140])]) ).

tff(f3012,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ( X0 = X2 )
      | ( $sum(X0,X1) != $sum(X2,X1) ) ),
    inference(cnf_transformation,[],[f1181]) ).

tff(f1181,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( X0 = X2 )
      | ( $sum(X0,X1) != $sum(X2,X1) ) ),
    inference(ennf_transformation,[],[f42]) ).

tff(f42,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( $sum(X0,X1) = $sum(X2,X1) )
     => ( X0 = X2 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom40) ).

tff(f4965,plain,
    spl310_139,
    inference(avatar_split_clause,[],[f3011,f4963]) ).

tff(f4963,plain,
    ( spl310_139
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ( X1 = X2 )
        | ( $sum(X0,X1) != $sum(X0,X2) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_139])]) ).

tff(f3011,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ( X1 = X2 )
      | ( $sum(X0,X1) != $sum(X0,X2) ) ),
    inference(cnf_transformation,[],[f1180]) ).

tff(f1180,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( X1 = X2 )
      | ( $sum(X0,X1) != $sum(X0,X2) ) ),
    inference(ennf_transformation,[],[f44]) ).

tff(f44,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( $sum(X0,X1) = $sum(X0,X2) )
     => ( X1 = X2 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom42) ).

tff(f4961,plain,
    spl310_138,
    inference(avatar_split_clause,[],[f3004,f4959]) ).

tff(f3004,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( $less($sum(X2,X0),$sum(X2,X1))
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f1170]) ).

tff(f1170,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $less($sum(X2,X0),$sum(X2,X1))
      | ~ $less(X0,X1) ),
    inference(ennf_transformation,[],[f506]) ).

tff(f506,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $less(X0,X1)
     => $less($sum(X2,X0),$sum(X2,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom504) ).

tff(f4957,plain,
    spl310_137,
    inference(avatar_split_clause,[],[f3001,f4955]) ).

tff(f3001,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less($sum(X2,X1),$sum(X2,X0))
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f1167]) ).

tff(f1167,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less($sum(X2,X1),$sum(X2,X0))
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f739]) ).

tff(f739,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less(X1,X0)
     => ~ $less($sum(X2,X1),$sum(X2,X0)) ),
    inference(theory_normalization,[],[f30]) ).

tff(f30,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $lesseq(X0,X1)
     => $lesseq($sum(X2,X0),$sum(X2,X1)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom28) ).

tff(f4953,plain,
    spl310_136,
    inference(avatar_split_clause,[],[f3000,f4951]) ).

tff(f3000,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less($sum(X1,X2),$sum(X0,X2))
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f1166]) ).

tff(f1166,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less($sum(X1,X2),$sum(X0,X2))
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f738]) ).

tff(f738,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less(X1,X0)
     => ~ $less($sum(X1,X2),$sum(X0,X2)) ),
    inference(theory_normalization,[],[f29]) ).

tff(f29,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( $lesseq(X0,X1)
     => $lesseq($sum(X0,X2),$sum(X1,X2)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom27) ).

tff(f4949,plain,
    spl310_135,
    inference(avatar_split_clause,[],[f2745,f4947]) ).

tff(f4947,plain,
    ( spl310_135
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ~ 'fun_app$c'('member$a'(sK112(X0,X1)),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_135])]) ).

tff(f2745,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | ~ 'fun_app$c'('member$a'(sK112(X0,X1)),X1) ),
    inference(cnf_transformation,[],[f1830]) ).

tff(f1830,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ( ~ 'fun_app$c'('member$a'(sK112(X0,X1)),X1)
          & 'fun_app$c'('member$a'(sK112(X0,X1)),X0) ) )
      & ( ! [X3: $int] :
            ( 'fun_app$c'('member$a'(X3),X1)
            | ~ 'fun_app$c'('member$a'(X3),X0) )
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK112])],[f1828,f1829]) ).

tff(f1829,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ? [X2: $int] :
          ( ~ 'fun_app$c'('member$a'(X2),X1)
          & 'fun_app$c'('member$a'(X2),X0) )
     => ( ~ 'fun_app$c'('member$a'(sK112(X0,X1)),X1)
        & 'fun_app$c'('member$a'(sK112(X0,X1)),X0) ) ),
    introduced(choice_axiom,[]) ).

tff(f1828,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ? [X2: $int] :
            ( ~ 'fun_app$c'('member$a'(X2),X1)
            & 'fun_app$c'('member$a'(X2),X0) ) )
      & ( ! [X3: $int] :
            ( 'fun_app$c'('member$a'(X3),X1)
            | ~ 'fun_app$c'('member$a'(X3),X0) )
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    inference(rectify,[],[f1827]) ).

tff(f1827,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ? [X2: $int] :
            ( ~ 'fun_app$c'('member$a'(X2),X1)
            & 'fun_app$c'('member$a'(X2),X0) ) )
      & ( ! [X2: $int] :
            ( 'fun_app$c'('member$a'(X2),X1)
            | ~ 'fun_app$c'('member$a'(X2),X0) )
        | ~ 'fun_app$c'('less_eq$a'(X0),X1) ) ),
    inference(nnf_transformation,[],[f1154]) ).

tff(f1154,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
    <=> ! [X2: $int] :
          ( 'fun_app$c'('member$a'(X2),X1)
          | ~ 'fun_app$c'('member$a'(X2),X0) ) ),
    inference(ennf_transformation,[],[f580]) ).

tff(f580,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
    <=> ! [X2: $int] :
          ( 'fun_app$c'('member$a'(X2),X0)
         => 'fun_app$c'('member$a'(X2),X1) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom578) ).

tff(f4945,plain,
    spl310_134,
    inference(avatar_split_clause,[],[f2744,f4943]) ).

tff(f4943,plain,
    ( spl310_134
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'(X0),X1)
        | 'fun_app$c'('member$a'(sK112(X0,X1)),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_134])]) ).

tff(f2744,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | 'fun_app$c'('member$a'(sK112(X0,X1)),X0) ),
    inference(cnf_transformation,[],[f1830]) ).

tff(f4941,plain,
    spl310_133,
    inference(avatar_split_clause,[],[f2742,f4939]) ).

tff(f4939,plain,
    ( spl310_133
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ~ 'fun_app$c'('member$a'(sK111(X0,X1)),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_133])]) ).

tff(f2742,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | ~ 'fun_app$c'('member$a'(sK111(X0,X1)),X1) ),
    inference(cnf_transformation,[],[f1826]) ).

tff(f4937,plain,
    spl310_132,
    inference(avatar_split_clause,[],[f2741,f4935]) ).

tff(f4935,plain,
    ( spl310_132
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'(X0),X1)
        | 'fun_app$c'('member$a'(sK111(X0,X1)),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_132])]) ).

tff(f2741,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | 'fun_app$c'('member$a'(sK111(X0,X1)),X0) ),
    inference(cnf_transformation,[],[f1826]) ).

tff(f4930,plain,
    spl310_131,
    inference(avatar_split_clause,[],[f2649,f4928]) ).

tff(f4928,plain,
    ( spl310_131
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'(X0),X1)
        | ~ 'fun_app$c'('member$a'(sK105(X0,X1)),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_131])]) ).

tff(f2649,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | ~ 'fun_app$c'('member$a'(sK105(X0,X1)),X1) ),
    inference(cnf_transformation,[],[f1760]) ).

tff(f1760,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | ( ~ 'fun_app$c'('member$a'(sK105(X0,X1)),X1)
        & 'fun_app$c'('member$a'(sK105(X0,X1)),X0) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK105])],[f1147,f1759]) ).

tff(f1759,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ? [X2: $int] :
          ( ~ 'fun_app$c'('member$a'(X2),X1)
          & 'fun_app$c'('member$a'(X2),X0) )
     => ( ~ 'fun_app$c'('member$a'(sK105(X0,X1)),X1)
        & 'fun_app$c'('member$a'(sK105(X0,X1)),X0) ) ),
    introduced(choice_axiom,[]) ).

tff(f1147,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | ? [X2: $int] :
          ( ~ 'fun_app$c'('member$a'(X2),X1)
          & 'fun_app$c'('member$a'(X2),X0) ) ),
    inference(ennf_transformation,[],[f570]) ).

tff(f570,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( ! [X2: $int] :
          ( 'fun_app$c'('member$a'(X2),X0)
         => 'fun_app$c'('member$a'(X2),X1) )
     => 'fun_app$c'('less_eq$a'(X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom568) ).

tff(f4926,plain,
    spl310_130,
    inference(avatar_split_clause,[],[f2648,f4924]) ).

tff(f4924,plain,
    ( spl310_130
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('less_eq$a'(X0),X1)
        | 'fun_app$c'('member$a'(sK105(X0,X1)),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_130])]) ).

tff(f2648,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('less_eq$a'(X0),X1)
      | 'fun_app$c'('member$a'(sK105(X0,X1)),X0) ),
    inference(cnf_transformation,[],[f1760]) ).

tff(f4922,plain,
    spl310_129,
    inference(avatar_split_clause,[],[f2576,f4920]) ).

tff(f4920,plain,
    ( spl310_129
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP1(X0,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_129])]) ).

tff(f2576,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP1(X0,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1589]) ).

tff(f1589,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( sP1(X0,X1)
        & sP0(X0,X1) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(definition_folding,[],[f1066,f1588,f1587]) ).

tff(f1587,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( 0 = 'fun_app$e'('of_nat$',X0) )
      | ( ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
        & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ( ~ $less('fun_app$e'('of_nat$',X1),0)
        & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
      | ~ sP0(X0,X1) ),
    introduced(predicate_definition_introduction,[new_symbols(naming,[sP0])]) ).

tff(f1066,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X1),0)
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( 0 = 'fun_app$e'('of_nat$',X0) )
          | ( ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less('fun_app$e'('of_nat$',X1),0)
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(flattening,[],[f1065]) ).

tff(f1065,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ( ( ( ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) )
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) )
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) )
          | ( ( $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
              | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
            & ( $less('fun_app$e'('of_nat$',X1),0)
              | ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
        & ( ( 0 = 'fun_app$e'('of_nat$',X0) )
          | ( ~ $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))
            & ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) )
          | ( ~ $less('fun_app$e'('of_nat$',X1),0)
            & $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ) )
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(ennf_transformation,[],[f876]) ).

tff(f876,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X1),0) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus($sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0))))) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X1),0) ) )
         => ( 0 = 'fun_app$e'('of_nat$',X0) ) ) ) ),
    inference(theory_normalization,[],[f677]) ).

tff(f677,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X1),0) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus($difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)))) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( 'fun_app$e'('of_nat$',X0) = $sum('fun_app$e'('of_nat$',X1),$uminus(0)) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X1),$sum('fun_app$e'('of_nat$',X1),$uminus('fun_app$e'('of_nat$',X0)))) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X1),0) ) )
         => ( 0 = 'fun_app$e'('of_nat$',X0) ) ) ) ),
    inference(theory_normalization,[],[f123]) ).

tff(f123,axiom,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( $lesseq('fun_app$e'('of_nat$',X0),'fun_app$e'('of_nat$',X1))
     => ( ( ~ ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X1),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
              & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
               => $less('fun_app$e'('of_nat$',X1),0) ) )
         => ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( 'fun_app$e'('of_nat$',X0) = $difference('fun_app$e'('of_nat$',X1),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) ) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => ( 'fun_app$e'('of_nat$',X0) = $difference('fun_app$e'('of_nat$',X1),0) ) ) ) )
        & ( ( ( ~ $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X1),$difference('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))) )
            & ( $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0))
             => $less('fun_app$e'('of_nat$',X1),0) ) )
         => ( 0 = 'fun_app$e'('of_nat$',X0) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom121) ).

tff(f4918,plain,
    spl310_128,
    inference(avatar_split_clause,[],[f2575,f4916]) ).

tff(f4916,plain,
    ( spl310_128
  <=> ! [X0: 'Nat$',X1: 'Nat$'] :
        ( sP0(X0,X1)
        | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_128])]) ).

tff(f2575,plain,
    ! [X0: 'Nat$',X1: 'Nat$'] :
      ( sP0(X0,X1)
      | $less('fun_app$e'('of_nat$',X1),'fun_app$e'('of_nat$',X0)) ),
    inference(cnf_transformation,[],[f1589]) ).

tff(f4914,plain,
    spl310_127,
    inference(avatar_split_clause,[],[f2534,f4912]) ).

tff(f2534,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f1039]) ).

tff(f1039,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f426]) ).

tff(f426,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),X1)
     => 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom424) ).

tff(f4910,plain,
    spl310_126,
    inference(avatar_split_clause,[],[f2532,f4908]) ).

tff(f2532,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(cnf_transformation,[],[f1037]) ).

tff(f1037,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | ~ 'fun_app$d'('fun_app$g'('less$',X0),X1) ),
    inference(ennf_transformation,[],[f284]) ).

tff(f284,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X0),X1)
     => ~ 'fun_app$d'('fun_app$g'('less$',X1),X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom282) ).

tff(f4906,plain,
    spl310_125,
    inference(avatar_split_clause,[],[f2530,f4904]) ).

tff(f2530,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f1035]) ).

tff(f1035,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( ~ 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | ~ 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(ennf_transformation,[],[f413]) ).

tff(f413,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X1)
     => ~ 'fun_app$d'('fun_app$g'('less$',X1),X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom411) ).

tff(f4902,plain,
    spl310_124,
    inference(avatar_split_clause,[],[f2478,f4900]) ).

tff(f2478,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f238]) ).

tff(f238,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom236) ).

tff(f4898,plain,
    spl310_123,
    inference(avatar_split_clause,[],[f2477,f4896]) ).

tff(f2477,plain,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    inference(cnf_transformation,[],[f499]) ).

tff(f499,axiom,
    ! [X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less$',X1),X0)
      | 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom497) ).

tff(f4894,plain,
    spl310_122,
    inference(avatar_split_clause,[],[f2419,f4892]) ).

tff(f4892,plain,
    ( spl310_122
  <=> ! [X0: 'Nat$'] :
        ( $less(0,'fun_app$e'('of_nat$',X0))
        | ( 0 = 'fun_app$e'('of_nat$',X0) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_122])]) ).

tff(f2419,plain,
    ! [X0: 'Nat$'] :
      ( $less(0,'fun_app$e'('of_nat$',X0))
      | ( 0 = 'fun_app$e'('of_nat$',X0) ) ),
    inference(cnf_transformation,[],[f1707]) ).

tff(f1707,plain,
    ! [X0: 'Nat$'] :
      ( ( ( 0 != 'fun_app$e'('of_nat$',X0) )
        | ~ $less(0,'fun_app$e'('of_nat$',X0)) )
      & ( $less(0,'fun_app$e'('of_nat$',X0))
        | ( 0 = 'fun_app$e'('of_nat$',X0) ) ) ),
    inference(nnf_transformation,[],[f609]) ).

tff(f609,axiom,
    ! [X0: 'Nat$'] :
      ( ( 0 != 'fun_app$e'('of_nat$',X0) )
    <=> $less(0,'fun_app$e'('of_nat$',X0)) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom607) ).

tff(f4890,plain,
    ( spl310_121
    | ~ spl310_58
    | ~ spl310_92 ),
    inference(avatar_split_clause,[],[f4859,f4696,f4490,f4888]) ).

tff(f4859,plain,
    ( ! [X0: $int] :
        ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),X0)
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    | ~ spl310_58
    | ~ spl310_92 ),
    inference(resolution,[],[f4697,f4492]) ).

tff(f4886,plain,
    spl310_120,
    inference(avatar_split_clause,[],[f872,f4884]) ).

tff(f4884,plain,
    ( spl310_120
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X0,X1)
        | $less($sum(X0,X2),$sum(X1,X2)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_120])]) ).

tff(f872,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X1)
      | $less($sum(X0,X2),$sum(X1,X2)) ),
    introduced(theory_axiom_145,[]) ).

tff(f4882,plain,
    spl310_119,
    inference(avatar_split_clause,[],[f867,f4880]) ).

tff(f4880,plain,
    ( spl310_119
  <=> ! [X0: $int,X1: $int] : ( $uminus($sum(X0,X1)) = $sum($uminus(X1),$uminus(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_119])]) ).

tff(f4867,plain,
    ( spl310_118
    | ~ spl310_58
    | ~ spl310_80 ),
    inference(avatar_split_clause,[],[f4842,f4642,f4490,f4865]) ).

tff(f4842,plain,
    ( ! [X0: $int] :
        ( ~ $less(X0,'fun_app$e'('of_nat$','to_nat$'('x$')))
        | $less(X0,'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    | ~ spl310_58
    | ~ spl310_80 ),
    inference(resolution,[],[f4492,f4643]) ).

tff(f4841,plain,
    ( spl310_1
    | spl310_117
    | spl310_58
    | ~ spl310_81 ),
    inference(avatar_split_clause,[],[f4805,f4652,f4490,f4838,f4231]) ).

tff(f4231,plain,
    ( spl310_1
  <=> ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = 'fun_app$e'('of_nat$','to_nat$'('i$')) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_1])]) ).

tff(f4652,plain,
    ( spl310_81
  <=> ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | $less(X1,X0)
        | ( X0 = X1 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_81])]) ).

tff(f4805,plain,
    ( $less('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','to_nat$'('x$')))
    | ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = 'fun_app$e'('of_nat$','to_nat$'('i$')) )
    | spl310_58
    | ~ spl310_81 ),
    inference(resolution,[],[f4653,f4491]) ).

tff(f4653,plain,
    ( ! [X0: $int,X1: $int] :
        ( $less(X1,X0)
        | $less(X0,X1)
        | ( X0 = X1 ) )
    | ~ spl310_81 ),
    inference(avatar_component_clause,[],[f4652]) ).

tff(f4794,plain,
    spl310_116,
    inference(avatar_split_clause,[],[f4223,f4792]) ).

tff(f4223,plain,
    ! [X0: $int,X1: $int] :
      ( ( 0 != $sum(X0,$uminus(X1)) )
      | ( X0 = X1 ) ),
    inference(forward_demodulation,[],[f3824,f868]) ).

tff(f868,plain,
    ! [X0: $int] : ( 0 = $sum(X0,$uminus(X0)) ),
    introduced(theory_axiom_140,[]) ).

tff(f3824,plain,
    ! [X3: $int,X0: $int,X1: $int] :
      ( ( X0 = X1 )
      | ( $sum(X0,$uminus(X1)) != $sum(X3,$uminus(X3)) ) ),
    inference(equality_resolution,[],[f3492]) ).

tff(f3492,plain,
    ! [X2: $int,X3: $int,X0: $int,X1: $int] :
      ( ( X0 = X1 )
      | ( X2 != X3 )
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(cnf_transformation,[],[f2201]) ).

tff(f2201,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ( ( X0 = X1 )
          | ( X2 != X3 ) )
        & ( ( X2 = X3 )
          | ( X0 != X1 ) ) )
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(nnf_transformation,[],[f1393]) ).

tff(f1393,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( ( X0 = X1 )
      <=> ( X2 = X3 ) )
      | ( $sum(X0,$uminus(X1)) != $sum(X2,$uminus(X3)) ) ),
    inference(ennf_transformation,[],[f814]) ).

tff(f814,plain,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $sum(X0,$uminus(X1)) = $sum(X2,$uminus(X3)) )
     => ( ( X0 = X1 )
      <=> ( X2 = X3 ) ) ),
    inference(theory_normalization,[],[f79]) ).

tff(f79,axiom,
    ! [X0: $int,X1: $int,X2: $int,X3: $int] :
      ( ( $difference(X0,X1) = $difference(X2,X3) )
     => ( ( X0 = X1 )
      <=> ( X2 = X3 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom77) ).

tff(f4790,plain,
    spl310_115,
    inference(avatar_split_clause,[],[f3919,f4788]) ).

tff(f4788,plain,
    ( spl310_115
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less(1,'fun_app$e'('of_nat$',X0))
        | ~ sP25(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_115])]) ).

tff(f3919,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(1,'fun_app$e'('of_nat$',X0))
      | ~ sP25(X0,X1,X2) ),
    inference(evaluation,[],[f3062]) ).

tff(f3062,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less($sum(0,1),'fun_app$e'('of_nat$',X0))
      | ~ sP25(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1976]) ).

tff(f4786,plain,
    spl310_114,
    inference(avatar_split_clause,[],[f3915,f4784]) ).

tff(f4784,plain,
    ( spl310_114
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( ~ $less('fun_app$e'('of_nat$',X0),1)
        | ~ sP28(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_114])]) ).

tff(f3915,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X0),1)
      | ~ sP28(X0,X1,X2) ),
    inference(evaluation,[],[f3075]) ).

tff(f3075,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less('fun_app$e'('of_nat$',X0),$sum(0,1))
      | ~ sP28(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1981]) ).

tff(f4782,plain,
    ( spl310_113
    | ~ spl310_66
    | ~ spl310_79 ),
    inference(avatar_split_clause,[],[f4650,f4626,f4524,f4780]) ).

tff(f4780,plain,
    ( spl310_113
  <=> ! [X0: $int] : $less($sum(X0,-1),X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_113])]) ).

tff(f4650,plain,
    ( ! [X0: $int] : $less($sum(X0,-1),X0)
    | ~ spl310_66
    | ~ spl310_79 ),
    inference(evaluation,[],[f4649]) ).

tff(f4649,plain,
    ( ! [X0: $int] : $less($sum(X0,$uminus(1)),X0)
    | ~ spl310_66
    | ~ spl310_79 ),
    inference(superposition,[],[f4627,f4525]) ).

tff(f4778,plain,
    spl310_112,
    inference(avatar_split_clause,[],[f3367,f4776]) ).

tff(f4776,plain,
    ( spl310_112
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP70(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_112])]) ).

tff(f3367,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP70(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2109]) ).

tff(f2109,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ~ sP70(X0,X1,X2) ),
    inference(rectify,[],[f2108]) ).

tff(f2108,plain,
    ! [X2: 'N$',X1: 'N$',X0: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
      | ~ sP70(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1678]) ).

tff(f4774,plain,
    spl310_111,
    inference(avatar_split_clause,[],[f3366,f4772]) ).

tff(f4772,plain,
    ( spl310_111
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        | ~ sP70(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_111])]) ).

tff(f3366,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
      | ~ sP70(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2109]) ).

tff(f4770,plain,
    spl310_110,
    inference(avatar_split_clause,[],[f3365,f4768]) ).

tff(f4768,plain,
    ( spl310_110
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP71(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_110])]) ).

tff(f3365,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP71(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2107]) ).

tff(f2107,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ~ sP71(X0,X1,X2) ),
    inference(rectify,[],[f2106]) ).

tff(f2106,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
      | ~ sP71(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1679]) ).

tff(f4766,plain,
    spl310_109,
    inference(avatar_split_clause,[],[f3364,f4764]) ).

tff(f4764,plain,
    ( spl310_109
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        | ~ sP71(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_109])]) ).

tff(f3364,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
      | ~ sP71(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2107]) ).

tff(f4762,plain,
    spl310_108,
    inference(avatar_split_clause,[],[f3363,f4760]) ).

tff(f4760,plain,
    ( spl310_108
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP72(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_108])]) ).

tff(f3363,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP72(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2105]) ).

tff(f2105,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ~ sP72(X0,X1,X2) ),
    inference(rectify,[],[f2104]) ).

tff(f2104,plain,
    ! [X1: 'N$',X2: 'N$',X0: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ~ sP72(X1,X2,X0) ),
    inference(nnf_transformation,[],[f1680]) ).

tff(f4758,plain,
    spl310_107,
    inference(avatar_split_clause,[],[f3362,f4756]) ).

tff(f4756,plain,
    ( spl310_107
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        | ~ sP72(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_107])]) ).

tff(f3362,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
      | ~ sP72(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2105]) ).

tff(f4754,plain,
    spl310_106,
    inference(avatar_split_clause,[],[f3361,f4752]) ).

tff(f4752,plain,
    ( spl310_106
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP73(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_106])]) ).

tff(f3361,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP73(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2103]) ).

tff(f2103,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ~ sP73(X0,X1,X2) ),
    inference(nnf_transformation,[],[f1681]) ).

tff(f4750,plain,
    spl310_105,
    inference(avatar_split_clause,[],[f3360,f4748]) ).

tff(f4748,plain,
    ( spl310_105
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        | ~ sP73(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_105])]) ).

tff(f3360,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
      | ~ sP73(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2103]) ).

tff(f4746,plain,
    spl310_104,
    inference(avatar_split_clause,[],[f3355,f4744]) ).

tff(f4744,plain,
    ( spl310_104
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP66(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_104])]) ).

tff(f3355,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP66(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2102]) ).

tff(f2102,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ~ sP66(X0,X1,X2) ),
    inference(rectify,[],[f2101]) ).

tff(f2101,plain,
    ! [X0: 'N$',X2: 'N$',X1: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X1) )
      | ~ sP66(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1673]) ).

tff(f4742,plain,
    spl310_103,
    inference(avatar_split_clause,[],[f3354,f4740]) ).

tff(f4740,plain,
    ( spl310_103
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        | ~ sP66(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_103])]) ).

tff(f3354,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
      | ~ sP66(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2102]) ).

tff(f4738,plain,
    ( spl310_102
    | ~ spl310_27
    | ~ spl310_79 ),
    inference(avatar_split_clause,[],[f4647,f4626,f4343,f4736]) ).

tff(f4736,plain,
    ( spl310_102
  <=> ! [X0: $int] : $less(X0,$sum(1,X0)) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_102])]) ).

tff(f4647,plain,
    ( ! [X0: $int] : $less(X0,$sum(1,X0))
    | ~ spl310_27
    | ~ spl310_79 ),
    inference(superposition,[],[f4627,f4344]) ).

tff(f4734,plain,
    spl310_101,
    inference(avatar_split_clause,[],[f3353,f4732]) ).

tff(f4732,plain,
    ( spl310_101
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP67(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_101])]) ).

tff(f3353,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP67(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2100]) ).

tff(f2100,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ~ sP67(X0,X1,X2) ),
    inference(rectify,[],[f2099]) ).

tff(f2099,plain,
    ! [X1: 'N$',X2: 'N$',X0: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X2),X1)
        & 'fun_app$d'('fun_app$g'('less_eq$',X1),X0) )
      | ~ sP67(X1,X2,X0) ),
    inference(nnf_transformation,[],[f1674]) ).

tff(f4730,plain,
    spl310_100,
    inference(avatar_split_clause,[],[f3352,f4728]) ).

tff(f4728,plain,
    ( spl310_100
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        | ~ sP67(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_100])]) ).

tff(f3352,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
      | ~ sP67(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2100]) ).

tff(f4726,plain,
    spl310_99,
    inference(avatar_split_clause,[],[f3351,f4724]) ).

tff(f4724,plain,
    ( spl310_99
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP68(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_99])]) ).

tff(f3351,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP68(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2098]) ).

tff(f2098,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ~ sP68(X0,X1,X2) ),
    inference(rectify,[],[f2097]) ).

tff(f2097,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        & 'fun_app$d'('fun_app$g'('less_eq$',X2),X1) )
      | ~ sP68(X2,X0,X1) ),
    inference(nnf_transformation,[],[f1675]) ).

tff(f4722,plain,
    spl310_98,
    inference(avatar_split_clause,[],[f3350,f4720]) ).

tff(f4720,plain,
    ( spl310_98
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        | ~ sP68(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_98])]) ).

tff(f3350,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
      | ~ sP68(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2098]) ).

tff(f4718,plain,
    spl310_97,
    inference(avatar_split_clause,[],[f3349,f4716]) ).

tff(f4716,plain,
    ( spl310_97
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        | ~ sP69(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_97])]) ).

tff(f3349,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
      | ~ sP69(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2096]) ).

tff(f2096,plain,
    ! [X0: 'N$',X1: 'N$',X2: 'N$'] :
      ( ( 'fun_app$d'('fun_app$g'('less_eq$',X1),X0)
        & 'fun_app$d'('fun_app$g'('less_eq$',X0),X2) )
      | ~ sP69(X0,X1,X2) ),
    inference(nnf_transformation,[],[f1676]) ).

tff(f4714,plain,
    spl310_96,
    inference(avatar_split_clause,[],[f3348,f4712]) ).

tff(f4712,plain,
    ( spl310_96
  <=> ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
        ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
        | ~ sP69(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_96])]) ).

tff(f3348,plain,
    ! [X2: 'N$',X0: 'N$',X1: 'N$'] :
      ( 'fun_app$d'('fun_app$g'('less_eq$',X0),X2)
      | ~ sP69(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2096]) ).

tff(f4710,plain,
    spl310_95,
    inference(avatar_split_clause,[],[f3321,f4708]) ).

tff(f4708,plain,
    ( spl310_95
  <=> ! [X0: 'Nat_nat_bool_fun_fun$'] :
        ( ~ 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK137(X0))
        | ~ sP60(X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_95])]) ).

tff(f3321,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( ~ 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK137(X0))
      | ~ sP60(X0) ),
    inference(cnf_transformation,[],[f2075]) ).

tff(f4706,plain,
    spl310_94,
    inference(avatar_split_clause,[],[f3320,f4704]) ).

tff(f4704,plain,
    ( spl310_94
  <=> ! [X0: 'Nat_nat_bool_fun_fun$'] :
        ( 'fun_app$j'('fun_app$q'(X0,sK136(X0)),sK137(X0))
        | ~ sP60(X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_94])]) ).

tff(f3320,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X0,sK136(X0)),sK137(X0))
      | ~ sP60(X0) ),
    inference(cnf_transformation,[],[f2075]) ).

tff(f4702,plain,
    spl310_93,
    inference(avatar_split_clause,[],[f3319,f4700]) ).

tff(f4700,plain,
    ( spl310_93
  <=> ! [X0: 'Nat_nat_bool_fun_fun$'] :
        ( 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK136(X0))
        | ~ sP60(X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_93])]) ).

tff(f3319,plain,
    ! [X0: 'Nat_nat_bool_fun_fun$'] :
      ( 'fun_app$j'('fun_app$q'(X0,sK135(X0)),sK136(X0))
      | ~ sP60(X0) ),
    inference(cnf_transformation,[],[f2075]) ).

tff(f4698,plain,
    spl310_92,
    inference(avatar_split_clause,[],[f3137,f4696]) ).

tff(f3137,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X2,X0)
      | $less(X2,X1)
      | $less(X1,X0) ),
    inference(cnf_transformation,[],[f1234]) ).

tff(f1234,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less(X2,X0)
      | $less(X2,X1)
      | $less(X1,X0) ),
    inference(flattening,[],[f1233]) ).

tff(f1233,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ~ $less(X2,X0)
      | $less(X2,X1)
      | $less(X1,X0) ),
    inference(ennf_transformation,[],[f768]) ).

tff(f768,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X2,X1)
        & ~ $less(X1,X0) )
     => ~ $less(X2,X0) ),
    inference(theory_normalization,[],[f221]) ).

tff(f221,axiom,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( $lesseq(X1,X2)
        & $lesseq(X0,X1) )
     => $lesseq(X0,X2) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom219) ).

tff(f4694,plain,
    ( spl310_91
    | ~ spl310_24
    | ~ spl310_79 ),
    inference(avatar_split_clause,[],[f4646,f4626,f4326,f4692]) ).

tff(f4692,plain,
    ( spl310_91
  <=> ! [X0: $int] : ~ $less($sum(X0,1),X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_91])]) ).

tff(f4646,plain,
    ( ! [X0: $int] : ~ $less($sum(X0,1),X0)
    | ~ spl310_24
    | ~ spl310_79 ),
    inference(resolution,[],[f4627,f4327]) ).

tff(f4690,plain,
    spl310_90,
    inference(avatar_split_clause,[],[f2947,f4688]) ).

tff(f2947,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP8(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1946]) ).

tff(f4686,plain,
    spl310_89,
    inference(avatar_split_clause,[],[f2919,f4684]) ).

tff(f2919,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( ~ $less(0,'fun_app$e'('of_nat$',X0))
      | ~ sP4(X0,X1,X2) ),
    inference(cnf_transformation,[],[f1934]) ).

tff(f4682,plain,
    spl310_88,
    inference(avatar_split_clause,[],[f2446,f4680]) ).

tff(f4680,plain,
    ( spl310_88
  <=> ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'('fun_app$f'('minus$'(X0),X1)),X1) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_88])]) ).

tff(f2446,plain,
    ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'('fun_app$f'('minus$'(X0),X1)),X1) = X0 ),
    inference(cnf_transformation,[],[f12]) ).

tff(f12,axiom,
    ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('plus$'('fun_app$f'('minus$'(X0),X1)),X1) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom10) ).

tff(f4678,plain,
    spl310_87,
    inference(avatar_split_clause,[],[f2445,f4676]) ).

tff(f2445,plain,
    ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X1)),X1) = X0 ),
    inference(cnf_transformation,[],[f10]) ).

tff(f10,axiom,
    ! [X0: 'N$',X1: 'N$'] : ( 'fun_app$f'('minus$'('fun_app$f'('plus$'(X0),X1)),X1) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom8) ).

tff(f4674,plain,
    spl310_86,
    inference(avatar_split_clause,[],[f2444,f4672]) ).

tff(f4672,plain,
    ( spl310_86
  <=> ! [X0: 'Int_set$',X1: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),X1) = 'fun_app$h'('plus$b'(X1),X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_86])]) ).

tff(f2444,plain,
    ! [X0: 'Int_set$',X1: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),X1) = 'fun_app$h'('plus$b'(X1),X0) ),
    inference(cnf_transformation,[],[f56]) ).

tff(f56,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),X1) = 'fun_app$h'('plus$b'(X1),X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom54) ).

tff(f4670,plain,
    spl310_85,
    inference(avatar_split_clause,[],[f2443,f4668]) ).

tff(f4668,plain,
    ( spl310_85
  <=> ! [X0: $int,X1: $int] : ( 'fun_app$b'('uud$'(X0),X1) = $sum(X0,$uminus(X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_85])]) ).

tff(f2443,plain,
    ! [X0: $int,X1: $int] : ( 'fun_app$b'('uud$'(X0),X1) = $sum(X0,$uminus(X1)) ),
    inference(cnf_transformation,[],[f645]) ).

tff(f645,plain,
    ! [X0: $int,X1: $int] : ( 'fun_app$b'('uud$'(X0),X1) = $sum(X0,$uminus(X1)) ),
    inference(theory_normalization,[],[f2]) ).

tff(f2,axiom,
    ! [X0: $int,X1: $int] : ( 'fun_app$b'('uud$'(X0),X1) = $difference(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom1) ).

tff(f4666,plain,
    spl310_84,
    inference(avatar_split_clause,[],[f2409,f4664]) ).

tff(f2409,plain,
    ! [X0: $int] :
      ( ( 0 = 'fun_app$e'('of_nat$','nat$'(X0)) )
      | ~ $less(X0,0) ),
    inference(cnf_transformation,[],[f984]) ).

tff(f984,plain,
    ! [X0: $int] :
      ( ( ( 0 = 'fun_app$e'('of_nat$','nat$'(X0)) )
        | ~ $less(X0,0) )
      & ( ( 'fun_app$e'('of_nat$','nat$'(X0)) = X0 )
        | $less(X0,0) ) ),
    inference(ennf_transformation,[],[f637]) ).

tff(f637,plain,
    ! [X0: $int] :
      ( ( $less(X0,0)
       => ( 0 = 'fun_app$e'('of_nat$','nat$'(X0)) ) )
      & ( ~ $less(X0,0)
       => ( 'fun_app$e'('of_nat$','nat$'(X0)) = X0 ) ) ),
    inference(theory_normalization,[],[f619]) ).

tff(f619,axiom,
    ! [X0: $int] :
      ( ( ~ $lesseq(0,X0)
       => ( 0 = 'fun_app$e'('of_nat$','nat$'(X0)) ) )
      & ( $lesseq(0,X0)
       => ( 'fun_app$e'('of_nat$','nat$'(X0)) = X0 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom617) ).

tff(f4662,plain,
    spl310_83,
    inference(avatar_split_clause,[],[f2408,f4660]) ).

tff(f2408,plain,
    ! [X0: $int] :
      ( ( 'fun_app$e'('of_nat$','nat$'(X0)) = X0 )
      | $less(X0,0) ),
    inference(cnf_transformation,[],[f984]) ).

tff(f4658,plain,
    spl310_82,
    inference(avatar_split_clause,[],[f2403,f4656]) ).

tff(f4656,plain,
    ( spl310_82
  <=> ! [X0: 'Nat$'] : ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X0),1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_82])]) ).

tff(f2403,plain,
    ! [X0: 'Nat$'] : ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X0),1) ),
    inference(cnf_transformation,[],[f394]) ).

tff(f394,axiom,
    ! [X0: 'Nat$'] : ( 'fun_app$e'('of_nat$',X0) != $sum('fun_app$e'('of_nat$',X0),1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom392) ).

tff(f4654,plain,
    spl310_81,
    inference(avatar_split_clause,[],[f871,f4652]) ).

tff(f871,plain,
    ! [X0: $int,X1: $int] :
      ( $less(X0,X1)
      | $less(X1,X0)
      | ( X0 = X1 ) ),
    introduced(theory_axiom_144,[]) ).

tff(f4644,plain,
    spl310_80,
    inference(avatar_split_clause,[],[f870,f4642]) ).

tff(f870,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X1)
      | ~ $less(X1,X2)
      | $less(X0,X2) ),
    introduced(theory_axiom_143,[]) ).

tff(f4628,plain,
    ( spl310_79
    | ~ spl310_2
    | ~ spl310_60 ),
    inference(avatar_split_clause,[],[f4575,f4499,f4236,f4626]) ).

tff(f4575,plain,
    ( ! [X0: $int] : $less(X0,$sum(X0,1))
    | ~ spl310_2
    | ~ spl310_60 ),
    inference(resolution,[],[f4500,f4237]) ).

tff(f4574,plain,
    spl310_78,
    inference(avatar_split_clause,[],[f3383,f4572]) ).

tff(f4572,plain,
    ( spl310_78
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'(X0,sK152(X0,X1))
        | ~ sP74(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_78])]) ).

tff(f3383,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'(X0,sK152(X0,X1))
      | ~ sP74(X0,X1) ),
    inference(cnf_transformation,[],[f2131]) ).

tff(f4570,plain,
    spl310_77,
    inference(avatar_split_clause,[],[f3325,f4568]) ).

tff(f4568,plain,
    ( spl310_77
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( ~ 'fun_app$j'(X0,sK139(X0,X1))
        | ~ sP61(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_77])]) ).

tff(f3325,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( ~ 'fun_app$j'(X0,sK139(X0,X1))
      | ~ sP61(X0,X1) ),
    inference(cnf_transformation,[],[f2082]) ).

tff(f4566,plain,
    spl310_76,
    inference(avatar_split_clause,[],[f2877,f4564]) ).

tff(f2877,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,sK118(X0,X1))
      | ~ sP3(X0,X1) ),
    inference(cnf_transformation,[],[f1925]) ).

tff(f4562,plain,
    spl310_75,
    inference(avatar_split_clause,[],[f2871,f4560]) ).

tff(f4560,plain,
    ( spl310_75
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( sP2(X0,X1)
        | ~ 'fun_app$j'(X0,sK116(X0,X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_75])]) ).

tff(f2871,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( sP2(X0,X1)
      | ~ 'fun_app$j'(X0,sK116(X0,X1)) ),
    inference(cnf_transformation,[],[f1916]) ).

tff(f4558,plain,
    spl310_74,
    inference(avatar_split_clause,[],[f2787,f4556]) ).

tff(f2787,plain,
    ! [X0: $int,X1: 'Int_bool_fun$'] :
      ( 'fun_app$c'('member$a'(X0),'collect$b'(X1))
      | ~ 'fun_app$'(X1,X0) ),
    inference(cnf_transformation,[],[f1854]) ).

tff(f1854,plain,
    ! [X0: $int,X1: 'Int_bool_fun$'] :
      ( ( 'fun_app$c'('member$a'(X0),'collect$b'(X1))
        | ~ 'fun_app$'(X1,X0) )
      & ( 'fun_app$'(X1,X0)
        | ~ 'fun_app$c'('member$a'(X0),'collect$b'(X1)) ) ),
    inference(nnf_transformation,[],[f50]) ).

tff(f50,axiom,
    ! [X0: $int,X1: 'Int_bool_fun$'] :
      ( 'fun_app$c'('member$a'(X0),'collect$b'(X1))
    <=> 'fun_app$'(X1,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom48) ).

tff(f4554,plain,
    spl310_73,
    inference(avatar_split_clause,[],[f2786,f4552]) ).

tff(f4552,plain,
    ( spl310_73
  <=> ! [X0: $int,X1: 'Int_bool_fun$'] :
        ( 'fun_app$'(X1,X0)
        | ~ 'fun_app$c'('member$a'(X0),'collect$b'(X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_73])]) ).

tff(f2786,plain,
    ! [X0: $int,X1: 'Int_bool_fun$'] :
      ( 'fun_app$'(X1,X0)
      | ~ 'fun_app$c'('member$a'(X0),'collect$b'(X1)) ),
    inference(cnf_transformation,[],[f1854]) ).

tff(f4550,plain,
    spl310_72,
    inference(avatar_split_clause,[],[f2785,f4548]) ).

tff(f4548,plain,
    ( spl310_72
  <=> ! [X0: $int,X1: $int] :
        ( 'fun_app$'('fun_app$a'('uuc$',X0),X1)
        | ~ $less(X0,X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_72])]) ).

tff(f2785,plain,
    ! [X0: $int,X1: $int] :
      ( 'fun_app$'('fun_app$a'('uuc$',X0),X1)
      | ~ $less(X0,X1) ),
    inference(cnf_transformation,[],[f1853]) ).

tff(f1853,plain,
    ! [X0: $int,X1: $int] :
      ( ( 'fun_app$'('fun_app$a'('uuc$',X0),X1)
        | ~ $less(X0,X1) )
      & ( $less(X0,X1)
        | ~ 'fun_app$'('fun_app$a'('uuc$',X0),X1) ) ),
    inference(nnf_transformation,[],[f1]) ).

tff(f1,axiom,
    ! [X0: $int,X1: $int] :
      ( 'fun_app$'('fun_app$a'('uuc$',X0),X1)
    <=> $less(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom0) ).

tff(f4546,plain,
    spl310_71,
    inference(avatar_split_clause,[],[f2784,f4544]) ).

tff(f4544,plain,
    ( spl310_71
  <=> ! [X0: $int,X1: $int] :
        ( $less(X0,X1)
        | ~ 'fun_app$'('fun_app$a'('uuc$',X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_71])]) ).

tff(f2784,plain,
    ! [X0: $int,X1: $int] :
      ( $less(X0,X1)
      | ~ 'fun_app$'('fun_app$a'('uuc$',X0),X1) ),
    inference(cnf_transformation,[],[f1853]) ).

tff(f4542,plain,
    spl310_70,
    inference(avatar_split_clause,[],[f2780,f4540]) ).

tff(f2780,plain,
    ! [X0: $int,X1: $int] :
      ( ( 0 = X0 )
      | ( $sum(X0,X1) != X1 ) ),
    inference(cnf_transformation,[],[f1851]) ).

tff(f1851,plain,
    ! [X0: $int,X1: $int] :
      ( ( ( $sum(X0,X1) = X1 )
        | ( 0 != X0 ) )
      & ( ( 0 = X0 )
        | ( $sum(X0,X1) != X1 ) ) ),
    inference(nnf_transformation,[],[f599]) ).

tff(f599,axiom,
    ! [X0: $int,X1: $int] :
      ( ( $sum(X0,X1) = X1 )
    <=> ( 0 = X0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom597) ).

tff(f4538,plain,
    spl310_69,
    inference(avatar_split_clause,[],[f2776,f4536]) ).

tff(f2776,plain,
    ! [X0: $int,X1: $int] :
      ( ( 0 = X1 )
      | ( $sum(X0,X1) != X0 ) ),
    inference(cnf_transformation,[],[f1849]) ).

tff(f1849,plain,
    ! [X0: $int,X1: $int] :
      ( ( ( $sum(X0,X1) = X0 )
        | ( 0 != X1 ) )
      & ( ( 0 = X1 )
        | ( $sum(X0,X1) != X0 ) ) ),
    inference(nnf_transformation,[],[f598]) ).

tff(f598,axiom,
    ! [X0: $int,X1: $int] :
      ( ( $sum(X0,X1) = X0 )
    <=> ( 0 = X1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom596) ).

tff(f4534,plain,
    spl310_68,
    inference(avatar_split_clause,[],[f2737,f4532]) ).

tff(f2737,plain,
    ! [X0: 'Int_set$',X1: $int] :
      ( 'fun_app$'('uub$'(X0),X1)
      | ~ 'fun_app$c'('member$a'(X1),X0) ),
    inference(cnf_transformation,[],[f1821]) ).

tff(f1821,plain,
    ! [X0: 'Int_set$',X1: $int] :
      ( ( 'fun_app$'('uub$'(X0),X1)
        | ~ 'fun_app$c'('member$a'(X1),X0) )
      & ( 'fun_app$c'('member$a'(X1),X0)
        | ~ 'fun_app$'('uub$'(X0),X1) ) ),
    inference(nnf_transformation,[],[f4]) ).

tff(f4,axiom,
    ! [X0: 'Int_set$',X1: $int] :
      ( 'fun_app$'('uub$'(X0),X1)
    <=> 'fun_app$c'('member$a'(X1),X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom3) ).

tff(f4530,plain,
    spl310_67,
    inference(avatar_split_clause,[],[f2736,f4528]) ).

tff(f4528,plain,
    ( spl310_67
  <=> ! [X0: 'Int_set$',X1: $int] :
        ( 'fun_app$c'('member$a'(X1),X0)
        | ~ 'fun_app$'('uub$'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_67])]) ).

tff(f2736,plain,
    ! [X0: 'Int_set$',X1: $int] :
      ( 'fun_app$c'('member$a'(X1),X0)
      | ~ 'fun_app$'('uub$'(X0),X1) ),
    inference(cnf_transformation,[],[f1821]) ).

tff(f4526,plain,
    spl310_66,
    inference(avatar_split_clause,[],[f2442,f4524]) ).

tff(f2442,plain,
    ! [X0: $int,X1: $int] : ( $sum($sum(X0,$uminus(X1)),X1) = X0 ),
    inference(cnf_transformation,[],[f644]) ).

tff(f644,plain,
    ! [X0: $int,X1: $int] : ( $sum($sum(X0,$uminus(X1)),X1) = X0 ),
    inference(theory_normalization,[],[f13]) ).

tff(f13,axiom,
    ! [X0: $int,X1: $int] : ( $sum($difference(X0,X1),X1) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom11) ).

tff(f4522,plain,
    spl310_65,
    inference(avatar_split_clause,[],[f2440,f4520]) ).

tff(f2440,plain,
    ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X0)) = X1 ),
    inference(cnf_transformation,[],[f642]) ).

tff(f642,plain,
    ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X0)) = X1 ),
    inference(theory_normalization,[],[f15]) ).

tff(f15,axiom,
    ! [X0: $int,X1: $int] : ( $difference($sum(X0,X1),X0) = X1 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom13) ).

tff(f4518,plain,
    spl310_64,
    inference(avatar_split_clause,[],[f2439,f4516]) ).

tff(f2439,plain,
    ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X1)) = X0 ),
    inference(cnf_transformation,[],[f641]) ).

tff(f641,plain,
    ! [X0: $int,X1: $int] : ( $sum($sum(X0,X1),$uminus(X1)) = X0 ),
    inference(theory_normalization,[],[f17]) ).

tff(f17,axiom,
    ! [X0: $int,X1: $int] : ( $difference($sum(X0,X1),X1) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom15) ).

tff(f4514,plain,
    spl310_63,
    inference(avatar_split_clause,[],[f2423,f4512]) ).

tff(f4512,plain,
    ( spl310_63
  <=> ! [X0: $int] :
        ( ( 0 = X0 )
        | ( 0 != $sum(X0,X0) ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_63])]) ).

tff(f2423,plain,
    ! [X0: $int] :
      ( ( 0 = X0 )
      | ( 0 != $sum(X0,X0) ) ),
    inference(cnf_transformation,[],[f1709]) ).

tff(f1709,plain,
    ! [X0: $int] :
      ( ( ( 0 = $sum(X0,X0) )
        | ( 0 != X0 ) )
      & ( ( 0 = X0 )
        | ( 0 != $sum(X0,X0) ) ) ),
    inference(nnf_transformation,[],[f600]) ).

tff(f600,axiom,
    ! [X0: $int] :
      ( ( 0 = $sum(X0,X0) )
    <=> ( 0 = X0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom598) ).

tff(f4510,plain,
    ~ spl310_62,
    inference(avatar_split_clause,[],[f2366,f4507]) ).

tff(f4507,plain,
    ( spl310_62
  <=> $less('fun_app$e'('of_nat$','a$'),'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_62])]) ).

tff(f2366,plain,
    ~ $less('fun_app$e'('of_nat$','a$'),'fun_app$e'('of_nat$','to_nat$'('i$'))),
    inference(cnf_transformation,[],[f623]) ).

tff(f623,plain,
    ~ $less('fun_app$e'('of_nat$','a$'),'fun_app$e'('of_nat$','to_nat$'('i$'))),
    inference(theory_normalization,[],[f120]) ).

tff(f120,axiom,
    $lesseq('fun_app$e'('of_nat$','to_nat$'('i$')),'fun_app$e'('of_nat$','a$')),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom118) ).

tff(f4505,plain,
    spl310_61,
    inference(avatar_split_clause,[],[f875,f4503]) ).

tff(f875,plain,
    ! [X0: $int,X1: $int] :
      ( ~ $less(X0,X1)
      | ~ $less(X1,$sum(X0,1)) ),
    introduced(theory_axiom_161,[]) ).

tff(f4501,plain,
    spl310_60,
    inference(avatar_split_clause,[],[f873,f4499]) ).

tff(f873,plain,
    ! [X0: $int,X1: $int] :
      ( $less(X0,X1)
      | $less(X1,$sum(X0,1)) ),
    introduced(theory_axiom_147,[]) ).

tff(f4497,plain,
    ( spl310_58
    | ~ spl310_59 ),
    inference(avatar_split_clause,[],[f2359,f4494,f4490]) ).

tff(f2359,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) )
    | $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) ),
    inference(cnf_transformation,[],[f982]) ).

tff(f982,plain,
    ( ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) )
      & ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) )
    | ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),0) )
      & $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$'))) ) ),
    inference(ennf_transformation,[],[f622]) ).

tff(f622,plain,
    ~ ( ( ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
       => ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) ) )
      & ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
       => ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),0) ) ) ),
    inference(theory_normalization,[],[f7]) ).

tff(f7,negated_conjecture,
    ~ ( ( ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
       => ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$difference('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))) ) )
      & ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
       => ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),0) ) ) ),
    inference(negated_conjecture,[],[f6]) ).

tff(f6,conjecture,
    ( ( ~ $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
     => ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$difference('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))) ) )
    & ( $less('fun_app$e'('of_nat$','to_nat$'('x$')),'fun_app$e'('of_nat$','to_nat$'('i$')))
     => ( 'fun_app$e'('of_nat$','to_nat$'('x$')) = $sum('fun_app$e'('of_nat$','to_nat$'('i$')),0) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conjecture5) ).

tff(f4468,plain,
    spl310_57,
    inference(avatar_split_clause,[],[f3343,f4466]) ).

tff(f4466,plain,
    ( spl310_57
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ sP62(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_57])]) ).

tff(f3343,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X1,X0)
      | ~ sP62(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2095]) ).

tff(f2095,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X1,X0)
        & ~ $less(X0,X2) )
      | ~ sP62(X0,X1,X2) ),
    inference(rectify,[],[f2094]) ).

tff(f2094,plain,
    ! [X1: $int,X2: $int,X0: $int] :
      ( ( ~ $less(X2,X1)
        & ~ $less(X1,X0) )
      | ~ sP62(X1,X2,X0) ),
    inference(nnf_transformation,[],[f1668]) ).

tff(f4464,plain,
    spl310_56,
    inference(avatar_split_clause,[],[f3342,f4462]) ).

tff(f4462,plain,
    ( spl310_56
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X0,X2)
        | ~ sP62(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_56])]) ).

tff(f3342,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X2)
      | ~ sP62(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2095]) ).

tff(f4460,plain,
    spl310_55,
    inference(avatar_split_clause,[],[f3341,f4458]) ).

tff(f4458,plain,
    ( spl310_55
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ sP63(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_55])]) ).

tff(f3341,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X1,X0)
      | ~ sP63(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2093]) ).

tff(f2093,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X1,X0)
        & ~ $less(X0,X2) )
      | ~ sP63(X0,X1,X2) ),
    inference(rectify,[],[f2092]) ).

tff(f2092,plain,
    ! [X0: $int,X2: $int,X1: $int] :
      ( ( ~ $less(X2,X0)
        & ~ $less(X0,X1) )
      | ~ sP63(X0,X2,X1) ),
    inference(nnf_transformation,[],[f1669]) ).

tff(f4456,plain,
    spl310_54,
    inference(avatar_split_clause,[],[f3340,f4454]) ).

tff(f4454,plain,
    ( spl310_54
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X0,X2)
        | ~ sP63(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_54])]) ).

tff(f3340,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X2)
      | ~ sP63(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2093]) ).

tff(f4452,plain,
    spl310_53,
    inference(avatar_split_clause,[],[f3339,f4450]) ).

tff(f4450,plain,
    ( spl310_53
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ sP64(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_53])]) ).

tff(f3339,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X1,X0)
      | ~ sP64(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2091]) ).

tff(f2091,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X1,X0)
        & ~ $less(X0,X2) )
      | ~ sP64(X0,X1,X2) ),
    inference(rectify,[],[f2090]) ).

tff(f2090,plain,
    ! [X2: $int,X1: $int,X0: $int] :
      ( ( ~ $less(X1,X2)
        & ~ $less(X2,X0) )
      | ~ sP64(X2,X1,X0) ),
    inference(nnf_transformation,[],[f1670]) ).

tff(f4448,plain,
    spl310_52,
    inference(avatar_split_clause,[],[f3338,f4446]) ).

tff(f4446,plain,
    ( spl310_52
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X0,X2)
        | ~ sP64(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_52])]) ).

tff(f3338,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X2)
      | ~ sP64(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2091]) ).

tff(f4444,plain,
    spl310_51,
    inference(avatar_split_clause,[],[f3337,f4442]) ).

tff(f4442,plain,
    ( spl310_51
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X1,X0)
        | ~ sP65(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_51])]) ).

tff(f3337,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X1,X0)
      | ~ sP65(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2089]) ).

tff(f2089,plain,
    ! [X0: $int,X1: $int,X2: $int] :
      ( ( ~ $less(X1,X0)
        & ~ $less(X0,X2) )
      | ~ sP65(X0,X1,X2) ),
    inference(rectify,[],[f2088]) ).

tff(f2088,plain,
    ! [X1: $int,X0: $int,X2: $int] :
      ( ( ~ $less(X0,X1)
        & ~ $less(X1,X2) )
      | ~ sP65(X1,X0,X2) ),
    inference(nnf_transformation,[],[f1671]) ).

tff(f4440,plain,
    spl310_50,
    inference(avatar_split_clause,[],[f3336,f4438]) ).

tff(f4438,plain,
    ( spl310_50
  <=> ! [X2: $int,X0: $int,X1: $int] :
        ( ~ $less(X0,X2)
        | ~ sP65(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_50])]) ).

tff(f3336,plain,
    ! [X2: $int,X0: $int,X1: $int] :
      ( ~ $less(X0,X2)
      | ~ sP65(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2089]) ).

tff(f4436,plain,
    spl310_49,
    inference(avatar_split_clause,[],[f3209,f4434]) ).

tff(f4434,plain,
    ( spl310_49
  <=> ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
        ( sP37(X1,X0)
        | ~ sP39(X0,X1,X2) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_49])]) ).

tff(f3209,plain,
    ! [X2: 'Nat$',X0: 'Nat$',X1: 'Nat$'] :
      ( sP37(X1,X0)
      | ~ sP39(X0,X1,X2) ),
    inference(cnf_transformation,[],[f2021]) ).

tff(f4432,plain,
    spl310_48,
    inference(avatar_split_clause,[],[f2747,f4430]) ).

tff(f4430,plain,
    ( spl310_48
  <=> ! [X0: 'N_set$',X1: 'N$'] :
        ( 'fun_app$d'('uu$'(X0),X1)
        | ~ 'member$b'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_48])]) ).

tff(f2747,plain,
    ! [X0: 'N_set$',X1: 'N$'] :
      ( 'fun_app$d'('uu$'(X0),X1)
      | ~ 'member$b'(X1,X0) ),
    inference(cnf_transformation,[],[f1831]) ).

tff(f1831,plain,
    ! [X0: 'N_set$',X1: 'N$'] :
      ( ( 'fun_app$d'('uu$'(X0),X1)
        | ~ 'member$b'(X1,X0) )
      & ( 'member$b'(X1,X0)
        | ~ 'fun_app$d'('uu$'(X0),X1) ) ),
    inference(nnf_transformation,[],[f5]) ).

tff(f5,axiom,
    ! [X0: 'N_set$',X1: 'N$'] :
      ( 'fun_app$d'('uu$'(X0),X1)
    <=> 'member$b'(X1,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom4) ).

tff(f4428,plain,
    spl310_47,
    inference(avatar_split_clause,[],[f2746,f4426]) ).

tff(f4426,plain,
    ( spl310_47
  <=> ! [X0: 'N_set$',X1: 'N$'] :
        ( 'member$b'(X1,X0)
        | ~ 'fun_app$d'('uu$'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_47])]) ).

tff(f2746,plain,
    ! [X0: 'N_set$',X1: 'N$'] :
      ( 'member$b'(X1,X0)
      | ~ 'fun_app$d'('uu$'(X0),X1) ),
    inference(cnf_transformation,[],[f1831]) ).

tff(f4424,plain,
    spl310_46,
    inference(avatar_split_clause,[],[f2739,f4422]) ).

tff(f4422,plain,
    ( spl310_46
  <=> ! [X0: 'Int_set_set$',X1: 'Int_set$'] :
        ( 'fun_app$c'('uua$'(X0),X1)
        | ~ 'member$'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_46])]) ).

tff(f2739,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('uua$'(X0),X1)
      | ~ 'member$'(X1,X0) ),
    inference(cnf_transformation,[],[f1822]) ).

tff(f1822,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set$'] :
      ( ( 'fun_app$c'('uua$'(X0),X1)
        | ~ 'member$'(X1,X0) )
      & ( 'member$'(X1,X0)
        | ~ 'fun_app$c'('uua$'(X0),X1) ) ),
    inference(nnf_transformation,[],[f3]) ).

tff(f3,axiom,
    ! [X0: 'Int_set_set$',X1: 'Int_set$'] :
      ( 'fun_app$c'('uua$'(X0),X1)
    <=> 'member$'(X1,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom2) ).

tff(f4420,plain,
    spl310_45,
    inference(avatar_split_clause,[],[f2738,f4418]) ).

tff(f4418,plain,
    ( spl310_45
  <=> ! [X0: 'Int_set_set$',X1: 'Int_set$'] :
        ( 'member$'(X1,X0)
        | ~ 'fun_app$c'('uua$'(X0),X1) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_45])]) ).

tff(f2738,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set$'] :
      ( 'member$'(X1,X0)
      | ~ 'fun_app$c'('uua$'(X0),X1) ),
    inference(cnf_transformation,[],[f1822]) ).

tff(f4416,plain,
    ( spl310_44
    | ~ spl310_6
    | ~ spl310_24 ),
    inference(avatar_split_clause,[],[f4341,f4326,f4253,f4414]) ).

tff(f4341,plain,
    ( ! [X0: $int] : ~ $less(sK96(X0),X0)
    | ~ spl310_6
    | ~ spl310_24 ),
    inference(resolution,[],[f4327,f4254]) ).

tff(f4412,plain,
    spl310_43,
    inference(avatar_split_clause,[],[f2735,f4410]) ).

tff(f4410,plain,
    ( spl310_43
  <=> ! [X0: 'N$',X1: 'N_bool_fun$'] :
        ( 'member$b'(X0,'collect$'(X1))
        | ~ 'fun_app$d'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_43])]) ).

tff(f2735,plain,
    ! [X0: 'N$',X1: 'N_bool_fun$'] :
      ( 'member$b'(X0,'collect$'(X1))
      | ~ 'fun_app$d'(X1,X0) ),
    inference(cnf_transformation,[],[f1820]) ).

tff(f1820,plain,
    ! [X0: 'N$',X1: 'N_bool_fun$'] :
      ( ( 'member$b'(X0,'collect$'(X1))
        | ~ 'fun_app$d'(X1,X0) )
      & ( 'fun_app$d'(X1,X0)
        | ~ 'member$b'(X0,'collect$'(X1)) ) ),
    inference(nnf_transformation,[],[f48]) ).

tff(f48,axiom,
    ! [X0: 'N$',X1: 'N_bool_fun$'] :
      ( 'member$b'(X0,'collect$'(X1))
    <=> 'fun_app$d'(X1,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom46) ).

tff(f4408,plain,
    spl310_42,
    inference(avatar_split_clause,[],[f2734,f4406]) ).

tff(f4406,plain,
    ( spl310_42
  <=> ! [X0: 'N$',X1: 'N_bool_fun$'] :
        ( 'fun_app$d'(X1,X0)
        | ~ 'member$b'(X0,'collect$'(X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_42])]) ).

tff(f2734,plain,
    ! [X0: 'N$',X1: 'N_bool_fun$'] :
      ( 'fun_app$d'(X1,X0)
      | ~ 'member$b'(X0,'collect$'(X1)) ),
    inference(cnf_transformation,[],[f1820]) ).

tff(f4404,plain,
    spl310_41,
    inference(avatar_split_clause,[],[f2733,f4402]) ).

tff(f4402,plain,
    ( spl310_41
  <=> ! [X0: 'Int_set$',X1: 'Int_set_bool_fun$'] :
        ( 'member$'(X0,'collect$a'(X1))
        | ~ 'fun_app$c'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_41])]) ).

tff(f2733,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_bool_fun$'] :
      ( 'member$'(X0,'collect$a'(X1))
      | ~ 'fun_app$c'(X1,X0) ),
    inference(cnf_transformation,[],[f1819]) ).

tff(f1819,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_bool_fun$'] :
      ( ( 'member$'(X0,'collect$a'(X1))
        | ~ 'fun_app$c'(X1,X0) )
      & ( 'fun_app$c'(X1,X0)
        | ~ 'member$'(X0,'collect$a'(X1)) ) ),
    inference(nnf_transformation,[],[f49]) ).

tff(f49,axiom,
    ! [X0: 'Int_set$',X1: 'Int_set_bool_fun$'] :
      ( 'member$'(X0,'collect$a'(X1))
    <=> 'fun_app$c'(X1,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom47) ).

tff(f4400,plain,
    spl310_40,
    inference(avatar_split_clause,[],[f2732,f4398]) ).

tff(f4398,plain,
    ( spl310_40
  <=> ! [X0: 'Int_set$',X1: 'Int_set_bool_fun$'] :
        ( 'fun_app$c'(X1,X0)
        | ~ 'member$'(X0,'collect$a'(X1)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_40])]) ).

tff(f2732,plain,
    ! [X0: 'Int_set$',X1: 'Int_set_bool_fun$'] :
      ( 'fun_app$c'(X1,X0)
      | ~ 'member$'(X0,'collect$a'(X1)) ),
    inference(cnf_transformation,[],[f1819]) ).

tff(f4396,plain,
    spl310_39,
    inference(avatar_split_clause,[],[f2657,f4394]) ).

tff(f4394,plain,
    ( spl310_39
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,X1)
        | ~ 'fun_app$j'(X0,sK109(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_39])]) ).

tff(f2657,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ~ 'fun_app$j'(X0,sK109(X0)) ),
    inference(cnf_transformation,[],[f1768]) ).

tff(f4392,plain,
    spl310_38,
    inference(avatar_split_clause,[],[f2655,f4390]) ).

tff(f4390,plain,
    ( spl310_38
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,X1)
        | ~ 'fun_app$j'(X0,sK108(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_38])]) ).

tff(f2655,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ~ 'fun_app$j'(X0,sK108(X0)) ),
    inference(cnf_transformation,[],[f1766]) ).

tff(f4388,plain,
    spl310_37,
    inference(avatar_split_clause,[],[f2653,f4386]) ).

tff(f4386,plain,
    ( spl310_37
  <=> ! [X0: 'N_bool_fun$',X1: 'N$'] :
        ( 'fun_app$d'(X0,X1)
        | ~ 'fun_app$d'(X0,sK107(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_37])]) ).

tff(f2653,plain,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | ~ 'fun_app$d'(X0,sK107(X0)) ),
    inference(cnf_transformation,[],[f1764]) ).

tff(f4384,plain,
    spl310_36,
    inference(avatar_split_clause,[],[f2651,f4382]) ).

tff(f4382,plain,
    ( spl310_36
  <=> ! [X0: 'N_bool_fun$',X1: 'N$'] :
        ( 'fun_app$d'(X0,X1)
        | ~ 'fun_app$d'(X0,sK106(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_36])]) ).

tff(f2651,plain,
    ! [X0: 'N_bool_fun$',X1: 'N$'] :
      ( 'fun_app$d'(X0,X1)
      | ~ 'fun_app$d'(X0,sK106(X0)) ),
    inference(cnf_transformation,[],[f1762]) ).

tff(f4380,plain,
    spl310_35,
    inference(avatar_split_clause,[],[f2646,f4378]) ).

tff(f4378,plain,
    ( spl310_35
  <=> ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
        ( 'fun_app$j'(X0,X1)
        | ~ 'fun_app$j'(X0,sK104(X0)) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_35])]) ).

tff(f2646,plain,
    ! [X0: 'Nat_bool_fun$',X1: 'Nat$'] :
      ( 'fun_app$j'(X0,X1)
      | ~ 'fun_app$j'(X0,sK104(X0)) ),
    inference(cnf_transformation,[],[f1758]) ).

tff(f4376,plain,
    spl310_34,
    inference(avatar_split_clause,[],[f2437,f4374]) ).

tff(f4374,plain,
    ( spl310_34
  <=> ! [X0: 'Int_set_set$',X1: 'Int_set_set$'] : ( 'plus$a'(X0,X1) = 'plus$a'(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_34])]) ).

tff(f2437,plain,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$'] : ( 'plus$a'(X0,X1) = 'plus$a'(X1,X0) ),
    inference(cnf_transformation,[],[f55]) ).

tff(f55,axiom,
    ! [X0: 'Int_set_set$',X1: 'Int_set_set$'] : ( 'plus$a'(X0,X1) = 'plus$a'(X1,X0) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom53) ).

tff(f4372,plain,
    ( spl310_33
    | ~ spl310_5
    | ~ spl310_24 ),
    inference(avatar_split_clause,[],[f4340,f4326,f4249,f4370]) ).

tff(f4370,plain,
    ( spl310_33
  <=> ! [X0: $int] : ~ $less(X0,sK95(X0)) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_33])]) ).

tff(f4340,plain,
    ( ! [X0: $int] : ~ $less(X0,sK95(X0))
    | ~ spl310_5
    | ~ spl310_24 ),
    inference(resolution,[],[f4327,f4250]) ).

tff(f4368,plain,
    spl310_32,
    inference(avatar_split_clause,[],[f2431,f4366]) ).

tff(f2431,plain,
    ! [X0: 'N_bool_fun$',X6: 'N$'] :
      ( 'fun_app$d'(X0,sK94(X0))
      | ~ 'fun_app$d'(X0,X6) ),
    inference(cnf_transformation,[],[f1721]) ).

tff(f4364,plain,
    spl310_31,
    inference(avatar_split_clause,[],[f2427,f4362]) ).

tff(f2427,plain,
    ! [X0: 'N_bool_fun$',X6: 'N$'] :
      ( 'fun_app$d'(X0,sK91(X0))
      | ~ 'fun_app$d'(X0,X6) ),
    inference(cnf_transformation,[],[f1715]) ).

tff(f4360,plain,
    spl310_30,
    inference(avatar_split_clause,[],[f2365,f4357]) ).

tff(f2365,plain,
    'fun_app$d'('fun_app$g'('less_eq$','i$'),'fun_app$i'('from_nat$','a$')),
    inference(cnf_transformation,[],[f121]) ).

tff(f121,axiom,
    'fun_app$d'('fun_app$g'('less_eq$','i$'),'fun_app$i'('from_nat$','a$')),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom119) ).

tff(f4355,plain,
    spl310_29,
    inference(avatar_split_clause,[],[f2364,f4352]) ).

tff(f2364,plain,
    'fun_app$d'('fun_app$g'('less$','x$'),'fun_app$i'('from_nat$','a$')),
    inference(cnf_transformation,[],[f244]) ).

tff(f244,axiom,
    'fun_app$d'('fun_app$g'('less$','x$'),'fun_app$i'('from_nat$','a$')),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom242) ).

tff(f4350,plain,
    ~ spl310_28,
    inference(avatar_split_clause,[],[f2363,f4347]) ).

tff(f2363,plain,
    ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'('from_nat$','a$')),'x$'),
    inference(cnf_transformation,[],[f116]) ).

tff(f116,axiom,
    ~ 'fun_app$d'('fun_app$g'('less_eq$','fun_app$i'('from_nat$','a$')),'x$'),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom114) ).

tff(f4345,plain,
    spl310_27,
    inference(avatar_split_clause,[],[f864,f4343]) ).

tff(f4336,plain,
    spl310_26,
    inference(avatar_split_clause,[],[f2881,f4334]) ).

tff(f2881,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( sP3(X1,X0)
      | ~ 'fun_app$j'(X1,X0) ),
    inference(cnf_transformation,[],[f1930]) ).

tff(f4332,plain,
    spl310_25,
    inference(avatar_split_clause,[],[f2872,f4330]) ).

tff(f4330,plain,
    ( spl310_25
  <=> ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
        ( 'fun_app$j'(X1,X0)
        | ~ sP2(X1,X0) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_25])]) ).

tff(f2872,plain,
    ! [X0: 'Nat$',X1: 'Nat_bool_fun$'] :
      ( 'fun_app$j'(X1,X0)
      | ~ sP2(X1,X0) ),
    inference(cnf_transformation,[],[f1921]) ).

tff(f4328,plain,
    spl310_24,
    inference(avatar_split_clause,[],[f2472,f4326]) ).

tff(f2472,plain,
    ! [X0: $int,X1: $int] :
      ( ~ $less(X0,X1)
      | ~ $less(X1,X0) ),
    inference(cnf_transformation,[],[f654]) ).

tff(f654,plain,
    ! [X0: $int,X1: $int] :
      ( ~ $less(X0,X1)
      | ~ $less(X1,X0) ),
    inference(theory_normalization,[],[f186]) ).

tff(f186,axiom,
    ! [X0: $int,X1: $int] :
      ( $lesseq(X1,X0)
      | $lesseq(X0,X1) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom184) ).

tff(f4324,plain,
    spl310_23,
    inference(avatar_split_clause,[],[f2414,f4322]) ).

tff(f4322,plain,
    ( spl310_23
  <=> ! [X0: tlbool] :
        ( ( tlfalse = X0 )
        | ( tltrue = X0 ) ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_23])]) ).

tff(f2414,plain,
    ! [X0: tlbool] :
      ( ( tlfalse = X0 )
      | ( tltrue = X0 ) ),
    inference(cnf_transformation,[],[f890]) ).

tff(f890,plain,
    ! [X0: tlbool] :
      ( ( tlfalse = X0 )
      | ( tltrue = X0 ) ),
    inference(rectify,[],[f620]) ).

tff(f620,axiom,
    ! [X6: tlbool] :
      ( ( tlfalse = X6 )
      | ( tltrue = X6 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',formula_619) ).

tff(f4320,plain,
    spl310_22,
    inference(avatar_split_clause,[],[f2400,f4318]) ).

tff(f2400,plain,
    ! [X0: 'N$'] : ( 'fun_app$f'('plus$'(X0),'zero$a') = X0 ),
    inference(cnf_transformation,[],[f602]) ).

tff(f602,axiom,
    ! [X0: 'N$'] : ( 'fun_app$f'('plus$'(X0),'zero$a') = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom600) ).

tff(f4316,plain,
    spl310_21,
    inference(avatar_split_clause,[],[f2399,f4314]) ).

tff(f2399,plain,
    ! [X0: 'N$'] : ( 'fun_app$f'('plus$'('zero$a'),X0) = X0 ),
    inference(cnf_transformation,[],[f594]) ).

tff(f594,axiom,
    ! [X0: 'N$'] : ( 'fun_app$f'('plus$'('zero$a'),X0) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom592) ).

tff(f4312,plain,
    spl310_20,
    inference(avatar_split_clause,[],[f2398,f4310]) ).

tff(f4310,plain,
    ( spl310_20
  <=> ! [X0: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'zero$') = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_20])]) ).

tff(f2398,plain,
    ! [X0: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'zero$') = X0 ),
    inference(cnf_transformation,[],[f601]) ).

tff(f601,axiom,
    ! [X0: 'Int_set$'] : ( 'fun_app$h'('plus$b'(X0),'zero$') = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom599) ).

tff(f4308,plain,
    spl310_19,
    inference(avatar_split_clause,[],[f2397,f4306]) ).

tff(f4306,plain,
    ( spl310_19
  <=> ! [X0: 'Int_set$'] : ( 'fun_app$h'('plus$b'('zero$'),X0) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_19])]) ).

tff(f2397,plain,
    ! [X0: 'Int_set$'] : ( 'fun_app$h'('plus$b'('zero$'),X0) = X0 ),
    inference(cnf_transformation,[],[f593]) ).

tff(f593,axiom,
    ! [X0: 'Int_set$'] : ( 'fun_app$h'('plus$b'('zero$'),X0) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom591) ).

tff(f4304,plain,
    spl310_18,
    inference(avatar_split_clause,[],[f2396,f4302]) ).

tff(f4302,plain,
    ( spl310_18
  <=> ! [X0: 'N$'] : ( 'fun_app$i'('from_nat$','to_nat$'(X0)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_18])]) ).

tff(f2396,plain,
    ! [X0: 'N$'] : ( 'fun_app$i'('from_nat$','to_nat$'(X0)) = X0 ),
    inference(cnf_transformation,[],[f124]) ).

tff(f124,axiom,
    ! [X0: 'N$'] : ( 'fun_app$i'('from_nat$','to_nat$'(X0)) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom122) ).

tff(f4300,plain,
    spl310_17,
    inference(avatar_split_clause,[],[f2395,f4298]) ).

tff(f2395,plain,
    ! [X0: 'Nat$'] : ( 'nat$'('fun_app$e'('of_nat$',X0)) = X0 ),
    inference(cnf_transformation,[],[f618]) ).

tff(f618,axiom,
    ! [X0: 'Nat$'] : ( 'nat$'('fun_app$e'('of_nat$',X0)) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom616) ).

tff(f4296,plain,
    spl310_16,
    inference(avatar_split_clause,[],[f868,f4294]) ).

tff(f4292,plain,
    spl310_15,
    inference(avatar_split_clause,[],[f2391,f4290]) ).

tff(f2391,plain,
    ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),X0),
    inference(cnf_transformation,[],[f576]) ).

tff(f576,axiom,
    ! [X0: 'N$'] : 'fun_app$d'('fun_app$g'('less_eq$',X0),X0),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom574) ).

tff(f4288,plain,
    spl310_14,
    inference(avatar_split_clause,[],[f2388,f4286]) ).

tff(f2388,plain,
    ! [X0: 'Nat$'] : ~ $less('fun_app$e'('of_nat$',X0),0),
    inference(cnf_transformation,[],[f632]) ).

tff(f632,plain,
    ! [X0: 'Nat$'] : ~ $less('fun_app$e'('of_nat$',X0),0),
    inference(theory_normalization,[],[f612]) ).

tff(f612,axiom,
    ! [X0: 'Nat$'] : $lesseq(0,'fun_app$e'('of_nat$',X0)),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom610) ).

tff(f4284,plain,
    spl310_13,
    inference(avatar_split_clause,[],[f2383,f4282]) ).

tff(f4282,plain,
    ( spl310_13
  <=> ! [X0: 'Int_set$'] : ( 'collect$b'('uub$'(X0)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_13])]) ).

tff(f2383,plain,
    ! [X0: 'Int_set$'] : ( 'collect$b'('uub$'(X0)) = X0 ),
    inference(cnf_transformation,[],[f53]) ).

tff(f53,axiom,
    ! [X0: 'Int_set$'] : ( 'collect$b'('uub$'(X0)) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom51) ).

tff(f4280,plain,
    spl310_12,
    inference(avatar_split_clause,[],[f2382,f4278]) ).

tff(f4278,plain,
    ( spl310_12
  <=> ! [X0: 'Int_set_set$'] : ( 'collect$a'('uua$'(X0)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_12])]) ).

tff(f2382,plain,
    ! [X0: 'Int_set_set$'] : ( 'collect$a'('uua$'(X0)) = X0 ),
    inference(cnf_transformation,[],[f52]) ).

tff(f52,axiom,
    ! [X0: 'Int_set_set$'] : ( 'collect$a'('uua$'(X0)) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom50) ).

tff(f4276,plain,
    spl310_11,
    inference(avatar_split_clause,[],[f2381,f4274]) ).

tff(f4274,plain,
    ( spl310_11
  <=> ! [X0: 'N_set$'] : ( 'collect$'('uu$'(X0)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_11])]) ).

tff(f2381,plain,
    ! [X0: 'N_set$'] : ( 'collect$'('uu$'(X0)) = X0 ),
    inference(cnf_transformation,[],[f51]) ).

tff(f51,axiom,
    ! [X0: 'N_set$'] : ( 'collect$'('uu$'(X0)) = X0 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom49) ).

tff(f4272,plain,
    spl310_10,
    inference(avatar_split_clause,[],[f2376,f4270]) ).

tff(f4270,plain,
    ( spl310_10
  <=> ! [X0: 'N$'] : ~ 'fun_app$d'('fun_app$g'('less$',X0),X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_10])]) ).

tff(f2376,plain,
    ! [X0: 'N$'] : ~ 'fun_app$d'('fun_app$g'('less$',X0),X0),
    inference(cnf_transformation,[],[f356]) ).

tff(f356,axiom,
    ! [X0: 'N$'] : ~ 'fun_app$d'('fun_app$g'('less$',X0),X0),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom354) ).

tff(f4268,plain,
    spl310_9,
    inference(avatar_split_clause,[],[f2362,f4265]) ).

tff(f2362,plain,
    'fun_app$d'('fun_app$g'('less_eq$','i$'),'x$'),
    inference(cnf_transformation,[],[f18]) ).

tff(f18,axiom,
    'fun_app$d'('fun_app$g'('less_eq$','i$'),'x$'),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom16) ).

tff(f4263,plain,
    spl310_8,
    inference(avatar_split_clause,[],[f874,f4261]) ).

tff(f4261,plain,
    ( spl310_8
  <=> ! [X0: $int] : ( $uminus($uminus(X0)) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_8])]) ).

tff(f874,plain,
    ! [X0: $int] : ( $uminus($uminus(X0)) = X0 ),
    introduced(theory_axiom_148,[]) ).

tff(f4259,plain,
    spl310_7,
    inference(avatar_split_clause,[],[f866,f4257]) ).

tff(f4257,plain,
    ( spl310_7
  <=> ! [X0: $int] : ( $sum(X0,0) = X0 ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_7])]) ).

tff(f866,plain,
    ! [X0: $int] : ( $sum(X0,0) = X0 ),
    introduced(theory_axiom_137,[]) ).

tff(f4255,plain,
    spl310_6,
    inference(avatar_split_clause,[],[f2436,f4253]) ).

tff(f2436,plain,
    ! [X0: $int] : $less(X0,sK96(X0)),
    inference(cnf_transformation,[],[f1725]) ).

tff(f1725,plain,
    ! [X0: $int] : $less(X0,sK96(X0)),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK96])],[f367,f1724]) ).

tff(f1724,plain,
    ! [X0: $int] :
      ( ? [X1: $int] : $less(X0,X1)
     => $less(X0,sK96(X0)) ),
    introduced(choice_axiom,[]) ).

tff(f367,axiom,
    ! [X0: $int] :
    ? [X1: $int] : $less(X0,X1),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom365) ).

tff(f4251,plain,
    spl310_5,
    inference(avatar_split_clause,[],[f2435,f4249]) ).

tff(f2435,plain,
    ! [X0: $int] : $less(sK95(X0),X0),
    inference(cnf_transformation,[],[f1723]) ).

tff(f1723,plain,
    ! [X0: $int] : $less(sK95(X0),X0),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK95])],[f368,f1722]) ).

tff(f1722,plain,
    ! [X0: $int] :
      ( ? [X1: $int] : $less(X1,X0)
     => $less(sK95(X0),X0) ),
    introduced(choice_axiom,[]) ).

tff(f368,axiom,
    ! [X0: $int] :
    ? [X1: $int] : $less(X1,X0),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom366) ).

tff(f4247,plain,
    spl310_4,
    inference(avatar_split_clause,[],[f2374,f4245]) ).

tff(f4245,plain,
    ( spl310_4
  <=> ! [X0: 'Int_set$'] : 'fun_app$c'('less_eq$a'(X0),X0) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_4])]) ).

tff(f2374,plain,
    ! [X0: 'Int_set$'] : 'fun_app$c'('less_eq$a'(X0),X0),
    inference(cnf_transformation,[],[f108]) ).

tff(f108,axiom,
    ! [X0: 'Int_set$'] : 'fun_app$c'('less_eq$a'(X0),X0),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom106) ).

tff(f4243,plain,
    ~ spl310_3,
    inference(avatar_split_clause,[],[f2361,f4240]) ).

tff(f4240,plain,
    ( spl310_3
  <=> ( tltrue = tlfalse ) ),
    introduced(avatar_definition,[new_symbols(naming,[spl310_3])]) ).

tff(f2361,plain,
    tltrue != tlfalse,
    inference(cnf_transformation,[],[f621]) ).

tff(f621,axiom,
    tltrue != tlfalse,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',formula_620) ).

tff(f4238,plain,
    spl310_2,
    inference(avatar_split_clause,[],[f869,f4236]) ).

tff(f869,plain,
    ! [X0: $int] : ~ $less(X0,X0),
    introduced(theory_axiom_142,[]) ).

tff(f4234,plain,
    ~ spl310_1,
    inference(avatar_split_clause,[],[f3975,f4231]) ).

tff(f3975,plain,
    'fun_app$e'('of_nat$','to_nat$'('x$')) != 'fun_app$e'('of_nat$','to_nat$'('i$')),
    inference(evaluation,[],[f3974]) ).

tff(f3974,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('i$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),0) )
    | ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != 'fun_app$e'('of_nat$','to_nat$'('i$')) ) ),
    inference(forward_demodulation,[],[f3973,f868]) ).

tff(f3973,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('i$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('i$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) )
    | ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != 'fun_app$e'('of_nat$','to_nat$'('i$')) ) ),
    inference(inner_rewriting,[],[f3972]) ).

tff(f3972,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) )
    | ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != 'fun_app$e'('of_nat$','to_nat$'('i$')) ) ),
    inference(evaluation,[],[f2360]) ).

tff(f2360,plain,
    ( ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),$sum('fun_app$e'('of_nat$','to_nat$'('x$')),$uminus('fun_app$e'('of_nat$','to_nat$'('i$'))))) )
    | ( 'fun_app$e'('of_nat$','to_nat$'('x$')) != $sum('fun_app$e'('of_nat$','to_nat$'('i$')),0) ) ),
    inference(cnf_transformation,[],[f982]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12  % Problem    : ITP001_1 : TPTP v8.1.2. Released v8.1.0.
% 0.12/0.14  % Command    : vampire --mode casc_sat -m 16384 --cores 7 -t %d %s
% 0.12/0.34  % Computer : n013.cluster.edu
% 0.12/0.34  % Model    : x86_64 x86_64
% 0.12/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34  % Memory   : 8042.1875MB
% 0.12/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34  % CPULimit   : 300
% 0.12/0.34  % WCLimit    : 300
% 0.12/0.34  % DateTime   : Fri May  3 19:00:38 EDT 2024
% 0.12/0.34  % CPUTime    : 
% 0.12/0.34  % (5444)Running in auto input_syntax mode. Trying TPTP
% 0.18/0.42  % (5445)fmb+10_1_bce=on:fmbas=function:fmbsr=1.2:fde=unused:nm=0_846 on theBenchmark for (846ds/0Mi)
% 0.18/0.42  % (5447)WARNING: value z3 for option sas not known
% 0.18/0.43  % (5449)ott+10_10:1_add=off:afr=on:amm=off:anc=all:bd=off:bs=on:fsr=off:irw=on:lma=on:msp=off:nm=4:nwc=4.0:sac=on:sp=reverse_frequency_531 on theBenchmark for (531ds/0Mi)
% 0.18/0.43  % (5447)dis+2_11_add=large:afr=on:amm=off:bd=off:bce=on:fsd=off:fde=none:gs=on:gsaa=full_model:gsem=off:irw=on:msp=off:nm=4:nwc=1.3:sas=z3:sims=off:sac=on:sp=reverse_arity_569 on theBenchmark for (569ds/0Mi)
% 0.18/0.43  % (5448)fmb+10_1_bce=on:fmbsr=1.5:nm=32_533 on theBenchmark for (533ds/0Mi)
% 0.18/0.43  % (5451)ott+1_64_av=off:bd=off:bce=on:fsd=off:fde=unused:gsp=on:irw=on:lcm=predicate:lma=on:nm=2:nwc=1.1:sims=off:urr=on_497 on theBenchmark for (497ds/0Mi)
% 0.18/0.43  % (5450)ott-10_8_av=off:bd=preordered:bs=on:fsd=off:fsr=off:fde=unused:irw=on:lcm=predicate:lma=on:nm=4:nwc=1.7:sp=frequency_522 on theBenchmark for (522ds/0Mi)
% 0.18/0.43  % (5446)fmb+10_1_bce=on:fmbdsb=on:fmbes=contour:fmbswr=3:fde=none:nm=0_793 on theBenchmark for (793ds/0Mi)
% 0.18/0.48  % (5445)WARNING: trying to run FMB on interpreted or otherwise provably infinite-domain problem!
% 0.18/0.48  % (5445)Terminated due to inappropriate strategy.
% 0.18/0.48  % (5445)------------------------------
% 0.18/0.48  % (5445)Version: Vampire 4.8 (commit 3a798227e on 2024-05-03 07:42:47 +0200)
% 0.18/0.48  % (5445)Termination reason: Inappropriate
% 0.18/0.48  
% 0.18/0.48  % (5445)Memory used [KB]: 2610
% 0.18/0.48  % (5445)Time elapsed: 0.061 s
% 0.18/0.48  % (5445)Instructions burned: 207 (million)
% 0.18/0.48  % (5445)------------------------------
% 0.18/0.48  % (5445)------------------------------
% 0.18/0.50  % (5448)WARNING: trying to run FMB on interpreted or otherwise provably infinite-domain problem!
% 0.18/0.50  % (5448)Terminated due to inappropriate strategy.
% 0.18/0.50  % (5448)------------------------------
% 0.18/0.50  % (5448)Version: Vampire 4.8 (commit 3a798227e on 2024-05-03 07:42:47 +0200)
% 0.18/0.50  % (5448)Termination reason: Inappropriate
% 0.18/0.50  
% 0.18/0.50  % (5448)Memory used [KB]: 2591
% 0.18/0.50  % (5448)Time elapsed: 0.075 s
% 0.18/0.50  % (5448)Instructions burned: 192 (million)
% 0.18/0.50  % (5448)------------------------------
% 0.18/0.50  % (5448)------------------------------
% 0.18/0.50  % (5452)fmb+10_1_fmbas=expand:fmbsr=1.1:gsp=on:nm=4_411 on theBenchmark for (411ds/0Mi)
% 0.18/0.51  % (5446)WARNING: trying to run FMB on interpreted or otherwise provably infinite-domain problem!
% 0.18/0.51  % (5446)Terminated due to inappropriate strategy.
% 0.18/0.51  % (5446)------------------------------
% 0.18/0.51  % (5446)Version: Vampire 4.8 (commit 3a798227e on 2024-05-03 07:42:47 +0200)
% 0.18/0.51  % (5446)Termination reason: Inappropriate
% 0.18/0.51  
% 0.18/0.51  % (5446)Memory used [KB]: 2611
% 0.18/0.51  % (5446)Time elapsed: 0.081 s
% 0.18/0.51  % (5446)Instructions burned: 207 (million)
% 0.18/0.51  % (5446)------------------------------
% 0.18/0.51  % (5446)------------------------------
% 0.18/0.52  % (5453)ott+1_9_av=off:bd=off:bs=on:gsp=on:lcm=predicate:nm=4:sp=weighted_frequency:urr=on_382 on theBenchmark for (382ds/0Mi)
% 0.18/0.52  % (5454)lrs-11_2:5_fsd=off:fde=none:nm=4:nwc=5.0:sims=off:sp=reverse_weighted_frequency:stl=62_367 on theBenchmark for (367ds/0Mi)
% 0.18/0.55  % (5452)WARNING: trying to run FMB on interpreted or otherwise provably infinite-domain problem!
% 0.18/0.55  % (5452)Terminated due to inappropriate strategy.
% 0.18/0.55  % (5452)------------------------------
% 0.18/0.55  % (5452)Version: Vampire 4.8 (commit 3a798227e on 2024-05-03 07:42:47 +0200)
% 0.18/0.55  % (5452)Termination reason: Inappropriate
% 0.18/0.55  
% 0.18/0.55  % (5452)Memory used [KB]: 2599
% 0.18/0.55  % (5452)Time elapsed: 0.050 s
% 0.18/0.55  % (5452)Instructions burned: 123 (million)
% 0.18/0.55  % (5452)------------------------------
% 0.18/0.55  % (5452)------------------------------
% 1.22/0.57  % (5455)ott+4_64_acc=on:anc=none:bs=on:bsr=on:fsd=off:gs=on:gsem=off:irw=on:msp=off:nwc=2.5:nicw=on:sims=off_354 on theBenchmark for (354ds/0Mi)
% 3.53/0.97  % (5449)First to succeed.
% 3.92/1.02  % (5449)Solution written to "/export/starexec/sandbox/tmp/vampire-proof-5444"
% 3.92/1.02  % (5449)Refutation found. Thanks to Tanya!
% 3.92/1.02  % SZS status Theorem for theBenchmark
% 3.92/1.02  % SZS output start Proof for theBenchmark
% See solution above
% 3.92/1.05  % (5449)------------------------------
% 3.92/1.05  % (5449)Version: Vampire 4.8 (commit 3a798227e on 2024-05-03 07:42:47 +0200)
% 3.92/1.05  % (5449)Termination reason: Refutation
% 3.92/1.05  
% 3.92/1.05  % (5449)Memory used [KB]: 11284
% 3.92/1.05  % (5449)Time elapsed: 0.597 s
% 3.92/1.05  % (5449)Instructions burned: 1978 (million)
% 3.92/1.05  % (5444)Success in time 0.7 s
%------------------------------------------------------------------------------