TPTP Problem File: NUM923_2.p
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%------------------------------------------------------------------------------
% File : NUM923_2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_500_tff_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.4.0, 0.67 v5.3.0
% Syntax : Number of formulae : 699 ( 191 unt; 52 typ; 0 def)
% Number of atoms : 1532 ( 426 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 1029 ( 144 ~; 41 |; 51 &)
% ( 130 <=>; 663 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 56 ( 27 >; 29 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 39 ( 39 usr; 13 con; 0-8 aty)
% Number of variables : 1591 (1567 !; 24 ?;1591 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:16:31
%------------------------------------------------------------------------------
%----Should-be-implicit typings (12)
tff(ty_ty_tc__HOL__Obool,type,
bool: $tType ).
tff(ty_ty_tc__Int__Oint,type,
int: $tType ).
tff(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
fun_int_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_J,type,
fun_int_fun_int_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Int__Oint_J,type,
fun_nat_int: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_J,type,
fun_nat_fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL__Obool_J,type,
fun_Pr974702441t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
fun_Pr1503267378t_bool: $tType ).
tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
produc393999548nt_int: $tType ).
%----Explicit typings (40)
tff(sy_c_Divides_Odivmod__int__rel,type,
divmod_int_rel: ( int * int ) > fun_Pr974702441t_bool ).
tff(sy_c_Divides_Opdivmod,type,
pdivmod: ( int * int ) > product_prod_int_int ).
tff(sy_c_Divides_OposDivAlg,type,
posDivAlg: ( int * int ) > product_prod_int_int ).
tff(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
abs_abs_int: int > int ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: ( int * int ) > int ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: ( int * int ) > int ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: ( int * int ) > int ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_IntFact_Ozfact,type,
zfact: int > int ).
tff(sy_c_IntPrimes_Oxzgcd,type,
xzgcd: ( int * int ) > produc393999548nt_int ).
tff(sy_c_IntPrimes_Oxzgcda,type,
xzgcda: ( int * int * int * int * int * int * int * int ) > produc393999548nt_int ).
tff(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: ( int * int ) > int ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: fun_int_fun_int_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: fun_int_fun_int_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
product_Pair_int_int: ( int * int ) > product_prod_int_int ).
tff(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__O,type,
produc282740534nt_int: ( int * product_prod_int_int ) > produc393999548nt_int ).
tff(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool,type,
produc262399358t_bool: ( fun_Pr974702441t_bool * int ) > fun_int_bool ).
tff(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__,type,
produc97574839t_bool: ( fun_Pr1503267378t_bool * int ) > fun_Pr974702441t_bool ).
tff(sy_c_TwoSquares__Mirabelle__xuacyfoabs_Ois__sum2sq,type,
twoSqu512355103sum2sq: fun_int_bool ).
tff(sy_c_TwoSquares__Mirabelle__xuacyfoabs_Osum2sq,type,
twoSqu2107342101sum2sq: product_prod_int_int > int ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,type,
hAPP_int_bool: ( fun_int_bool * int ) > bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
hAPP_nat_int: ( fun_nat_int * nat ) > int ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
hAPP_n1699378549t_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool,type,
hAPP_P603027463t_bool: ( fun_Pr974702441t_bool * product_prod_int_int ) > bool ).
tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
hAPP_P178748804t_bool: ( fun_Pr1503267378t_bool * produc393999548nt_int ) > bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_v_a,type,
a: int ).
tff(sy_v_b,type,
b: int ).
tff(sy_v_p,type,
p: int ).
tff(sy_v_q,type,
q: int ).
%----Relevant facts (646)
tff(fact_0_xzgcda__linear__aux1,axiom,
! [A_19: int,R_1: int,B: int,M: int,C_31: int,D: int,N: int] : plus_plus_int(times_times_int(minus_minus_int(A_19,times_times_int(R_1,B)),M),times_times_int(minus_minus_int(C_31,times_times_int(R_1,D)),N)) = minus_minus_int(plus_plus_int(times_times_int(A_19,M),times_times_int(C_31,N)),times_times_int(R_1,plus_plus_int(times_times_int(B,M),times_times_int(D,N)))) ).
tff(fact_1_mult__diff__mult,axiom,
! [X_44: int,Y_43: int,A_146: int,B_112: int] : minus_minus_int(times_times_int(X_44,Y_43),times_times_int(A_146,B_112)) = plus_plus_int(times_times_int(X_44,minus_minus_int(Y_43,B_112)),times_times_int(minus_minus_int(X_44,A_146),B_112)) ).
tff(fact_2_eq__add__iff2,axiom,
! [Aa: int,E: int,C_4: int,Ba: int,D_3: int] :
( ( plus_plus_int(times_times_int(Aa,E),C_4) = plus_plus_int(times_times_int(Ba,E),D_3) )
<=> ( C_4 = plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_3) ) ) ).
tff(fact_3_eq__add__iff1,axiom,
! [Aa: int,E: int,C_4: int,Ba: int,D_3: int] :
( ( plus_plus_int(times_times_int(Aa,E),C_4) = plus_plus_int(times_times_int(Ba,E),D_3) )
<=> ( plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C_4) = D_3 ) ) ).
tff(fact_4_is__sum2sq__def,axiom,
! [X_1: int] :
( hBOOL(hAPP_int_bool(twoSqu512355103sum2sq,X_1))
<=> ? [A_88: int,B_92: int] : twoSqu2107342101sum2sq(product_Pair_int_int(A_88,B_92)) = X_1 ) ).
tff(fact_5_Int2_Oaux1,axiom,
! [A_19: int,B: int,C_31: int] :
( ( minus_minus_int(A_19,B) = C_31 )
=> ( A_19 = plus_plus_int(C_31,B) ) ) ).
tff(fact_6_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : times_times_int(W,minus_minus_int(Z1,Z2)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
tff(fact_7_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : times_times_int(minus_minus_int(Z1,Z2),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
tff(fact_8_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
tff(fact_9_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : times_times_int(plus_plus_int(Z1,Z2),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
tff(fact_10_diff__add__cancel,axiom,
! [A_145: int,B_111: int] : plus_plus_int(minus_minus_int(A_145,B_111),B_111) = A_145 ).
tff(fact_11_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_144: int,B_110: int,C_67: int] : times_times_int(times_times_int(A_144,B_110),C_67) = times_times_int(A_144,times_times_int(B_110,C_67)) ).
tff(fact_12_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_144: nat,B_110: nat,C_67: nat] : times_times_nat(times_times_nat(A_144,B_110),C_67) = times_times_nat(A_144,times_times_nat(B_110,C_67)) ).
tff(fact_13_add__right__imp__eq,axiom,
! [B_109: int,A_143: int,C_66: int] :
( ( plus_plus_int(B_109,A_143) = plus_plus_int(C_66,A_143) )
=> ( B_109 = C_66 ) ) ).
tff(fact_14_add__right__imp__eq,axiom,
! [B_109: nat,A_143: nat,C_66: nat] :
( ( plus_plus_nat(B_109,A_143) = plus_plus_nat(C_66,A_143) )
=> ( B_109 = C_66 ) ) ).
tff(fact_15_add__imp__eq,axiom,
! [A_142: int,B_108: int,C_65: int] :
( ( plus_plus_int(A_142,B_108) = plus_plus_int(A_142,C_65) )
=> ( B_108 = C_65 ) ) ).
tff(fact_16_add__imp__eq,axiom,
! [A_142: nat,B_108: nat,C_65: nat] :
( ( plus_plus_nat(A_142,B_108) = plus_plus_nat(A_142,C_65) )
=> ( B_108 = C_65 ) ) ).
tff(fact_17_add__left__imp__eq,axiom,
! [A_141: int,B_107: int,C_64: int] :
( ( plus_plus_int(A_141,B_107) = plus_plus_int(A_141,C_64) )
=> ( B_107 = C_64 ) ) ).
tff(fact_18_add__left__imp__eq,axiom,
! [A_141: nat,B_107: nat,C_64: nat] :
( ( plus_plus_nat(A_141,B_107) = plus_plus_nat(A_141,C_64) )
=> ( B_107 = C_64 ) ) ).
tff(fact_19_add__right__cancel,axiom,
! [Ba: int,Aa: int,C_4: int] :
( ( plus_plus_int(Ba,Aa) = plus_plus_int(C_4,Aa) )
<=> ( Ba = C_4 ) ) ).
tff(fact_20_add__right__cancel,axiom,
! [Ba: nat,Aa: nat,C_4: nat] :
( ( plus_plus_nat(Ba,Aa) = plus_plus_nat(C_4,Aa) )
<=> ( Ba = C_4 ) ) ).
tff(fact_21_add__left__cancel,axiom,
! [Aa: int,Ba: int,C_4: int] :
( ( plus_plus_int(Aa,Ba) = plus_plus_int(Aa,C_4) )
<=> ( Ba = C_4 ) ) ).
tff(fact_22_add__left__cancel,axiom,
! [Aa: nat,Ba: nat,C_4: nat] :
( ( plus_plus_nat(Aa,Ba) = plus_plus_nat(Aa,C_4) )
<=> ( Ba = C_4 ) ) ).
tff(fact_23_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_140: int,B_106: int,C_63: int] : plus_plus_int(plus_plus_int(A_140,B_106),C_63) = plus_plus_int(A_140,plus_plus_int(B_106,C_63)) ).
tff(fact_24_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_140: nat,B_106: nat,C_63: nat] : plus_plus_nat(plus_plus_nat(A_140,B_106),C_63) = plus_plus_nat(A_140,plus_plus_nat(B_106,C_63)) ).
tff(fact_25_diff__eq__diff__eq,axiom,
! [Aa: int,Ba: int,C_4: int,D_3: int] :
( ( minus_minus_int(Aa,Ba) = minus_minus_int(C_4,D_3) )
=> ( ( Aa = Ba )
<=> ( C_4 = D_3 ) ) ) ).
tff(fact_26_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3)) ).
tff(fact_27_zmult__commute,axiom,
! [Z: int,W: int] : times_times_int(Z,W) = times_times_int(W,Z) ).
tff(fact_28_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3)) ).
tff(fact_29_zadd__left__commute,axiom,
! [X: int,Y: int,Z: int] : plus_plus_int(X,plus_plus_int(Y,Z)) = plus_plus_int(Y,plus_plus_int(X,Z)) ).
tff(fact_30_zadd__commute,axiom,
! [Z: int,W: int] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
tff(fact_31_combine__common__factor,axiom,
! [A_139: int,E_1: int,B_105: int,C_62: int] : plus_plus_int(times_times_int(A_139,E_1),plus_plus_int(times_times_int(B_105,E_1),C_62)) = plus_plus_int(times_times_int(plus_plus_int(A_139,B_105),E_1),C_62) ).
tff(fact_32_combine__common__factor,axiom,
! [A_139: nat,E_1: nat,B_105: nat,C_62: nat] : plus_plus_nat(times_times_nat(A_139,E_1),plus_plus_nat(times_times_nat(B_105,E_1),C_62)) = plus_plus_nat(times_times_nat(plus_plus_nat(A_139,B_105),E_1),C_62) ).
tff(fact_33_comm__semiring__class_Odistrib,axiom,
! [A_138: int,B_104: int,C_61: int] : times_times_int(plus_plus_int(A_138,B_104),C_61) = plus_plus_int(times_times_int(A_138,C_61),times_times_int(B_104,C_61)) ).
tff(fact_34_comm__semiring__class_Odistrib,axiom,
! [A_138: nat,B_104: nat,C_61: nat] : times_times_nat(plus_plus_nat(A_138,B_104),C_61) = plus_plus_nat(times_times_nat(A_138,C_61),times_times_nat(B_104,C_61)) ).
tff(fact_35_add__diff__add,axiom,
! [A_137: int,C_60: int,B_103: int,D_18: int] : minus_minus_int(plus_plus_int(A_137,C_60),plus_plus_int(B_103,D_18)) = plus_plus_int(minus_minus_int(A_137,B_103),minus_minus_int(C_60,D_18)) ).
tff(fact_36_add__diff__cancel,axiom,
! [A_136: int,B_102: int] : minus_minus_int(plus_plus_int(A_136,B_102),B_102) = A_136 ).
tff(fact_37_crossproduct__eq,axiom,
! [W_1: int,Y_3: int,X_1: int,Z_1: int] :
( ( plus_plus_int(times_times_int(W_1,Y_3),times_times_int(X_1,Z_1)) = plus_plus_int(times_times_int(W_1,Z_1),times_times_int(X_1,Y_3)) )
<=> ( ( W_1 = X_1 )
| ( Y_3 = Z_1 ) ) ) ).
tff(fact_38_crossproduct__eq,axiom,
! [W_1: nat,Y_3: nat,X_1: nat,Z_1: nat] :
( ( plus_plus_nat(times_times_nat(W_1,Y_3),times_times_nat(X_1,Z_1)) = plus_plus_nat(times_times_nat(W_1,Z_1),times_times_nat(X_1,Y_3)) )
<=> ( ( W_1 = X_1 )
| ( Y_3 = Z_1 ) ) ) ).
tff(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_135: int,M_6: int,B_101: int] : plus_plus_int(times_times_int(A_135,M_6),times_times_int(B_101,M_6)) = times_times_int(plus_plus_int(A_135,B_101),M_6) ).
tff(fact_40_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_135: nat,M_6: nat,B_101: nat] : plus_plus_nat(times_times_nat(A_135,M_6),times_times_nat(B_101,M_6)) = times_times_nat(plus_plus_nat(A_135,B_101),M_6) ).
tff(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_134: int,B_100: int,C_59: int] : times_times_int(plus_plus_int(A_134,B_100),C_59) = plus_plus_int(times_times_int(A_134,C_59),times_times_int(B_100,C_59)) ).
tff(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_134: nat,B_100: nat,C_59: nat] : times_times_nat(plus_plus_nat(A_134,B_100),C_59) = plus_plus_nat(times_times_nat(A_134,C_59),times_times_nat(B_100,C_59)) ).
tff(fact_43_crossproduct__noteq,axiom,
! [C_4: int,D_3: int,Aa: int,Ba: int] :
( ( ( Aa != Ba )
& ( C_4 != D_3 ) )
<=> ( plus_plus_int(times_times_int(Aa,C_4),times_times_int(Ba,D_3)) != plus_plus_int(times_times_int(Aa,D_3),times_times_int(Ba,C_4)) ) ) ).
tff(fact_44_crossproduct__noteq,axiom,
! [C_4: nat,D_3: nat,Aa: nat,Ba: nat] :
( ( ( Aa != Ba )
& ( C_4 != D_3 ) )
<=> ( plus_plus_nat(times_times_nat(Aa,C_4),times_times_nat(Ba,D_3)) != plus_plus_nat(times_times_nat(Aa,D_3),times_times_nat(Ba,C_4)) ) ) ).
tff(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: int,Y_42: int,Z_12: int] : times_times_int(X_43,plus_plus_int(Y_42,Z_12)) = plus_plus_int(times_times_int(X_43,Y_42),times_times_int(X_43,Z_12)) ).
tff(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: nat,Y_42: nat,Z_12: nat] : times_times_nat(X_43,plus_plus_nat(Y_42,Z_12)) = plus_plus_nat(times_times_nat(X_43,Y_42),times_times_nat(X_43,Z_12)) ).
tff(fact_47_Pair__inject,axiom,
! [A_133: int,B_99: int,A_132: int,B_98: int] :
( ( product_Pair_int_int(A_133,B_99) = product_Pair_int_int(A_132,B_98) )
=> ~ ( ( A_133 = A_132 )
=> ( B_99 != B_98 ) ) ) ).
tff(fact_48_Pair__inject,axiom,
! [A_133: int,B_99: product_prod_int_int,A_132: int,B_98: product_prod_int_int] :
( ( produc282740534nt_int(A_133,B_99) = produc282740534nt_int(A_132,B_98) )
=> ~ ( ( A_133 = A_132 )
=> ( B_99 != B_98 ) ) ) ).
tff(fact_49_Pair__eq,axiom,
! [Aa: int,Ba: int,A_131: int,B_97: int] :
( ( product_Pair_int_int(Aa,Ba) = product_Pair_int_int(A_131,B_97) )
<=> ( ( Aa = A_131 )
& ( Ba = B_97 ) ) ) ).
tff(fact_50_Pair__eq,axiom,
! [Aa: int,Ba: product_prod_int_int,A_131: int,B_97: product_prod_int_int] :
( ( produc282740534nt_int(Aa,Ba) = produc282740534nt_int(A_131,B_97) )
<=> ( ( Aa = A_131 )
& ( Ba = B_97 ) ) ) ).
tff(fact_51_split__paired__All,axiom,
! [P: fun_Pr974702441t_bool] :
( ! [X1: product_prod_int_int] : hBOOL(hAPP_P603027463t_bool(P,X1))
<=> ! [A_88: int,B_92: int] : hBOOL(hAPP_P603027463t_bool(P,product_Pair_int_int(A_88,B_92))) ) ).
tff(fact_52_split__paired__All,axiom,
! [P: fun_Pr1503267378t_bool] :
( ! [X1: produc393999548nt_int] : hBOOL(hAPP_P178748804t_bool(P,X1))
<=> ! [A_88: int,B_92: product_prod_int_int] : hBOOL(hAPP_P178748804t_bool(P,produc282740534nt_int(A_88,B_92))) ) ).
tff(fact_53_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_130: int,B_96: int,C_58: int,D_17: int] : plus_plus_int(plus_plus_int(A_130,B_96),plus_plus_int(C_58,D_17)) = plus_plus_int(plus_plus_int(A_130,C_58),plus_plus_int(B_96,D_17)) ).
tff(fact_54_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_130: nat,B_96: nat,C_58: nat,D_17: nat] : plus_plus_nat(plus_plus_nat(A_130,B_96),plus_plus_nat(C_58,D_17)) = plus_plus_nat(plus_plus_nat(A_130,C_58),plus_plus_nat(B_96,D_17)) ).
tff(fact_55_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_129: int,B_95: int] : times_times_int(A_129,B_95) = times_times_int(B_95,A_129) ).
tff(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_129: nat,B_95: nat] : times_times_nat(A_129,B_95) = times_times_nat(B_95,A_129) ).
tff(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] : times_times_int(Lx_6,times_times_int(Rx_6,Ry_4)) = times_times_int(Rx_6,times_times_int(Lx_6,Ry_4)) ).
tff(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: nat,Rx_6: nat,Ry_4: nat] : times_times_nat(Lx_6,times_times_nat(Rx_6,Ry_4)) = times_times_nat(Rx_6,times_times_nat(Lx_6,Ry_4)) ).
tff(fact_59_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] : times_times_int(Lx_5,times_times_int(Rx_5,Ry_3)) = times_times_int(times_times_int(Lx_5,Rx_5),Ry_3) ).
tff(fact_60_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: nat,Rx_5: nat,Ry_3: nat] : times_times_nat(Lx_5,times_times_nat(Rx_5,Ry_3)) = times_times_nat(times_times_nat(Lx_5,Rx_5),Ry_3) ).
tff(fact_61_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_4: int] : times_times_int(times_times_int(Lx_4,Ly_4),Rx_4) = times_times_int(Lx_4,times_times_int(Ly_4,Rx_4)) ).
tff(fact_62_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: nat,Ly_4: nat,Rx_4: nat] : times_times_nat(times_times_nat(Lx_4,Ly_4),Rx_4) = times_times_nat(Lx_4,times_times_nat(Ly_4,Rx_4)) ).
tff(fact_63_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: int,Rx_3: int] : times_times_int(times_times_int(Lx_3,Ly_3),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_3) ).
tff(fact_64_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_3: nat,Rx_3: nat] : times_times_nat(times_times_nat(Lx_3,Ly_3),Rx_3) = times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_3) ).
tff(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] : times_times_int(times_times_int(Lx_2,Ly_2),times_times_int(Rx_2,Ry_2)) = times_times_int(Lx_2,times_times_int(Ly_2,times_times_int(Rx_2,Ry_2))) ).
tff(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: nat,Ly_2: nat,Rx_2: nat,Ry_2: nat] : times_times_nat(times_times_nat(Lx_2,Ly_2),times_times_nat(Rx_2,Ry_2)) = times_times_nat(Lx_2,times_times_nat(Ly_2,times_times_nat(Rx_2,Ry_2))) ).
tff(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] : times_times_int(times_times_int(Lx_1,Ly_1),times_times_int(Rx_1,Ry_1)) = times_times_int(Rx_1,times_times_int(times_times_int(Lx_1,Ly_1),Ry_1)) ).
tff(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] : times_times_nat(times_times_nat(Lx_1,Ly_1),times_times_nat(Rx_1,Ry_1)) = times_times_nat(Rx_1,times_times_nat(times_times_nat(Lx_1,Ly_1),Ry_1)) ).
tff(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] : times_times_int(times_times_int(Lx,Ly),times_times_int(Rx,Ry)) = times_times_int(times_times_int(Lx,Rx),times_times_int(Ly,Ry)) ).
tff(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] : times_times_nat(times_times_nat(Lx,Ly),times_times_nat(Rx,Ry)) = times_times_nat(times_times_nat(Lx,Rx),times_times_nat(Ly,Ry)) ).
tff(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_128: int,C_57: int] : plus_plus_int(A_128,C_57) = plus_plus_int(C_57,A_128) ).
tff(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_128: nat,C_57: nat] : plus_plus_nat(A_128,C_57) = plus_plus_nat(C_57,A_128) ).
tff(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_127: int,C_56: int,D_16: int] : plus_plus_int(A_127,plus_plus_int(C_56,D_16)) = plus_plus_int(C_56,plus_plus_int(A_127,D_16)) ).
tff(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_127: nat,C_56: nat,D_16: nat] : plus_plus_nat(A_127,plus_plus_nat(C_56,D_16)) = plus_plus_nat(C_56,plus_plus_nat(A_127,D_16)) ).
tff(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_126: int,C_55: int,D_15: int] : plus_plus_int(A_126,plus_plus_int(C_55,D_15)) = plus_plus_int(plus_plus_int(A_126,C_55),D_15) ).
tff(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_126: nat,C_55: nat,D_15: nat] : plus_plus_nat(A_126,plus_plus_nat(C_55,D_15)) = plus_plus_nat(plus_plus_nat(A_126,C_55),D_15) ).
tff(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_125: int,B_94: int,C_54: int] : plus_plus_int(plus_plus_int(A_125,B_94),C_54) = plus_plus_int(A_125,plus_plus_int(B_94,C_54)) ).
tff(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_125: nat,B_94: nat,C_54: nat] : plus_plus_nat(plus_plus_nat(A_125,B_94),C_54) = plus_plus_nat(A_125,plus_plus_nat(B_94,C_54)) ).
tff(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_124: int,B_93: int,C_53: int] : plus_plus_int(plus_plus_int(A_124,B_93),C_53) = plus_plus_int(plus_plus_int(A_124,C_53),B_93) ).
tff(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_124: nat,B_93: nat,C_53: nat] : plus_plus_nat(plus_plus_nat(A_124,B_93),C_53) = plus_plus_nat(plus_plus_nat(A_124,C_53),B_93) ).
tff(fact_81_split__paired__Ex,axiom,
! [P: fun_Pr974702441t_bool] :
( ? [X1: product_prod_int_int] : hBOOL(hAPP_P603027463t_bool(P,X1))
<=> ? [A_88: int,B_92: int] : hBOOL(hAPP_P603027463t_bool(P,product_Pair_int_int(A_88,B_92))) ) ).
tff(fact_82_split__paired__Ex,axiom,
! [P: fun_Pr1503267378t_bool] :
( ? [X1: produc393999548nt_int] : hBOOL(hAPP_P178748804t_bool(P,X1))
<=> ? [A_88: int,B_92: product_prod_int_int] : hBOOL(hAPP_P178748804t_bool(P,produc282740534nt_int(A_88,B_92))) ) ).
tff(fact_83_prod__induct3,axiom,
! [X_1: produc393999548nt_int,P: fun_Pr1503267378t_bool] :
( ! [A_88: int,B_92: int,C_52: int] : hBOOL(hAPP_P178748804t_bool(P,produc282740534nt_int(A_88,product_Pair_int_int(B_92,C_52))))
=> hBOOL(hAPP_P178748804t_bool(P,X_1)) ) ).
tff(fact_84_prod__cases3,axiom,
! [Y_41: produc393999548nt_int] :
~ ! [A_88: int,B_92: int,C_52: int] : Y_41 != produc282740534nt_int(A_88,product_Pair_int_int(B_92,C_52)) ).
tff(fact_85_prod_Oexhaust,axiom,
! [Y_40: product_prod_int_int] :
~ ! [A_88: int,B_92: int] : Y_40 != product_Pair_int_int(A_88,B_92) ).
tff(fact_86_prod_Oexhaust,axiom,
! [Y_40: produc393999548nt_int] :
~ ! [A_88: int,B_92: product_prod_int_int] : Y_40 != produc282740534nt_int(A_88,B_92) ).
tff(fact_87_PairE,axiom,
! [P_2: product_prod_int_int] :
~ ! [X_3: int,Y_39: int] : P_2 != product_Pair_int_int(X_3,Y_39) ).
tff(fact_88_PairE,axiom,
! [P_2: produc393999548nt_int] :
~ ! [X_3: int,Y_39: product_prod_int_int] : P_2 != produc282740534nt_int(X_3,Y_39) ).
tff(fact_89_curryI,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba)))
=> hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba)) ) ).
tff(fact_90_curryI,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba)))
=> hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba)) ) ).
tff(fact_91_le__add__iff1,axiom,
! [Aa: int,E: int,C_4: int,Ba: int,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C_4)),D_3)) ) ).
tff(fact_92_le__add__iff2,axiom,
! [Aa: int,E: int,C_4: int,Ba: int,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_4),plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_3))) ) ).
tff(fact_93_real__squared__diff__one__factored,axiom,
! [X_42: int] : minus_minus_int(times_times_int(X_42,X_42),one_one_int) = times_times_int(plus_plus_int(X_42,one_one_int),minus_minus_int(X_42,one_one_int)) ).
tff(fact_94_less__add__iff1,axiom,
! [Aa: int,E: int,C_4: int,Ba: int,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C_4)),D_3)) ) ).
tff(fact_95_less__add__iff2,axiom,
! [Aa: int,E: int,C_4: int,Ba: int,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_3))) ) ).
tff(fact_96_one__reorient,axiom,
! [X_1: int] :
( ( one_one_int = X_1 )
<=> ( X_1 = one_one_int ) ) ).
tff(fact_97_one__reorient,axiom,
! [X_1: nat] :
( ( one_one_nat = X_1 )
<=> ( X_1 = one_one_nat ) ) ).
tff(fact_98_zle__add1__eq__le,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),plus_plus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1)) ) ).
tff(fact_99_add1__zle__eq,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(W_1,one_one_int)),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) ) ).
tff(fact_100_linorder__neqE__linordered__idom,axiom,
! [X_41: int,Y_38: int] :
( ( X_41 != Y_38 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_41),Y_38))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_38),X_41)) ) ) ).
tff(fact_101_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(W,one_one_int)),Z)) ) ).
tff(fact_102_order__le__neq__implies__less,axiom,
! [X_40: int,Y_37: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_40),Y_37))
=> ( ( X_40 != Y_37 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_40),Y_37)) ) ) ).
tff(fact_103_order__le__neq__implies__less,axiom,
! [X_40: nat,Y_37: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_40),Y_37))
=> ( ( X_40 != Y_37 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_40),Y_37)) ) ) ).
tff(fact_104_zless__add1__eq,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),plus_plus_int(Z_1,one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1))
| ( W_1 = Z_1 ) ) ) ).
tff(fact_105_zadd__zless__mono,axiom,
! [Z_11: int,Z: int,W_2: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_11),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(W_2,Z_11)),plus_plus_int(W,Z))) ) ) ).
tff(fact_106_add__less__le__mono,axiom,
! [C_51: int,D_14: int,A_123: int,B_91: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_123),B_91))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_51),D_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_123,C_51)),plus_plus_int(B_91,D_14))) ) ) ).
tff(fact_107_add__less__le__mono,axiom,
! [C_51: nat,D_14: nat,A_123: nat,B_91: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_123),B_91))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_51),D_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_123,C_51)),plus_plus_nat(B_91,D_14))) ) ) ).
tff(fact_108_add__le__less__mono,axiom,
! [C_50: int,D_13: int,A_122: int,B_90: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_122),B_90))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_50),D_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_122,C_50)),plus_plus_int(B_90,D_13))) ) ) ).
tff(fact_109_add__le__less__mono,axiom,
! [C_50: nat,D_13: nat,A_122: nat,B_90: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_122),B_90))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_50),D_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_122,C_50)),plus_plus_nat(B_90,D_13))) ) ) ).
tff(fact_110_less__add__one,axiom,
! [A_121: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_121),plus_plus_int(A_121,one_one_int))) ).
tff(fact_111_less__add__one,axiom,
! [A_121: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_121),plus_plus_nat(A_121,one_one_nat))) ).
tff(fact_112_less__1__mult,axiom,
! [N_2: int,M_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),times_times_int(M_5,N_2))) ) ) ).
tff(fact_113_less__1__mult,axiom,
! [N_2: nat,M_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),times_times_nat(M_5,N_2))) ) ) ).
tff(fact_114_zle__diff1__eq,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),minus_minus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) ) ).
tff(fact_115_add__less__cancel__right,axiom,
! [Aa: int,C_4: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(Aa,C_4)),plus_plus_int(Ba,C_4)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
tff(fact_116_add__less__cancel__right,axiom,
! [Aa: nat,C_4: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(Aa,C_4)),plus_plus_nat(Ba,C_4)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_117_add__less__cancel__left,axiom,
! [C_4: int,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_4,Aa)),plus_plus_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
tff(fact_118_add__less__cancel__left,axiom,
! [C_4: nat,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_4,Aa)),plus_plus_nat(C_4,Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_119_add__strict__right__mono,axiom,
! [C_49: int,A_120: int,B_89: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_120),B_89))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_120,C_49)),plus_plus_int(B_89,C_49))) ) ).
tff(fact_120_add__strict__right__mono,axiom,
! [C_49: nat,A_120: nat,B_89: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_120),B_89))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_120,C_49)),plus_plus_nat(B_89,C_49))) ) ).
tff(fact_121_add__strict__left__mono,axiom,
! [C_48: int,A_119: int,B_88: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_119),B_88))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_48,A_119)),plus_plus_int(C_48,B_88))) ) ).
tff(fact_122_add__strict__left__mono,axiom,
! [C_48: nat,A_119: nat,B_88: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_119),B_88))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_48,A_119)),plus_plus_nat(C_48,B_88))) ) ).
tff(fact_123_add__strict__mono,axiom,
! [C_47: int,D_12: int,A_118: int,B_87: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_118),B_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_47),D_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_118,C_47)),plus_plus_int(B_87,D_12))) ) ) ).
tff(fact_124_add__strict__mono,axiom,
! [C_47: nat,D_12: nat,A_118: nat,B_87: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_118),B_87))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_47),D_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_118,C_47)),plus_plus_nat(B_87,D_12))) ) ) ).
tff(fact_125_add__less__imp__less__right,axiom,
! [A_117: int,C_46: int,B_86: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_117,C_46)),plus_plus_int(B_86,C_46)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_117),B_86)) ) ).
tff(fact_126_add__less__imp__less__right,axiom,
! [A_117: nat,C_46: nat,B_86: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_117,C_46)),plus_plus_nat(B_86,C_46)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_117),B_86)) ) ).
tff(fact_127_add__less__imp__less__left,axiom,
! [C_45: int,A_116: int,B_85: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_45,A_116)),plus_plus_int(C_45,B_85)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_116),B_85)) ) ).
tff(fact_128_add__less__imp__less__left,axiom,
! [C_45: nat,A_116: nat,B_85: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_45,A_116)),plus_plus_nat(C_45,B_85)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_116),B_85)) ) ).
tff(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_115: int] : times_times_int(one_one_int,A_115) = A_115 ).
tff(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_115: nat] : times_times_nat(one_one_nat,A_115) = A_115 ).
tff(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_114: int] : times_times_int(A_114,one_one_int) = A_114 ).
tff(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_114: nat] : times_times_nat(A_114,one_one_nat) = A_114 ).
tff(fact_133_mult__1__left,axiom,
! [A_113: int] : times_times_int(one_one_int,A_113) = A_113 ).
tff(fact_134_mult__1__left,axiom,
! [A_113: nat] : times_times_nat(one_one_nat,A_113) = A_113 ).
tff(fact_135_mult__1,axiom,
! [A_112: int] : times_times_int(one_one_int,A_112) = A_112 ).
tff(fact_136_mult__1,axiom,
! [A_112: nat] : times_times_nat(one_one_nat,A_112) = A_112 ).
tff(fact_137_mult__1__right,axiom,
! [A_111: int] : times_times_int(A_111,one_one_int) = A_111 ).
tff(fact_138_mult__1__right,axiom,
! [A_111: nat] : times_times_nat(A_111,one_one_nat) = A_111 ).
tff(fact_139_mult_Ocomm__neutral,axiom,
! [A_110: int] : times_times_int(A_110,one_one_int) = A_110 ).
tff(fact_140_mult_Ocomm__neutral,axiom,
! [A_110: nat] : times_times_nat(A_110,one_one_nat) = A_110 ).
tff(fact_141_diff__eq__diff__less,axiom,
! [Aa: int,Ba: int,C_4: int,D_3: int] :
( ( minus_minus_int(Aa,Ba) = minus_minus_int(C_4,D_3) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),D_3)) ) ) ).
tff(fact_142_add__le__cancel__right,axiom,
! [Aa: int,C_4: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(Aa,C_4)),plus_plus_int(Ba,C_4)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
tff(fact_143_add__le__cancel__right,axiom,
! [Aa: nat,C_4: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(Aa,C_4)),plus_plus_nat(Ba,C_4)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
tff(fact_144_add__le__cancel__left,axiom,
! [C_4: int,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_4,Aa)),plus_plus_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
tff(fact_145_add__le__cancel__left,axiom,
! [C_4: nat,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_4,Aa)),plus_plus_nat(C_4,Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
tff(fact_146_add__right__mono,axiom,
! [C_44: int,A_109: int,B_84: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_109),B_84))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_109,C_44)),plus_plus_int(B_84,C_44))) ) ).
tff(fact_147_add__right__mono,axiom,
! [C_44: nat,A_109: nat,B_84: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_109),B_84))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_109,C_44)),plus_plus_nat(B_84,C_44))) ) ).
tff(fact_148_add__left__mono,axiom,
! [C_43: int,A_108: int,B_83: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_108),B_83))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_43,A_108)),plus_plus_int(C_43,B_83))) ) ).
tff(fact_149_add__left__mono,axiom,
! [C_43: nat,A_108: nat,B_83: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_108),B_83))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_43,A_108)),plus_plus_nat(C_43,B_83))) ) ).
tff(fact_150_add__mono,axiom,
! [C_42: int,D_11: int,A_107: int,B_82: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_107),B_82))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_42),D_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_107,C_42)),plus_plus_int(B_82,D_11))) ) ) ).
tff(fact_151_add__mono,axiom,
! [C_42: nat,D_11: nat,A_107: nat,B_82: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_107),B_82))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_42),D_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_107,C_42)),plus_plus_nat(B_82,D_11))) ) ) ).
tff(fact_152_add__le__imp__le__right,axiom,
! [A_106: int,C_41: int,B_81: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_106,C_41)),plus_plus_int(B_81,C_41)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_106),B_81)) ) ).
tff(fact_153_add__le__imp__le__right,axiom,
! [A_106: nat,C_41: nat,B_81: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_106,C_41)),plus_plus_nat(B_81,C_41)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_106),B_81)) ) ).
tff(fact_154_add__le__imp__le__left,axiom,
! [C_40: int,A_105: int,B_80: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_40,A_105)),plus_plus_int(C_40,B_80)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_105),B_80)) ) ).
tff(fact_155_add__le__imp__le__left,axiom,
! [C_40: nat,A_105: nat,B_80: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_40,A_105)),plus_plus_nat(C_40,B_80)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_105),B_80)) ) ).
tff(fact_156_diff__eq__diff__less__eq,axiom,
! [Aa: int,Ba: int,C_4: int,D_3: int] :
( ( minus_minus_int(Aa,Ba) = minus_minus_int(C_4,D_3) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_4),D_3)) ) ) ).
tff(fact_157_zadd__strict__right__mono,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(I,K)),plus_plus_int(J,K))) ) ).
tff(fact_158_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(K,I)),plus_plus_int(K,J))) ) ).
tff(fact_159_zmult__1,axiom,
! [Z: int] : times_times_int(one_one_int,Z) = Z ).
tff(fact_160_zmult__1__right,axiom,
! [Z: int] : times_times_int(Z,one_one_int) = Z ).
tff(fact_161_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4: int] : plus_plus_int(M_4,M_4) = times_times_int(plus_plus_int(one_one_int,one_one_int),M_4) ).
tff(fact_162_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4: nat] : plus_plus_nat(M_4,M_4) = times_times_nat(plus_plus_nat(one_one_nat,one_one_nat),M_4) ).
tff(fact_163_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_3: nat,A_104: nat] : plus_plus_nat(M_3,times_times_nat(A_104,M_3)) = times_times_nat(plus_plus_nat(A_104,one_one_nat),M_3) ).
tff(fact_164_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_3: int,A_104: int] : plus_plus_int(M_3,times_times_int(A_104,M_3)) = times_times_int(plus_plus_int(A_104,one_one_int),M_3) ).
tff(fact_165_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_103: nat,M_2: nat] : plus_plus_nat(times_times_nat(A_103,M_2),M_2) = times_times_nat(plus_plus_nat(A_103,one_one_nat),M_2) ).
tff(fact_166_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_103: int,M_2: int] : plus_plus_int(times_times_int(A_103,M_2),M_2) = times_times_int(plus_plus_int(A_103,one_one_int),M_2) ).
tff(fact_167_curryD,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
tff(fact_168_curryD,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
tff(fact_169_curryE,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
tff(fact_170_curryE,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
tff(fact_171_curry__conv,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
<=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
tff(fact_172_curry__conv,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
<=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
tff(fact_173_order__refl,axiom,
! [X_39: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_39),X_39)) ).
tff(fact_174_order__refl,axiom,
! [X_39: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_39),X_39)) ).
tff(fact_175_xt1_I8_J,axiom,
! [Z_10: nat,Y_36: nat,X_38: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_36),X_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),Y_36))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),X_38)) ) ) ).
tff(fact_176_xt1_I8_J,axiom,
! [Z_10: int,Y_36: int,X_38: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_36),X_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),Y_36))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),X_38)) ) ) ).
tff(fact_177_order__le__less__trans,axiom,
! [Z_9: nat,X_37: nat,Y_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_37),Y_35))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_35),Z_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_37),Z_9)) ) ) ).
tff(fact_178_order__le__less__trans,axiom,
! [Z_9: int,X_37: int,Y_35: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_37),Y_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_35),Z_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),Z_9)) ) ) ).
tff(fact_179_xt1_I7_J,axiom,
! [Z_8: nat,Y_34: nat,X_36: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_34),X_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),Y_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_8),X_36)) ) ) ).
tff(fact_180_xt1_I7_J,axiom,
! [Z_8: int,Y_34: int,X_36: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_34),X_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),Y_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_8),X_36)) ) ) ).
tff(fact_181_order__less__le__trans,axiom,
! [Z_7: nat,X_35: nat,Y_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Y_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_33),Z_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Z_7)) ) ) ).
tff(fact_182_order__less__le__trans,axiom,
! [Z_7: int,X_35: int,Y_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Y_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_33),Z_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Z_7)) ) ) ).
tff(fact_183_xt1_I11_J,axiom,
! [B_79: nat,A_102: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_79),A_102))
=> ( ( A_102 != B_79 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_79),A_102)) ) ) ).
tff(fact_184_xt1_I11_J,axiom,
! [B_79: int,A_102: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_79),A_102))
=> ( ( A_102 != B_79 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_79),A_102)) ) ) ).
tff(fact_185_order__le__neq__trans,axiom,
! [A_101: nat,B_78: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_101),B_78))
=> ( ( A_101 != B_78 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_101),B_78)) ) ) ).
tff(fact_186_order__le__neq__trans,axiom,
! [A_101: int,B_78: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_101),B_78))
=> ( ( A_101 != B_78 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_101),B_78)) ) ) ).
tff(fact_187_order__le__imp__less__or__eq,axiom,
! [X_34: nat,Y_32: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_34),Y_32))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_34),Y_32))
| ( X_34 = Y_32 ) ) ) ).
tff(fact_188_order__le__imp__less__or__eq,axiom,
! [X_34: int,Y_32: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_34),Y_32))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_34),Y_32))
| ( X_34 = Y_32 ) ) ) ).
tff(fact_189_zle__refl,axiom,
! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
tff(fact_190_zle__linear,axiom,
! [Z: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).
tff(fact_191_zless__le,axiom,
! [Z_1: int,W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),W_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1))
& ( Z_1 != W_1 ) ) ) ).
tff(fact_192_zless__linear,axiom,
! [X: int,Y: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
| ( X = Y )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ).
tff(fact_193_zle__trans,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K)) ) ) ).
tff(fact_194_zle__antisym,axiom,
! [Z: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
=> ( Z = W ) ) ) ).
tff(fact_195_linorder__linear,axiom,
! [X_33: nat,Y_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_33),Y_31))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_31),X_33)) ) ).
tff(fact_196_linorder__linear,axiom,
! [X_33: int,Y_31: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_33),Y_31))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_31),X_33)) ) ).
tff(fact_197_order__eq__iff,axiom,
! [X_1: nat,Y_3: nat] :
( ( X_1 = Y_3 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1)) ) ) ).
tff(fact_198_order__eq__iff,axiom,
! [X_1: int,Y_3: int] :
( ( X_1 = Y_3 )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1)) ) ) ).
tff(fact_199_order__eq__refl,axiom,
! [X_32: nat,Y_30: nat] :
( ( X_32 = Y_30 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_32),Y_30)) ) ).
tff(fact_200_order__eq__refl,axiom,
! [X_32: int,Y_30: int] :
( ( X_32 = Y_30 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_32),Y_30)) ) ).
tff(fact_201_order__antisym__conv,axiom,
! [Y_3: nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_202_order__antisym__conv,axiom,
! [Y_3: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_203_ord__eq__le__trans,axiom,
! [C_39: nat,A_100: nat,B_77: nat] :
( ( A_100 = B_77 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_77),C_39))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_100),C_39)) ) ) ).
tff(fact_204_ord__eq__le__trans,axiom,
! [C_39: int,A_100: int,B_77: int] :
( ( A_100 = B_77 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_77),C_39))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_100),C_39)) ) ) ).
tff(fact_205_xt1_I3_J,axiom,
! [C_38: nat,A_99: nat,B_76: nat] :
( ( A_99 = B_76 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_38),B_76))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_38),A_99)) ) ) ).
tff(fact_206_xt1_I3_J,axiom,
! [C_38: int,A_99: int,B_76: int] :
( ( A_99 = B_76 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_38),B_76))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_38),A_99)) ) ) ).
tff(fact_207_ord__le__eq__trans,axiom,
! [C_37: nat,A_98: nat,B_75: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_98),B_75))
=> ( ( B_75 = C_37 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_98),C_37)) ) ) ).
tff(fact_208_ord__le__eq__trans,axiom,
! [C_37: int,A_98: int,B_75: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_98),B_75))
=> ( ( B_75 = C_37 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_98),C_37)) ) ) ).
tff(fact_209_xt1_I4_J,axiom,
! [C_36: nat,B_74: nat,A_97: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_74),A_97))
=> ( ( B_74 = C_36 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_36),A_97)) ) ) ).
tff(fact_210_xt1_I4_J,axiom,
! [C_36: int,B_74: int,A_97: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_74),A_97))
=> ( ( B_74 = C_36 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_36),A_97)) ) ) ).
tff(fact_211_order__antisym,axiom,
! [X_31: nat,Y_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_31),Y_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_29),X_31))
=> ( X_31 = Y_29 ) ) ) ).
tff(fact_212_order__antisym,axiom,
! [X_31: int,Y_29: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_31),Y_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_29),X_31))
=> ( X_31 = Y_29 ) ) ) ).
tff(fact_213_order__trans,axiom,
! [Z_6: nat,X_30: nat,Y_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_28),Z_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Z_6)) ) ) ).
tff(fact_214_order__trans,axiom,
! [Z_6: int,X_30: int,Y_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Y_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_28),Z_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Z_6)) ) ) ).
tff(fact_215_xt1_I5_J,axiom,
! [Y_27: nat,X_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_27),X_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_27))
=> ( X_29 = Y_27 ) ) ) ).
tff(fact_216_xt1_I5_J,axiom,
! [Y_27: int,X_29: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_27),X_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_29),Y_27))
=> ( X_29 = Y_27 ) ) ) ).
tff(fact_217_xt1_I6_J,axiom,
! [Z_5: nat,Y_26: nat,X_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_26),X_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),Y_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),X_28)) ) ) ).
tff(fact_218_xt1_I6_J,axiom,
! [Z_5: int,Y_26: int,X_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_26),X_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),Y_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),X_28)) ) ) ).
tff(fact_219_linorder__le__cases,axiom,
! [X_27: nat,Y_25: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),Y_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_27)) ) ).
tff(fact_220_linorder__le__cases,axiom,
! [X_27: int,Y_25: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_27),Y_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_25),X_27)) ) ).
tff(fact_221_order__less__irrefl,axiom,
! [X_26: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_26),X_26)) ).
tff(fact_222_order__less__irrefl,axiom,
! [X_26: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_26),X_26)) ).
tff(fact_223_linorder__neq__iff,axiom,
! [X_1: nat,Y_3: nat] :
( ( X_1 != Y_3 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1)) ) ) ).
tff(fact_224_linorder__neq__iff,axiom,
! [X_1: int,Y_3: int] :
( ( X_1 != Y_3 )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1)) ) ) ).
tff(fact_225_not__less__iff__gr__or__eq,axiom,
! [X_1: nat,Y_3: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1))
| ( X_1 = Y_3 ) ) ) ).
tff(fact_226_not__less__iff__gr__or__eq,axiom,
! [X_1: int,Y_3: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1))
| ( X_1 = Y_3 ) ) ) ).
tff(fact_227_linorder__less__linear,axiom,
! [X_25: nat,Y_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_25),Y_24))
| ( X_25 = Y_24 )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_24),X_25)) ) ).
tff(fact_228_linorder__less__linear,axiom,
! [X_25: int,Y_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_25),Y_24))
| ( X_25 = Y_24 )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),X_25)) ) ).
tff(fact_229_linorder__antisym__conv3,axiom,
! [Y_3: nat,X_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_230_linorder__antisym__conv3,axiom,
! [Y_3: int,X_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_231_linorder__neqE,axiom,
! [X_24: nat,Y_23: nat] :
( ( X_24 != Y_23 )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_24),Y_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_23),X_24)) ) ) ).
tff(fact_232_linorder__neqE,axiom,
! [X_24: int,Y_23: int] :
( ( X_24 != Y_23 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_24),Y_23))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_23),X_24)) ) ) ).
tff(fact_233_less__imp__neq,axiom,
! [X_23: nat,Y_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_23),Y_22))
=> ( X_23 != Y_22 ) ) ).
tff(fact_234_less__imp__neq,axiom,
! [X_23: int,Y_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_23),Y_22))
=> ( X_23 != Y_22 ) ) ).
tff(fact_235_order__less__not__sym,axiom,
! [X_22: nat,Y_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Y_21))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_21),X_22)) ) ).
tff(fact_236_order__less__not__sym,axiom,
! [X_22: int,Y_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Y_21))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_21),X_22)) ) ).
tff(fact_237_order__less__imp__not__less,axiom,
! [X_21: nat,Y_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_20))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_20),X_21)) ) ).
tff(fact_238_order__less__imp__not__less,axiom,
! [X_21: int,Y_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_20))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_20),X_21)) ) ).
tff(fact_239_order__less__imp__not__eq,axiom,
! [X_20: nat,Y_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_19))
=> ( X_20 != Y_19 ) ) ).
tff(fact_240_order__less__imp__not__eq,axiom,
! [X_20: int,Y_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_20),Y_19))
=> ( X_20 != Y_19 ) ) ).
tff(fact_241_order__less__imp__not__eq2,axiom,
! [X_19: nat,Y_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Y_18))
=> ( Y_18 != X_19 ) ) ).
tff(fact_242_order__less__imp__not__eq2,axiom,
! [X_19: int,Y_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Y_18))
=> ( Y_18 != X_19 ) ) ).
tff(fact_243_order__less__imp__triv,axiom,
! [P: bool,X_1: nat,Y_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1))
=> hBOOL(P) ) ) ).
tff(fact_244_order__less__imp__triv,axiom,
! [P: bool,X_1: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1))
=> hBOOL(P) ) ) ).
tff(fact_245_order__less__asym_H,axiom,
! [A_96: nat,B_73: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_73))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_73),A_96)) ) ).
tff(fact_246_order__less__asym_H,axiom,
! [A_96: int,B_73: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_73))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_73),A_96)) ) ).
tff(fact_247_xt1_I9_J,axiom,
! [B_72: nat,A_95: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_72),A_95))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_72)) ) ).
tff(fact_248_xt1_I9_J,axiom,
! [B_72: int,A_95: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_72),A_95))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_72)) ) ).
tff(fact_249_ord__eq__less__trans,axiom,
! [C_35: nat,A_94: nat,B_71: nat] :
( ( A_94 = B_71 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_71),C_35))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_94),C_35)) ) ) ).
tff(fact_250_ord__eq__less__trans,axiom,
! [C_35: int,A_94: int,B_71: int] :
( ( A_94 = B_71 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_71),C_35))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_94),C_35)) ) ) ).
tff(fact_251_xt1_I1_J,axiom,
! [C_34: nat,A_93: nat,B_70: nat] :
( ( A_93 = B_70 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),B_70))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),A_93)) ) ) ).
tff(fact_252_xt1_I1_J,axiom,
! [C_34: int,A_93: int,B_70: int] :
( ( A_93 = B_70 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),B_70))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),A_93)) ) ) ).
tff(fact_253_ord__less__eq__trans,axiom,
! [C_33: nat,A_92: nat,B_69: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_92),B_69))
=> ( ( B_69 = C_33 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_92),C_33)) ) ) ).
tff(fact_254_ord__less__eq__trans,axiom,
! [C_33: int,A_92: int,B_69: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_92),B_69))
=> ( ( B_69 = C_33 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_92),C_33)) ) ) ).
tff(fact_255_xt1_I2_J,axiom,
! [C_32: nat,B_68: nat,A_91: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_68),A_91))
=> ( ( B_68 = C_32 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_32),A_91)) ) ) ).
tff(fact_256_xt1_I2_J,axiom,
! [C_32: int,B_68: int,A_91: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_68),A_91))
=> ( ( B_68 = C_32 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_32),A_91)) ) ) ).
tff(fact_257_order__less__trans,axiom,
! [Z_4: nat,X_18: nat,Y_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Y_17))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_17),Z_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Z_4)) ) ) ).
tff(fact_258_order__less__trans,axiom,
! [Z_4: int,X_18: int,Y_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Y_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_17),Z_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Z_4)) ) ) ).
tff(fact_259_xt1_I10_J,axiom,
! [Z_3: nat,Y_16: nat,X_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_16),X_17))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_3),Y_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_3),X_17)) ) ) ).
tff(fact_260_xt1_I10_J,axiom,
! [Z_3: int,Y_16: int,X_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_16),X_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),Y_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),X_17)) ) ) ).
tff(fact_261_order__less__asym,axiom,
! [X_16: nat,Y_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_16),Y_15))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_15),X_16)) ) ).
tff(fact_262_order__less__asym,axiom,
! [X_16: int,Y_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_16),Y_15))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_15),X_16)) ) ).
tff(fact_263_linorder__cases,axiom,
! [X_15: nat,Y_14: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_15),Y_14))
=> ( ( X_15 != Y_14 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_14),X_15)) ) ) ).
tff(fact_264_linorder__cases,axiom,
! [X_15: int,Y_14: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Y_14))
=> ( ( X_15 != Y_14 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_14),X_15)) ) ) ).
tff(fact_265_linorder__not__less,axiom,
! [X_1: nat,Y_3: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1)) ) ).
tff(fact_266_linorder__not__less,axiom,
! [X_1: int,Y_3: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1)) ) ).
tff(fact_267_linorder__not__le,axiom,
! [X_1: nat,Y_3: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1)) ) ).
tff(fact_268_linorder__not__le,axiom,
! [X_1: int,Y_3: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1)) ) ).
tff(fact_269_linorder__le__less__linear,axiom,
! [X_14: nat,Y_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),Y_13))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_13),X_14)) ) ).
tff(fact_270_linorder__le__less__linear,axiom,
! [X_14: int,Y_13: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_14),Y_13))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_13),X_14)) ) ).
tff(fact_271_order__less__le,axiom,
! [X_1: nat,Y_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
& ( X_1 != Y_3 ) ) ) ).
tff(fact_272_order__less__le,axiom,
! [X_1: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
& ( X_1 != Y_3 ) ) ) ).
tff(fact_273_less__le__not__le,axiom,
! [X_1: nat,Y_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1)) ) ) ).
tff(fact_274_less__le__not__le,axiom,
! [X_1: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
& ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1)) ) ) ).
tff(fact_275_order__le__less,axiom,
! [X_1: nat,Y_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
| ( X_1 = Y_3 ) ) ) ).
tff(fact_276_order__le__less,axiom,
! [X_1: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
| ( X_1 = Y_3 ) ) ) ).
tff(fact_277_leI,axiom,
! [X_13: nat,Y_12: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_12),X_13)) ) ).
tff(fact_278_leI,axiom,
! [X_13: int,Y_12: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_13),Y_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_12),X_13)) ) ).
tff(fact_279_not__leE,axiom,
! [Y_11: nat,X_12: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_11),X_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_12),Y_11)) ) ).
tff(fact_280_not__leE,axiom,
! [Y_11: int,X_12: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_11),X_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_12),Y_11)) ) ).
tff(fact_281_linorder__antisym__conv1,axiom,
! [X_1: nat,Y_3: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_282_linorder__antisym__conv1,axiom,
! [X_1: int,Y_3: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_283_order__neq__le__trans,axiom,
! [A_90: nat,B_67: nat] :
( ( A_90 != B_67 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_90),B_67))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_90),B_67)) ) ) ).
tff(fact_284_order__neq__le__trans,axiom,
! [A_90: int,B_67: int] :
( ( A_90 != B_67 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_90),B_67))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_90),B_67)) ) ) ).
tff(fact_285_xt1_I12_J,axiom,
! [A_89: nat,B_66: nat] :
( ( A_89 != B_66 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_66),A_89))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_66),A_89)) ) ) ).
tff(fact_286_xt1_I12_J,axiom,
! [A_89: int,B_66: int] :
( ( A_89 != B_66 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_66),A_89))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_66),A_89)) ) ) ).
tff(fact_287_leD,axiom,
! [Y_10: nat,X_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_10),X_11))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_11),Y_10)) ) ).
tff(fact_288_leD,axiom,
! [Y_10: int,X_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_10),X_11))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_11),Y_10)) ) ).
tff(fact_289_order__less__imp__le,axiom,
! [X_10: nat,Y_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_10),Y_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_10),Y_9)) ) ).
tff(fact_290_order__less__imp__le,axiom,
! [X_10: int,Y_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_10),Y_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_10),Y_9)) ) ).
tff(fact_291_linorder__antisym__conv2,axiom,
! [X_1: nat,Y_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_292_linorder__antisym__conv2,axiom,
! [X_1: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( X_1 = Y_3 ) ) ) ).
tff(fact_293_norR__mem__unique__aux,axiom,
! [A_19: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),minus_minus_int(B,one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_19),B)) ) ).
tff(fact_294_int__le__induct,axiom,
! [P: fun_int_bool,I_1: int,K_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P,K_1))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
tff(fact_295_int__less__induct,axiom,
! [P: fun_int_bool,I_1: int,K_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P,minus_minus_int(K_1,one_one_int)))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
tff(fact_296_int__ge__induct,axiom,
! [P: fun_int_bool,K_1: int,I_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P,K_1))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
tff(fact_297_int__gr__induct,axiom,
! [P: fun_int_bool,K_1: int,I_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P,plus_plus_int(K_1,one_one_int)))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
tff(fact_298_int__induct,axiom,
! [I_1: int,P: fun_int_bool,K_1: int] :
( hBOOL(hAPP_int_bool(P,K_1))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(I_2,one_one_int))) ) )
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
tff(fact_299_Euler_Oaux2,axiom,
! [B: int,A_19: int,C_31: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_19),C_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),C_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),B))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A_19)) ) ) ) ).
tff(fact_300_d22set__induct__old,axiom,
! [X_1: int,P: fun_int_bool] :
( ! [A_88: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_88))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(A_88,one_one_int))) )
=> hBOOL(hAPP_int_bool(P,A_88)) )
=> hBOOL(hAPP_int_bool(P,X_1)) ) ).
tff(fact_301_convex__bound__lt,axiom,
! [V_1: int,U_1: int,Y_8: int,X_9: int,A_87: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),A_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_8),A_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_1))
=> ( ( plus_plus_int(U_1,V_1) = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(U_1,X_9),times_times_int(V_1,Y_8))),A_87)) ) ) ) ) ) ).
tff(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_86: nat] : times_times_nat(zero_zero_nat,A_86) = zero_zero_nat ).
tff(fact_303_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_86: int] : times_times_int(zero_zero_int,A_86) = zero_zero_int ).
tff(fact_304_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_85: nat] : times_times_nat(A_85,zero_zero_nat) = zero_zero_nat ).
tff(fact_305_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_85: int] : times_times_int(A_85,zero_zero_int) = zero_zero_int ).
tff(fact_306_mult__zero__left,axiom,
! [A_84: nat] : times_times_nat(zero_zero_nat,A_84) = zero_zero_nat ).
tff(fact_307_mult__zero__left,axiom,
! [A_84: int] : times_times_int(zero_zero_int,A_84) = zero_zero_int ).
tff(fact_308_mult__zero__right,axiom,
! [A_83: nat] : times_times_nat(A_83,zero_zero_nat) = zero_zero_nat ).
tff(fact_309_mult__zero__right,axiom,
! [A_83: int] : times_times_int(A_83,zero_zero_int) = zero_zero_int ).
tff(fact_310_mult__eq__0__iff,axiom,
! [Aa: int,Ba: int] :
( ( times_times_int(Aa,Ba) = zero_zero_int )
<=> ( ( Aa = zero_zero_int )
| ( Ba = zero_zero_int ) ) ) ).
tff(fact_311_no__zero__divisors,axiom,
! [B_65: nat,A_82: nat] :
( ( A_82 != zero_zero_nat )
=> ( ( B_65 != zero_zero_nat )
=> ( times_times_nat(A_82,B_65) != zero_zero_nat ) ) ) ).
tff(fact_312_no__zero__divisors,axiom,
! [B_65: int,A_82: int] :
( ( A_82 != zero_zero_int )
=> ( ( B_65 != zero_zero_int )
=> ( times_times_int(A_82,B_65) != zero_zero_int ) ) ) ).
tff(fact_313_divisors__zero,axiom,
! [A_81: nat,B_64: nat] :
( ( times_times_nat(A_81,B_64) = zero_zero_nat )
=> ( ( A_81 = zero_zero_nat )
| ( B_64 = zero_zero_nat ) ) ) ).
tff(fact_314_divisors__zero,axiom,
! [A_81: int,B_64: int] :
( ( times_times_int(A_81,B_64) = zero_zero_int )
=> ( ( A_81 = zero_zero_int )
| ( B_64 = zero_zero_int ) ) ) ).
tff(fact_315_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_80: nat] : plus_plus_nat(zero_zero_nat,A_80) = A_80 ).
tff(fact_316_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_80: int] : plus_plus_int(zero_zero_int,A_80) = A_80 ).
tff(fact_317_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_79: nat] : plus_plus_nat(A_79,zero_zero_nat) = A_79 ).
tff(fact_318_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_79: int] : plus_plus_int(A_79,zero_zero_int) = A_79 ).
tff(fact_319_add__0__iff,axiom,
! [Ba: nat,Aa: nat] :
( ( Ba = plus_plus_nat(Ba,Aa) )
<=> ( Aa = zero_zero_nat ) ) ).
tff(fact_320_add__0__iff,axiom,
! [Ba: int,Aa: int] :
( ( Ba = plus_plus_int(Ba,Aa) )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_321_add__0__left,axiom,
! [A_78: nat] : plus_plus_nat(zero_zero_nat,A_78) = A_78 ).
tff(fact_322_add__0__left,axiom,
! [A_78: int] : plus_plus_int(zero_zero_int,A_78) = A_78 ).
tff(fact_323_add__0,axiom,
! [A_77: nat] : plus_plus_nat(zero_zero_nat,A_77) = A_77 ).
tff(fact_324_add__0,axiom,
! [A_77: int] : plus_plus_int(zero_zero_int,A_77) = A_77 ).
tff(fact_325_double__zero__sym,axiom,
! [Aa: int] :
( ( zero_zero_int = plus_plus_int(Aa,Aa) )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_326_add__0__right,axiom,
! [A_76: nat] : plus_plus_nat(A_76,zero_zero_nat) = A_76 ).
tff(fact_327_add__0__right,axiom,
! [A_76: int] : plus_plus_int(A_76,zero_zero_int) = A_76 ).
tff(fact_328_add_Ocomm__neutral,axiom,
! [A_75: nat] : plus_plus_nat(A_75,zero_zero_nat) = A_75 ).
tff(fact_329_add_Ocomm__neutral,axiom,
! [A_75: int] : plus_plus_int(A_75,zero_zero_int) = A_75 ).
tff(fact_330_double__eq__0__iff,axiom,
! [Aa: int] :
( ( plus_plus_int(Aa,Aa) = zero_zero_int )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_331_zero__reorient,axiom,
! [X_1: nat] :
( ( zero_zero_nat = X_1 )
<=> ( X_1 = zero_zero_nat ) ) ).
tff(fact_332_zero__reorient,axiom,
! [X_1: int] :
( ( zero_zero_int = X_1 )
<=> ( X_1 = zero_zero_int ) ) ).
tff(fact_333_diff__0__right,axiom,
! [A_74: int] : minus_minus_int(A_74,zero_zero_int) = A_74 ).
tff(fact_334_diff__self,axiom,
! [A_73: int] : minus_minus_int(A_73,A_73) = zero_zero_int ).
tff(fact_335_eq__iff__diff__eq__0,axiom,
! [Aa: int,Ba: int] :
( ( Aa = Ba )
<=> ( minus_minus_int(Aa,Ba) = zero_zero_int ) ) ).
tff(fact_336_right__minus__eq,axiom,
! [Aa: int,Ba: int] :
( ( minus_minus_int(Aa,Ba) = zero_zero_int )
<=> ( Aa = Ba ) ) ).
tff(fact_337_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
tff(fact_338_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
tff(fact_339_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
tff(fact_340_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
tff(fact_341_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_342_zadd__0__right,axiom,
! [Z: int] : plus_plus_int(Z,zero_zero_int) = Z ).
tff(fact_343_zadd__0,axiom,
! [Z: int] : plus_plus_int(zero_zero_int,Z) = Z ).
tff(fact_344_split__mult__neg__le,axiom,
! [B_63: nat,A_72: nat] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_72))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_63),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_63)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_72,B_63)),zero_zero_nat)) ) ).
tff(fact_345_split__mult__neg__le,axiom,
! [B_63: int,A_72: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_72))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_63),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_63)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_72,B_63)),zero_zero_int)) ) ).
tff(fact_346_split__mult__pos__le,axiom,
! [B_62: int,A_71: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_71))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_62)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_71),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_62),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_71,B_62))) ) ).
tff(fact_347_mult__mono,axiom,
! [C_30: nat,D_10: nat,A_70: nat,B_61: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_70),B_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_30),D_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_30))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_70,C_30)),times_times_nat(B_61,D_10))) ) ) ) ) ).
tff(fact_348_mult__mono,axiom,
! [C_30: int,D_10: int,A_70: int,B_61: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_70),B_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_30),D_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_30))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_70,C_30)),times_times_int(B_61,D_10))) ) ) ) ) ).
tff(fact_349_mult__mono_H,axiom,
! [C_29: nat,D_9: nat,A_69: nat,B_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_69),B_60))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_29),D_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_69))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_69,C_29)),times_times_nat(B_60,D_9))) ) ) ) ) ).
tff(fact_350_mult__mono_H,axiom,
! [C_29: int,D_9: int,A_69: int,B_60: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_69),B_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_29),D_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_69))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_69,C_29)),times_times_int(B_60,D_9))) ) ) ) ) ).
tff(fact_351_mult__left__mono__neg,axiom,
! [C_28: int,B_59: int,A_68: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_59),A_68))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_28),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_28,A_68)),times_times_int(C_28,B_59))) ) ) ).
tff(fact_352_mult__right__mono__neg,axiom,
! [C_27: int,B_58: int,A_67: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_58),A_67))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_27),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_67,C_27)),times_times_int(B_58,C_27))) ) ) ).
tff(fact_353_comm__mult__left__mono,axiom,
! [C_26: nat,A_66: nat,B_57: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_66),B_57))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_26,A_66)),times_times_nat(C_26,B_57))) ) ) ).
tff(fact_354_comm__mult__left__mono,axiom,
! [C_26: int,A_66: int,B_57: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_66),B_57))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_26,A_66)),times_times_int(C_26,B_57))) ) ) ).
tff(fact_355_mult__left__mono,axiom,
! [C_25: nat,A_65: nat,B_56: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_65),B_56))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_25,A_65)),times_times_nat(C_25,B_56))) ) ) ).
tff(fact_356_mult__left__mono,axiom,
! [C_25: int,A_65: int,B_56: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_65),B_56))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_25,A_65)),times_times_int(C_25,B_56))) ) ) ).
tff(fact_357_mult__right__mono,axiom,
! [C_24: nat,A_64: nat,B_55: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_64),B_55))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_64,C_24)),times_times_nat(B_55,C_24))) ) ) ).
tff(fact_358_mult__right__mono,axiom,
! [C_24: int,A_64: int,B_55: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_64),B_55))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_64,C_24)),times_times_int(B_55,C_24))) ) ) ).
tff(fact_359_mult__nonpos__nonpos,axiom,
! [B_54: int,A_63: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_63),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_54),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_63,B_54))) ) ) ).
tff(fact_360_mult__nonpos__nonneg,axiom,
! [B_53: nat,A_62: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_62),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_53))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_62,B_53)),zero_zero_nat)) ) ) ).
tff(fact_361_mult__nonpos__nonneg,axiom,
! [B_53: int,A_62: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_62),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_53))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_62,B_53)),zero_zero_int)) ) ) ).
tff(fact_362_mult__nonneg__nonpos2,axiom,
! [B_52: nat,A_61: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_52),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(B_52,A_61)),zero_zero_nat)) ) ) ).
tff(fact_363_mult__nonneg__nonpos2,axiom,
! [B_52: int,A_61: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_52),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(B_52,A_61)),zero_zero_int)) ) ) ).
tff(fact_364_mult__nonneg__nonpos,axiom,
! [B_51: nat,A_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_60))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_51),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_60,B_51)),zero_zero_nat)) ) ) ).
tff(fact_365_mult__nonneg__nonpos,axiom,
! [B_51: int,A_60: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_51),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_60,B_51)),zero_zero_int)) ) ) ).
tff(fact_366_mult__nonneg__nonneg,axiom,
! [B_50: nat,A_59: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_59))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),times_times_nat(A_59,B_50))) ) ) ).
tff(fact_367_mult__nonneg__nonneg,axiom,
! [B_50: int,A_59: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_59))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_59,B_50))) ) ) ).
tff(fact_368_mult__le__0__iff,axiom,
! [Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(Aa,Ba)),zero_zero_int))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) ) ) ) ).
tff(fact_369_zero__le__mult__iff,axiom,
! [Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(Aa,Ba)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) ) ) ) ).
tff(fact_370_zero__le__square,axiom,
! [A_58: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_58,A_58))) ).
tff(fact_371_mult__strict__left__mono__neg,axiom,
! [C_23: int,B_49: int,A_57: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_49),A_57))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_23),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_23,A_57)),times_times_int(C_23,B_49))) ) ) ).
tff(fact_372_mult__strict__right__mono__neg,axiom,
! [C_22: int,B_48: int,A_56: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_48),A_56))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_22),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_56,C_22)),times_times_int(B_48,C_22))) ) ) ).
tff(fact_373_comm__mult__strict__left__mono,axiom,
! [C_21: nat,A_55: nat,B_47: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_55),B_47))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_21,A_55)),times_times_nat(C_21,B_47))) ) ) ).
tff(fact_374_comm__mult__strict__left__mono,axiom,
! [C_21: int,A_55: int,B_47: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_55),B_47))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_21,A_55)),times_times_int(C_21,B_47))) ) ) ).
tff(fact_375_mult__strict__left__mono,axiom,
! [C_20: nat,A_54: nat,B_46: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_54),B_46))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_20,A_54)),times_times_nat(C_20,B_46))) ) ) ).
tff(fact_376_mult__strict__left__mono,axiom,
! [C_20: int,A_54: int,B_46: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_54),B_46))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_20,A_54)),times_times_int(C_20,B_46))) ) ) ).
tff(fact_377_mult__strict__right__mono,axiom,
! [C_19: nat,A_53: nat,B_45: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_53),B_45))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_53,C_19)),times_times_nat(B_45,C_19))) ) ) ).
tff(fact_378_mult__strict__right__mono,axiom,
! [C_19: int,A_53: int,B_45: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_53),B_45))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_53,C_19)),times_times_int(B_45,C_19))) ) ) ).
tff(fact_379_mult__neg__neg,axiom,
! [B_44: int,A_52: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_52),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_44),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_52,B_44))) ) ) ).
tff(fact_380_mult__neg__pos,axiom,
! [B_43: nat,A_51: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_51),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_43))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_51,B_43)),zero_zero_nat)) ) ) ).
tff(fact_381_mult__neg__pos,axiom,
! [B_43: int,A_51: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_51),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_43))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_51,B_43)),zero_zero_int)) ) ) ).
tff(fact_382_mult__less__cancel__left__neg,axiom,
! [Aa: int,Ba: int,C_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ).
tff(fact_383_zero__less__mult__pos2,axiom,
! [B_42: nat,A_50: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(B_42,A_50)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_42)) ) ) ).
tff(fact_384_zero__less__mult__pos2,axiom,
! [B_42: int,A_50: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(B_42,A_50)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_42)) ) ) ).
tff(fact_385_zero__less__mult__pos,axiom,
! [A_49: nat,B_41: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(A_49,B_41)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_49))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_41)) ) ) ).
tff(fact_386_zero__less__mult__pos,axiom,
! [A_49: int,B_41: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_49,B_41)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_49))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_41)) ) ) ).
tff(fact_387_mult__pos__neg2,axiom,
! [B_40: nat,A_48: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_48))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_40),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(B_40,A_48)),zero_zero_nat)) ) ) ).
tff(fact_388_mult__pos__neg2,axiom,
! [B_40: int,A_48: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_40),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(B_40,A_48)),zero_zero_int)) ) ) ).
tff(fact_389_mult__pos__neg,axiom,
! [B_39: nat,A_47: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_47))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_39),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_47,B_39)),zero_zero_nat)) ) ) ).
tff(fact_390_mult__pos__neg,axiom,
! [B_39: int,A_47: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_47))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_39),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_47,B_39)),zero_zero_int)) ) ) ).
tff(fact_391_mult__pos__pos,axiom,
! [B_38: nat,A_46: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_46))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_38))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(A_46,B_38))) ) ) ).
tff(fact_392_mult__pos__pos,axiom,
! [B_38: int,A_46: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_46))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_38))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_46,B_38))) ) ) ).
tff(fact_393_mult__less__cancel__left__pos,axiom,
! [Aa: int,Ba: int,C_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ) ).
tff(fact_394_mult__less__cancel__left__disj,axiom,
! [C_4: int,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ) ).
tff(fact_395_mult__less__cancel__right__disj,axiom,
! [Aa: int,C_4: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(Aa,C_4)),times_times_int(Ba,C_4)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ) ).
tff(fact_396_not__square__less__zero,axiom,
! [A_45: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_45,A_45)),zero_zero_int)) ).
tff(fact_397_add__nonpos__nonpos,axiom,
! [B_37: nat,A_44: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_44),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_37),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_44,B_37)),zero_zero_nat)) ) ) ).
tff(fact_398_add__nonpos__nonpos,axiom,
! [B_37: int,A_44: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_44),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_37),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_44,B_37)),zero_zero_int)) ) ) ).
tff(fact_399_add__increasing2,axiom,
! [B_36: nat,A_43: nat,C_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_36),A_43))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_36),plus_plus_nat(A_43,C_18))) ) ) ).
tff(fact_400_add__increasing2,axiom,
! [B_36: int,A_43: int,C_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_36),A_43))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_36),plus_plus_int(A_43,C_18))) ) ) ).
tff(fact_401_add__increasing,axiom,
! [B_35: nat,C_17: nat,A_42: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_42))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_35),C_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_35),plus_plus_nat(A_42,C_17))) ) ) ).
tff(fact_402_add__increasing,axiom,
! [B_35: int,C_17: int,A_42: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_42))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_35),C_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_35),plus_plus_int(A_42,C_17))) ) ) ).
tff(fact_403_add__nonneg__eq__0__iff,axiom,
! [Y_3: nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_3))
=> ( ( plus_plus_nat(X_1,Y_3) = zero_zero_nat )
<=> ( ( X_1 = zero_zero_nat )
& ( Y_3 = zero_zero_nat ) ) ) ) ) ).
tff(fact_404_add__nonneg__eq__0__iff,axiom,
! [Y_3: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_3))
=> ( ( plus_plus_int(X_1,Y_3) = zero_zero_int )
<=> ( ( X_1 = zero_zero_int )
& ( Y_3 = zero_zero_int ) ) ) ) ) ).
tff(fact_405_add__nonneg__nonneg,axiom,
! [B_34: nat,A_41: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_41))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),plus_plus_nat(A_41,B_34))) ) ) ).
tff(fact_406_add__nonneg__nonneg,axiom,
! [B_34: int,A_41: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_41))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(A_41,B_34))) ) ) ).
tff(fact_407_double__add__le__zero__iff__single__add__le__zero,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(Aa,Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ).
tff(fact_408_zero__le__double__add__iff__zero__le__single__add,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(Aa,Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ).
tff(fact_409_add__neg__neg,axiom,
! [B_33: nat,A_40: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_40),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_33),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_40,B_33)),zero_zero_nat)) ) ) ).
tff(fact_410_add__neg__neg,axiom,
! [B_33: int,A_40: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_40),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_33),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_40,B_33)),zero_zero_int)) ) ) ).
tff(fact_411_pos__add__strict,axiom,
! [B_32: nat,C_16: nat,A_39: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_32),C_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_32),plus_plus_nat(A_39,C_16))) ) ) ).
tff(fact_412_pos__add__strict,axiom,
! [B_32: int,C_16: int,A_39: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_32),C_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_32),plus_plus_int(A_39,C_16))) ) ) ).
tff(fact_413_add__pos__pos,axiom,
! [B_31: nat,A_38: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_38,B_31))) ) ) ).
tff(fact_414_add__pos__pos,axiom,
! [B_31: int,A_38: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_38,B_31))) ) ) ).
tff(fact_415_double__add__less__zero__iff__single__add__less__zero,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(Aa,Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).
tff(fact_416_even__less__0__iff,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(Aa,Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).
tff(fact_417_zero__less__double__add__iff__zero__less__single__add,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(Aa,Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).
tff(fact_418_zero__le__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_419_zero__le__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
tff(fact_420_not__one__le__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_421_not__one__le__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
tff(fact_422_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_423_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_424_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_425_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
tff(fact_426_le__iff__diff__le__0,axiom,
! [Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(Aa,Ba)),zero_zero_int)) ) ).
tff(fact_427_add__scale__eq__noteq,axiom,
! [C_15: nat,D_8: nat,A_37: nat,B_30: nat,R_3: nat] :
( ( R_3 != zero_zero_nat )
=> ( ( ( A_37 = B_30 )
& ( C_15 != D_8 ) )
=> ( plus_plus_nat(A_37,times_times_nat(R_3,C_15)) != plus_plus_nat(B_30,times_times_nat(R_3,D_8)) ) ) ) ).
tff(fact_428_add__scale__eq__noteq,axiom,
! [C_15: int,D_8: int,A_37: int,B_30: int,R_3: int] :
( ( R_3 != zero_zero_int )
=> ( ( ( A_37 = B_30 )
& ( C_15 != D_8 ) )
=> ( plus_plus_int(A_37,times_times_int(R_3,C_15)) != plus_plus_int(B_30,times_times_int(R_3,D_8)) ) ) ) ).
tff(fact_429_less__iff__diff__less__0,axiom,
! [Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(Aa,Ba)),zero_zero_int)) ) ).
tff(fact_430_Euler_Oaux1,axiom,
! [A_19: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A_19))
=> ( ( X != minus_minus_int(A_19,one_one_int) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),minus_minus_int(A_19,one_one_int))) ) ) ) ).
tff(fact_431_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_432_zmult__zless__mono2,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(K,I)),times_times_int(K,J))) ) ) ).
tff(fact_433_odd__nonzero,axiom,
! [Z: int] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).
tff(fact_434_less__bin__lemma,axiom,
! [K_1: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(K_1,L_1)),zero_zero_int)) ) ).
tff(fact_435_mult__left__le__imp__le,axiom,
! [C_14: nat,A_36: nat,B_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_14,A_36)),times_times_nat(C_14,B_29)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_36),B_29)) ) ) ).
tff(fact_436_mult__left__le__imp__le,axiom,
! [C_14: int,A_36: int,B_29: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_14,A_36)),times_times_int(C_14,B_29)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_36),B_29)) ) ) ).
tff(fact_437_mult__right__le__imp__le,axiom,
! [A_35: nat,C_13: nat,B_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_35,C_13)),times_times_nat(B_28,C_13)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_35),B_28)) ) ) ).
tff(fact_438_mult__right__le__imp__le,axiom,
! [A_35: int,C_13: int,B_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_35,C_13)),times_times_int(B_28,C_13)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_35),B_28)) ) ) ).
tff(fact_439_mult__less__imp__less__left,axiom,
! [C_12: nat,A_34: nat,B_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_12,A_34)),times_times_nat(C_12,B_27)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_34),B_27)) ) ) ).
tff(fact_440_mult__less__imp__less__left,axiom,
! [C_12: int,A_34: int,B_27: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_12,A_34)),times_times_int(C_12,B_27)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_34),B_27)) ) ) ).
tff(fact_441_mult__left__less__imp__less,axiom,
! [C_11: nat,A_33: nat,B_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_11,A_33)),times_times_nat(C_11,B_26)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_33),B_26)) ) ) ).
tff(fact_442_mult__left__less__imp__less,axiom,
! [C_11: int,A_33: int,B_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_11,A_33)),times_times_int(C_11,B_26)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_33),B_26)) ) ) ).
tff(fact_443_mult__less__imp__less__right,axiom,
! [A_32: nat,C_10: nat,B_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_32,C_10)),times_times_nat(B_25,C_10)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_32),B_25)) ) ) ).
tff(fact_444_mult__less__imp__less__right,axiom,
! [A_32: int,C_10: int,B_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_32,C_10)),times_times_int(B_25,C_10)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_32),B_25)) ) ) ).
tff(fact_445_mult__right__less__imp__less,axiom,
! [A_31: nat,C_9: nat,B_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_31,C_9)),times_times_nat(B_24,C_9)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_31),B_24)) ) ) ).
tff(fact_446_mult__right__less__imp__less,axiom,
! [A_31: int,C_9: int,B_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_31,C_9)),times_times_int(B_24,C_9)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_31),B_24)) ) ) ).
tff(fact_447_mult__le__less__imp__less,axiom,
! [C_8: nat,D_7: nat,A_30: nat,B_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_30),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_8),D_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_30))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_30,C_8)),times_times_nat(B_23,D_7))) ) ) ) ) ).
tff(fact_448_mult__le__less__imp__less,axiom,
! [C_8: int,D_7: int,A_30: int,B_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_30),B_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_8),D_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_30))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_30,C_8)),times_times_int(B_23,D_7))) ) ) ) ) ).
tff(fact_449_mult__less__le__imp__less,axiom,
! [C_7: nat,D_6: nat,A_29: nat,B_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_29),B_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_7),D_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_29,C_7)),times_times_nat(B_22,D_6))) ) ) ) ) ).
tff(fact_450_mult__less__le__imp__less,axiom,
! [C_7: int,D_6: int,A_29: int,B_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_29),B_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_7),D_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_29,C_7)),times_times_int(B_22,D_6))) ) ) ) ) ).
tff(fact_451_mult__strict__mono_H,axiom,
! [C_6: nat,D_5: nat,A_28: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_28),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_6),D_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_28,C_6)),times_times_nat(B_21,D_5))) ) ) ) ) ).
tff(fact_452_mult__strict__mono_H,axiom,
! [C_6: int,D_5: int,A_28: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_28),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_6),D_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_28,C_6)),times_times_int(B_21,D_5))) ) ) ) ) ).
tff(fact_453_mult__strict__mono,axiom,
! [C_5: nat,D_4: nat,A_27: nat,B_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_27),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_5),D_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_27,C_5)),times_times_nat(B_20,D_4))) ) ) ) ) ).
tff(fact_454_mult__strict__mono,axiom,
! [C_5: int,D_4: int,A_27: int,B_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_27),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_5),D_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_27,C_5)),times_times_int(B_20,D_4))) ) ) ) ) ).
tff(fact_455_mult__le__cancel__left__neg,axiom,
! [Aa: int,Ba: int,C_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa)) ) ) ).
tff(fact_456_mult__le__cancel__left__pos,axiom,
! [Aa: int,Ba: int,C_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ) ).
tff(fact_457_add__nonpos__neg,axiom,
! [B_19: nat,A_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_19),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_26,B_19)),zero_zero_nat)) ) ) ).
tff(fact_458_add__nonpos__neg,axiom,
! [B_19: int,A_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_19),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_26,B_19)),zero_zero_int)) ) ) ).
tff(fact_459_add__neg__nonpos,axiom,
! [B_18: nat,A_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_25),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_25,B_18)),zero_zero_nat)) ) ) ).
tff(fact_460_add__neg__nonpos,axiom,
! [B_18: int,A_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_25),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_25,B_18)),zero_zero_int)) ) ) ).
tff(fact_461_add__strict__increasing2,axiom,
! [B_17: nat,C_3: nat,A_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_17),C_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_17),plus_plus_nat(A_24,C_3))) ) ) ).
tff(fact_462_add__strict__increasing2,axiom,
! [B_17: int,C_3: int,A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_17),C_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_17),plus_plus_int(A_24,C_3))) ) ) ).
tff(fact_463_add__strict__increasing,axiom,
! [B_16: nat,C_2: nat,A_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_16),C_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_16),plus_plus_nat(A_23,C_2))) ) ) ).
tff(fact_464_add__strict__increasing,axiom,
! [B_16: int,C_2: int,A_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_16),C_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_16),plus_plus_int(A_23,C_2))) ) ) ).
tff(fact_465_add__nonneg__pos,axiom,
! [B_15: nat,A_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_22,B_15))) ) ) ).
tff(fact_466_add__nonneg__pos,axiom,
! [B_15: int,A_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_22,B_15))) ) ) ).
tff(fact_467_add__pos__nonneg,axiom,
! [B_14: nat,A_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_21,B_14))) ) ) ).
tff(fact_468_add__pos__nonneg,axiom,
! [B_14: int,A_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_21,B_14))) ) ) ).
tff(fact_469_mult__left__le__one__le,axiom,
! [Y_7: int,X_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(Y_7,X_8)),X_8)) ) ) ) ).
tff(fact_470_mult__right__le__one__le,axiom,
! [Y_6: int,X_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(X_7,Y_6)),X_7)) ) ) ) ).
tff(fact_471_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(one_one_nat,one_one_nat))) ).
tff(fact_472_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,one_one_int))) ).
tff(fact_473_int__one__le__iff__zero__less,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) ) ).
tff(fact_474_pos__zmult__eq__1__iff,axiom,
! [N_1: int,M_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
=> ( ( times_times_int(M_1,N_1) = one_one_int )
<=> ( ( M_1 = one_one_int )
& ( N_1 = one_one_int ) ) ) ) ).
tff(fact_475_odd__less__0,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(plus_plus_int(one_one_int,Z_1),Z_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) ) ).
tff(fact_476_convex__bound__le,axiom,
! [V: int,U: int,Y_5: int,X_6: int,A_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_6),A_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),A_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V))
=> ( ( plus_plus_int(U,V) = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(U,X_6),times_times_int(V,Y_5))),A_20)) ) ) ) ) ) ).
tff(fact_477_le__imp__0__less,axiom,
! [Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,Z))) ) ).
tff(fact_478_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_19))
=> ( ( A_19 = plus_plus_int(R_1,times_times_int(A_19,Q)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),A_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q)) ) ) ) ).
tff(fact_479_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_19))
=> ( ( A_19 = plus_plus_int(R_1,times_times_int(A_19,Q)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),one_one_int)) ) ) ) ).
tff(fact_480_q__pos__lemma,axiom,
! [B_13: int,Q_1: int,R_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_13,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).
tff(fact_481_q__neg__lemma,axiom,
! [B_13: int,Q_1: int,R_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_13,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).
tff(fact_482_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R_1: int,B_13: int,Q_1: int,R_2: int] :
( ( plus_plus_int(times_times_int(B,Q),R_1) = plus_plus_int(times_times_int(B_13,Q_1),R_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_13,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_13),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ) ) ).
tff(fact_483_unique__quotient__lemma__neg,axiom,
! [B: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B,Q_1),R_2)),plus_plus_int(times_times_int(B,Q),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ).
tff(fact_484_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R_1: int,B_13: int,Q_1: int,R_2: int] :
( ( plus_plus_int(times_times_int(B,Q),R_1) = plus_plus_int(times_times_int(B_13,Q_1),R_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_13,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_13),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ) ) ).
tff(fact_485_unique__quotient__lemma,axiom,
! [B: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B,Q_1),R_2)),plus_plus_int(times_times_int(B,Q),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ).
tff(fact_486_xzgcd__linear,axiom,
! [M: int,R_1: int,S: int,T: int,N: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( ( xzgcd(M,N) = produc282740534nt_int(R_1,product_Pair_int_int(S,T)) )
=> ( R_1 = plus_plus_int(times_times_int(S,M),times_times_int(T,N)) ) ) ) ).
tff(fact_487_not__sum__squares__lt__zero,axiom,
! [X_5: int,Y_4: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(X_5,X_5),times_times_int(Y_4,Y_4))),zero_zero_int)) ).
tff(fact_488_sum__squares__gt__zero__iff,axiom,
! [X_1: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_3,Y_3))))
<=> ( ( X_1 != zero_zero_int )
| ( Y_3 != zero_zero_int ) ) ) ).
tff(fact_489_sum__squares__eq__zero__iff,axiom,
! [X_1: int,Y_3: int] :
( ( plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_3,Y_3)) = zero_zero_int )
<=> ( ( X_1 = zero_zero_int )
& ( Y_3 = zero_zero_int ) ) ) ).
tff(fact_490_sum__squares__le__zero__iff,axiom,
! [X_1: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_3,Y_3))),zero_zero_int))
<=> ( ( X_1 = zero_zero_int )
& ( Y_3 = zero_zero_int ) ) ) ).
tff(fact_491_sum__squares__ge__zero,axiom,
! [X_4: int,Y_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(X_4,X_4),times_times_int(Y_2,Y_2)))) ).
tff(fact_492_xzgcda__linear,axiom,
! [M: int,N: int,R_2: int,S_1: int,S: int,T_5: int,T: int,Rn: int,Sn: int,Tn: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_1))
=> ( ( xzgcda(M,N,R_2,R_1,S_1,S,T_5,T) = produc282740534nt_int(Rn,product_Pair_int_int(Sn,Tn)) )
=> ( ( R_2 = plus_plus_int(times_times_int(S_1,M),times_times_int(T_5,N)) )
=> ( ( R_1 = plus_plus_int(times_times_int(S,M),times_times_int(T,N)) )
=> ( Rn = plus_plus_int(times_times_int(Sn,M),times_times_int(Tn,N)) ) ) ) ) ) ).
tff(fact_493_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(X,Y))) ) ) ).
tff(fact_494_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(X,Y))) ) ) ).
tff(fact_495_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).
tff(fact_496_xzgcd__def,axiom,
! [M: int,N: int] : xzgcd(M,N) = xzgcda(M,N,M,N,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).
tff(fact_497_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
tff(fact_498_pos__zmult__pos,axiom,
! [B: int,A_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_19,B)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ) ).
tff(fact_499_decr__mult__lemma,axiom,
! [K_1: int,P: fun_int_bool,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3: int] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(X_3,D_3))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_3: int] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(X_3,times_times_int(K_1,D_3)))) ) ) ) ) ).
tff(fact_500_incr__mult__lemma,axiom,
! [K_1: int,P: fun_int_bool,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3: int] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(X_3,D_3))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_3: int] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(X_3,times_times_int(K_1,D_3)))) ) ) ) ) ).
tff(fact_501_imp__le__cong,axiom,
! [P_1: bool,P: bool,X_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P_1) ) ) ) ).
tff(fact_502_conj__le__cong,axiom,
! [P_1: bool,P: bool,X_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P_1) ) ) ) ).
tff(fact_503_minf_I8_J,axiom,
! [T_4: nat] :
? [Z_2: nat] :
! [X_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_3),Z_2))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,T_4),X_3)) ) ).
tff(fact_504_minf_I8_J,axiom,
! [T_4: int] :
? [Z_2: int] :
! [X_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),Z_2))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T_4),X_3)) ) ).
tff(fact_505_pinf_I8_J,axiom,
! [T_3: nat] :
? [Z_2: nat] :
! [X_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_2),X_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,T_3),X_3)) ) ).
tff(fact_506_pinf_I8_J,axiom,
! [T_3: int] :
? [Z_2: int] :
! [X_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),X_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T_3),X_3)) ) ).
tff(fact_507_pinf_I6_J,axiom,
! [T_2: nat] :
? [Z_2: nat] :
! [X_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_2),X_3))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),T_2)) ) ).
tff(fact_508_pinf_I6_J,axiom,
! [T_2: int] :
? [Z_2: int] :
! [X_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),X_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),T_2)) ) ).
tff(fact_509_minf_I6_J,axiom,
! [T_1: nat] :
? [Z_2: nat] :
! [X_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_3),Z_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),T_1)) ) ).
tff(fact_510_minf_I6_J,axiom,
! [T_1: int] :
? [Z_2: int] :
! [X_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),Z_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),T_1)) ) ).
tff(fact_511_divmod__int__relI,axiom,
! [A_19: int,B: int,Q: int,R_1: int] :
( ( A_19 = plus_plus_int(times_times_int(B,Q),R_1) )
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int)) ) ) )
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q,R_1))) ) ) ).
tff(fact_512_divmod__int__rel__0,axiom,
! [B: int] :
( ( B != zero_zero_int )
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B),product_Pair_int_int(zero_zero_int,zero_zero_int))) ) ).
tff(fact_513_self__remainder,axiom,
! [A_19: int,Q: int,R_1: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,A_19),product_Pair_int_int(Q,R_1)))
=> ( ( A_19 != zero_zero_int )
=> ( R_1 = zero_zero_int ) ) ) ).
tff(fact_514_unique__quotient,axiom,
! [Q_1: int,R_2: int,A_19: int,B: int,Q: int,R_1: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q,R_1)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q_1,R_2)))
=> ( ( B != zero_zero_int )
=> ( Q = Q_1 ) ) ) ) ).
tff(fact_515_unique__remainder,axiom,
! [Q_1: int,R_2: int,A_19: int,B: int,Q: int,R_1: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q,R_1)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q_1,R_2)))
=> ( ( B != zero_zero_int )
=> ( R_1 = R_2 ) ) ) ) ).
tff(fact_516_self__quotient,axiom,
! [A_19: int,Q: int,R_1: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,A_19),product_Pair_int_int(Q,R_1)))
=> ( ( A_19 != zero_zero_int )
=> ( Q = one_one_int ) ) ) ).
tff(fact_517_minusinfinity,axiom,
! [P: fun_int_bool,P1: fun_int_bool,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3: int,K_2: int] :
( hBOOL(hAPP_int_bool(P1,X_3))
<=> hBOOL(hAPP_int_bool(P1,minus_minus_int(X_3,times_times_int(K_2,D_3)))) )
=> ( ? [Z_2: int] :
! [X_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),Z_2))
=> ( hBOOL(hAPP_int_bool(P,X_3))
<=> hBOOL(hAPP_int_bool(P1,X_3)) ) )
=> ( ? [X1: int] : hBOOL(hAPP_int_bool(P1,X1))
=> ? [X1: int] : hBOOL(hAPP_int_bool(P,X1)) ) ) ) ) ).
tff(fact_518_plusinfinity,axiom,
! [P: fun_int_bool,P_1: fun_int_bool,D_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3: int,K_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_3))
<=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_3,times_times_int(K_2,D_3)))) )
=> ( ? [Z_2: int] :
! [X_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),X_3))
=> ( hBOOL(hAPP_int_bool(P,X_3))
<=> hBOOL(hAPP_int_bool(P_1,X_3)) ) )
=> ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
=> ? [X1: int] : hBOOL(hAPP_int_bool(P,X1)) ) ) ) ) ).
tff(fact_519_tsub__def,axiom,
! [Y: int,X: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( nat_tsub(X,Y) = minus_minus_int(X,Y) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( nat_tsub(X,Y) = zero_zero_int ) ) ) ).
tff(fact_520_zfact_Osimps,axiom,
! [N: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
=> ( zfact(N) = one_one_int ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
=> ( zfact(N) = times_times_int(N,zfact(minus_minus_int(N,one_one_int))) ) ) ) ).
tff(fact_521_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),nat_tsub(X,Y))) ) ) ).
tff(fact_522_tsub__eq,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( nat_tsub(X,Y) = minus_minus_int(X,Y) ) ) ).
tff(fact_523_posDivAlg__correct,axiom,
! [B: int,A_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),posDivAlg(A_19,B))) ) ) ).
tff(fact_524_decr__lemma,axiom,
! [X: int,Z: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D))),Z)) ) ).
tff(fact_525_abs__eq__0,axiom,
! [Aa: int] :
( ( abs_abs_int(Aa) = zero_zero_int )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_526_abs__zero,axiom,
abs_abs_int(zero_zero_int) = zero_zero_int ).
tff(fact_527_abs__of__nonneg,axiom,
! [A_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_18))
=> ( abs_abs_int(A_18) = A_18 ) ) ).
tff(fact_528_abs__le__zero__iff,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(Aa)),zero_zero_int))
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_529_abs__ge__zero,axiom,
! [A_17: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),abs_abs_int(A_17))) ).
tff(fact_530_abs__of__pos,axiom,
! [A_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_16))
=> ( abs_abs_int(A_16) = A_16 ) ) ).
tff(fact_531_zero__less__abs__iff,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),abs_abs_int(Aa)))
<=> ( Aa != zero_zero_int ) ) ).
tff(fact_532_abs__not__less__zero,axiom,
! [A_15: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(A_15)),zero_zero_int)) ).
tff(fact_533_abs__mult__less,axiom,
! [B_12: int,D_2: int,A_14: int,C_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(A_14)),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(B_12)),D_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(abs_abs_int(A_14),abs_abs_int(B_12))),times_times_int(C_1,D_2))) ) ) ).
tff(fact_534_abs__triangle__ineq,axiom,
! [A_13: int,B_11: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(plus_plus_int(A_13,B_11))),plus_plus_int(abs_abs_int(A_13),abs_abs_int(B_11)))) ).
tff(fact_535_abs__triangle__ineq3,axiom,
! [A_12: int,B_10: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(abs_abs_int(A_12),abs_abs_int(B_10)))),abs_abs_int(minus_minus_int(A_12,B_10)))) ).
tff(fact_536_abs__triangle__ineq2,axiom,
! [A_11: int,B_9: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(abs_abs_int(A_11),abs_abs_int(B_9))),abs_abs_int(minus_minus_int(A_11,B_9)))) ).
tff(fact_537_abs__triangle__ineq2__sym,axiom,
! [A_10: int,B_8: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(abs_abs_int(A_10),abs_abs_int(B_8))),abs_abs_int(minus_minus_int(B_8,A_10)))) ).
tff(fact_538_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( abs_abs_int(times_times_int(M,N)) = one_one_int )
=> ( abs_abs_int(M) = one_one_int ) ) ).
tff(fact_539_abs__ge__self,axiom,
! [A_9: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),abs_abs_int(A_9))) ).
tff(fact_540_abs__le__D1,axiom,
! [A_8: int,B_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(A_8)),B_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_8),B_7)) ) ).
tff(fact_541_abs__mult,axiom,
! [A_7: int,B_6: int] : abs_abs_int(times_times_int(A_7,B_6)) = times_times_int(abs_abs_int(A_7),abs_abs_int(B_6)) ).
tff(fact_542_abs__mult__self,axiom,
! [A_6: int] : times_times_int(abs_abs_int(A_6),abs_abs_int(A_6)) = times_times_int(A_6,A_6) ).
tff(fact_543_abs__add__abs,axiom,
! [A_5: int,B_5: int] : abs_abs_int(plus_plus_int(abs_abs_int(A_5),abs_abs_int(B_5))) = plus_plus_int(abs_abs_int(A_5),abs_abs_int(B_5)) ).
tff(fact_544_abs__one,axiom,
abs_abs_int(one_one_int) = one_one_int ).
tff(fact_545_abs__minus__commute,axiom,
! [A_4: int,B_4: int] : abs_abs_int(minus_minus_int(A_4,B_4)) = abs_abs_int(minus_minus_int(B_4,A_4)) ).
tff(fact_546_abs__idempotent,axiom,
! [A_3: int] : abs_abs_int(abs_abs_int(A_3)) = abs_abs_int(A_3) ).
tff(fact_547_abs__mult__pos,axiom,
! [Y_1: int,X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( times_times_int(abs_abs_int(Y_1),X_2) = abs_abs_int(times_times_int(Y_1,X_2)) ) ) ).
tff(fact_548_abs__eq__mult,axiom,
! [B_3: int,A_2: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_3))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),zero_zero_int)) ) )
=> ( abs_abs_int(times_times_int(A_2,B_3)) = times_times_int(abs_abs_int(A_2),abs_abs_int(B_3)) ) ) ).
tff(fact_549_abs__diff__triangle__ineq,axiom,
! [A_1: int,B_2: int,C: int,D_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(plus_plus_int(A_1,B_2),plus_plus_int(C,D_1)))),plus_plus_int(abs_abs_int(minus_minus_int(A_1,C)),abs_abs_int(minus_minus_int(B_2,D_1))))) ).
tff(fact_550_abs__triangle__ineq4,axiom,
! [A: int,B_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(A,B_1))),plus_plus_int(abs_abs_int(A),abs_abs_int(B_1)))) ).
tff(fact_551_zabs__less__one__iff,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z_1)),one_one_int))
<=> ( Z_1 = zero_zero_int ) ) ).
tff(fact_552_posDivAlg__0,axiom,
! [B: int] : posDivAlg(zero_zero_int,B) = product_Pair_int_int(zero_zero_int,zero_zero_int) ).
tff(fact_553_incr__lemma,axiom,
! [Z: int,X: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),plus_plus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D)))) ) ).
tff(fact_554_abs__diff__less__iff,axiom,
! [X_1: int,Aa: int,R: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(minus_minus_int(X_1,Aa))),R))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(Aa,R)),X_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),plus_plus_int(Aa,R))) ) ) ).
tff(fact_555_pdivmod__posDivAlg,axiom,
! [K: int,L: int] :
( ( ( L = zero_zero_int )
=> ( pdivmod(K,L) = product_Pair_int_int(zero_zero_int,abs_abs_int(K)) ) )
& ( ( L != zero_zero_int )
=> ( pdivmod(K,L) = posDivAlg(abs_abs_int(K),abs_abs_int(L)) ) ) ) ).
tff(fact_556_nat0__intermed__int__val,axiom,
! [K_1: int,F: fun_nat_int,N_1: nat] :
( ! [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I_2,one_one_nat)),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1)))
=> ? [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))
& ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).
tff(fact_557_int__val__lemma,axiom,
! [K_1: int,F: fun_nat_int,N_1: nat] :
( ! [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I_2,one_one_nat)),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1)))
=> ? [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))
& ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).
tff(fact_558_mult__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( times_times_nat(M,N) = zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( times_times_nat(M,N) = plus_plus_nat(N,times_times_nat(minus_minus_nat(M,one_one_nat),N)) ) ) ) ).
tff(fact_559_le0,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
tff(fact_560_less__zeroE,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_561_nat__mult__commute,axiom,
! [M: nat,N: nat] : times_times_nat(M,N) = times_times_nat(N,M) ).
tff(fact_562_nat__mult__assoc,axiom,
! [M: nat,N: nat,K: nat] : times_times_nat(times_times_nat(M,N),K) = times_times_nat(M,times_times_nat(N,K)) ).
tff(fact_563_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] : times_times_nat(minus_minus_nat(M,N),K) = minus_minus_nat(times_times_nat(M,K),times_times_nat(N,K)) ).
tff(fact_564_diff__commute,axiom,
! [I: nat,J: nat,K: nat] : minus_minus_nat(minus_minus_nat(I,J),K) = minus_minus_nat(minus_minus_nat(I,K),J) ).
tff(fact_565_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] : times_times_nat(K,minus_minus_nat(M,N)) = minus_minus_nat(times_times_nat(K,M),times_times_nat(K,N)) ).
tff(fact_566_mult__0,axiom,
! [N: nat] : times_times_nat(zero_zero_nat,N) = zero_zero_nat ).
tff(fact_567_mult__0__right,axiom,
! [M: nat] : times_times_nat(M,zero_zero_nat) = zero_zero_nat ).
tff(fact_568_mult__is__0,axiom,
! [M_1: nat,N_1: nat] :
( ( times_times_nat(M_1,N_1) = zero_zero_nat )
<=> ( ( M_1 = zero_zero_nat )
| ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_569_mult__cancel1,axiom,
! [K_1: nat,M_1: nat,N_1: nat] :
( ( times_times_nat(K_1,M_1) = times_times_nat(K_1,N_1) )
<=> ( ( M_1 = N_1 )
| ( K_1 = zero_zero_nat ) ) ) ).
tff(fact_570_mult__cancel2,axiom,
! [M_1: nat,K_1: nat,N_1: nat] :
( ( times_times_nat(M_1,K_1) = times_times_nat(N_1,K_1) )
<=> ( ( M_1 = N_1 )
| ( K_1 = zero_zero_nat ) ) ) ).
tff(fact_571_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat(M,N) = zero_zero_nat )
=> ( ( minus_minus_nat(N,M) = zero_zero_nat )
=> ( M = N ) ) ) ).
tff(fact_572_diff__self__eq__0,axiom,
! [M: nat] : minus_minus_nat(M,M) = zero_zero_nat ).
tff(fact_573_minus__nat_Odiff__0,axiom,
! [M: nat] : minus_minus_nat(M,zero_zero_nat) = M ).
tff(fact_574_diff__0__eq__0,axiom,
! [N: nat] : minus_minus_nat(zero_zero_nat,N) = zero_zero_nat ).
tff(fact_575_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(J,N)),K)) ) ).
tff(fact_576_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ) ).
tff(fact_577_less__not__refl,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_578_nat__neq__iff,axiom,
! [M_1: nat,N_1: nat] :
( ( M_1 != N_1 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1)) ) ) ).
tff(fact_579_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).
tff(fact_580_less__irrefl__nat,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_581_less__not__refl2,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( M != N ) ) ).
tff(fact_582_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_583_nat__less__cases,axiom,
! [P: fun_nat_fun_nat_bool,M_1: nat,N_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) )
=> ( ( ( M_1 = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) ) ) ) ).
tff(fact_584_nat__add__commute,axiom,
! [M: nat,N: nat] : plus_plus_nat(M,N) = plus_plus_nat(N,M) ).
tff(fact_585_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] : plus_plus_nat(X,plus_plus_nat(Y,Z)) = plus_plus_nat(Y,plus_plus_nat(X,Z)) ).
tff(fact_586_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] : times_times_nat(K,plus_plus_nat(M,N)) = plus_plus_nat(times_times_nat(K,M),times_times_nat(K,N)) ).
tff(fact_587_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] : plus_plus_nat(plus_plus_nat(M,N),K) = plus_plus_nat(M,plus_plus_nat(N,K)) ).
tff(fact_588_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] : times_times_nat(plus_plus_nat(M,N),K) = plus_plus_nat(times_times_nat(M,K),times_times_nat(N,K)) ).
tff(fact_589_nat__add__left__cancel,axiom,
! [K_1: nat,M_1: nat,N_1: nat] :
( ( plus_plus_nat(K_1,M_1) = plus_plus_nat(K_1,N_1) )
<=> ( M_1 = N_1 ) ) ).
tff(fact_590_nat__add__right__cancel,axiom,
! [M_1: nat,K_1: nat,N_1: nat] :
( ( plus_plus_nat(M_1,K_1) = plus_plus_nat(N_1,K_1) )
<=> ( M_1 = N_1 ) ) ).
tff(fact_591_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] : minus_minus_nat(plus_plus_nat(M,K),plus_plus_nat(N,K)) = minus_minus_nat(M,N) ).
tff(fact_592_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] : minus_minus_nat(plus_plus_nat(K,M),plus_plus_nat(K,N)) = minus_minus_nat(M,N) ).
tff(fact_593_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] : minus_minus_nat(minus_minus_nat(I,J),K) = minus_minus_nat(I,plus_plus_nat(J,K)) ).
tff(fact_594_diff__add__inverse,axiom,
! [N: nat,M: nat] : minus_minus_nat(plus_plus_nat(N,M),N) = M ).
tff(fact_595_diff__add__inverse2,axiom,
! [M: nat,N: nat] : minus_minus_nat(plus_plus_nat(M,N),N) = M ).
tff(fact_596_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,N)),M)) ).
tff(fact_597_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ).
tff(fact_598_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,L)),minus_minus_nat(N,L))) ) ).
tff(fact_599_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
=> ( minus_minus_nat(N,minus_minus_nat(N,I)) = I ) ) ).
tff(fact_600_eq__diff__iff,axiom,
! [N_1: nat,K_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> ( ( minus_minus_nat(M_1,K_1) = minus_minus_nat(N_1,K_1) )
<=> ( M_1 = N_1 ) ) ) ) ).
tff(fact_601_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> ( minus_minus_nat(minus_minus_nat(M,K),minus_minus_nat(N,K)) = minus_minus_nat(M,N) ) ) ) ).
tff(fact_602_le__diff__iff,axiom,
! [N_1: nat,K_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M_1,K_1)),minus_minus_nat(N_1,K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ) ) ).
tff(fact_603_le__refl,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
tff(fact_604_nat__le__linear,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
tff(fact_605_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_606_le__trans,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K)) ) ) ).
tff(fact_607_le__antisym,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( M = N ) ) ) ).
tff(fact_608_mult__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(I,K)),times_times_nat(J,L))) ) ) ).
tff(fact_609_mult__le__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K,I)),times_times_nat(K,J))) ) ).
tff(fact_610_mult__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(I,K)),times_times_nat(J,K))) ) ).
tff(fact_611_le__cube,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,times_times_nat(M,M)))) ).
tff(fact_612_le__square,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,M))) ).
tff(fact_613_nat__mult__1,axiom,
! [N: nat] : times_times_nat(one_one_nat,N) = N ).
tff(fact_614_nat__1__eq__mult__iff,axiom,
! [M_1: nat,N_1: nat] :
( ( one_one_nat = times_times_nat(M_1,N_1) )
<=> ( ( M_1 = one_one_nat )
& ( N_1 = one_one_nat ) ) ) ).
tff(fact_615_nat__mult__1__right,axiom,
! [N: nat] : times_times_nat(N,one_one_nat) = N ).
tff(fact_616_nat__mult__eq__1__iff,axiom,
! [M_1: nat,N_1: nat] :
( ( times_times_nat(M_1,N_1) = one_one_nat )
<=> ( ( M_1 = one_one_nat )
& ( N_1 = one_one_nat ) ) ) ).
tff(fact_617_zero__less__diff,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),minus_minus_nat(N_1,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_618_diff__less,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(M,N)),M)) ) ) ).
tff(fact_619_not__less0,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_620_neq0__conv,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
tff(fact_621_less__nat__zero__code,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_622_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( N != zero_zero_nat ) ) ).
tff(fact_623_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
tff(fact_624_mult__less__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K,I)),times_times_nat(K,J))) ) ) ).
tff(fact_625_mult__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(I,K)),times_times_nat(J,K))) ) ) ).
tff(fact_626_mult__less__cancel2,axiom,
! [M_1: nat,K_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(M_1,K_1)),times_times_nat(N_1,K_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).
tff(fact_627_mult__less__cancel1,axiom,
! [K_1: nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K_1,M_1)),times_times_nat(K_1,N_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).
tff(fact_628_nat__0__less__mult__iff,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(M_1,N_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ) ).
tff(fact_629_plus__nat_Oadd__0,axiom,
! [N: nat] : plus_plus_nat(zero_zero_nat,N) = N ).
tff(fact_630_Nat_Oadd__0__right,axiom,
! [M: nat] : plus_plus_nat(M,zero_zero_nat) = M ).
tff(fact_631_add__is__0,axiom,
! [M_1: nat,N_1: nat] :
( ( plus_plus_nat(M_1,N_1) = zero_zero_nat )
<=> ( ( M_1 = zero_zero_nat )
& ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_632_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat(M,N) = M )
=> ( N = zero_zero_nat ) ) ).
tff(fact_633_diff__add__0,axiom,
! [N: nat,M: nat] : minus_minus_nat(N,plus_plus_nat(N,M)) = zero_zero_nat ).
tff(fact_634_diff__is__0__eq,axiom,
! [M_1: nat,N_1: nat] :
( ( minus_minus_nat(M_1,N_1) = zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_635_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( minus_minus_nat(M,N) = zero_zero_nat ) ) ).
tff(fact_636_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
tff(fact_637_le__0__eq,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),zero_zero_nat))
<=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_638_less__diff__conv,axiom,
! [I_1: nat,J_1: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),minus_minus_nat(J_1,K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_1,K_1)),J_1)) ) ).
tff(fact_639_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( plus_plus_nat(N,minus_minus_nat(M,N)) = M ) ) ).
tff(fact_640_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I,J)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).
tff(fact_641_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> ( ( plus_plus_nat(M,L) = plus_plus_nat(K,N) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_642_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I,K)),plus_plus_nat(J,L))) ) ) ).
tff(fact_643_add__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I,K)),plus_plus_nat(J,K))) ) ).
tff(fact_644_trans__less__add2,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),plus_plus_nat(M,J))) ) ).
tff(fact_645_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),plus_plus_nat(J,M))) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
times_times_int(twoSqu2107342101sum2sq(product_Pair_int_int(a,b)),twoSqu2107342101sum2sq(product_Pair_int_int(p,q))) = twoSqu2107342101sum2sq(product_Pair_int_int(plus_plus_int(times_times_int(a,p),times_times_int(b,q)),minus_minus_int(times_times_int(a,q),times_times_int(b,p)))) ).
%------------------------------------------------------------------------------