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_thf_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 685 ( 204 unt; 38 typ; 0 def)
% Number of atoms : 1463 ( 426 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 5342 ( 144 ~; 41 |; 51 &;4307 @)
% ( 130 <=>; 663 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 88 ( 88 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 34 usr; 9 con; 0-8 aty)
% ( 0 !!; 6 ??; 0 @@+; 0 @@-)
% Number of variables : 1583 ( 0 ^;1565 !; 18 ?;1583 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:28:22
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_ty_tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
produc393999548nt_int: $tType ).
%----Explicit typings (37)
thf(sy_c_All2,type,
all2: ( produc393999548nt_int > $o ) > $o ).
thf(sy_c_All1,type,
all1: ( product_prod_int_int > $o ) > $o ).
thf(sy_c_Divides_Odivmod__int__rel,type,
divmod_int_rel: int > int > product_prod_int_int > $o ).
thf(sy_c_Divides_Opdivmod,type,
pdivmod: int > int > product_prod_int_int ).
thf(sy_c_Divides_OposDivAlg,type,
posDivAlg: int > int > product_prod_int_int ).
thf(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_IntFact_Ozfact,type,
zfact: int > int ).
thf(sy_c_IntPrimes_Oxzgcd,type,
xzgcd: int > int > produc393999548nt_int ).
thf(sy_c_IntPrimes_Oxzgcda,type,
xzgcda: int > int > int > int > int > int > int > int > produc393999548nt_int ).
thf(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: int > int > int ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(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 ).
thf(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__Int__Oint_000_Eo,type,
produc176579150_int_o: ( product_prod_int_int > $o ) > int > int > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__,type,
produc122891093_int_o: ( produc393999548nt_int > $o ) > int > product_prod_int_int > $o ).
thf(sy_c_TwoSquares__Mirabelle__hgbmwyaznu_Ois__sum2sq,type,
twoSqu362149276sum2sq: int > $o ).
thf(sy_c_TwoSquares__Mirabelle__hgbmwyaznu_Osum2sq,type,
twoSqu1078207634sum2sq: product_prod_int_int > int ).
thf(sy_v_a,type,
a: int ).
thf(sy_v_b,type,
b: int ).
thf(sy_v_p,type,
p: int ).
thf(sy_v_q,type,
q: int ).
%----Relevant facts (646)
thf(fact_0_xzgcda__linear__aux1,axiom,
! [A_23: int,R_1: int,B: int,M: int,C_36: int,D: int,N: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_23 @ ( times_times_int @ R_1 @ B ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C_36 @ ( times_times_int @ R_1 @ D ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A_23 @ M ) @ ( times_times_int @ C_36 @ N ) ) @ ( times_times_int @ R_1 @ ( plus_plus_int @ ( times_times_int @ B @ M ) @ ( times_times_int @ D @ N ) ) ) ) ) ).
thf(fact_1_mult__diff__mult,axiom,
! [X_65: int,Y_60: int,A_192: int,B_147: int] :
( ( minus_minus_int @ ( times_times_int @ X_65 @ Y_60 ) @ ( times_times_int @ A_192 @ B_147 ) )
= ( plus_plus_int @ ( times_times_int @ X_65 @ ( minus_minus_int @ Y_60 @ B_147 ) ) @ ( times_times_int @ ( minus_minus_int @ X_65 @ A_192 ) @ B_147 ) ) ) ).
thf(fact_2_eq__add__iff2,axiom,
! [A_191: int,E_6: int,C_88: int,B_146: int,D_27: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_191 @ E_6 ) @ C_88 )
= ( plus_plus_int @ ( times_times_int @ B_146 @ E_6 ) @ D_27 ) )
<=> ( C_88
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_146 @ A_191 ) @ E_6 ) @ D_27 ) ) ) ).
thf(fact_3_eq__add__iff1,axiom,
! [A_190: int,E_5: int,C_87: int,B_145: int,D_26: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_190 @ E_5 ) @ C_87 )
= ( plus_plus_int @ ( times_times_int @ B_145 @ E_5 ) @ D_26 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_190 @ B_145 ) @ E_5 ) @ C_87 )
= D_26 ) ) ).
thf(fact_4_is__sum2sq__def,axiom,
! [X: int] :
( ( twoSqu362149276sum2sq @ X )
<=> ? [A_113: int,B_120: int] :
( ( twoSqu1078207634sum2sq @ ( product_Pair_int_int @ A_113 @ B_120 ) )
= X ) ) ).
thf(fact_5_Int2_Oaux1,axiom,
! [A_23: int,B: int,C_36: int] :
( ( ( minus_minus_int @ A_23 @ B )
= C_36 )
=> ( A_23
= ( plus_plus_int @ C_36 @ B ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(fact_10_diff__add__cancel,axiom,
! [A_189: int,B_144: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_189 @ B_144 ) @ B_144 )
= A_189 ) ).
thf(fact_11_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_188: int,B_143: int,C_86: int] :
( ( times_times_int @ ( times_times_int @ A_188 @ B_143 ) @ C_86 )
= ( times_times_int @ A_188 @ ( times_times_int @ B_143 @ C_86 ) ) ) ).
thf(fact_12_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_188: nat,B_143: nat,C_86: nat] :
( ( times_times_nat @ ( times_times_nat @ A_188 @ B_143 ) @ C_86 )
= ( times_times_nat @ A_188 @ ( times_times_nat @ B_143 @ C_86 ) ) ) ).
thf(fact_13_add__right__imp__eq,axiom,
! [B_142: int,A_187: int,C_85: int] :
( ( ( plus_plus_int @ B_142 @ A_187 )
= ( plus_plus_int @ C_85 @ A_187 ) )
=> ( B_142 = C_85 ) ) ).
thf(fact_14_add__right__imp__eq,axiom,
! [B_142: nat,A_187: nat,C_85: nat] :
( ( ( plus_plus_nat @ B_142 @ A_187 )
= ( plus_plus_nat @ C_85 @ A_187 ) )
=> ( B_142 = C_85 ) ) ).
thf(fact_15_add__imp__eq,axiom,
! [A_186: int,B_141: int,C_84: int] :
( ( ( plus_plus_int @ A_186 @ B_141 )
= ( plus_plus_int @ A_186 @ C_84 ) )
=> ( B_141 = C_84 ) ) ).
thf(fact_16_add__imp__eq,axiom,
! [A_186: nat,B_141: nat,C_84: nat] :
( ( ( plus_plus_nat @ A_186 @ B_141 )
= ( plus_plus_nat @ A_186 @ C_84 ) )
=> ( B_141 = C_84 ) ) ).
thf(fact_17_add__left__imp__eq,axiom,
! [A_185: int,B_140: int,C_83: int] :
( ( ( plus_plus_int @ A_185 @ B_140 )
= ( plus_plus_int @ A_185 @ C_83 ) )
=> ( B_140 = C_83 ) ) ).
thf(fact_18_add__left__imp__eq,axiom,
! [A_185: nat,B_140: nat,C_83: nat] :
( ( ( plus_plus_nat @ A_185 @ B_140 )
= ( plus_plus_nat @ A_185 @ C_83 ) )
=> ( B_140 = C_83 ) ) ).
thf(fact_19_add__right__cancel,axiom,
! [B_139: int,A_184: int,C_82: int] :
( ( ( plus_plus_int @ B_139 @ A_184 )
= ( plus_plus_int @ C_82 @ A_184 ) )
<=> ( B_139 = C_82 ) ) ).
thf(fact_20_add__right__cancel,axiom,
! [B_139: nat,A_184: nat,C_82: nat] :
( ( ( plus_plus_nat @ B_139 @ A_184 )
= ( plus_plus_nat @ C_82 @ A_184 ) )
<=> ( B_139 = C_82 ) ) ).
thf(fact_21_add__left__cancel,axiom,
! [A_183: int,B_138: int,C_81: int] :
( ( ( plus_plus_int @ A_183 @ B_138 )
= ( plus_plus_int @ A_183 @ C_81 ) )
<=> ( B_138 = C_81 ) ) ).
thf(fact_22_add__left__cancel,axiom,
! [A_183: nat,B_138: nat,C_81: nat] :
( ( ( plus_plus_nat @ A_183 @ B_138 )
= ( plus_plus_nat @ A_183 @ C_81 ) )
<=> ( B_138 = C_81 ) ) ).
thf(fact_23_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_182: int,B_137: int,C_80: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_182 @ B_137 ) @ C_80 )
= ( plus_plus_int @ A_182 @ ( plus_plus_int @ B_137 @ C_80 ) ) ) ).
thf(fact_24_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_182: nat,B_137: nat,C_80: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_182 @ B_137 ) @ C_80 )
= ( plus_plus_nat @ A_182 @ ( plus_plus_nat @ B_137 @ C_80 ) ) ) ).
thf(fact_25_diff__eq__diff__eq,axiom,
! [A_181: int,B_136: int,C_79: int,D_25: int] :
( ( ( minus_minus_int @ A_181 @ B_136 )
= ( minus_minus_int @ C_79 @ D_25 ) )
=> ( ( A_181 = B_136 )
<=> ( C_79 = D_25 ) ) ) ).
thf(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 ) ) ) ).
thf(fact_27_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(fact_30_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_31_combine__common__factor,axiom,
! [A_180: int,E_4: int,B_135: int,C_78: int] :
( ( plus_plus_int @ ( times_times_int @ A_180 @ E_4 ) @ ( plus_plus_int @ ( times_times_int @ B_135 @ E_4 ) @ C_78 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_180 @ B_135 ) @ E_4 ) @ C_78 ) ) ).
thf(fact_32_combine__common__factor,axiom,
! [A_180: nat,E_4: nat,B_135: nat,C_78: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_180 @ E_4 ) @ ( plus_plus_nat @ ( times_times_nat @ B_135 @ E_4 ) @ C_78 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_180 @ B_135 ) @ E_4 ) @ C_78 ) ) ).
thf(fact_33_comm__semiring__class_Odistrib,axiom,
! [A_179: int,B_134: int,C_77: int] :
( ( times_times_int @ ( plus_plus_int @ A_179 @ B_134 ) @ C_77 )
= ( plus_plus_int @ ( times_times_int @ A_179 @ C_77 ) @ ( times_times_int @ B_134 @ C_77 ) ) ) ).
thf(fact_34_comm__semiring__class_Odistrib,axiom,
! [A_179: nat,B_134: nat,C_77: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_179 @ B_134 ) @ C_77 )
= ( plus_plus_nat @ ( times_times_nat @ A_179 @ C_77 ) @ ( times_times_nat @ B_134 @ C_77 ) ) ) ).
thf(fact_35_add__diff__add,axiom,
! [A_178: int,C_76: int,B_133: int,D_24: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_178 @ C_76 ) @ ( plus_plus_int @ B_133 @ D_24 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_178 @ B_133 ) @ ( minus_minus_int @ C_76 @ D_24 ) ) ) ).
thf(fact_36_add__diff__cancel,axiom,
! [A_177: int,B_132: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_177 @ B_132 ) @ B_132 )
= A_177 ) ).
thf(fact_37_crossproduct__eq,axiom,
! [W_2: int,Y_59: int,X_64: int,Z_12: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_2 @ Y_59 ) @ ( times_times_int @ X_64 @ Z_12 ) )
= ( plus_plus_int @ ( times_times_int @ W_2 @ Z_12 ) @ ( times_times_int @ X_64 @ Y_59 ) ) )
<=> ( ( W_2 = X_64 )
| ( Y_59 = Z_12 ) ) ) ).
thf(fact_38_crossproduct__eq,axiom,
! [W_2: nat,Y_59: nat,X_64: nat,Z_12: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_2 @ Y_59 ) @ ( times_times_nat @ X_64 @ Z_12 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_2 @ Z_12 ) @ ( times_times_nat @ X_64 @ Y_59 ) ) )
<=> ( ( W_2 = X_64 )
| ( Y_59 = Z_12 ) ) ) ).
thf(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_176: int,M_5: int,B_131: int] :
( ( plus_plus_int @ ( times_times_int @ A_176 @ M_5 ) @ ( times_times_int @ B_131 @ M_5 ) )
= ( times_times_int @ ( plus_plus_int @ A_176 @ B_131 ) @ M_5 ) ) ).
thf(fact_40_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_176: nat,M_5: nat,B_131: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_176 @ M_5 ) @ ( times_times_nat @ B_131 @ M_5 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_176 @ B_131 ) @ M_5 ) ) ).
thf(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_175: int,B_130: int,C_75: int] :
( ( times_times_int @ ( plus_plus_int @ A_175 @ B_130 ) @ C_75 )
= ( plus_plus_int @ ( times_times_int @ A_175 @ C_75 ) @ ( times_times_int @ B_130 @ C_75 ) ) ) ).
thf(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_175: nat,B_130: nat,C_75: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_175 @ B_130 ) @ C_75 )
= ( plus_plus_nat @ ( times_times_nat @ A_175 @ C_75 ) @ ( times_times_nat @ B_130 @ C_75 ) ) ) ).
thf(fact_43_crossproduct__noteq,axiom,
! [C_74: int,D_23: int,A_174: int,B_129: int] :
( ( ( A_174 != B_129 )
& ( C_74 != D_23 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_174 @ C_74 ) @ ( times_times_int @ B_129 @ D_23 ) )
!= ( plus_plus_int @ ( times_times_int @ A_174 @ D_23 ) @ ( times_times_int @ B_129 @ C_74 ) ) ) ) ).
thf(fact_44_crossproduct__noteq,axiom,
! [C_74: nat,D_23: nat,A_174: nat,B_129: nat] :
( ( ( A_174 != B_129 )
& ( C_74 != D_23 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_174 @ C_74 ) @ ( times_times_nat @ B_129 @ D_23 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_174 @ D_23 ) @ ( times_times_nat @ B_129 @ C_74 ) ) ) ) ).
thf(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_63: int,Y_58: int,Z_11: int] :
( ( times_times_int @ X_63 @ ( plus_plus_int @ Y_58 @ Z_11 ) )
= ( plus_plus_int @ ( times_times_int @ X_63 @ Y_58 ) @ ( times_times_int @ X_63 @ Z_11 ) ) ) ).
thf(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_63: nat,Y_58: nat,Z_11: nat] :
( ( times_times_nat @ X_63 @ ( plus_plus_nat @ Y_58 @ Z_11 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_63 @ Y_58 ) @ ( times_times_nat @ X_63 @ Z_11 ) ) ) ).
thf(fact_47_Pair__inject,axiom,
! [A_173: int,B_128: int,A_172: int,B_127: int] :
( ( ( product_Pair_int_int @ A_173 @ B_128 )
= ( product_Pair_int_int @ A_172 @ B_127 ) )
=> ~ ( ( A_173 = A_172 )
=> ( B_128 != B_127 ) ) ) ).
thf(fact_48_Pair__inject,axiom,
! [A_173: int,B_128: product_prod_int_int,A_172: int,B_127: product_prod_int_int] :
( ( ( produc282740534nt_int @ A_173 @ B_128 )
= ( produc282740534nt_int @ A_172 @ B_127 ) )
=> ~ ( ( A_173 = A_172 )
=> ( B_128 != B_127 ) ) ) ).
thf(fact_49_Pair__eq,axiom,
! [A_171: int,B_126: int,A_170: int,B_125: int] :
( ( ( product_Pair_int_int @ A_171 @ B_126 )
= ( product_Pair_int_int @ A_170 @ B_125 ) )
<=> ( ( A_171 = A_170 )
& ( B_126 = B_125 ) ) ) ).
thf(fact_50_Pair__eq,axiom,
! [A_171: int,B_126: product_prod_int_int,A_170: int,B_125: product_prod_int_int] :
( ( ( produc282740534nt_int @ A_171 @ B_126 )
= ( produc282740534nt_int @ A_170 @ B_125 ) )
<=> ( ( A_171 = A_170 )
& ( B_126 = B_125 ) ) ) ).
thf(fact_51_split__paired__All,axiom,
! [P_6: product_prod_int_int > $o] :
( ( all1 @ P_6 )
<=> ! [A_113: int,B_120: int] : ( P_6 @ ( product_Pair_int_int @ A_113 @ B_120 ) ) ) ).
thf(fact_52_split__paired__All,axiom,
! [P_6: produc393999548nt_int > $o] :
( ( all2 @ P_6 )
<=> ! [A_113: int,B_120: product_prod_int_int] : ( P_6 @ ( produc282740534nt_int @ A_113 @ B_120 ) ) ) ).
thf(fact_53_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_169: int,B_124: int,C_73: int,D_22: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_169 @ B_124 ) @ ( plus_plus_int @ C_73 @ D_22 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_169 @ C_73 ) @ ( plus_plus_int @ B_124 @ D_22 ) ) ) ).
thf(fact_54_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_169: nat,B_124: nat,C_73: nat,D_22: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_169 @ B_124 ) @ ( plus_plus_nat @ C_73 @ D_22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_169 @ C_73 ) @ ( plus_plus_nat @ B_124 @ D_22 ) ) ) ).
thf(fact_55_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_168: int,B_123: int] :
( ( times_times_int @ A_168 @ B_123 )
= ( times_times_int @ B_123 @ A_168 ) ) ).
thf(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_168: nat,B_123: nat] :
( ( times_times_nat @ A_168 @ B_123 )
= ( times_times_nat @ B_123 @ A_168 ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ).
thf(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 ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ).
thf(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 ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_167: int,C_72: int] :
( ( plus_plus_int @ A_167 @ C_72 )
= ( plus_plus_int @ C_72 @ A_167 ) ) ).
thf(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_167: nat,C_72: nat] :
( ( plus_plus_nat @ A_167 @ C_72 )
= ( plus_plus_nat @ C_72 @ A_167 ) ) ).
thf(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_166: int,C_71: int,D_21: int] :
( ( plus_plus_int @ A_166 @ ( plus_plus_int @ C_71 @ D_21 ) )
= ( plus_plus_int @ C_71 @ ( plus_plus_int @ A_166 @ D_21 ) ) ) ).
thf(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_166: nat,C_71: nat,D_21: nat] :
( ( plus_plus_nat @ A_166 @ ( plus_plus_nat @ C_71 @ D_21 ) )
= ( plus_plus_nat @ C_71 @ ( plus_plus_nat @ A_166 @ D_21 ) ) ) ).
thf(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_165: int,C_70: int,D_20: int] :
( ( plus_plus_int @ A_165 @ ( plus_plus_int @ C_70 @ D_20 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_165 @ C_70 ) @ D_20 ) ) ).
thf(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_165: nat,C_70: nat,D_20: nat] :
( ( plus_plus_nat @ A_165 @ ( plus_plus_nat @ C_70 @ D_20 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_165 @ C_70 ) @ D_20 ) ) ).
thf(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_164: int,B_122: int,C_69: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_164 @ B_122 ) @ C_69 )
= ( plus_plus_int @ A_164 @ ( plus_plus_int @ B_122 @ C_69 ) ) ) ).
thf(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_164: nat,B_122: nat,C_69: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_164 @ B_122 ) @ C_69 )
= ( plus_plus_nat @ A_164 @ ( plus_plus_nat @ B_122 @ C_69 ) ) ) ).
thf(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_163: int,B_121: int,C_68: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_163 @ B_121 ) @ C_68 )
= ( plus_plus_int @ ( plus_plus_int @ A_163 @ C_68 ) @ B_121 ) ) ).
thf(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_163: nat,B_121: nat,C_68: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_163 @ B_121 ) @ C_68 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_163 @ C_68 ) @ B_121 ) ) ).
thf(fact_81_split__paired__Ex,axiom,
! [P_5: product_prod_int_int > $o] :
( ( ?? @ product_prod_int_int @ P_5 )
<=> ? [A_113: int,B_120: int] : ( P_5 @ ( product_Pair_int_int @ A_113 @ B_120 ) ) ) ).
thf(fact_82_split__paired__Ex,axiom,
! [P_5: produc393999548nt_int > $o] :
( ( ?? @ produc393999548nt_int @ P_5 )
<=> ? [A_113: int,B_120: product_prod_int_int] : ( P_5 @ ( produc282740534nt_int @ A_113 @ B_120 ) ) ) ).
thf(fact_83_prod__induct3,axiom,
! [X_62: produc393999548nt_int,P_4: produc393999548nt_int > $o] :
( ! [A_113: int,B_120: int,C_67: int] : ( P_4 @ ( produc282740534nt_int @ A_113 @ ( product_Pair_int_int @ B_120 @ C_67 ) ) )
=> ( P_4 @ X_62 ) ) ).
thf(fact_84_prod__cases3,axiom,
! [Y_57: produc393999548nt_int] :
~ ! [A_113: int,B_120: int,C_67: int] :
( Y_57
!= ( produc282740534nt_int @ A_113 @ ( product_Pair_int_int @ B_120 @ C_67 ) ) ) ).
thf(fact_85_prod_Oexhaust,axiom,
! [Y_56: product_prod_int_int] :
~ ! [A_113: int,B_120: int] :
( Y_56
!= ( product_Pair_int_int @ A_113 @ B_120 ) ) ).
thf(fact_86_prod_Oexhaust,axiom,
! [Y_56: produc393999548nt_int] :
~ ! [A_113: int,B_120: product_prod_int_int] :
( Y_56
!= ( produc282740534nt_int @ A_113 @ B_120 ) ) ).
thf(fact_87_PairE,axiom,
! [P_3: product_prod_int_int] :
~ ! [X_3: int,Y_55: int] :
( P_3
!= ( product_Pair_int_int @ X_3 @ Y_55 ) ) ).
thf(fact_88_PairE,axiom,
! [P_3: produc393999548nt_int] :
~ ! [X_3: int,Y_55: product_prod_int_int] :
( P_3
!= ( produc282740534nt_int @ X_3 @ Y_55 ) ) ).
thf(fact_89_curryI,axiom,
! [F_4: product_prod_int_int > $o,A_162: int,B_119: int] :
( ( F_4 @ ( product_Pair_int_int @ A_162 @ B_119 ) )
=> ( produc176579150_int_o @ F_4 @ A_162 @ B_119 ) ) ).
thf(fact_90_curryI,axiom,
! [F_4: produc393999548nt_int > $o,A_162: int,B_119: product_prod_int_int] :
( ( F_4 @ ( produc282740534nt_int @ A_162 @ B_119 ) )
=> ( produc122891093_int_o @ F_4 @ A_162 @ B_119 ) ) ).
thf(fact_91_le__add__iff1,axiom,
! [A_161: int,E_3: int,C_66: int,B_118: int,D_19: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_161 @ E_3 ) @ C_66 ) @ ( plus_plus_int @ ( times_times_int @ B_118 @ E_3 ) @ D_19 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_161 @ B_118 ) @ E_3 ) @ C_66 ) @ D_19 ) ) ).
thf(fact_92_le__add__iff2,axiom,
! [A_160: int,E_2: int,C_65: int,B_117: int,D_18: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_160 @ E_2 ) @ C_65 ) @ ( plus_plus_int @ ( times_times_int @ B_117 @ E_2 ) @ D_18 ) )
<=> ( ord_less_eq_int @ C_65 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_117 @ A_160 ) @ E_2 ) @ D_18 ) ) ) ).
thf(fact_93_real__squared__diff__one__factored,axiom,
! [X_61: int] :
( ( minus_minus_int @ ( times_times_int @ X_61 @ X_61 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X_61 @ one_one_int ) @ ( minus_minus_int @ X_61 @ one_one_int ) ) ) ).
thf(fact_94_less__add__iff1,axiom,
! [A_159: int,E_1: int,C_64: int,B_116: int,D_17: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_159 @ E_1 ) @ C_64 ) @ ( plus_plus_int @ ( times_times_int @ B_116 @ E_1 ) @ D_17 ) )
<=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_159 @ B_116 ) @ E_1 ) @ C_64 ) @ D_17 ) ) ).
thf(fact_95_less__add__iff2,axiom,
! [A_158: int,E: int,C_63: int,B_115: int,D_16: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_158 @ E ) @ C_63 ) @ ( plus_plus_int @ ( times_times_int @ B_115 @ E ) @ D_16 ) )
<=> ( ord_less_int @ C_63 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_115 @ A_158 ) @ E ) @ D_16 ) ) ) ).
thf(fact_96_one__reorient,axiom,
! [X_60: int] :
( ( one_one_int = X_60 )
<=> ( X_60 = one_one_int ) ) ).
thf(fact_97_one__reorient,axiom,
! [X_60: nat] :
( ( one_one_nat = X_60 )
<=> ( X_60 = one_one_nat ) ) ).
thf(fact_98_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_99_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_100_linorder__neqE__linordered__idom,axiom,
! [X_59: int,Y_54: int] :
( ( X_59 != Y_54 )
=> ( ~ ( ord_less_int @ X_59 @ Y_54 )
=> ( ord_less_int @ Y_54 @ X_59 ) ) ) ).
thf(fact_101_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
thf(fact_102_order__le__neq__implies__less,axiom,
! [X_58: int,Y_53: int] :
( ( ord_less_eq_int @ X_58 @ Y_53 )
=> ( ( X_58 != Y_53 )
=> ( ord_less_int @ X_58 @ Y_53 ) ) ) ).
thf(fact_103_order__le__neq__implies__less,axiom,
! [X_58: nat,Y_53: nat] :
( ( ord_less_eq_nat @ X_58 @ Y_53 )
=> ( ( X_58 != Y_53 )
=> ( ord_less_nat @ X_58 @ Y_53 ) ) ) ).
thf(fact_104_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
thf(fact_105_zadd__zless__mono,axiom,
! [Z_10: int,Z: int,W_1: int,W: int] :
( ( ord_less_int @ W_1 @ W )
=> ( ( ord_less_eq_int @ Z_10 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_1 @ Z_10 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_106_add__less__le__mono,axiom,
! [C_62: int,D_15: int,A_157: int,B_114: int] :
( ( ord_less_int @ A_157 @ B_114 )
=> ( ( ord_less_eq_int @ C_62 @ D_15 )
=> ( ord_less_int @ ( plus_plus_int @ A_157 @ C_62 ) @ ( plus_plus_int @ B_114 @ D_15 ) ) ) ) ).
thf(fact_107_add__less__le__mono,axiom,
! [C_62: nat,D_15: nat,A_157: nat,B_114: nat] :
( ( ord_less_nat @ A_157 @ B_114 )
=> ( ( ord_less_eq_nat @ C_62 @ D_15 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_157 @ C_62 ) @ ( plus_plus_nat @ B_114 @ D_15 ) ) ) ) ).
thf(fact_108_add__le__less__mono,axiom,
! [C_61: int,D_14: int,A_156: int,B_113: int] :
( ( ord_less_eq_int @ A_156 @ B_113 )
=> ( ( ord_less_int @ C_61 @ D_14 )
=> ( ord_less_int @ ( plus_plus_int @ A_156 @ C_61 ) @ ( plus_plus_int @ B_113 @ D_14 ) ) ) ) ).
thf(fact_109_add__le__less__mono,axiom,
! [C_61: nat,D_14: nat,A_156: nat,B_113: nat] :
( ( ord_less_eq_nat @ A_156 @ B_113 )
=> ( ( ord_less_nat @ C_61 @ D_14 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_156 @ C_61 ) @ ( plus_plus_nat @ B_113 @ D_14 ) ) ) ) ).
thf(fact_110_less__add__one,axiom,
! [A_155: int] : ( ord_less_int @ A_155 @ ( plus_plus_int @ A_155 @ one_one_int ) ) ).
thf(fact_111_less__add__one,axiom,
! [A_155: nat] : ( ord_less_nat @ A_155 @ ( plus_plus_nat @ A_155 @ one_one_nat ) ) ).
thf(fact_112_less__1__mult,axiom,
! [N_1: int,M_4: int] :
( ( ord_less_int @ one_one_int @ M_4 )
=> ( ( ord_less_int @ one_one_int @ N_1 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M_4 @ N_1 ) ) ) ) ).
thf(fact_113_less__1__mult,axiom,
! [N_1: nat,M_4: nat] :
( ( ord_less_nat @ one_one_nat @ M_4 )
=> ( ( ord_less_nat @ one_one_nat @ N_1 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M_4 @ N_1 ) ) ) ) ).
thf(fact_114_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_115_add__less__cancel__right,axiom,
! [A_154: int,C_60: int,B_112: int] :
( ( ord_less_int @ ( plus_plus_int @ A_154 @ C_60 ) @ ( plus_plus_int @ B_112 @ C_60 ) )
<=> ( ord_less_int @ A_154 @ B_112 ) ) ).
thf(fact_116_add__less__cancel__right,axiom,
! [A_154: nat,C_60: nat,B_112: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_154 @ C_60 ) @ ( plus_plus_nat @ B_112 @ C_60 ) )
<=> ( ord_less_nat @ A_154 @ B_112 ) ) ).
thf(fact_117_add__less__cancel__left,axiom,
! [C_59: int,A_153: int,B_111: int] :
( ( ord_less_int @ ( plus_plus_int @ C_59 @ A_153 ) @ ( plus_plus_int @ C_59 @ B_111 ) )
<=> ( ord_less_int @ A_153 @ B_111 ) ) ).
thf(fact_118_add__less__cancel__left,axiom,
! [C_59: nat,A_153: nat,B_111: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_59 @ A_153 ) @ ( plus_plus_nat @ C_59 @ B_111 ) )
<=> ( ord_less_nat @ A_153 @ B_111 ) ) ).
thf(fact_119_add__strict__right__mono,axiom,
! [C_58: int,A_152: int,B_110: int] :
( ( ord_less_int @ A_152 @ B_110 )
=> ( ord_less_int @ ( plus_plus_int @ A_152 @ C_58 ) @ ( plus_plus_int @ B_110 @ C_58 ) ) ) ).
thf(fact_120_add__strict__right__mono,axiom,
! [C_58: nat,A_152: nat,B_110: nat] :
( ( ord_less_nat @ A_152 @ B_110 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_152 @ C_58 ) @ ( plus_plus_nat @ B_110 @ C_58 ) ) ) ).
thf(fact_121_add__strict__left__mono,axiom,
! [C_57: int,A_151: int,B_109: int] :
( ( ord_less_int @ A_151 @ B_109 )
=> ( ord_less_int @ ( plus_plus_int @ C_57 @ A_151 ) @ ( plus_plus_int @ C_57 @ B_109 ) ) ) ).
thf(fact_122_add__strict__left__mono,axiom,
! [C_57: nat,A_151: nat,B_109: nat] :
( ( ord_less_nat @ A_151 @ B_109 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_57 @ A_151 ) @ ( plus_plus_nat @ C_57 @ B_109 ) ) ) ).
thf(fact_123_add__strict__mono,axiom,
! [C_56: int,D_13: int,A_150: int,B_108: int] :
( ( ord_less_int @ A_150 @ B_108 )
=> ( ( ord_less_int @ C_56 @ D_13 )
=> ( ord_less_int @ ( plus_plus_int @ A_150 @ C_56 ) @ ( plus_plus_int @ B_108 @ D_13 ) ) ) ) ).
thf(fact_124_add__strict__mono,axiom,
! [C_56: nat,D_13: nat,A_150: nat,B_108: nat] :
( ( ord_less_nat @ A_150 @ B_108 )
=> ( ( ord_less_nat @ C_56 @ D_13 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_150 @ C_56 ) @ ( plus_plus_nat @ B_108 @ D_13 ) ) ) ) ).
thf(fact_125_add__less__imp__less__right,axiom,
! [A_149: int,C_55: int,B_107: int] :
( ( ord_less_int @ ( plus_plus_int @ A_149 @ C_55 ) @ ( plus_plus_int @ B_107 @ C_55 ) )
=> ( ord_less_int @ A_149 @ B_107 ) ) ).
thf(fact_126_add__less__imp__less__right,axiom,
! [A_149: nat,C_55: nat,B_107: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_149 @ C_55 ) @ ( plus_plus_nat @ B_107 @ C_55 ) )
=> ( ord_less_nat @ A_149 @ B_107 ) ) ).
thf(fact_127_add__less__imp__less__left,axiom,
! [C_54: int,A_148: int,B_106: int] :
( ( ord_less_int @ ( plus_plus_int @ C_54 @ A_148 ) @ ( plus_plus_int @ C_54 @ B_106 ) )
=> ( ord_less_int @ A_148 @ B_106 ) ) ).
thf(fact_128_add__less__imp__less__left,axiom,
! [C_54: nat,A_148: nat,B_106: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_54 @ A_148 ) @ ( plus_plus_nat @ C_54 @ B_106 ) )
=> ( ord_less_nat @ A_148 @ B_106 ) ) ).
thf(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_147: int] :
( ( times_times_int @ one_one_int @ A_147 )
= A_147 ) ).
thf(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_147: nat] :
( ( times_times_nat @ one_one_nat @ A_147 )
= A_147 ) ).
thf(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_146: int] :
( ( times_times_int @ A_146 @ one_one_int )
= A_146 ) ).
thf(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_146: nat] :
( ( times_times_nat @ A_146 @ one_one_nat )
= A_146 ) ).
thf(fact_133_mult__1__left,axiom,
! [A_145: int] :
( ( times_times_int @ one_one_int @ A_145 )
= A_145 ) ).
thf(fact_134_mult__1__left,axiom,
! [A_145: nat] :
( ( times_times_nat @ one_one_nat @ A_145 )
= A_145 ) ).
thf(fact_135_mult__1,axiom,
! [A_144: int] :
( ( times_times_int @ one_one_int @ A_144 )
= A_144 ) ).
thf(fact_136_mult__1,axiom,
! [A_144: nat] :
( ( times_times_nat @ one_one_nat @ A_144 )
= A_144 ) ).
thf(fact_137_mult__1__right,axiom,
! [A_143: int] :
( ( times_times_int @ A_143 @ one_one_int )
= A_143 ) ).
thf(fact_138_mult__1__right,axiom,
! [A_143: nat] :
( ( times_times_nat @ A_143 @ one_one_nat )
= A_143 ) ).
thf(fact_139_mult_Ocomm__neutral,axiom,
! [A_142: int] :
( ( times_times_int @ A_142 @ one_one_int )
= A_142 ) ).
thf(fact_140_mult_Ocomm__neutral,axiom,
! [A_142: nat] :
( ( times_times_nat @ A_142 @ one_one_nat )
= A_142 ) ).
thf(fact_141_diff__eq__diff__less,axiom,
! [A_141: int,B_105: int,C_53: int,D_12: int] :
( ( ( minus_minus_int @ A_141 @ B_105 )
= ( minus_minus_int @ C_53 @ D_12 ) )
=> ( ( ord_less_int @ A_141 @ B_105 )
<=> ( ord_less_int @ C_53 @ D_12 ) ) ) ).
thf(fact_142_add__le__cancel__right,axiom,
! [A_140: int,C_52: int,B_104: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_140 @ C_52 ) @ ( plus_plus_int @ B_104 @ C_52 ) )
<=> ( ord_less_eq_int @ A_140 @ B_104 ) ) ).
thf(fact_143_add__le__cancel__right,axiom,
! [A_140: nat,C_52: nat,B_104: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_140 @ C_52 ) @ ( plus_plus_nat @ B_104 @ C_52 ) )
<=> ( ord_less_eq_nat @ A_140 @ B_104 ) ) ).
thf(fact_144_add__le__cancel__left,axiom,
! [C_51: int,A_139: int,B_103: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_51 @ A_139 ) @ ( plus_plus_int @ C_51 @ B_103 ) )
<=> ( ord_less_eq_int @ A_139 @ B_103 ) ) ).
thf(fact_145_add__le__cancel__left,axiom,
! [C_51: nat,A_139: nat,B_103: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_51 @ A_139 ) @ ( plus_plus_nat @ C_51 @ B_103 ) )
<=> ( ord_less_eq_nat @ A_139 @ B_103 ) ) ).
thf(fact_146_add__right__mono,axiom,
! [C_50: int,A_138: int,B_102: int] :
( ( ord_less_eq_int @ A_138 @ B_102 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_138 @ C_50 ) @ ( plus_plus_int @ B_102 @ C_50 ) ) ) ).
thf(fact_147_add__right__mono,axiom,
! [C_50: nat,A_138: nat,B_102: nat] :
( ( ord_less_eq_nat @ A_138 @ B_102 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_138 @ C_50 ) @ ( plus_plus_nat @ B_102 @ C_50 ) ) ) ).
thf(fact_148_add__left__mono,axiom,
! [C_49: int,A_137: int,B_101: int] :
( ( ord_less_eq_int @ A_137 @ B_101 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_49 @ A_137 ) @ ( plus_plus_int @ C_49 @ B_101 ) ) ) ).
thf(fact_149_add__left__mono,axiom,
! [C_49: nat,A_137: nat,B_101: nat] :
( ( ord_less_eq_nat @ A_137 @ B_101 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_49 @ A_137 ) @ ( plus_plus_nat @ C_49 @ B_101 ) ) ) ).
thf(fact_150_add__mono,axiom,
! [C_48: int,D_11: int,A_136: int,B_100: int] :
( ( ord_less_eq_int @ A_136 @ B_100 )
=> ( ( ord_less_eq_int @ C_48 @ D_11 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_136 @ C_48 ) @ ( plus_plus_int @ B_100 @ D_11 ) ) ) ) ).
thf(fact_151_add__mono,axiom,
! [C_48: nat,D_11: nat,A_136: nat,B_100: nat] :
( ( ord_less_eq_nat @ A_136 @ B_100 )
=> ( ( ord_less_eq_nat @ C_48 @ D_11 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_136 @ C_48 ) @ ( plus_plus_nat @ B_100 @ D_11 ) ) ) ) ).
thf(fact_152_add__le__imp__le__right,axiom,
! [A_135: int,C_47: int,B_99: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_135 @ C_47 ) @ ( plus_plus_int @ B_99 @ C_47 ) )
=> ( ord_less_eq_int @ A_135 @ B_99 ) ) ).
thf(fact_153_add__le__imp__le__right,axiom,
! [A_135: nat,C_47: nat,B_99: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_135 @ C_47 ) @ ( plus_plus_nat @ B_99 @ C_47 ) )
=> ( ord_less_eq_nat @ A_135 @ B_99 ) ) ).
thf(fact_154_add__le__imp__le__left,axiom,
! [C_46: int,A_134: int,B_98: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_46 @ A_134 ) @ ( plus_plus_int @ C_46 @ B_98 ) )
=> ( ord_less_eq_int @ A_134 @ B_98 ) ) ).
thf(fact_155_add__le__imp__le__left,axiom,
! [C_46: nat,A_134: nat,B_98: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_46 @ A_134 ) @ ( plus_plus_nat @ C_46 @ B_98 ) )
=> ( ord_less_eq_nat @ A_134 @ B_98 ) ) ).
thf(fact_156_diff__eq__diff__less__eq,axiom,
! [A_133: int,B_97: int,C_45: int,D_10: int] :
( ( ( minus_minus_int @ A_133 @ B_97 )
= ( minus_minus_int @ C_45 @ D_10 ) )
=> ( ( ord_less_eq_int @ A_133 @ B_97 )
<=> ( ord_less_eq_int @ C_45 @ D_10 ) ) ) ).
thf(fact_157_zadd__strict__right__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
thf(fact_158_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_159_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_160_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_161_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_3: int] :
( ( plus_plus_int @ M_3 @ M_3 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_3 ) ) ).
thf(fact_162_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_3: nat] :
( ( plus_plus_nat @ M_3 @ M_3 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_3 ) ) ).
thf(fact_163_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_2: nat,A_132: nat] :
( ( plus_plus_nat @ M_2 @ ( times_times_nat @ A_132 @ M_2 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_132 @ one_one_nat ) @ M_2 ) ) ).
thf(fact_164_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_2: int,A_132: int] :
( ( plus_plus_int @ M_2 @ ( times_times_int @ A_132 @ M_2 ) )
= ( times_times_int @ ( plus_plus_int @ A_132 @ one_one_int ) @ M_2 ) ) ).
thf(fact_165_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_131: nat,M_1: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_131 @ M_1 ) @ M_1 )
= ( times_times_nat @ ( plus_plus_nat @ A_131 @ one_one_nat ) @ M_1 ) ) ).
thf(fact_166_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_131: int,M_1: int] :
( ( plus_plus_int @ ( times_times_int @ A_131 @ M_1 ) @ M_1 )
= ( times_times_int @ ( plus_plus_int @ A_131 @ one_one_int ) @ M_1 ) ) ).
thf(fact_167_curryD,axiom,
! [F_3: produc393999548nt_int > $o,A_130: int,B_96: product_prod_int_int] :
( ( produc122891093_int_o @ F_3 @ A_130 @ B_96 )
=> ( F_3 @ ( produc282740534nt_int @ A_130 @ B_96 ) ) ) ).
thf(fact_168_curryD,axiom,
! [F_3: product_prod_int_int > $o,A_130: int,B_96: int] :
( ( produc176579150_int_o @ F_3 @ A_130 @ B_96 )
=> ( F_3 @ ( product_Pair_int_int @ A_130 @ B_96 ) ) ) ).
thf(fact_169_curryE,axiom,
! [F_2: produc393999548nt_int > $o,A_129: int,B_95: product_prod_int_int] :
( ( produc122891093_int_o @ F_2 @ A_129 @ B_95 )
=> ( F_2 @ ( produc282740534nt_int @ A_129 @ B_95 ) ) ) ).
thf(fact_170_curryE,axiom,
! [F_2: product_prod_int_int > $o,A_129: int,B_95: int] :
( ( produc176579150_int_o @ F_2 @ A_129 @ B_95 )
=> ( F_2 @ ( product_Pair_int_int @ A_129 @ B_95 ) ) ) ).
thf(fact_171_curry__conv,axiom,
! [F_1: produc393999548nt_int > $o,A_128: int,B_94: product_prod_int_int] :
( ( produc122891093_int_o @ F_1 @ A_128 @ B_94 )
<=> ( F_1 @ ( produc282740534nt_int @ A_128 @ B_94 ) ) ) ).
thf(fact_172_curry__conv,axiom,
! [F_1: product_prod_int_int > $o,A_128: int,B_94: int] :
( ( produc176579150_int_o @ F_1 @ A_128 @ B_94 )
<=> ( F_1 @ ( product_Pair_int_int @ A_128 @ B_94 ) ) ) ).
thf(fact_173_order__refl,axiom,
! [X_57: nat] : ( ord_less_eq_nat @ X_57 @ X_57 ) ).
thf(fact_174_order__refl,axiom,
! [X_57: int] : ( ord_less_eq_int @ X_57 @ X_57 ) ).
thf(fact_175_xt1_I8_J,axiom,
! [Z_9: nat,Y_52: nat,X_56: nat] :
( ( ord_less_eq_nat @ Y_52 @ X_56 )
=> ( ( ord_less_nat @ Z_9 @ Y_52 )
=> ( ord_less_nat @ Z_9 @ X_56 ) ) ) ).
thf(fact_176_xt1_I8_J,axiom,
! [Z_9: int,Y_52: int,X_56: int] :
( ( ord_less_eq_int @ Y_52 @ X_56 )
=> ( ( ord_less_int @ Z_9 @ Y_52 )
=> ( ord_less_int @ Z_9 @ X_56 ) ) ) ).
thf(fact_177_order__le__less__trans,axiom,
! [Z_8: nat,X_55: nat,Y_51: nat] :
( ( ord_less_eq_nat @ X_55 @ Y_51 )
=> ( ( ord_less_nat @ Y_51 @ Z_8 )
=> ( ord_less_nat @ X_55 @ Z_8 ) ) ) ).
thf(fact_178_order__le__less__trans,axiom,
! [Z_8: int,X_55: int,Y_51: int] :
( ( ord_less_eq_int @ X_55 @ Y_51 )
=> ( ( ord_less_int @ Y_51 @ Z_8 )
=> ( ord_less_int @ X_55 @ Z_8 ) ) ) ).
thf(fact_179_xt1_I7_J,axiom,
! [Z_7: nat,Y_50: nat,X_54: nat] :
( ( ord_less_nat @ Y_50 @ X_54 )
=> ( ( ord_less_eq_nat @ Z_7 @ Y_50 )
=> ( ord_less_nat @ Z_7 @ X_54 ) ) ) ).
thf(fact_180_xt1_I7_J,axiom,
! [Z_7: int,Y_50: int,X_54: int] :
( ( ord_less_int @ Y_50 @ X_54 )
=> ( ( ord_less_eq_int @ Z_7 @ Y_50 )
=> ( ord_less_int @ Z_7 @ X_54 ) ) ) ).
thf(fact_181_order__less__le__trans,axiom,
! [Z_6: nat,X_53: nat,Y_49: nat] :
( ( ord_less_nat @ X_53 @ Y_49 )
=> ( ( ord_less_eq_nat @ Y_49 @ Z_6 )
=> ( ord_less_nat @ X_53 @ Z_6 ) ) ) ).
thf(fact_182_order__less__le__trans,axiom,
! [Z_6: int,X_53: int,Y_49: int] :
( ( ord_less_int @ X_53 @ Y_49 )
=> ( ( ord_less_eq_int @ Y_49 @ Z_6 )
=> ( ord_less_int @ X_53 @ Z_6 ) ) ) ).
thf(fact_183_xt1_I11_J,axiom,
! [B_93: nat,A_127: nat] :
( ( ord_less_eq_nat @ B_93 @ A_127 )
=> ( ( A_127 != B_93 )
=> ( ord_less_nat @ B_93 @ A_127 ) ) ) ).
thf(fact_184_xt1_I11_J,axiom,
! [B_93: int,A_127: int] :
( ( ord_less_eq_int @ B_93 @ A_127 )
=> ( ( A_127 != B_93 )
=> ( ord_less_int @ B_93 @ A_127 ) ) ) ).
thf(fact_185_order__le__neq__trans,axiom,
! [A_126: nat,B_92: nat] :
( ( ord_less_eq_nat @ A_126 @ B_92 )
=> ( ( A_126 != B_92 )
=> ( ord_less_nat @ A_126 @ B_92 ) ) ) ).
thf(fact_186_order__le__neq__trans,axiom,
! [A_126: int,B_92: int] :
( ( ord_less_eq_int @ A_126 @ B_92 )
=> ( ( A_126 != B_92 )
=> ( ord_less_int @ A_126 @ B_92 ) ) ) ).
thf(fact_187_order__le__imp__less__or__eq,axiom,
! [X_52: nat,Y_48: nat] :
( ( ord_less_eq_nat @ X_52 @ Y_48 )
=> ( ( ord_less_nat @ X_52 @ Y_48 )
| ( X_52 = Y_48 ) ) ) ).
thf(fact_188_order__le__imp__less__or__eq,axiom,
! [X_52: int,Y_48: int] :
( ( ord_less_eq_int @ X_52 @ Y_48 )
=> ( ( ord_less_int @ X_52 @ Y_48 )
| ( X_52 = Y_48 ) ) ) ).
thf(fact_189_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_190_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_191_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_192_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_193_zle__trans,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I @ K ) ) ) ).
thf(fact_194_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_195_linorder__linear,axiom,
! [X_51: nat,Y_47: nat] :
( ( ord_less_eq_nat @ X_51 @ Y_47 )
| ( ord_less_eq_nat @ Y_47 @ X_51 ) ) ).
thf(fact_196_linorder__linear,axiom,
! [X_51: int,Y_47: int] :
( ( ord_less_eq_int @ X_51 @ Y_47 )
| ( ord_less_eq_int @ Y_47 @ X_51 ) ) ).
thf(fact_197_order__eq__iff,axiom,
! [X_50: nat,Y_46: nat] :
( ( X_50 = Y_46 )
<=> ( ( ord_less_eq_nat @ X_50 @ Y_46 )
& ( ord_less_eq_nat @ Y_46 @ X_50 ) ) ) ).
thf(fact_198_order__eq__iff,axiom,
! [X_50: int,Y_46: int] :
( ( X_50 = Y_46 )
<=> ( ( ord_less_eq_int @ X_50 @ Y_46 )
& ( ord_less_eq_int @ Y_46 @ X_50 ) ) ) ).
thf(fact_199_order__eq__refl,axiom,
! [X_49: nat,Y_45: nat] :
( ( X_49 = Y_45 )
=> ( ord_less_eq_nat @ X_49 @ Y_45 ) ) ).
thf(fact_200_order__eq__refl,axiom,
! [X_49: int,Y_45: int] :
( ( X_49 = Y_45 )
=> ( ord_less_eq_int @ X_49 @ Y_45 ) ) ).
thf(fact_201_order__antisym__conv,axiom,
! [Y_44: nat,X_48: nat] :
( ( ord_less_eq_nat @ Y_44 @ X_48 )
=> ( ( ord_less_eq_nat @ X_48 @ Y_44 )
<=> ( X_48 = Y_44 ) ) ) ).
thf(fact_202_order__antisym__conv,axiom,
! [Y_44: int,X_48: int] :
( ( ord_less_eq_int @ Y_44 @ X_48 )
=> ( ( ord_less_eq_int @ X_48 @ Y_44 )
<=> ( X_48 = Y_44 ) ) ) ).
thf(fact_203_ord__eq__le__trans,axiom,
! [C_44: nat,A_125: nat,B_91: nat] :
( ( A_125 = B_91 )
=> ( ( ord_less_eq_nat @ B_91 @ C_44 )
=> ( ord_less_eq_nat @ A_125 @ C_44 ) ) ) ).
thf(fact_204_ord__eq__le__trans,axiom,
! [C_44: int,A_125: int,B_91: int] :
( ( A_125 = B_91 )
=> ( ( ord_less_eq_int @ B_91 @ C_44 )
=> ( ord_less_eq_int @ A_125 @ C_44 ) ) ) ).
thf(fact_205_xt1_I3_J,axiom,
! [C_43: nat,A_124: nat,B_90: nat] :
( ( A_124 = B_90 )
=> ( ( ord_less_eq_nat @ C_43 @ B_90 )
=> ( ord_less_eq_nat @ C_43 @ A_124 ) ) ) ).
thf(fact_206_xt1_I3_J,axiom,
! [C_43: int,A_124: int,B_90: int] :
( ( A_124 = B_90 )
=> ( ( ord_less_eq_int @ C_43 @ B_90 )
=> ( ord_less_eq_int @ C_43 @ A_124 ) ) ) ).
thf(fact_207_ord__le__eq__trans,axiom,
! [C_42: nat,A_123: nat,B_89: nat] :
( ( ord_less_eq_nat @ A_123 @ B_89 )
=> ( ( B_89 = C_42 )
=> ( ord_less_eq_nat @ A_123 @ C_42 ) ) ) ).
thf(fact_208_ord__le__eq__trans,axiom,
! [C_42: int,A_123: int,B_89: int] :
( ( ord_less_eq_int @ A_123 @ B_89 )
=> ( ( B_89 = C_42 )
=> ( ord_less_eq_int @ A_123 @ C_42 ) ) ) ).
thf(fact_209_xt1_I4_J,axiom,
! [C_41: nat,B_88: nat,A_122: nat] :
( ( ord_less_eq_nat @ B_88 @ A_122 )
=> ( ( B_88 = C_41 )
=> ( ord_less_eq_nat @ C_41 @ A_122 ) ) ) ).
thf(fact_210_xt1_I4_J,axiom,
! [C_41: int,B_88: int,A_122: int] :
( ( ord_less_eq_int @ B_88 @ A_122 )
=> ( ( B_88 = C_41 )
=> ( ord_less_eq_int @ C_41 @ A_122 ) ) ) ).
thf(fact_211_order__antisym,axiom,
! [X_47: nat,Y_43: nat] :
( ( ord_less_eq_nat @ X_47 @ Y_43 )
=> ( ( ord_less_eq_nat @ Y_43 @ X_47 )
=> ( X_47 = Y_43 ) ) ) ).
thf(fact_212_order__antisym,axiom,
! [X_47: int,Y_43: int] :
( ( ord_less_eq_int @ X_47 @ Y_43 )
=> ( ( ord_less_eq_int @ Y_43 @ X_47 )
=> ( X_47 = Y_43 ) ) ) ).
thf(fact_213_order__trans,axiom,
! [Z_5: nat,X_46: nat,Y_42: nat] :
( ( ord_less_eq_nat @ X_46 @ Y_42 )
=> ( ( ord_less_eq_nat @ Y_42 @ Z_5 )
=> ( ord_less_eq_nat @ X_46 @ Z_5 ) ) ) ).
thf(fact_214_order__trans,axiom,
! [Z_5: int,X_46: int,Y_42: int] :
( ( ord_less_eq_int @ X_46 @ Y_42 )
=> ( ( ord_less_eq_int @ Y_42 @ Z_5 )
=> ( ord_less_eq_int @ X_46 @ Z_5 ) ) ) ).
thf(fact_215_xt1_I5_J,axiom,
! [Y_41: nat,X_45: nat] :
( ( ord_less_eq_nat @ Y_41 @ X_45 )
=> ( ( ord_less_eq_nat @ X_45 @ Y_41 )
=> ( X_45 = Y_41 ) ) ) ).
thf(fact_216_xt1_I5_J,axiom,
! [Y_41: int,X_45: int] :
( ( ord_less_eq_int @ Y_41 @ X_45 )
=> ( ( ord_less_eq_int @ X_45 @ Y_41 )
=> ( X_45 = Y_41 ) ) ) ).
thf(fact_217_xt1_I6_J,axiom,
! [Z_4: nat,Y_40: nat,X_44: nat] :
( ( ord_less_eq_nat @ Y_40 @ X_44 )
=> ( ( ord_less_eq_nat @ Z_4 @ Y_40 )
=> ( ord_less_eq_nat @ Z_4 @ X_44 ) ) ) ).
thf(fact_218_xt1_I6_J,axiom,
! [Z_4: int,Y_40: int,X_44: int] :
( ( ord_less_eq_int @ Y_40 @ X_44 )
=> ( ( ord_less_eq_int @ Z_4 @ Y_40 )
=> ( ord_less_eq_int @ Z_4 @ X_44 ) ) ) ).
thf(fact_219_linorder__le__cases,axiom,
! [X_43: nat,Y_39: nat] :
( ~ ( ord_less_eq_nat @ X_43 @ Y_39 )
=> ( ord_less_eq_nat @ Y_39 @ X_43 ) ) ).
thf(fact_220_linorder__le__cases,axiom,
! [X_43: int,Y_39: int] :
( ~ ( ord_less_eq_int @ X_43 @ Y_39 )
=> ( ord_less_eq_int @ Y_39 @ X_43 ) ) ).
thf(fact_221_order__less__irrefl,axiom,
! [X_42: nat] :
~ ( ord_less_nat @ X_42 @ X_42 ) ).
thf(fact_222_order__less__irrefl,axiom,
! [X_42: int] :
~ ( ord_less_int @ X_42 @ X_42 ) ).
thf(fact_223_linorder__neq__iff,axiom,
! [X_41: nat,Y_38: nat] :
( ( X_41 != Y_38 )
<=> ( ( ord_less_nat @ X_41 @ Y_38 )
| ( ord_less_nat @ Y_38 @ X_41 ) ) ) ).
thf(fact_224_linorder__neq__iff,axiom,
! [X_41: int,Y_38: int] :
( ( X_41 != Y_38 )
<=> ( ( ord_less_int @ X_41 @ Y_38 )
| ( ord_less_int @ Y_38 @ X_41 ) ) ) ).
thf(fact_225_not__less__iff__gr__or__eq,axiom,
! [X_40: nat,Y_37: nat] :
( ~ ( ord_less_nat @ X_40 @ Y_37 )
<=> ( ( ord_less_nat @ Y_37 @ X_40 )
| ( X_40 = Y_37 ) ) ) ).
thf(fact_226_not__less__iff__gr__or__eq,axiom,
! [X_40: int,Y_37: int] :
( ~ ( ord_less_int @ X_40 @ Y_37 )
<=> ( ( ord_less_int @ Y_37 @ X_40 )
| ( X_40 = Y_37 ) ) ) ).
thf(fact_227_linorder__less__linear,axiom,
! [X_39: nat,Y_36: nat] :
( ( ord_less_nat @ X_39 @ Y_36 )
| ( X_39 = Y_36 )
| ( ord_less_nat @ Y_36 @ X_39 ) ) ).
thf(fact_228_linorder__less__linear,axiom,
! [X_39: int,Y_36: int] :
( ( ord_less_int @ X_39 @ Y_36 )
| ( X_39 = Y_36 )
| ( ord_less_int @ Y_36 @ X_39 ) ) ).
thf(fact_229_linorder__antisym__conv3,axiom,
! [Y_35: nat,X_38: nat] :
( ~ ( ord_less_nat @ Y_35 @ X_38 )
=> ( ~ ( ord_less_nat @ X_38 @ Y_35 )
<=> ( X_38 = Y_35 ) ) ) ).
thf(fact_230_linorder__antisym__conv3,axiom,
! [Y_35: int,X_38: int] :
( ~ ( ord_less_int @ Y_35 @ X_38 )
=> ( ~ ( ord_less_int @ X_38 @ Y_35 )
<=> ( X_38 = Y_35 ) ) ) ).
thf(fact_231_linorder__neqE,axiom,
! [X_37: nat,Y_34: nat] :
( ( X_37 != Y_34 )
=> ( ~ ( ord_less_nat @ X_37 @ Y_34 )
=> ( ord_less_nat @ Y_34 @ X_37 ) ) ) ).
thf(fact_232_linorder__neqE,axiom,
! [X_37: int,Y_34: int] :
( ( X_37 != Y_34 )
=> ( ~ ( ord_less_int @ X_37 @ Y_34 )
=> ( ord_less_int @ Y_34 @ X_37 ) ) ) ).
thf(fact_233_less__imp__neq,axiom,
! [X_36: nat,Y_33: nat] :
( ( ord_less_nat @ X_36 @ Y_33 )
=> ( X_36 != Y_33 ) ) ).
thf(fact_234_less__imp__neq,axiom,
! [X_36: int,Y_33: int] :
( ( ord_less_int @ X_36 @ Y_33 )
=> ( X_36 != Y_33 ) ) ).
thf(fact_235_order__less__not__sym,axiom,
! [X_35: nat,Y_32: nat] :
( ( ord_less_nat @ X_35 @ Y_32 )
=> ~ ( ord_less_nat @ Y_32 @ X_35 ) ) ).
thf(fact_236_order__less__not__sym,axiom,
! [X_35: int,Y_32: int] :
( ( ord_less_int @ X_35 @ Y_32 )
=> ~ ( ord_less_int @ Y_32 @ X_35 ) ) ).
thf(fact_237_order__less__imp__not__less,axiom,
! [X_34: nat,Y_31: nat] :
( ( ord_less_nat @ X_34 @ Y_31 )
=> ~ ( ord_less_nat @ Y_31 @ X_34 ) ) ).
thf(fact_238_order__less__imp__not__less,axiom,
! [X_34: int,Y_31: int] :
( ( ord_less_int @ X_34 @ Y_31 )
=> ~ ( ord_less_int @ Y_31 @ X_34 ) ) ).
thf(fact_239_order__less__imp__not__eq,axiom,
! [X_33: nat,Y_30: nat] :
( ( ord_less_nat @ X_33 @ Y_30 )
=> ( X_33 != Y_30 ) ) ).
thf(fact_240_order__less__imp__not__eq,axiom,
! [X_33: int,Y_30: int] :
( ( ord_less_int @ X_33 @ Y_30 )
=> ( X_33 != Y_30 ) ) ).
thf(fact_241_order__less__imp__not__eq2,axiom,
! [X_32: nat,Y_29: nat] :
( ( ord_less_nat @ X_32 @ Y_29 )
=> ( Y_29 != X_32 ) ) ).
thf(fact_242_order__less__imp__not__eq2,axiom,
! [X_32: int,Y_29: int] :
( ( ord_less_int @ X_32 @ Y_29 )
=> ( Y_29 != X_32 ) ) ).
thf(fact_243_order__less__imp__triv,axiom,
! [P_2: $o,X_31: nat,Y_28: nat] :
( ( ord_less_nat @ X_31 @ Y_28 )
=> ( ( ord_less_nat @ Y_28 @ X_31 )
=> P_2 ) ) ).
thf(fact_244_order__less__imp__triv,axiom,
! [P_2: $o,X_31: int,Y_28: int] :
( ( ord_less_int @ X_31 @ Y_28 )
=> ( ( ord_less_int @ Y_28 @ X_31 )
=> P_2 ) ) ).
thf(fact_245_order__less__asym_H,axiom,
! [A_121: nat,B_87: nat] :
( ( ord_less_nat @ A_121 @ B_87 )
=> ~ ( ord_less_nat @ B_87 @ A_121 ) ) ).
thf(fact_246_order__less__asym_H,axiom,
! [A_121: int,B_87: int] :
( ( ord_less_int @ A_121 @ B_87 )
=> ~ ( ord_less_int @ B_87 @ A_121 ) ) ).
thf(fact_247_xt1_I9_J,axiom,
! [B_86: nat,A_120: nat] :
( ( ord_less_nat @ B_86 @ A_120 )
=> ~ ( ord_less_nat @ A_120 @ B_86 ) ) ).
thf(fact_248_xt1_I9_J,axiom,
! [B_86: int,A_120: int] :
( ( ord_less_int @ B_86 @ A_120 )
=> ~ ( ord_less_int @ A_120 @ B_86 ) ) ).
thf(fact_249_ord__eq__less__trans,axiom,
! [C_40: nat,A_119: nat,B_85: nat] :
( ( A_119 = B_85 )
=> ( ( ord_less_nat @ B_85 @ C_40 )
=> ( ord_less_nat @ A_119 @ C_40 ) ) ) ).
thf(fact_250_ord__eq__less__trans,axiom,
! [C_40: int,A_119: int,B_85: int] :
( ( A_119 = B_85 )
=> ( ( ord_less_int @ B_85 @ C_40 )
=> ( ord_less_int @ A_119 @ C_40 ) ) ) ).
thf(fact_251_xt1_I1_J,axiom,
! [C_39: nat,A_118: nat,B_84: nat] :
( ( A_118 = B_84 )
=> ( ( ord_less_nat @ C_39 @ B_84 )
=> ( ord_less_nat @ C_39 @ A_118 ) ) ) ).
thf(fact_252_xt1_I1_J,axiom,
! [C_39: int,A_118: int,B_84: int] :
( ( A_118 = B_84 )
=> ( ( ord_less_int @ C_39 @ B_84 )
=> ( ord_less_int @ C_39 @ A_118 ) ) ) ).
thf(fact_253_ord__less__eq__trans,axiom,
! [C_38: nat,A_117: nat,B_83: nat] :
( ( ord_less_nat @ A_117 @ B_83 )
=> ( ( B_83 = C_38 )
=> ( ord_less_nat @ A_117 @ C_38 ) ) ) ).
thf(fact_254_ord__less__eq__trans,axiom,
! [C_38: int,A_117: int,B_83: int] :
( ( ord_less_int @ A_117 @ B_83 )
=> ( ( B_83 = C_38 )
=> ( ord_less_int @ A_117 @ C_38 ) ) ) ).
thf(fact_255_xt1_I2_J,axiom,
! [C_37: nat,B_82: nat,A_116: nat] :
( ( ord_less_nat @ B_82 @ A_116 )
=> ( ( B_82 = C_37 )
=> ( ord_less_nat @ C_37 @ A_116 ) ) ) ).
thf(fact_256_xt1_I2_J,axiom,
! [C_37: int,B_82: int,A_116: int] :
( ( ord_less_int @ B_82 @ A_116 )
=> ( ( B_82 = C_37 )
=> ( ord_less_int @ C_37 @ A_116 ) ) ) ).
thf(fact_257_order__less__trans,axiom,
! [Z_3: nat,X_30: nat,Y_27: nat] :
( ( ord_less_nat @ X_30 @ Y_27 )
=> ( ( ord_less_nat @ Y_27 @ Z_3 )
=> ( ord_less_nat @ X_30 @ Z_3 ) ) ) ).
thf(fact_258_order__less__trans,axiom,
! [Z_3: int,X_30: int,Y_27: int] :
( ( ord_less_int @ X_30 @ Y_27 )
=> ( ( ord_less_int @ Y_27 @ Z_3 )
=> ( ord_less_int @ X_30 @ Z_3 ) ) ) ).
thf(fact_259_xt1_I10_J,axiom,
! [Z_2: nat,Y_26: nat,X_29: nat] :
( ( ord_less_nat @ Y_26 @ X_29 )
=> ( ( ord_less_nat @ Z_2 @ Y_26 )
=> ( ord_less_nat @ Z_2 @ X_29 ) ) ) ).
thf(fact_260_xt1_I10_J,axiom,
! [Z_2: int,Y_26: int,X_29: int] :
( ( ord_less_int @ Y_26 @ X_29 )
=> ( ( ord_less_int @ Z_2 @ Y_26 )
=> ( ord_less_int @ Z_2 @ X_29 ) ) ) ).
thf(fact_261_order__less__asym,axiom,
! [X_28: nat,Y_25: nat] :
( ( ord_less_nat @ X_28 @ Y_25 )
=> ~ ( ord_less_nat @ Y_25 @ X_28 ) ) ).
thf(fact_262_order__less__asym,axiom,
! [X_28: int,Y_25: int] :
( ( ord_less_int @ X_28 @ Y_25 )
=> ~ ( ord_less_int @ Y_25 @ X_28 ) ) ).
thf(fact_263_linorder__cases,axiom,
! [X_27: nat,Y_24: nat] :
( ~ ( ord_less_nat @ X_27 @ Y_24 )
=> ( ( X_27 != Y_24 )
=> ( ord_less_nat @ Y_24 @ X_27 ) ) ) ).
thf(fact_264_linorder__cases,axiom,
! [X_27: int,Y_24: int] :
( ~ ( ord_less_int @ X_27 @ Y_24 )
=> ( ( X_27 != Y_24 )
=> ( ord_less_int @ Y_24 @ X_27 ) ) ) ).
thf(fact_265_linorder__not__less,axiom,
! [X_26: nat,Y_23: nat] :
( ~ ( ord_less_nat @ X_26 @ Y_23 )
<=> ( ord_less_eq_nat @ Y_23 @ X_26 ) ) ).
thf(fact_266_linorder__not__less,axiom,
! [X_26: int,Y_23: int] :
( ~ ( ord_less_int @ X_26 @ Y_23 )
<=> ( ord_less_eq_int @ Y_23 @ X_26 ) ) ).
thf(fact_267_linorder__not__le,axiom,
! [X_25: nat,Y_22: nat] :
( ~ ( ord_less_eq_nat @ X_25 @ Y_22 )
<=> ( ord_less_nat @ Y_22 @ X_25 ) ) ).
thf(fact_268_linorder__not__le,axiom,
! [X_25: int,Y_22: int] :
( ~ ( ord_less_eq_int @ X_25 @ Y_22 )
<=> ( ord_less_int @ Y_22 @ X_25 ) ) ).
thf(fact_269_linorder__le__less__linear,axiom,
! [X_24: nat,Y_21: nat] :
( ( ord_less_eq_nat @ X_24 @ Y_21 )
| ( ord_less_nat @ Y_21 @ X_24 ) ) ).
thf(fact_270_linorder__le__less__linear,axiom,
! [X_24: int,Y_21: int] :
( ( ord_less_eq_int @ X_24 @ Y_21 )
| ( ord_less_int @ Y_21 @ X_24 ) ) ).
thf(fact_271_order__less__le,axiom,
! [X_23: nat,Y_20: nat] :
( ( ord_less_nat @ X_23 @ Y_20 )
<=> ( ( ord_less_eq_nat @ X_23 @ Y_20 )
& ( X_23 != Y_20 ) ) ) ).
thf(fact_272_order__less__le,axiom,
! [X_23: int,Y_20: int] :
( ( ord_less_int @ X_23 @ Y_20 )
<=> ( ( ord_less_eq_int @ X_23 @ Y_20 )
& ( X_23 != Y_20 ) ) ) ).
thf(fact_273_less__le__not__le,axiom,
! [X_22: nat,Y_19: nat] :
( ( ord_less_nat @ X_22 @ Y_19 )
<=> ( ( ord_less_eq_nat @ X_22 @ Y_19 )
& ~ ( ord_less_eq_nat @ Y_19 @ X_22 ) ) ) ).
thf(fact_274_less__le__not__le,axiom,
! [X_22: int,Y_19: int] :
( ( ord_less_int @ X_22 @ Y_19 )
<=> ( ( ord_less_eq_int @ X_22 @ Y_19 )
& ~ ( ord_less_eq_int @ Y_19 @ X_22 ) ) ) ).
thf(fact_275_order__le__less,axiom,
! [X_21: nat,Y_18: nat] :
( ( ord_less_eq_nat @ X_21 @ Y_18 )
<=> ( ( ord_less_nat @ X_21 @ Y_18 )
| ( X_21 = Y_18 ) ) ) ).
thf(fact_276_order__le__less,axiom,
! [X_21: int,Y_18: int] :
( ( ord_less_eq_int @ X_21 @ Y_18 )
<=> ( ( ord_less_int @ X_21 @ Y_18 )
| ( X_21 = Y_18 ) ) ) ).
thf(fact_277_leI,axiom,
! [X_20: nat,Y_17: nat] :
( ~ ( ord_less_nat @ X_20 @ Y_17 )
=> ( ord_less_eq_nat @ Y_17 @ X_20 ) ) ).
thf(fact_278_leI,axiom,
! [X_20: int,Y_17: int] :
( ~ ( ord_less_int @ X_20 @ Y_17 )
=> ( ord_less_eq_int @ Y_17 @ X_20 ) ) ).
thf(fact_279_not__leE,axiom,
! [Y_16: nat,X_19: nat] :
( ~ ( ord_less_eq_nat @ Y_16 @ X_19 )
=> ( ord_less_nat @ X_19 @ Y_16 ) ) ).
thf(fact_280_not__leE,axiom,
! [Y_16: int,X_19: int] :
( ~ ( ord_less_eq_int @ Y_16 @ X_19 )
=> ( ord_less_int @ X_19 @ Y_16 ) ) ).
thf(fact_281_linorder__antisym__conv1,axiom,
! [X_18: nat,Y_15: nat] :
( ~ ( ord_less_nat @ X_18 @ Y_15 )
=> ( ( ord_less_eq_nat @ X_18 @ Y_15 )
<=> ( X_18 = Y_15 ) ) ) ).
thf(fact_282_linorder__antisym__conv1,axiom,
! [X_18: int,Y_15: int] :
( ~ ( ord_less_int @ X_18 @ Y_15 )
=> ( ( ord_less_eq_int @ X_18 @ Y_15 )
<=> ( X_18 = Y_15 ) ) ) ).
thf(fact_283_order__neq__le__trans,axiom,
! [A_115: nat,B_81: nat] :
( ( A_115 != B_81 )
=> ( ( ord_less_eq_nat @ A_115 @ B_81 )
=> ( ord_less_nat @ A_115 @ B_81 ) ) ) ).
thf(fact_284_order__neq__le__trans,axiom,
! [A_115: int,B_81: int] :
( ( A_115 != B_81 )
=> ( ( ord_less_eq_int @ A_115 @ B_81 )
=> ( ord_less_int @ A_115 @ B_81 ) ) ) ).
thf(fact_285_xt1_I12_J,axiom,
! [A_114: nat,B_80: nat] :
( ( A_114 != B_80 )
=> ( ( ord_less_eq_nat @ B_80 @ A_114 )
=> ( ord_less_nat @ B_80 @ A_114 ) ) ) ).
thf(fact_286_xt1_I12_J,axiom,
! [A_114: int,B_80: int] :
( ( A_114 != B_80 )
=> ( ( ord_less_eq_int @ B_80 @ A_114 )
=> ( ord_less_int @ B_80 @ A_114 ) ) ) ).
thf(fact_287_leD,axiom,
! [Y_14: nat,X_17: nat] :
( ( ord_less_eq_nat @ Y_14 @ X_17 )
=> ~ ( ord_less_nat @ X_17 @ Y_14 ) ) ).
thf(fact_288_leD,axiom,
! [Y_14: int,X_17: int] :
( ( ord_less_eq_int @ Y_14 @ X_17 )
=> ~ ( ord_less_int @ X_17 @ Y_14 ) ) ).
thf(fact_289_order__less__imp__le,axiom,
! [X_16: nat,Y_13: nat] :
( ( ord_less_nat @ X_16 @ Y_13 )
=> ( ord_less_eq_nat @ X_16 @ Y_13 ) ) ).
thf(fact_290_order__less__imp__le,axiom,
! [X_16: int,Y_13: int] :
( ( ord_less_int @ X_16 @ Y_13 )
=> ( ord_less_eq_int @ X_16 @ Y_13 ) ) ).
thf(fact_291_linorder__antisym__conv2,axiom,
! [X_15: nat,Y_12: nat] :
( ( ord_less_eq_nat @ X_15 @ Y_12 )
=> ( ~ ( ord_less_nat @ X_15 @ Y_12 )
<=> ( X_15 = Y_12 ) ) ) ).
thf(fact_292_linorder__antisym__conv2,axiom,
! [X_15: int,Y_12: int] :
( ( ord_less_eq_int @ X_15 @ Y_12 )
=> ( ~ ( ord_less_int @ X_15 @ Y_12 )
<=> ( X_15 = Y_12 ) ) ) ).
thf(fact_293_norR__mem__unique__aux,axiom,
! [A_23: int,B: int] :
( ( ord_less_eq_int @ A_23 @ ( minus_minus_int @ B @ one_one_int ) )
=> ( ord_less_int @ A_23 @ B ) ) ).
thf(fact_294_int__le__induct,axiom,
! [P: int > $o,I: int,K: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_295_int__less__induct,axiom,
! [P: int > $o,I: int,K: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_296_int__ge__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_297_int__gr__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_298_int__induct,axiom,
! [I: int,P: int > $o,K: int] :
( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_299_Euler_Oaux2,axiom,
! [B: int,A_23: int,C_36: int] :
( ( ord_less_int @ A_23 @ C_36 )
=> ( ( ord_less_int @ B @ C_36 )
=> ( ( ord_less_eq_int @ A_23 @ B )
| ( ord_less_eq_int @ B @ A_23 ) ) ) ) ).
thf(fact_300_d22set__induct__old,axiom,
! [X: int,P: int > $o] :
( ! [A_113: int] :
( ( ( ord_less_int @ one_one_int @ A_113 )
=> ( P @ ( minus_minus_int @ A_113 @ one_one_int ) ) )
=> ( P @ A_113 ) )
=> ( P @ X ) ) ).
thf(fact_301_convex__bound__lt,axiom,
! [V_1: int,U_1: int,Y_11: int,X_14: int,A_112: int] :
( ( ord_less_int @ X_14 @ A_112 )
=> ( ( ord_less_int @ Y_11 @ A_112 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_1 )
=> ( ( ( plus_plus_int @ U_1 @ V_1 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U_1 @ X_14 ) @ ( times_times_int @ V_1 @ Y_11 ) ) @ A_112 ) ) ) ) ) ) ).
thf(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_111: nat] :
( ( times_times_nat @ zero_zero_nat @ A_111 )
= zero_zero_nat ) ).
thf(fact_303_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_111: int] :
( ( times_times_int @ zero_zero_int @ A_111 )
= zero_zero_int ) ).
thf(fact_304_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_110: nat] :
( ( times_times_nat @ A_110 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_305_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_110: int] :
( ( times_times_int @ A_110 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_306_mult__zero__left,axiom,
! [A_109: nat] :
( ( times_times_nat @ zero_zero_nat @ A_109 )
= zero_zero_nat ) ).
thf(fact_307_mult__zero__left,axiom,
! [A_109: int] :
( ( times_times_int @ zero_zero_int @ A_109 )
= zero_zero_int ) ).
thf(fact_308_mult__zero__right,axiom,
! [A_108: nat] :
( ( times_times_nat @ A_108 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_309_mult__zero__right,axiom,
! [A_108: int] :
( ( times_times_int @ A_108 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_310_mult__eq__0__iff,axiom,
! [A_107: int,B_79: int] :
( ( ( times_times_int @ A_107 @ B_79 )
= zero_zero_int )
<=> ( ( A_107 = zero_zero_int )
| ( B_79 = zero_zero_int ) ) ) ).
thf(fact_311_no__zero__divisors,axiom,
! [B_78: nat,A_106: nat] :
( ( A_106 != zero_zero_nat )
=> ( ( B_78 != zero_zero_nat )
=> ( ( times_times_nat @ A_106 @ B_78 )
!= zero_zero_nat ) ) ) ).
thf(fact_312_no__zero__divisors,axiom,
! [B_78: int,A_106: int] :
( ( A_106 != zero_zero_int )
=> ( ( B_78 != zero_zero_int )
=> ( ( times_times_int @ A_106 @ B_78 )
!= zero_zero_int ) ) ) ).
thf(fact_313_divisors__zero,axiom,
! [A_105: nat,B_77: nat] :
( ( ( times_times_nat @ A_105 @ B_77 )
= zero_zero_nat )
=> ( ( A_105 = zero_zero_nat )
| ( B_77 = zero_zero_nat ) ) ) ).
thf(fact_314_divisors__zero,axiom,
! [A_105: int,B_77: int] :
( ( ( times_times_int @ A_105 @ B_77 )
= zero_zero_int )
=> ( ( A_105 = zero_zero_int )
| ( B_77 = zero_zero_int ) ) ) ).
thf(fact_315_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_104: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_104 )
= A_104 ) ).
thf(fact_316_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_104: int] :
( ( plus_plus_int @ zero_zero_int @ A_104 )
= A_104 ) ).
thf(fact_317_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_103: nat] :
( ( plus_plus_nat @ A_103 @ zero_zero_nat )
= A_103 ) ).
thf(fact_318_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_103: int] :
( ( plus_plus_int @ A_103 @ zero_zero_int )
= A_103 ) ).
thf(fact_319_add__0__iff,axiom,
! [B_76: nat,A_102: nat] :
( ( B_76
= ( plus_plus_nat @ B_76 @ A_102 ) )
<=> ( A_102 = zero_zero_nat ) ) ).
thf(fact_320_add__0__iff,axiom,
! [B_76: int,A_102: int] :
( ( B_76
= ( plus_plus_int @ B_76 @ A_102 ) )
<=> ( A_102 = zero_zero_int ) ) ).
thf(fact_321_add__0__left,axiom,
! [A_101: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_101 )
= A_101 ) ).
thf(fact_322_add__0__left,axiom,
! [A_101: int] :
( ( plus_plus_int @ zero_zero_int @ A_101 )
= A_101 ) ).
thf(fact_323_add__0,axiom,
! [A_100: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_100 )
= A_100 ) ).
thf(fact_324_add__0,axiom,
! [A_100: int] :
( ( plus_plus_int @ zero_zero_int @ A_100 )
= A_100 ) ).
thf(fact_325_double__zero__sym,axiom,
! [A_99: int] :
( ( zero_zero_int
= ( plus_plus_int @ A_99 @ A_99 ) )
<=> ( A_99 = zero_zero_int ) ) ).
thf(fact_326_add__0__right,axiom,
! [A_98: nat] :
( ( plus_plus_nat @ A_98 @ zero_zero_nat )
= A_98 ) ).
thf(fact_327_add__0__right,axiom,
! [A_98: int] :
( ( plus_plus_int @ A_98 @ zero_zero_int )
= A_98 ) ).
thf(fact_328_add_Ocomm__neutral,axiom,
! [A_97: nat] :
( ( plus_plus_nat @ A_97 @ zero_zero_nat )
= A_97 ) ).
thf(fact_329_add_Ocomm__neutral,axiom,
! [A_97: int] :
( ( plus_plus_int @ A_97 @ zero_zero_int )
= A_97 ) ).
thf(fact_330_double__eq__0__iff,axiom,
! [A_96: int] :
( ( ( plus_plus_int @ A_96 @ A_96 )
= zero_zero_int )
<=> ( A_96 = zero_zero_int ) ) ).
thf(fact_331_zero__reorient,axiom,
! [X_13: nat] :
( ( zero_zero_nat = X_13 )
<=> ( X_13 = zero_zero_nat ) ) ).
thf(fact_332_zero__reorient,axiom,
! [X_13: int] :
( ( zero_zero_int = X_13 )
<=> ( X_13 = zero_zero_int ) ) ).
thf(fact_333_diff__0__right,axiom,
! [A_95: int] :
( ( minus_minus_int @ A_95 @ zero_zero_int )
= A_95 ) ).
thf(fact_334_diff__self,axiom,
! [A_94: int] :
( ( minus_minus_int @ A_94 @ A_94 )
= zero_zero_int ) ).
thf(fact_335_eq__iff__diff__eq__0,axiom,
! [A_93: int,B_75: int] :
( ( A_93 = B_75 )
<=> ( ( minus_minus_int @ A_93 @ B_75 )
= zero_zero_int ) ) ).
thf(fact_336_right__minus__eq,axiom,
! [A_92: int,B_74: int] :
( ( ( minus_minus_int @ A_92 @ B_74 )
= zero_zero_int )
<=> ( A_92 = B_74 ) ) ).
thf(fact_337_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_338_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_339_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_340_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_341_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_342_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_343_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_344_split__mult__neg__le,axiom,
! [B_73: nat,A_91: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_91 )
& ( ord_less_eq_nat @ B_73 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_91 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_73 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_91 @ B_73 ) @ zero_zero_nat ) ) ).
thf(fact_345_split__mult__neg__le,axiom,
! [B_73: int,A_91: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_91 )
& ( ord_less_eq_int @ B_73 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_91 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_73 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_91 @ B_73 ) @ zero_zero_int ) ) ).
thf(fact_346_split__mult__pos__le,axiom,
! [B_72: int,A_90: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_90 )
& ( ord_less_eq_int @ zero_zero_int @ B_72 ) )
| ( ( ord_less_eq_int @ A_90 @ zero_zero_int )
& ( ord_less_eq_int @ B_72 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_90 @ B_72 ) ) ) ).
thf(fact_347_mult__mono,axiom,
! [C_35: nat,D_9: nat,A_89: nat,B_71: nat] :
( ( ord_less_eq_nat @ A_89 @ B_71 )
=> ( ( ord_less_eq_nat @ C_35 @ D_9 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_71 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_35 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_89 @ C_35 ) @ ( times_times_nat @ B_71 @ D_9 ) ) ) ) ) ) ).
thf(fact_348_mult__mono,axiom,
! [C_35: int,D_9: int,A_89: int,B_71: int] :
( ( ord_less_eq_int @ A_89 @ B_71 )
=> ( ( ord_less_eq_int @ C_35 @ D_9 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_71 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_35 )
=> ( ord_less_eq_int @ ( times_times_int @ A_89 @ C_35 ) @ ( times_times_int @ B_71 @ D_9 ) ) ) ) ) ) ).
thf(fact_349_mult__mono_H,axiom,
! [C_34: nat,D_8: nat,A_88: nat,B_70: nat] :
( ( ord_less_eq_nat @ A_88 @ B_70 )
=> ( ( ord_less_eq_nat @ C_34 @ D_8 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_88 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_34 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_88 @ C_34 ) @ ( times_times_nat @ B_70 @ D_8 ) ) ) ) ) ) ).
thf(fact_350_mult__mono_H,axiom,
! [C_34: int,D_8: int,A_88: int,B_70: int] :
( ( ord_less_eq_int @ A_88 @ B_70 )
=> ( ( ord_less_eq_int @ C_34 @ D_8 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_88 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_34 )
=> ( ord_less_eq_int @ ( times_times_int @ A_88 @ C_34 ) @ ( times_times_int @ B_70 @ D_8 ) ) ) ) ) ) ).
thf(fact_351_mult__left__mono__neg,axiom,
! [C_33: int,B_69: int,A_87: int] :
( ( ord_less_eq_int @ B_69 @ A_87 )
=> ( ( ord_less_eq_int @ C_33 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_33 @ A_87 ) @ ( times_times_int @ C_33 @ B_69 ) ) ) ) ).
thf(fact_352_mult__right__mono__neg,axiom,
! [C_32: int,B_68: int,A_86: int] :
( ( ord_less_eq_int @ B_68 @ A_86 )
=> ( ( ord_less_eq_int @ C_32 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_86 @ C_32 ) @ ( times_times_int @ B_68 @ C_32 ) ) ) ) ).
thf(fact_353_comm__mult__left__mono,axiom,
! [C_31: nat,A_85: nat,B_67: nat] :
( ( ord_less_eq_nat @ A_85 @ B_67 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_31 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_31 @ A_85 ) @ ( times_times_nat @ C_31 @ B_67 ) ) ) ) ).
thf(fact_354_comm__mult__left__mono,axiom,
! [C_31: int,A_85: int,B_67: int] :
( ( ord_less_eq_int @ A_85 @ B_67 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_31 )
=> ( ord_less_eq_int @ ( times_times_int @ C_31 @ A_85 ) @ ( times_times_int @ C_31 @ B_67 ) ) ) ) ).
thf(fact_355_mult__left__mono,axiom,
! [C_30: nat,A_84: nat,B_66: nat] :
( ( ord_less_eq_nat @ A_84 @ B_66 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_30 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_30 @ A_84 ) @ ( times_times_nat @ C_30 @ B_66 ) ) ) ) ).
thf(fact_356_mult__left__mono,axiom,
! [C_30: int,A_84: int,B_66: int] :
( ( ord_less_eq_int @ A_84 @ B_66 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_30 )
=> ( ord_less_eq_int @ ( times_times_int @ C_30 @ A_84 ) @ ( times_times_int @ C_30 @ B_66 ) ) ) ) ).
thf(fact_357_mult__right__mono,axiom,
! [C_29: nat,A_83: nat,B_65: nat] :
( ( ord_less_eq_nat @ A_83 @ B_65 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_29 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_83 @ C_29 ) @ ( times_times_nat @ B_65 @ C_29 ) ) ) ) ).
thf(fact_358_mult__right__mono,axiom,
! [C_29: int,A_83: int,B_65: int] :
( ( ord_less_eq_int @ A_83 @ B_65 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_29 )
=> ( ord_less_eq_int @ ( times_times_int @ A_83 @ C_29 ) @ ( times_times_int @ B_65 @ C_29 ) ) ) ) ).
thf(fact_359_mult__nonpos__nonpos,axiom,
! [B_64: int,A_82: int] :
( ( ord_less_eq_int @ A_82 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_64 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_82 @ B_64 ) ) ) ) ).
thf(fact_360_mult__nonpos__nonneg,axiom,
! [B_63: nat,A_81: nat] :
( ( ord_less_eq_nat @ A_81 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_63 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_81 @ B_63 ) @ zero_zero_nat ) ) ) ).
thf(fact_361_mult__nonpos__nonneg,axiom,
! [B_63: int,A_81: int] :
( ( ord_less_eq_int @ A_81 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_63 )
=> ( ord_less_eq_int @ ( times_times_int @ A_81 @ B_63 ) @ zero_zero_int ) ) ) ).
thf(fact_362_mult__nonneg__nonpos2,axiom,
! [B_62: nat,A_80: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_80 )
=> ( ( ord_less_eq_nat @ B_62 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_62 @ A_80 ) @ zero_zero_nat ) ) ) ).
thf(fact_363_mult__nonneg__nonpos2,axiom,
! [B_62: int,A_80: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_80 )
=> ( ( ord_less_eq_int @ B_62 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_62 @ A_80 ) @ zero_zero_int ) ) ) ).
thf(fact_364_mult__nonneg__nonpos,axiom,
! [B_61: nat,A_79: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_79 )
=> ( ( ord_less_eq_nat @ B_61 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_79 @ B_61 ) @ zero_zero_nat ) ) ) ).
thf(fact_365_mult__nonneg__nonpos,axiom,
! [B_61: int,A_79: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_79 )
=> ( ( ord_less_eq_int @ B_61 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_79 @ B_61 ) @ zero_zero_int ) ) ) ).
thf(fact_366_mult__nonneg__nonneg,axiom,
! [B_60: nat,A_78: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_78 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_60 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_78 @ B_60 ) ) ) ) ).
thf(fact_367_mult__nonneg__nonneg,axiom,
! [B_60: int,A_78: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_78 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_60 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_78 @ B_60 ) ) ) ) ).
thf(fact_368_mult__le__0__iff,axiom,
! [A_77: int,B_59: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_77 @ B_59 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_77 )
& ( ord_less_eq_int @ B_59 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_77 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_59 ) ) ) ) ).
thf(fact_369_zero__le__mult__iff,axiom,
! [A_76: int,B_58: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_76 @ B_58 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_76 )
& ( ord_less_eq_int @ zero_zero_int @ B_58 ) )
| ( ( ord_less_eq_int @ A_76 @ zero_zero_int )
& ( ord_less_eq_int @ B_58 @ zero_zero_int ) ) ) ) ).
thf(fact_370_zero__le__square,axiom,
! [A_75: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_75 @ A_75 ) ) ).
thf(fact_371_mult__strict__left__mono__neg,axiom,
! [C_28: int,B_57: int,A_74: int] :
( ( ord_less_int @ B_57 @ A_74 )
=> ( ( ord_less_int @ C_28 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C_28 @ A_74 ) @ ( times_times_int @ C_28 @ B_57 ) ) ) ) ).
thf(fact_372_mult__strict__right__mono__neg,axiom,
! [C_27: int,B_56: int,A_73: int] :
( ( ord_less_int @ B_56 @ A_73 )
=> ( ( ord_less_int @ C_27 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_73 @ C_27 ) @ ( times_times_int @ B_56 @ C_27 ) ) ) ) ).
thf(fact_373_comm__mult__strict__left__mono,axiom,
! [C_26: nat,A_72: nat,B_55: nat] :
( ( ord_less_nat @ A_72 @ B_55 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_26 )
=> ( ord_less_nat @ ( times_times_nat @ C_26 @ A_72 ) @ ( times_times_nat @ C_26 @ B_55 ) ) ) ) ).
thf(fact_374_comm__mult__strict__left__mono,axiom,
! [C_26: int,A_72: int,B_55: int] :
( ( ord_less_int @ A_72 @ B_55 )
=> ( ( ord_less_int @ zero_zero_int @ C_26 )
=> ( ord_less_int @ ( times_times_int @ C_26 @ A_72 ) @ ( times_times_int @ C_26 @ B_55 ) ) ) ) ).
thf(fact_375_mult__strict__left__mono,axiom,
! [C_25: nat,A_71: nat,B_54: nat] :
( ( ord_less_nat @ A_71 @ B_54 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_25 )
=> ( ord_less_nat @ ( times_times_nat @ C_25 @ A_71 ) @ ( times_times_nat @ C_25 @ B_54 ) ) ) ) ).
thf(fact_376_mult__strict__left__mono,axiom,
! [C_25: int,A_71: int,B_54: int] :
( ( ord_less_int @ A_71 @ B_54 )
=> ( ( ord_less_int @ zero_zero_int @ C_25 )
=> ( ord_less_int @ ( times_times_int @ C_25 @ A_71 ) @ ( times_times_int @ C_25 @ B_54 ) ) ) ) ).
thf(fact_377_mult__strict__right__mono,axiom,
! [C_24: nat,A_70: nat,B_53: nat] :
( ( ord_less_nat @ A_70 @ B_53 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_24 )
=> ( ord_less_nat @ ( times_times_nat @ A_70 @ C_24 ) @ ( times_times_nat @ B_53 @ C_24 ) ) ) ) ).
thf(fact_378_mult__strict__right__mono,axiom,
! [C_24: int,A_70: int,B_53: int] :
( ( ord_less_int @ A_70 @ B_53 )
=> ( ( ord_less_int @ zero_zero_int @ C_24 )
=> ( ord_less_int @ ( times_times_int @ A_70 @ C_24 ) @ ( times_times_int @ B_53 @ C_24 ) ) ) ) ).
thf(fact_379_mult__neg__neg,axiom,
! [B_52: int,A_69: int] :
( ( ord_less_int @ A_69 @ zero_zero_int )
=> ( ( ord_less_int @ B_52 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_69 @ B_52 ) ) ) ) ).
thf(fact_380_mult__neg__pos,axiom,
! [B_51: nat,A_68: nat] :
( ( ord_less_nat @ A_68 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B_51 )
=> ( ord_less_nat @ ( times_times_nat @ A_68 @ B_51 ) @ zero_zero_nat ) ) ) ).
thf(fact_381_mult__neg__pos,axiom,
! [B_51: int,A_68: int] :
( ( ord_less_int @ A_68 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_51 )
=> ( ord_less_int @ ( times_times_int @ A_68 @ B_51 ) @ zero_zero_int ) ) ) ).
thf(fact_382_mult__less__cancel__left__neg,axiom,
! [A_67: int,B_50: int,C_23: int] :
( ( ord_less_int @ C_23 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C_23 @ A_67 ) @ ( times_times_int @ C_23 @ B_50 ) )
<=> ( ord_less_int @ B_50 @ A_67 ) ) ) ).
thf(fact_383_zero__less__mult__pos2,axiom,
! [B_49: nat,A_66: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B_49 @ A_66 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_66 )
=> ( ord_less_nat @ zero_zero_nat @ B_49 ) ) ) ).
thf(fact_384_zero__less__mult__pos2,axiom,
! [B_49: int,A_66: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B_49 @ A_66 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_66 )
=> ( ord_less_int @ zero_zero_int @ B_49 ) ) ) ).
thf(fact_385_zero__less__mult__pos,axiom,
! [A_65: nat,B_48: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_65 @ B_48 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_65 )
=> ( ord_less_nat @ zero_zero_nat @ B_48 ) ) ) ).
thf(fact_386_zero__less__mult__pos,axiom,
! [A_65: int,B_48: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_65 @ B_48 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_65 )
=> ( ord_less_int @ zero_zero_int @ B_48 ) ) ) ).
thf(fact_387_mult__pos__neg2,axiom,
! [B_47: nat,A_64: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_64 )
=> ( ( ord_less_nat @ B_47 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B_47 @ A_64 ) @ zero_zero_nat ) ) ) ).
thf(fact_388_mult__pos__neg2,axiom,
! [B_47: int,A_64: int] :
( ( ord_less_int @ zero_zero_int @ A_64 )
=> ( ( ord_less_int @ B_47 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_47 @ A_64 ) @ zero_zero_int ) ) ) ).
thf(fact_389_mult__pos__neg,axiom,
! [B_46: nat,A_63: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_63 )
=> ( ( ord_less_nat @ B_46 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_63 @ B_46 ) @ zero_zero_nat ) ) ) ).
thf(fact_390_mult__pos__neg,axiom,
! [B_46: int,A_63: int] :
( ( ord_less_int @ zero_zero_int @ A_63 )
=> ( ( ord_less_int @ B_46 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_63 @ B_46 ) @ zero_zero_int ) ) ) ).
thf(fact_391_mult__pos__pos,axiom,
! [B_45: nat,A_62: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_62 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_45 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_62 @ B_45 ) ) ) ) ).
thf(fact_392_mult__pos__pos,axiom,
! [B_45: int,A_62: int] :
( ( ord_less_int @ zero_zero_int @ A_62 )
=> ( ( ord_less_int @ zero_zero_int @ B_45 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_62 @ B_45 ) ) ) ) ).
thf(fact_393_mult__less__cancel__left__pos,axiom,
! [A_61: int,B_44: int,C_22: int] :
( ( ord_less_int @ zero_zero_int @ C_22 )
=> ( ( ord_less_int @ ( times_times_int @ C_22 @ A_61 ) @ ( times_times_int @ C_22 @ B_44 ) )
<=> ( ord_less_int @ A_61 @ B_44 ) ) ) ).
thf(fact_394_mult__less__cancel__left__disj,axiom,
! [C_21: int,A_60: int,B_43: int] :
( ( ord_less_int @ ( times_times_int @ C_21 @ A_60 ) @ ( times_times_int @ C_21 @ B_43 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_21 )
& ( ord_less_int @ A_60 @ B_43 ) )
| ( ( ord_less_int @ C_21 @ zero_zero_int )
& ( ord_less_int @ B_43 @ A_60 ) ) ) ) ).
thf(fact_395_mult__less__cancel__right__disj,axiom,
! [A_59: int,C_20: int,B_42: int] :
( ( ord_less_int @ ( times_times_int @ A_59 @ C_20 ) @ ( times_times_int @ B_42 @ C_20 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_20 )
& ( ord_less_int @ A_59 @ B_42 ) )
| ( ( ord_less_int @ C_20 @ zero_zero_int )
& ( ord_less_int @ B_42 @ A_59 ) ) ) ) ).
thf(fact_396_not__square__less__zero,axiom,
! [A_58: int] :
~ ( ord_less_int @ ( times_times_int @ A_58 @ A_58 ) @ zero_zero_int ) ).
thf(fact_397_add__nonpos__nonpos,axiom,
! [B_41: nat,A_57: nat] :
( ( ord_less_eq_nat @ A_57 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_41 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_57 @ B_41 ) @ zero_zero_nat ) ) ) ).
thf(fact_398_add__nonpos__nonpos,axiom,
! [B_41: int,A_57: int] :
( ( ord_less_eq_int @ A_57 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_41 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_57 @ B_41 ) @ zero_zero_int ) ) ) ).
thf(fact_399_add__increasing2,axiom,
! [B_40: nat,A_56: nat,C_19: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C_19 )
=> ( ( ord_less_eq_nat @ B_40 @ A_56 )
=> ( ord_less_eq_nat @ B_40 @ ( plus_plus_nat @ A_56 @ C_19 ) ) ) ) ).
thf(fact_400_add__increasing2,axiom,
! [B_40: int,A_56: int,C_19: int] :
( ( ord_less_eq_int @ zero_zero_int @ C_19 )
=> ( ( ord_less_eq_int @ B_40 @ A_56 )
=> ( ord_less_eq_int @ B_40 @ ( plus_plus_int @ A_56 @ C_19 ) ) ) ) ).
thf(fact_401_add__increasing,axiom,
! [B_39: nat,C_18: nat,A_55: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_55 )
=> ( ( ord_less_eq_nat @ B_39 @ C_18 )
=> ( ord_less_eq_nat @ B_39 @ ( plus_plus_nat @ A_55 @ C_18 ) ) ) ) ).
thf(fact_402_add__increasing,axiom,
! [B_39: int,C_18: int,A_55: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_55 )
=> ( ( ord_less_eq_int @ B_39 @ C_18 )
=> ( ord_less_eq_int @ B_39 @ ( plus_plus_int @ A_55 @ C_18 ) ) ) ) ).
thf(fact_403_add__nonneg__eq__0__iff,axiom,
! [Y_10: nat,X_12: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X_12 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_10 )
=> ( ( ( plus_plus_nat @ X_12 @ Y_10 )
= zero_zero_nat )
<=> ( ( X_12 = zero_zero_nat )
& ( Y_10 = zero_zero_nat ) ) ) ) ) ).
thf(fact_404_add__nonneg__eq__0__iff,axiom,
! [Y_10: int,X_12: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_12 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_10 )
=> ( ( ( plus_plus_int @ X_12 @ Y_10 )
= zero_zero_int )
<=> ( ( X_12 = zero_zero_int )
& ( Y_10 = zero_zero_int ) ) ) ) ) ).
thf(fact_405_add__nonneg__nonneg,axiom,
! [B_38: nat,A_54: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_54 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_38 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A_54 @ B_38 ) ) ) ) ).
thf(fact_406_add__nonneg__nonneg,axiom,
! [B_38: int,A_54: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_54 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_38 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_54 @ B_38 ) ) ) ) ).
thf(fact_407_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_53: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_53 @ A_53 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ A_53 @ zero_zero_int ) ) ).
thf(fact_408_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_52: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_52 @ A_52 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A_52 ) ) ).
thf(fact_409_add__neg__neg,axiom,
! [B_37: nat,A_51: nat] :
( ( ord_less_nat @ A_51 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_37 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_51 @ B_37 ) @ zero_zero_nat ) ) ) ).
thf(fact_410_add__neg__neg,axiom,
! [B_37: int,A_51: int] :
( ( ord_less_int @ A_51 @ zero_zero_int )
=> ( ( ord_less_int @ B_37 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_51 @ B_37 ) @ zero_zero_int ) ) ) ).
thf(fact_411_pos__add__strict,axiom,
! [B_36: nat,C_17: nat,A_50: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_50 )
=> ( ( ord_less_nat @ B_36 @ C_17 )
=> ( ord_less_nat @ B_36 @ ( plus_plus_nat @ A_50 @ C_17 ) ) ) ) ).
thf(fact_412_pos__add__strict,axiom,
! [B_36: int,C_17: int,A_50: int] :
( ( ord_less_int @ zero_zero_int @ A_50 )
=> ( ( ord_less_int @ B_36 @ C_17 )
=> ( ord_less_int @ B_36 @ ( plus_plus_int @ A_50 @ C_17 ) ) ) ) ).
thf(fact_413_add__pos__pos,axiom,
! [B_35: nat,A_49: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_49 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_35 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_49 @ B_35 ) ) ) ) ).
thf(fact_414_add__pos__pos,axiom,
! [B_35: int,A_49: int] :
( ( ord_less_int @ zero_zero_int @ A_49 )
=> ( ( ord_less_int @ zero_zero_int @ B_35 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_49 @ B_35 ) ) ) ) ).
thf(fact_415_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_48: int] :
( ( ord_less_int @ ( plus_plus_int @ A_48 @ A_48 ) @ zero_zero_int )
<=> ( ord_less_int @ A_48 @ zero_zero_int ) ) ).
thf(fact_416_even__less__0__iff,axiom,
! [A_47: int] :
( ( ord_less_int @ ( plus_plus_int @ A_47 @ A_47 ) @ zero_zero_int )
<=> ( ord_less_int @ A_47 @ zero_zero_int ) ) ).
thf(fact_417_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_46: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_46 @ A_46 ) )
<=> ( ord_less_int @ zero_zero_int @ A_46 ) ) ).
thf(fact_418_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_419_zero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_420_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_421_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
thf(fact_422_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_423_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_424_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_425_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
thf(fact_426_le__iff__diff__le__0,axiom,
! [A_45: int,B_34: int] :
( ( ord_less_eq_int @ A_45 @ B_34 )
<=> ( ord_less_eq_int @ ( minus_minus_int @ A_45 @ B_34 ) @ zero_zero_int ) ) ).
thf(fact_427_add__scale__eq__noteq,axiom,
! [C_16: nat,D_7: nat,A_44: nat,B_33: nat,R_3: nat] :
( ( R_3 != zero_zero_nat )
=> ( ( ( A_44 = B_33 )
& ( C_16 != D_7 ) )
=> ( ( plus_plus_nat @ A_44 @ ( times_times_nat @ R_3 @ C_16 ) )
!= ( plus_plus_nat @ B_33 @ ( times_times_nat @ R_3 @ D_7 ) ) ) ) ) ).
thf(fact_428_add__scale__eq__noteq,axiom,
! [C_16: int,D_7: int,A_44: int,B_33: int,R_3: int] :
( ( R_3 != zero_zero_int )
=> ( ( ( A_44 = B_33 )
& ( C_16 != D_7 ) )
=> ( ( plus_plus_int @ A_44 @ ( times_times_int @ R_3 @ C_16 ) )
!= ( plus_plus_int @ B_33 @ ( times_times_int @ R_3 @ D_7 ) ) ) ) ) ).
thf(fact_429_less__iff__diff__less__0,axiom,
! [A_43: int,B_32: int] :
( ( ord_less_int @ A_43 @ B_32 )
<=> ( ord_less_int @ ( minus_minus_int @ A_43 @ B_32 ) @ zero_zero_int ) ) ).
thf(fact_430_Euler_Oaux1,axiom,
! [A_23: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ A_23 )
=> ( ( X
!= ( minus_minus_int @ A_23 @ one_one_int ) )
=> ( ord_less_int @ X @ ( minus_minus_int @ A_23 @ one_one_int ) ) ) ) ) ).
thf(fact_431_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_432_zmult__zless__mono2,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
thf(fact_433_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_434_less__bin__lemma,axiom,
! [K: int,L: int] :
( ( ord_less_int @ K @ L )
<=> ( ord_less_int @ ( minus_minus_int @ K @ L ) @ zero_zero_int ) ) ).
thf(fact_435_mult__left__le__imp__le,axiom,
! [C_15: nat,A_42: nat,B_31: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C_15 @ A_42 ) @ ( times_times_nat @ C_15 @ B_31 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_15 )
=> ( ord_less_eq_nat @ A_42 @ B_31 ) ) ) ).
thf(fact_436_mult__left__le__imp__le,axiom,
! [C_15: int,A_42: int,B_31: int] :
( ( ord_less_eq_int @ ( times_times_int @ C_15 @ A_42 ) @ ( times_times_int @ C_15 @ B_31 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_15 )
=> ( ord_less_eq_int @ A_42 @ B_31 ) ) ) ).
thf(fact_437_mult__right__le__imp__le,axiom,
! [A_41: nat,C_14: nat,B_30: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A_41 @ C_14 ) @ ( times_times_nat @ B_30 @ C_14 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_14 )
=> ( ord_less_eq_nat @ A_41 @ B_30 ) ) ) ).
thf(fact_438_mult__right__le__imp__le,axiom,
! [A_41: int,C_14: int,B_30: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_41 @ C_14 ) @ ( times_times_int @ B_30 @ C_14 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_14 )
=> ( ord_less_eq_int @ A_41 @ B_30 ) ) ) ).
thf(fact_439_mult__less__imp__less__left,axiom,
! [C_13: nat,A_40: nat,B_29: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_13 @ A_40 ) @ ( times_times_nat @ C_13 @ B_29 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_13 )
=> ( ord_less_nat @ A_40 @ B_29 ) ) ) ).
thf(fact_440_mult__less__imp__less__left,axiom,
! [C_13: int,A_40: int,B_29: int] :
( ( ord_less_int @ ( times_times_int @ C_13 @ A_40 ) @ ( times_times_int @ C_13 @ B_29 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_13 )
=> ( ord_less_int @ A_40 @ B_29 ) ) ) ).
thf(fact_441_mult__left__less__imp__less,axiom,
! [C_12: nat,A_39: nat,B_28: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_12 @ A_39 ) @ ( times_times_nat @ C_12 @ B_28 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_12 )
=> ( ord_less_nat @ A_39 @ B_28 ) ) ) ).
thf(fact_442_mult__left__less__imp__less,axiom,
! [C_12: int,A_39: int,B_28: int] :
( ( ord_less_int @ ( times_times_int @ C_12 @ A_39 ) @ ( times_times_int @ C_12 @ B_28 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_12 )
=> ( ord_less_int @ A_39 @ B_28 ) ) ) ).
thf(fact_443_mult__less__imp__less__right,axiom,
! [A_38: nat,C_11: nat,B_27: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_38 @ C_11 ) @ ( times_times_nat @ B_27 @ C_11 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_11 )
=> ( ord_less_nat @ A_38 @ B_27 ) ) ) ).
thf(fact_444_mult__less__imp__less__right,axiom,
! [A_38: int,C_11: int,B_27: int] :
( ( ord_less_int @ ( times_times_int @ A_38 @ C_11 ) @ ( times_times_int @ B_27 @ C_11 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_11 )
=> ( ord_less_int @ A_38 @ B_27 ) ) ) ).
thf(fact_445_mult__right__less__imp__less,axiom,
! [A_37: nat,C_10: nat,B_26: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_37 @ C_10 ) @ ( times_times_nat @ B_26 @ C_10 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_10 )
=> ( ord_less_nat @ A_37 @ B_26 ) ) ) ).
thf(fact_446_mult__right__less__imp__less,axiom,
! [A_37: int,C_10: int,B_26: int] :
( ( ord_less_int @ ( times_times_int @ A_37 @ C_10 ) @ ( times_times_int @ B_26 @ C_10 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_10 )
=> ( ord_less_int @ A_37 @ B_26 ) ) ) ).
thf(fact_447_mult__le__less__imp__less,axiom,
! [C_9: nat,D_6: nat,A_36: nat,B_25: nat] :
( ( ord_less_eq_nat @ A_36 @ B_25 )
=> ( ( ord_less_nat @ C_9 @ D_6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_36 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_9 )
=> ( ord_less_nat @ ( times_times_nat @ A_36 @ C_9 ) @ ( times_times_nat @ B_25 @ D_6 ) ) ) ) ) ) ).
thf(fact_448_mult__le__less__imp__less,axiom,
! [C_9: int,D_6: int,A_36: int,B_25: int] :
( ( ord_less_eq_int @ A_36 @ B_25 )
=> ( ( ord_less_int @ C_9 @ D_6 )
=> ( ( ord_less_int @ zero_zero_int @ A_36 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_9 )
=> ( ord_less_int @ ( times_times_int @ A_36 @ C_9 ) @ ( times_times_int @ B_25 @ D_6 ) ) ) ) ) ) ).
thf(fact_449_mult__less__le__imp__less,axiom,
! [C_8: nat,D_5: nat,A_35: nat,B_24: nat] :
( ( ord_less_nat @ A_35 @ B_24 )
=> ( ( ord_less_eq_nat @ C_8 @ D_5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_35 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_8 )
=> ( ord_less_nat @ ( times_times_nat @ A_35 @ C_8 ) @ ( times_times_nat @ B_24 @ D_5 ) ) ) ) ) ) ).
thf(fact_450_mult__less__le__imp__less,axiom,
! [C_8: int,D_5: int,A_35: int,B_24: int] :
( ( ord_less_int @ A_35 @ B_24 )
=> ( ( ord_less_eq_int @ C_8 @ D_5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_35 )
=> ( ( ord_less_int @ zero_zero_int @ C_8 )
=> ( ord_less_int @ ( times_times_int @ A_35 @ C_8 ) @ ( times_times_int @ B_24 @ D_5 ) ) ) ) ) ) ).
thf(fact_451_mult__strict__mono_H,axiom,
! [C_7: nat,D_4: nat,A_34: nat,B_23: nat] :
( ( ord_less_nat @ A_34 @ B_23 )
=> ( ( ord_less_nat @ C_7 @ D_4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_34 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_7 )
=> ( ord_less_nat @ ( times_times_nat @ A_34 @ C_7 ) @ ( times_times_nat @ B_23 @ D_4 ) ) ) ) ) ) ).
thf(fact_452_mult__strict__mono_H,axiom,
! [C_7: int,D_4: int,A_34: int,B_23: int] :
( ( ord_less_int @ A_34 @ B_23 )
=> ( ( ord_less_int @ C_7 @ D_4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_34 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_7 )
=> ( ord_less_int @ ( times_times_int @ A_34 @ C_7 ) @ ( times_times_int @ B_23 @ D_4 ) ) ) ) ) ) ).
thf(fact_453_mult__strict__mono,axiom,
! [C_6: nat,D_3: nat,A_33: nat,B_22: nat] :
( ( ord_less_nat @ A_33 @ B_22 )
=> ( ( ord_less_nat @ C_6 @ D_3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_22 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_6 )
=> ( ord_less_nat @ ( times_times_nat @ A_33 @ C_6 ) @ ( times_times_nat @ B_22 @ D_3 ) ) ) ) ) ) ).
thf(fact_454_mult__strict__mono,axiom,
! [C_6: int,D_3: int,A_33: int,B_22: int] :
( ( ord_less_int @ A_33 @ B_22 )
=> ( ( ord_less_int @ C_6 @ D_3 )
=> ( ( ord_less_int @ zero_zero_int @ B_22 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_6 )
=> ( ord_less_int @ ( times_times_int @ A_33 @ C_6 ) @ ( times_times_int @ B_22 @ D_3 ) ) ) ) ) ) ).
thf(fact_455_mult__le__cancel__left__neg,axiom,
! [A_32: int,B_21: int,C_5: int] :
( ( ord_less_int @ C_5 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_5 @ A_32 ) @ ( times_times_int @ C_5 @ B_21 ) )
<=> ( ord_less_eq_int @ B_21 @ A_32 ) ) ) ).
thf(fact_456_mult__le__cancel__left__pos,axiom,
! [A_31: int,B_20: int,C_4: int] :
( ( ord_less_int @ zero_zero_int @ C_4 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_4 @ A_31 ) @ ( times_times_int @ C_4 @ B_20 ) )
<=> ( ord_less_eq_int @ A_31 @ B_20 ) ) ) ).
thf(fact_457_add__nonpos__neg,axiom,
! [B_19: nat,A_30: nat] :
( ( ord_less_eq_nat @ A_30 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_19 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_30 @ B_19 ) @ zero_zero_nat ) ) ) ).
thf(fact_458_add__nonpos__neg,axiom,
! [B_19: int,A_30: int] :
( ( ord_less_eq_int @ A_30 @ zero_zero_int )
=> ( ( ord_less_int @ B_19 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_30 @ B_19 ) @ zero_zero_int ) ) ) ).
thf(fact_459_add__neg__nonpos,axiom,
! [B_18: nat,A_29: nat] :
( ( ord_less_nat @ A_29 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_18 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_29 @ B_18 ) @ zero_zero_nat ) ) ) ).
thf(fact_460_add__neg__nonpos,axiom,
! [B_18: int,A_29: int] :
( ( ord_less_int @ A_29 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_18 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_29 @ B_18 ) @ zero_zero_int ) ) ) ).
thf(fact_461_add__strict__increasing2,axiom,
! [B_17: nat,C_3: nat,A_28: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_28 )
=> ( ( ord_less_nat @ B_17 @ C_3 )
=> ( ord_less_nat @ B_17 @ ( plus_plus_nat @ A_28 @ C_3 ) ) ) ) ).
thf(fact_462_add__strict__increasing2,axiom,
! [B_17: int,C_3: int,A_28: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_28 )
=> ( ( ord_less_int @ B_17 @ C_3 )
=> ( ord_less_int @ B_17 @ ( plus_plus_int @ A_28 @ C_3 ) ) ) ) ).
thf(fact_463_add__strict__increasing,axiom,
! [B_16: nat,C_2: nat,A_27: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_27 )
=> ( ( ord_less_eq_nat @ B_16 @ C_2 )
=> ( ord_less_nat @ B_16 @ ( plus_plus_nat @ A_27 @ C_2 ) ) ) ) ).
thf(fact_464_add__strict__increasing,axiom,
! [B_16: int,C_2: int,A_27: int] :
( ( ord_less_int @ zero_zero_int @ A_27 )
=> ( ( ord_less_eq_int @ B_16 @ C_2 )
=> ( ord_less_int @ B_16 @ ( plus_plus_int @ A_27 @ C_2 ) ) ) ) ).
thf(fact_465_add__nonneg__pos,axiom,
! [B_15: nat,A_26: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_26 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_15 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_26 @ B_15 ) ) ) ) ).
thf(fact_466_add__nonneg__pos,axiom,
! [B_15: int,A_26: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_26 )
=> ( ( ord_less_int @ zero_zero_int @ B_15 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_26 @ B_15 ) ) ) ) ).
thf(fact_467_add__pos__nonneg,axiom,
! [B_14: nat,A_25: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_25 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_14 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_25 @ B_14 ) ) ) ) ).
thf(fact_468_add__pos__nonneg,axiom,
! [B_14: int,A_25: int] :
( ( ord_less_int @ zero_zero_int @ A_25 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_14 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_25 @ B_14 ) ) ) ) ).
thf(fact_469_mult__left__le__one__le,axiom,
! [Y_9: int,X_11: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_11 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_9 )
=> ( ( ord_less_eq_int @ Y_9 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y_9 @ X_11 ) @ X_11 ) ) ) ) ).
thf(fact_470_mult__right__le__one__le,axiom,
! [Y_8: int,X_10: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_10 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_8 )
=> ( ( ord_less_eq_int @ Y_8 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X_10 @ Y_8 ) @ X_10 ) ) ) ) ).
thf(fact_471_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
thf(fact_472_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
thf(fact_473_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
<=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
thf(fact_474_pos__zmult__eq__1__iff,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
thf(fact_475_odd__less__0,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
<=> ( ord_less_int @ Z @ zero_zero_int ) ) ).
thf(fact_476_convex__bound__le,axiom,
! [V: int,U: int,Y_7: int,X_9: int,A_24: int] :
( ( ord_less_eq_int @ X_9 @ A_24 )
=> ( ( ord_less_eq_int @ Y_7 @ A_24 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X_9 ) @ ( times_times_int @ V @ Y_7 ) ) @ A_24 ) ) ) ) ) ) ).
thf(fact_477_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
thf(fact_478_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A_23: int] :
( ( ord_less_int @ zero_zero_int @ A_23 )
=> ( ( A_23
= ( plus_plus_int @ R_1 @ ( times_times_int @ A_23 @ Q ) ) )
=> ( ( ord_less_int @ R_1 @ A_23 )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_479_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A_23: int] :
( ( ord_less_int @ zero_zero_int @ A_23 )
=> ( ( A_23
= ( plus_plus_int @ R_1 @ ( times_times_int @ A_23 @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_480_q__pos__lemma,axiom,
! [B_13: int,Q_1: int,R_2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_13 @ Q_1 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_13 )
=> ( ( ord_less_int @ zero_zero_int @ B_13 )
=> ( ord_less_eq_int @ zero_zero_int @ Q_1 ) ) ) ) ).
thf(fact_481_q__neg__lemma,axiom,
! [B_13: int,Q_1: int,R_2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_13 @ Q_1 ) @ R_2 ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ zero_zero_int @ B_13 )
=> ( ord_less_eq_int @ Q_1 @ zero_zero_int ) ) ) ) ).
thf(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 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_13 @ Q_1 ) @ R_2 ) @ zero_zero_int )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ zero_zero_int @ B_13 )
=> ( ( ord_less_eq_int @ B_13 @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
thf(fact_483_unique__quotient__lemma__neg,axiom,
! [B: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( ( 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 ) )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_int @ B @ R_2 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
thf(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 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_13 @ Q_1 ) @ R_2 ) )
=> ( ( ord_less_int @ R_2 @ B_13 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_13 )
=> ( ( ord_less_eq_int @ B_13 @ B )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
thf(fact_485_unique__quotient__lemma,axiom,
! [B: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( ( 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 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_2 )
=> ( ( ord_less_int @ R_2 @ B )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
thf(fact_486_xzgcd__linear,axiom,
! [M: int,R_1: int,S: int,T: int,N: int] :
( ( 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 ) ) ) ) ) ).
thf(fact_487_not__sum__squares__lt__zero,axiom,
! [X_8: int,Y_6: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X_8 @ X_8 ) @ ( times_times_int @ Y_6 @ Y_6 ) ) @ zero_zero_int ) ).
thf(fact_488_sum__squares__gt__zero__iff,axiom,
! [X_7: int,Y_5: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_7 @ X_7 ) @ ( times_times_int @ Y_5 @ Y_5 ) ) )
<=> ( ( X_7 != zero_zero_int )
| ( Y_5 != zero_zero_int ) ) ) ).
thf(fact_489_sum__squares__eq__zero__iff,axiom,
! [X_6: int,Y_4: int] :
( ( ( plus_plus_int @ ( times_times_int @ X_6 @ X_6 ) @ ( times_times_int @ Y_4 @ Y_4 ) )
= zero_zero_int )
<=> ( ( X_6 = zero_zero_int )
& ( Y_4 = zero_zero_int ) ) ) ).
thf(fact_490_sum__squares__le__zero__iff,axiom,
! [X_5: int,Y_3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_5 @ X_5 ) @ ( times_times_int @ Y_3 @ Y_3 ) ) @ zero_zero_int )
<=> ( ( X_5 = zero_zero_int )
& ( Y_3 = zero_zero_int ) ) ) ).
thf(fact_491_sum__squares__ge__zero,axiom,
! [X_4: int,Y_2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_4 @ X_4 ) @ ( times_times_int @ Y_2 @ Y_2 ) ) ) ).
thf(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] :
( ( 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 ) ) ) ) ) ) ) ).
thf(fact_493_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_494_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_495_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(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 ) ) ).
thf(fact_497_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_498_pos__zmult__pos,axiom,
! [B: int,A_23: int] :
( ( ord_less_int @ zero_zero_int @ A_23 )
=> ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_23 @ B ) )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
thf(fact_499_decr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_3: int] :
( ( P @ X_3 )
=> ( P @ ( minus_minus_int @ X_3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_3: int] :
( ( P @ X_3 )
=> ( P @ ( minus_minus_int @ X_3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_500_incr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_3: int] :
( ( P @ X_3 )
=> ( P @ ( plus_plus_int @ X_3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_3: int] :
( ( P @ X_3 )
=> ( P @ ( plus_plus_int @ X_3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_501_imp__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P_1 ) ) ) ).
thf(fact_502_conj__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
& P_1 ) ) ) ).
thf(fact_503_minf_I8_J,axiom,
! [T_4: nat] :
? [Z_1: nat] :
! [X_3: nat] :
( ( ord_less_nat @ X_3 @ Z_1 )
=> ~ ( ord_less_eq_nat @ T_4 @ X_3 ) ) ).
thf(fact_504_minf_I8_J,axiom,
! [T_4: int] :
? [Z_1: int] :
! [X_3: int] :
( ( ord_less_int @ X_3 @ Z_1 )
=> ~ ( ord_less_eq_int @ T_4 @ X_3 ) ) ).
thf(fact_505_pinf_I8_J,axiom,
! [T_3: nat] :
? [Z_1: nat] :
! [X_3: nat] :
( ( ord_less_nat @ Z_1 @ X_3 )
=> ( ord_less_eq_nat @ T_3 @ X_3 ) ) ).
thf(fact_506_pinf_I8_J,axiom,
! [T_3: int] :
? [Z_1: int] :
! [X_3: int] :
( ( ord_less_int @ Z_1 @ X_3 )
=> ( ord_less_eq_int @ T_3 @ X_3 ) ) ).
thf(fact_507_pinf_I6_J,axiom,
! [T_2: nat] :
? [Z_1: nat] :
! [X_3: nat] :
( ( ord_less_nat @ Z_1 @ X_3 )
=> ~ ( ord_less_eq_nat @ X_3 @ T_2 ) ) ).
thf(fact_508_pinf_I6_J,axiom,
! [T_2: int] :
? [Z_1: int] :
! [X_3: int] :
( ( ord_less_int @ Z_1 @ X_3 )
=> ~ ( ord_less_eq_int @ X_3 @ T_2 ) ) ).
thf(fact_509_minf_I6_J,axiom,
! [T_1: nat] :
? [Z_1: nat] :
! [X_3: nat] :
( ( ord_less_nat @ X_3 @ Z_1 )
=> ( ord_less_eq_nat @ X_3 @ T_1 ) ) ).
thf(fact_510_minf_I6_J,axiom,
! [T_1: int] :
? [Z_1: int] :
! [X_3: int] :
( ( ord_less_int @ X_3 @ Z_1 )
=> ( ord_less_eq_int @ X_3 @ T_1 ) ) ).
thf(fact_511_divmod__int__relI,axiom,
! [A_23: int,B: int,Q: int,R_1: int] :
( ( A_23
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
& ( ord_less_int @ R_1 @ B ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ R_1 )
& ( ord_less_eq_int @ R_1 @ zero_zero_int ) ) ) )
=> ( divmod_int_rel @ A_23 @ B @ ( product_Pair_int_int @ Q @ R_1 ) ) ) ) ).
thf(fact_512_divmod__int__rel__0,axiom,
! [B: int] :
( ( B != zero_zero_int )
=> ( divmod_int_rel @ zero_zero_int @ B @ ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ) ).
thf(fact_513_self__remainder,axiom,
! [A_23: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_23 @ A_23 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A_23 != zero_zero_int )
=> ( R_1 = zero_zero_int ) ) ) ).
thf(fact_514_unique__quotient,axiom,
! [Q_1: int,R_2: int,A_23: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_23 @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A_23 @ B @ ( product_Pair_int_int @ Q_1 @ R_2 ) )
=> ( ( B != zero_zero_int )
=> ( Q = Q_1 ) ) ) ) ).
thf(fact_515_unique__remainder,axiom,
! [Q_1: int,R_2: int,A_23: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_23 @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A_23 @ B @ ( product_Pair_int_int @ Q_1 @ R_2 ) )
=> ( ( B != zero_zero_int )
=> ( R_1 = R_2 ) ) ) ) ).
thf(fact_516_self__quotient,axiom,
! [A_23: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_23 @ A_23 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A_23 != zero_zero_int )
=> ( Q = one_one_int ) ) ) ).
thf(fact_517_minusinfinity,axiom,
! [P: int > $o,P1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_3: int,K_1: int] :
( ( P1 @ X_3 )
<=> ( P1 @ ( minus_minus_int @ X_3 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ? [Z_1: int] :
! [X_3: int] :
( ( ord_less_int @ X_3 @ Z_1 )
=> ( ( P @ X_3 )
<=> ( P1 @ X_3 ) ) )
=> ( ( ?? @ int @ P1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_518_plusinfinity,axiom,
! [P: int > $o,P_1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_3: int,K_1: int] :
( ( P_1 @ X_3 )
<=> ( P_1 @ ( minus_minus_int @ X_3 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ? [Z_1: int] :
! [X_3: int] :
( ( ord_less_int @ Z_1 @ X_3 )
=> ( ( P @ X_3 )
<=> ( P_1 @ X_3 ) ) )
=> ( ( ?? @ int @ P_1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_519_tsub__def,axiom,
! [Y: int,X: int] :
( ( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) )
& ( ~ ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= zero_zero_int ) ) ) ).
thf(fact_520_zfact_Osimps,axiom,
! [N: int] :
( ( ( ord_less_eq_int @ N @ zero_zero_int )
=> ( ( zfact @ N )
= one_one_int ) )
& ( ~ ( ord_less_eq_int @ N @ zero_zero_int )
=> ( ( zfact @ N )
= ( times_times_int @ N @ ( zfact @ ( minus_minus_int @ N @ one_one_int ) ) ) ) ) ) ).
thf(fact_521_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( nat_tsub @ X @ Y ) ) ) ) ).
thf(fact_522_tsub__eq,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_523_posDivAlg__correct,axiom,
! [B: int,A_23: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_23 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( divmod_int_rel @ A_23 @ B @ ( posDivAlg @ A_23 @ B ) ) ) ) ).
thf(fact_524_decr__lemma,axiom,
! [X: int,Z: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( 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 ) ) ).
thf(fact_525_abs__eq__0,axiom,
! [A_22: int] :
( ( ( abs_abs_int @ A_22 )
= zero_zero_int )
<=> ( A_22 = zero_zero_int ) ) ).
thf(fact_526_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_527_abs__of__nonneg,axiom,
! [A_21: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_21 )
=> ( ( abs_abs_int @ A_21 )
= A_21 ) ) ).
thf(fact_528_abs__le__zero__iff,axiom,
! [A_20: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_20 ) @ zero_zero_int )
<=> ( A_20 = zero_zero_int ) ) ).
thf(fact_529_abs__ge__zero,axiom,
! [A_19: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A_19 ) ) ).
thf(fact_530_abs__of__pos,axiom,
! [A_18: int] :
( ( ord_less_int @ zero_zero_int @ A_18 )
=> ( ( abs_abs_int @ A_18 )
= A_18 ) ) ).
thf(fact_531_zero__less__abs__iff,axiom,
! [A_17: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A_17 ) )
<=> ( A_17 != zero_zero_int ) ) ).
thf(fact_532_abs__not__less__zero,axiom,
! [A_16: int] :
~ ( ord_less_int @ ( abs_abs_int @ A_16 ) @ zero_zero_int ) ).
thf(fact_533_abs__mult__less,axiom,
! [B_12: int,D_2: int,A_15: int,C_1: int] :
( ( ord_less_int @ ( abs_abs_int @ A_15 ) @ C_1 )
=> ( ( ord_less_int @ ( abs_abs_int @ B_12 ) @ D_2 )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A_15 ) @ ( abs_abs_int @ B_12 ) ) @ ( times_times_int @ C_1 @ D_2 ) ) ) ) ).
thf(fact_534_abs__triangle__ineq,axiom,
! [A_14: int,B_11: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A_14 @ B_11 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A_14 ) @ ( abs_abs_int @ B_11 ) ) ) ).
thf(fact_535_abs__triangle__ineq3,axiom,
! [A_13: int,B_10: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A_13 ) @ ( abs_abs_int @ B_10 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A_13 @ B_10 ) ) ) ).
thf(fact_536_abs__triangle__ineq2,axiom,
! [A_12: int,B_9: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A_12 ) @ ( abs_abs_int @ B_9 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A_12 @ B_9 ) ) ) ).
thf(fact_537_abs__triangle__ineq2__sym,axiom,
! [A_11: int,B_8: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A_11 ) @ ( abs_abs_int @ B_8 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B_8 @ A_11 ) ) ) ).
thf(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 ) ) ).
thf(fact_539_abs__ge__self,axiom,
! [A_10: int] : ( ord_less_eq_int @ A_10 @ ( abs_abs_int @ A_10 ) ) ).
thf(fact_540_abs__le__D1,axiom,
! [A_9: int,B_7: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_9 ) @ B_7 )
=> ( ord_less_eq_int @ A_9 @ B_7 ) ) ).
thf(fact_541_abs__mult,axiom,
! [A_8: int,B_6: int] :
( ( abs_abs_int @ ( times_times_int @ A_8 @ B_6 ) )
= ( times_times_int @ ( abs_abs_int @ A_8 ) @ ( abs_abs_int @ B_6 ) ) ) ).
thf(fact_542_abs__mult__self,axiom,
! [A_7: int] :
( ( times_times_int @ ( abs_abs_int @ A_7 ) @ ( abs_abs_int @ A_7 ) )
= ( times_times_int @ A_7 @ A_7 ) ) ).
thf(fact_543_abs__add__abs,axiom,
! [A_6: int,B_5: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A_6 ) @ ( abs_abs_int @ B_5 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A_6 ) @ ( abs_abs_int @ B_5 ) ) ) ).
thf(fact_544_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
thf(fact_545_abs__minus__commute,axiom,
! [A_5: int,B_4: int] :
( ( abs_abs_int @ ( minus_minus_int @ A_5 @ B_4 ) )
= ( abs_abs_int @ ( minus_minus_int @ B_4 @ A_5 ) ) ) ).
thf(fact_546_abs__idempotent,axiom,
! [A_4: int] :
( ( abs_abs_int @ ( abs_abs_int @ A_4 ) )
= ( abs_abs_int @ A_4 ) ) ).
thf(fact_547_abs__mult__pos,axiom,
! [Y_1: int,X_2: int] :
( ( 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 ) ) ) ) ).
thf(fact_548_abs__eq__mult,axiom,
! [B_3: int,A_3: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_3 )
| ( ord_less_eq_int @ A_3 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B_3 )
| ( ord_less_eq_int @ B_3 @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A_3 @ B_3 ) )
= ( times_times_int @ ( abs_abs_int @ A_3 ) @ ( abs_abs_int @ B_3 ) ) ) ) ).
thf(fact_549_abs__diff__triangle__ineq,axiom,
! [A_2: int,B_2: int,C: int,D_1: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A_2 @ B_2 ) @ ( plus_plus_int @ C @ D_1 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A_2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B_2 @ D_1 ) ) ) ) ).
thf(fact_550_abs__triangle__ineq4,axiom,
! [A_1: int,B_1: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A_1 @ B_1 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A_1 ) @ ( abs_abs_int @ B_1 ) ) ) ).
thf(fact_551_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
<=> ( Z = zero_zero_int ) ) ).
thf(fact_552_posDivAlg__0,axiom,
! [B: int] :
( ( posDivAlg @ zero_zero_int @ B )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ).
thf(fact_553_incr__lemma,axiom,
! [Z: int,X: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( 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 ) ) ) ) ).
thf(fact_554_abs__diff__less__iff,axiom,
! [X_1: int,A: int,R: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X_1 @ A ) ) @ R )
<=> ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X_1 )
& ( ord_less_int @ X_1 @ ( plus_plus_int @ A @ R ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(fact_556_nat0__intermed__int__val,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K ) ) ) ) ) ).
thf(fact_557_int__val__lemma,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(fact_559_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_560_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_561_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ).
thf(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 ) ) ) ).
thf(fact_566_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_567_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_568_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_569_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_570_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(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 ) ) ) ).
thf(fact_572_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_573_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_574_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_575_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
thf(fact_576_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_577_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_578_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_579_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_580_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_581_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_582_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_583_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( P @ N @ M ) )
=> ( ( ( M = N )
=> ( P @ N @ M ) )
=> ( ( ( ord_less_nat @ N @ M )
=> ( P @ N @ M ) )
=> ( P @ N @ M ) ) ) ) ).
thf(fact_584_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(fact_589_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N ) )
<=> ( M = N ) ) ).
thf(fact_590_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N @ K ) )
<=> ( M = N ) ) ).
thf(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 ) ) ).
thf(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 ) ) ).
thf(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 ) ) ) ).
thf(fact_594_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_595_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_596_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_597_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_598_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
thf(fact_599_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
thf(fact_600_eq__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
<=> ( M = N ) ) ) ) ).
thf(fact_601_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_602_le__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_603_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_604_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_605_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_606_le__trans,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
thf(fact_607_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_608_mult__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_609_mult__le__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
thf(fact_610_mult__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
thf(fact_611_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_612_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_613_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_614_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_615_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_616_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_617_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_618_diff__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
thf(fact_619_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_620_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_621_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_622_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_623_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_624_mult__less__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
thf(fact_625_mult__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
thf(fact_626_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_627_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_628_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_629_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_630_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_631_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_632_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_633_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_634_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_635_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_636_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_637_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_638_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
thf(fact_639_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_640_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
thf(fact_641_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_642_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_643_add__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_644_trans__less__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_645_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_int @ ( twoSqu1078207634sum2sq @ ( product_Pair_int_int @ a @ b ) ) @ ( twoSqu1078207634sum2sq @ ( product_Pair_int_int @ p @ q ) ) )
= ( twoSqu1078207634sum2sq @ ( 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 ) ) ) ) ) ).
%------------------------------------------------------------------------------