TPTP Problem File: NUM923^3.p
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%------------------------------------------------------------------------------
% File : NUM923^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_1000_thf_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 1269 ( 406 unt; 66 typ; 0 def)
% Number of atoms : 2570 ( 857 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9543 ( 246 ~; 69 |; 116 &;7836 @)
% ( 307 <=>; 963 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 61 usr; 14 con; 0-8 aty)
% ( 0 !!; 6 ??; 0 @@+; 0 @@-)
% Number of variables : 2739 ( 0 ^;2700 !; 39 ?;2739 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:39:02
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__RealDef__Oreal,type,
real: $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 (64)
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_Archimedean__Field_Oceiling_000tc__RealDef__Oreal,type,
archim856651990g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal,type,
archim1246769320r_real: real > int ).
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_Oabs__class_Oabs_000tc__RealDef__Oreal,type,
abs_abs_real: real > real ).
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_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
minus_minus_real: real > real > real ).
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_Oone__class_Oone_000tc__RealDef__Oreal,type,
one_one_real: real ).
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_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: real > real > real ).
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_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
times_times_real: real > real > real ).
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_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_000tc__Nat__Onat,type,
if_nat: $o > nat > 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_Int_OPls,type,
pls: int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
number267125858f_real: int > real ).
thf(sy_c_Int_Opred,type,
pred: int > int ).
thf(sy_c_Int_Oring__1__class_OInts_000tc__RealDef__Oreal,type,
ring_1_Ints_real: real > $o ).
thf(sy_c_Int_Osucc,type,
succ: int > int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
semiri1621563631at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__RealDef__Oreal,type,
semiri132038758t_real: nat > real ).
thf(sy_c_Nat__Numeral_Oneg,type,
nat_neg: int > $o ).
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_000tc__RealDef__Oreal,type,
ord_less_real: real > real > $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_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
ord_less_eq_real: real > real > $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_RComplete_Onatceiling,type,
natceiling: real > nat ).
thf(sy_c_RComplete_Onatfloor,type,
natfloor: real > nat ).
thf(sy_c_RealDef_Oreal_000tc__Int__Oint,type,
real_int: int > real ).
thf(sy_c_RealDef_Oreal_000tc__Nat__Onat,type,
real_nat: nat > real ).
thf(sy_c_TwoSquares__Mirabelle__dzzvbppuls_Ois__sum2sq,type,
twoSqu919416604sum2sq: int > $o ).
thf(sy_c_TwoSquares__Mirabelle__dzzvbppuls_Osum2sq,type,
twoSqu2057625106sum2sq: product_prod_int_int > int ).
thf(sy_c_member_000tc__RealDef__Oreal,type,
member_real: real > ( real > $o ) > $o ).
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 (1199)
thf(fact_0_xzgcda__linear__aux1,axiom,
! [A: int,R: int,B: int,M: int,C: int,D_1: int,N_1: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ ( times_times_int @ R @ B ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C @ ( times_times_int @ R @ D_1 ) ) @ N_1 ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A @ M ) @ ( times_times_int @ C @ N_1 ) ) @ ( times_times_int @ R @ ( plus_plus_int @ ( times_times_int @ B @ M ) @ ( times_times_int @ D_1 @ N_1 ) ) ) ) ) ).
thf(fact_1_mult__diff__mult,axiom,
! [X_75: int,Y_64: int,A_153: int,B_123: int] :
( ( minus_minus_int @ ( times_times_int @ X_75 @ Y_64 ) @ ( times_times_int @ A_153 @ B_123 ) )
= ( plus_plus_int @ ( times_times_int @ X_75 @ ( minus_minus_int @ Y_64 @ B_123 ) ) @ ( times_times_int @ ( minus_minus_int @ X_75 @ A_153 ) @ B_123 ) ) ) ).
thf(fact_2_mult__diff__mult,axiom,
! [X_75: real,Y_64: real,A_153: real,B_123: real] :
( ( minus_minus_real @ ( times_times_real @ X_75 @ Y_64 ) @ ( times_times_real @ A_153 @ B_123 ) )
= ( plus_plus_real @ ( times_times_real @ X_75 @ ( minus_minus_real @ Y_64 @ B_123 ) ) @ ( times_times_real @ ( minus_minus_real @ X_75 @ A_153 ) @ B_123 ) ) ) ).
thf(fact_3_mult_Oprod__diff__prod,axiom,
! [X_74: real,Y_63: real,A_152: real,B_122: real] :
( ( minus_minus_real @ ( times_times_real @ X_74 @ Y_63 ) @ ( times_times_real @ A_152 @ B_122 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ X_74 @ A_152 ) @ ( minus_minus_real @ Y_63 @ B_122 ) ) @ ( times_times_real @ ( minus_minus_real @ X_74 @ A_152 ) @ B_122 ) ) @ ( times_times_real @ A_152 @ ( minus_minus_real @ Y_63 @ B_122 ) ) ) ) ).
thf(fact_4_eq__add__iff2,axiom,
! [A_151: int,E_7: int,C_73: int,B_121: int,D_21: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_151 @ E_7 ) @ C_73 )
= ( plus_plus_int @ ( times_times_int @ B_121 @ E_7 ) @ D_21 ) )
<=> ( C_73
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_121 @ A_151 ) @ E_7 ) @ D_21 ) ) ) ).
thf(fact_5_eq__add__iff2,axiom,
! [A_151: real,E_7: real,C_73: real,B_121: real,D_21: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_151 @ E_7 ) @ C_73 )
= ( plus_plus_real @ ( times_times_real @ B_121 @ E_7 ) @ D_21 ) )
<=> ( C_73
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_121 @ A_151 ) @ E_7 ) @ D_21 ) ) ) ).
thf(fact_6_eq__add__iff1,axiom,
! [A_150: int,E_6: int,C_72: int,B_120: int,D_20: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_150 @ E_6 ) @ C_72 )
= ( plus_plus_int @ ( times_times_int @ B_120 @ E_6 ) @ D_20 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_150 @ B_120 ) @ E_6 ) @ C_72 )
= D_20 ) ) ).
thf(fact_7_eq__add__iff1,axiom,
! [A_150: real,E_6: real,C_72: real,B_120: real,D_20: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_150 @ E_6 ) @ C_72 )
= ( plus_plus_real @ ( times_times_real @ B_120 @ E_6 ) @ D_20 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_150 @ B_120 ) @ E_6 ) @ C_72 )
= D_20 ) ) ).
thf(fact_8_is__sum2sq__def,axiom,
! [X: int] :
( ( twoSqu919416604sum2sq @ X )
<=> ? [A_64: int,B_89: int] :
( ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ A_64 @ B_89 ) )
= X ) ) ).
thf(fact_9_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
=> ( A
= ( plus_plus_int @ C @ B ) ) ) ).
thf(fact_10_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_11_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_12_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_13_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_14_diff__add__cancel,axiom,
! [A_149: int,B_119: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_149 @ B_119 ) @ B_119 )
= A_149 ) ).
thf(fact_15_diff__add__cancel,axiom,
! [A_149: real,B_119: real] :
( ( plus_plus_real @ ( minus_minus_real @ A_149 @ B_119 ) @ B_119 )
= A_149 ) ).
thf(fact_16_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_148: int,B_118: int,C_71: int] :
( ( times_times_int @ ( times_times_int @ A_148 @ B_118 ) @ C_71 )
= ( times_times_int @ A_148 @ ( times_times_int @ B_118 @ C_71 ) ) ) ).
thf(fact_17_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_148: real,B_118: real,C_71: real] :
( ( times_times_real @ ( times_times_real @ A_148 @ B_118 ) @ C_71 )
= ( times_times_real @ A_148 @ ( times_times_real @ B_118 @ C_71 ) ) ) ).
thf(fact_18_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_148: nat,B_118: nat,C_71: nat] :
( ( times_times_nat @ ( times_times_nat @ A_148 @ B_118 ) @ C_71 )
= ( times_times_nat @ A_148 @ ( times_times_nat @ B_118 @ C_71 ) ) ) ).
thf(fact_19_add__right__imp__eq,axiom,
! [B_117: int,A_147: int,C_70: int] :
( ( ( plus_plus_int @ B_117 @ A_147 )
= ( plus_plus_int @ C_70 @ A_147 ) )
=> ( B_117 = C_70 ) ) ).
thf(fact_20_add__right__imp__eq,axiom,
! [B_117: real,A_147: real,C_70: real] :
( ( ( plus_plus_real @ B_117 @ A_147 )
= ( plus_plus_real @ C_70 @ A_147 ) )
=> ( B_117 = C_70 ) ) ).
thf(fact_21_add__right__imp__eq,axiom,
! [B_117: nat,A_147: nat,C_70: nat] :
( ( ( plus_plus_nat @ B_117 @ A_147 )
= ( plus_plus_nat @ C_70 @ A_147 ) )
=> ( B_117 = C_70 ) ) ).
thf(fact_22_add__imp__eq,axiom,
! [A_146: int,B_116: int,C_69: int] :
( ( ( plus_plus_int @ A_146 @ B_116 )
= ( plus_plus_int @ A_146 @ C_69 ) )
=> ( B_116 = C_69 ) ) ).
thf(fact_23_add__imp__eq,axiom,
! [A_146: real,B_116: real,C_69: real] :
( ( ( plus_plus_real @ A_146 @ B_116 )
= ( plus_plus_real @ A_146 @ C_69 ) )
=> ( B_116 = C_69 ) ) ).
thf(fact_24_add__imp__eq,axiom,
! [A_146: nat,B_116: nat,C_69: nat] :
( ( ( plus_plus_nat @ A_146 @ B_116 )
= ( plus_plus_nat @ A_146 @ C_69 ) )
=> ( B_116 = C_69 ) ) ).
thf(fact_25_add__left__imp__eq,axiom,
! [A_145: int,B_115: int,C_68: int] :
( ( ( plus_plus_int @ A_145 @ B_115 )
= ( plus_plus_int @ A_145 @ C_68 ) )
=> ( B_115 = C_68 ) ) ).
thf(fact_26_add__left__imp__eq,axiom,
! [A_145: real,B_115: real,C_68: real] :
( ( ( plus_plus_real @ A_145 @ B_115 )
= ( plus_plus_real @ A_145 @ C_68 ) )
=> ( B_115 = C_68 ) ) ).
thf(fact_27_add__left__imp__eq,axiom,
! [A_145: nat,B_115: nat,C_68: nat] :
( ( ( plus_plus_nat @ A_145 @ B_115 )
= ( plus_plus_nat @ A_145 @ C_68 ) )
=> ( B_115 = C_68 ) ) ).
thf(fact_28_add__right__cancel,axiom,
! [B_114: int,A_144: int,C_67: int] :
( ( ( plus_plus_int @ B_114 @ A_144 )
= ( plus_plus_int @ C_67 @ A_144 ) )
<=> ( B_114 = C_67 ) ) ).
thf(fact_29_add__right__cancel,axiom,
! [B_114: real,A_144: real,C_67: real] :
( ( ( plus_plus_real @ B_114 @ A_144 )
= ( plus_plus_real @ C_67 @ A_144 ) )
<=> ( B_114 = C_67 ) ) ).
thf(fact_30_add__right__cancel,axiom,
! [B_114: nat,A_144: nat,C_67: nat] :
( ( ( plus_plus_nat @ B_114 @ A_144 )
= ( plus_plus_nat @ C_67 @ A_144 ) )
<=> ( B_114 = C_67 ) ) ).
thf(fact_31_add__left__cancel,axiom,
! [A_143: int,B_113: int,C_66: int] :
( ( ( plus_plus_int @ A_143 @ B_113 )
= ( plus_plus_int @ A_143 @ C_66 ) )
<=> ( B_113 = C_66 ) ) ).
thf(fact_32_add__left__cancel,axiom,
! [A_143: real,B_113: real,C_66: real] :
( ( ( plus_plus_real @ A_143 @ B_113 )
= ( plus_plus_real @ A_143 @ C_66 ) )
<=> ( B_113 = C_66 ) ) ).
thf(fact_33_add__left__cancel,axiom,
! [A_143: nat,B_113: nat,C_66: nat] :
( ( ( plus_plus_nat @ A_143 @ B_113 )
= ( plus_plus_nat @ A_143 @ C_66 ) )
<=> ( B_113 = C_66 ) ) ).
thf(fact_34_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_142: int,B_112: int,C_65: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_142 @ B_112 ) @ C_65 )
= ( plus_plus_int @ A_142 @ ( plus_plus_int @ B_112 @ C_65 ) ) ) ).
thf(fact_35_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_142: real,B_112: real,C_65: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_142 @ B_112 ) @ C_65 )
= ( plus_plus_real @ A_142 @ ( plus_plus_real @ B_112 @ C_65 ) ) ) ).
thf(fact_36_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_142: nat,B_112: nat,C_65: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_142 @ B_112 ) @ C_65 )
= ( plus_plus_nat @ A_142 @ ( plus_plus_nat @ B_112 @ C_65 ) ) ) ).
thf(fact_37_diff__eq__diff__eq,axiom,
! [A_141: int,B_111: int,C_64: int,D_19: int] :
( ( ( minus_minus_int @ A_141 @ B_111 )
= ( minus_minus_int @ C_64 @ D_19 ) )
=> ( ( A_141 = B_111 )
<=> ( C_64 = D_19 ) ) ) ).
thf(fact_38_diff__eq__diff__eq,axiom,
! [A_141: real,B_111: real,C_64: real,D_19: real] :
( ( ( minus_minus_real @ A_141 @ B_111 )
= ( minus_minus_real @ C_64 @ D_19 ) )
=> ( ( A_141 = B_111 )
<=> ( C_64 = D_19 ) ) ) ).
thf(fact_39_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_40_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_41_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_42_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_43_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_44_combine__common__factor,axiom,
! [A_140: int,E_5: int,B_110: int,C_63: int] :
( ( plus_plus_int @ ( times_times_int @ A_140 @ E_5 ) @ ( plus_plus_int @ ( times_times_int @ B_110 @ E_5 ) @ C_63 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_140 @ B_110 ) @ E_5 ) @ C_63 ) ) ).
thf(fact_45_combine__common__factor,axiom,
! [A_140: real,E_5: real,B_110: real,C_63: real] :
( ( plus_plus_real @ ( times_times_real @ A_140 @ E_5 ) @ ( plus_plus_real @ ( times_times_real @ B_110 @ E_5 ) @ C_63 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A_140 @ B_110 ) @ E_5 ) @ C_63 ) ) ).
thf(fact_46_combine__common__factor,axiom,
! [A_140: nat,E_5: nat,B_110: nat,C_63: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_140 @ E_5 ) @ ( plus_plus_nat @ ( times_times_nat @ B_110 @ E_5 ) @ C_63 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_140 @ B_110 ) @ E_5 ) @ C_63 ) ) ).
thf(fact_47_mult__left_Oadd,axiom,
! [X_73: real,Y_62: real,Ya_1: real] :
( ( times_times_real @ ( plus_plus_real @ X_73 @ Y_62 ) @ Ya_1 )
= ( plus_plus_real @ ( times_times_real @ X_73 @ Ya_1 ) @ ( times_times_real @ Y_62 @ Ya_1 ) ) ) ).
thf(fact_48_mult_Oadd__left,axiom,
! [A_139: real,A_138: real,B_109: real] :
( ( times_times_real @ ( plus_plus_real @ A_139 @ A_138 ) @ B_109 )
= ( plus_plus_real @ ( times_times_real @ A_139 @ B_109 ) @ ( times_times_real @ A_138 @ B_109 ) ) ) ).
thf(fact_49_comm__semiring__class_Odistrib,axiom,
! [A_137: int,B_108: int,C_62: int] :
( ( times_times_int @ ( plus_plus_int @ A_137 @ B_108 ) @ C_62 )
= ( plus_plus_int @ ( times_times_int @ A_137 @ C_62 ) @ ( times_times_int @ B_108 @ C_62 ) ) ) ).
thf(fact_50_comm__semiring__class_Odistrib,axiom,
! [A_137: real,B_108: real,C_62: real] :
( ( times_times_real @ ( plus_plus_real @ A_137 @ B_108 ) @ C_62 )
= ( plus_plus_real @ ( times_times_real @ A_137 @ C_62 ) @ ( times_times_real @ B_108 @ C_62 ) ) ) ).
thf(fact_51_comm__semiring__class_Odistrib,axiom,
! [A_137: nat,B_108: nat,C_62: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_137 @ B_108 ) @ C_62 )
= ( plus_plus_nat @ ( times_times_nat @ A_137 @ C_62 ) @ ( times_times_nat @ B_108 @ C_62 ) ) ) ).
thf(fact_52_mult__right_Oadd,axiom,
! [Xa_1: real,X_72: real,Y_61: real] :
( ( times_times_real @ Xa_1 @ ( plus_plus_real @ X_72 @ Y_61 ) )
= ( plus_plus_real @ ( times_times_real @ Xa_1 @ X_72 ) @ ( times_times_real @ Xa_1 @ Y_61 ) ) ) ).
thf(fact_53_mult_Oadd__right,axiom,
! [A_136: real,B_107: real,B_106: real] :
( ( times_times_real @ A_136 @ ( plus_plus_real @ B_107 @ B_106 ) )
= ( plus_plus_real @ ( times_times_real @ A_136 @ B_107 ) @ ( times_times_real @ A_136 @ B_106 ) ) ) ).
thf(fact_54_mult__left_Odiff,axiom,
! [X_71: real,Y_60: real,Ya: real] :
( ( times_times_real @ ( minus_minus_real @ X_71 @ Y_60 ) @ Ya )
= ( minus_minus_real @ ( times_times_real @ X_71 @ Ya ) @ ( times_times_real @ Y_60 @ Ya ) ) ) ).
thf(fact_55_mult_Odiff__left,axiom,
! [A_135: real,A_134: real,B_105: real] :
( ( times_times_real @ ( minus_minus_real @ A_135 @ A_134 ) @ B_105 )
= ( minus_minus_real @ ( times_times_real @ A_135 @ B_105 ) @ ( times_times_real @ A_134 @ B_105 ) ) ) ).
thf(fact_56_mult__right_Odiff,axiom,
! [Xa: real,X_70: real,Y_59: real] :
( ( times_times_real @ Xa @ ( minus_minus_real @ X_70 @ Y_59 ) )
= ( minus_minus_real @ ( times_times_real @ Xa @ X_70 ) @ ( times_times_real @ Xa @ Y_59 ) ) ) ).
thf(fact_57_mult_Odiff__right,axiom,
! [A_133: real,B_104: real,B_103: real] :
( ( times_times_real @ A_133 @ ( minus_minus_real @ B_104 @ B_103 ) )
= ( minus_minus_real @ ( times_times_real @ A_133 @ B_104 ) @ ( times_times_real @ A_133 @ B_103 ) ) ) ).
thf(fact_58_add__diff__add,axiom,
! [A_132: int,C_61: int,B_102: int,D_18: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_132 @ C_61 ) @ ( plus_plus_int @ B_102 @ D_18 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_132 @ B_102 ) @ ( minus_minus_int @ C_61 @ D_18 ) ) ) ).
thf(fact_59_add__diff__add,axiom,
! [A_132: real,C_61: real,B_102: real,D_18: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_132 @ C_61 ) @ ( plus_plus_real @ B_102 @ D_18 ) )
= ( plus_plus_real @ ( minus_minus_real @ A_132 @ B_102 ) @ ( minus_minus_real @ C_61 @ D_18 ) ) ) ).
thf(fact_60_add__diff__cancel,axiom,
! [A_131: int,B_101: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_131 @ B_101 ) @ B_101 )
= A_131 ) ).
thf(fact_61_add__diff__cancel,axiom,
! [A_131: real,B_101: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_131 @ B_101 ) @ B_101 )
= A_131 ) ).
thf(fact_62_crossproduct__eq,axiom,
! [W_13: int,Y_58: int,X_69: int,Z_17: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_13 @ Y_58 ) @ ( times_times_int @ X_69 @ Z_17 ) )
= ( plus_plus_int @ ( times_times_int @ W_13 @ Z_17 ) @ ( times_times_int @ X_69 @ Y_58 ) ) )
<=> ( ( W_13 = X_69 )
| ( Y_58 = Z_17 ) ) ) ).
thf(fact_63_crossproduct__eq,axiom,
! [W_13: real,Y_58: real,X_69: real,Z_17: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_13 @ Y_58 ) @ ( times_times_real @ X_69 @ Z_17 ) )
= ( plus_plus_real @ ( times_times_real @ W_13 @ Z_17 ) @ ( times_times_real @ X_69 @ Y_58 ) ) )
<=> ( ( W_13 = X_69 )
| ( Y_58 = Z_17 ) ) ) ).
thf(fact_64_crossproduct__eq,axiom,
! [W_13: nat,Y_58: nat,X_69: nat,Z_17: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_13 @ Y_58 ) @ ( times_times_nat @ X_69 @ Z_17 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_13 @ Z_17 ) @ ( times_times_nat @ X_69 @ Y_58 ) ) )
<=> ( ( W_13 = X_69 )
| ( Y_58 = Z_17 ) ) ) ).
thf(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_130: int,M_6: int,B_100: int] :
( ( plus_plus_int @ ( times_times_int @ A_130 @ M_6 ) @ ( times_times_int @ B_100 @ M_6 ) )
= ( times_times_int @ ( plus_plus_int @ A_130 @ B_100 ) @ M_6 ) ) ).
thf(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_130: real,M_6: real,B_100: real] :
( ( plus_plus_real @ ( times_times_real @ A_130 @ M_6 ) @ ( times_times_real @ B_100 @ M_6 ) )
= ( times_times_real @ ( plus_plus_real @ A_130 @ B_100 ) @ M_6 ) ) ).
thf(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_130: nat,M_6: nat,B_100: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_130 @ M_6 ) @ ( times_times_nat @ B_100 @ M_6 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_130 @ B_100 ) @ M_6 ) ) ).
thf(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_129: int,B_99: int,C_60: int] :
( ( times_times_int @ ( plus_plus_int @ A_129 @ B_99 ) @ C_60 )
= ( plus_plus_int @ ( times_times_int @ A_129 @ C_60 ) @ ( times_times_int @ B_99 @ C_60 ) ) ) ).
thf(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_129: real,B_99: real,C_60: real] :
( ( times_times_real @ ( plus_plus_real @ A_129 @ B_99 ) @ C_60 )
= ( plus_plus_real @ ( times_times_real @ A_129 @ C_60 ) @ ( times_times_real @ B_99 @ C_60 ) ) ) ).
thf(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_129: nat,B_99: nat,C_60: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_129 @ B_99 ) @ C_60 )
= ( plus_plus_nat @ ( times_times_nat @ A_129 @ C_60 ) @ ( times_times_nat @ B_99 @ C_60 ) ) ) ).
thf(fact_71_crossproduct__noteq,axiom,
! [C_59: int,D_17: int,A_128: int,B_98: int] :
( ( ( A_128 != B_98 )
& ( C_59 != D_17 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_128 @ C_59 ) @ ( times_times_int @ B_98 @ D_17 ) )
!= ( plus_plus_int @ ( times_times_int @ A_128 @ D_17 ) @ ( times_times_int @ B_98 @ C_59 ) ) ) ) ).
thf(fact_72_crossproduct__noteq,axiom,
! [C_59: real,D_17: real,A_128: real,B_98: real] :
( ( ( A_128 != B_98 )
& ( C_59 != D_17 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_128 @ C_59 ) @ ( times_times_real @ B_98 @ D_17 ) )
!= ( plus_plus_real @ ( times_times_real @ A_128 @ D_17 ) @ ( times_times_real @ B_98 @ C_59 ) ) ) ) ).
thf(fact_73_crossproduct__noteq,axiom,
! [C_59: nat,D_17: nat,A_128: nat,B_98: nat] :
( ( ( A_128 != B_98 )
& ( C_59 != D_17 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_128 @ C_59 ) @ ( times_times_nat @ B_98 @ D_17 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_128 @ D_17 ) @ ( times_times_nat @ B_98 @ C_59 ) ) ) ) ).
thf(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_68: int,Y_57: int,Z_16: int] :
( ( times_times_int @ X_68 @ ( plus_plus_int @ Y_57 @ Z_16 ) )
= ( plus_plus_int @ ( times_times_int @ X_68 @ Y_57 ) @ ( times_times_int @ X_68 @ Z_16 ) ) ) ).
thf(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_68: real,Y_57: real,Z_16: real] :
( ( times_times_real @ X_68 @ ( plus_plus_real @ Y_57 @ Z_16 ) )
= ( plus_plus_real @ ( times_times_real @ X_68 @ Y_57 ) @ ( times_times_real @ X_68 @ Z_16 ) ) ) ).
thf(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_68: nat,Y_57: nat,Z_16: nat] :
( ( times_times_nat @ X_68 @ ( plus_plus_nat @ Y_57 @ Z_16 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_68 @ Y_57 ) @ ( times_times_nat @ X_68 @ Z_16 ) ) ) ).
thf(fact_77_Pair__inject,axiom,
! [A_127: int,B_97: int,A_126: int,B_96: int] :
( ( ( product_Pair_int_int @ A_127 @ B_97 )
= ( product_Pair_int_int @ A_126 @ B_96 ) )
=> ~ ( ( A_127 = A_126 )
=> ( B_97 != B_96 ) ) ) ).
thf(fact_78_Pair__inject,axiom,
! [A_127: int,B_97: product_prod_int_int,A_126: int,B_96: product_prod_int_int] :
( ( ( produc282740534nt_int @ A_127 @ B_97 )
= ( produc282740534nt_int @ A_126 @ B_96 ) )
=> ~ ( ( A_127 = A_126 )
=> ( B_97 != B_96 ) ) ) ).
thf(fact_79_Pair__eq,axiom,
! [A_125: int,B_95: int,A_124: int,B_94: int] :
( ( ( product_Pair_int_int @ A_125 @ B_95 )
= ( product_Pair_int_int @ A_124 @ B_94 ) )
<=> ( ( A_125 = A_124 )
& ( B_95 = B_94 ) ) ) ).
thf(fact_80_Pair__eq,axiom,
! [A_125: int,B_95: product_prod_int_int,A_124: int,B_94: product_prod_int_int] :
( ( ( produc282740534nt_int @ A_125 @ B_95 )
= ( produc282740534nt_int @ A_124 @ B_94 ) )
<=> ( ( A_125 = A_124 )
& ( B_95 = B_94 ) ) ) ).
thf(fact_81_split__paired__All,axiom,
! [P_6: product_prod_int_int > $o] :
( ( all1 @ P_6 )
<=> ! [A_64: int,B_89: int] : ( P_6 @ ( product_Pair_int_int @ A_64 @ B_89 ) ) ) ).
thf(fact_82_split__paired__All,axiom,
! [P_6: produc393999548nt_int > $o] :
( ( all2 @ P_6 )
<=> ! [A_64: int,B_89: product_prod_int_int] : ( P_6 @ ( produc282740534nt_int @ A_64 @ B_89 ) ) ) ).
thf(fact_83_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_123: int,B_93: int,C_58: int,D_16: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_123 @ B_93 ) @ ( plus_plus_int @ C_58 @ D_16 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_123 @ C_58 ) @ ( plus_plus_int @ B_93 @ D_16 ) ) ) ).
thf(fact_84_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_123: real,B_93: real,C_58: real,D_16: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_123 @ B_93 ) @ ( plus_plus_real @ C_58 @ D_16 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_123 @ C_58 ) @ ( plus_plus_real @ B_93 @ D_16 ) ) ) ).
thf(fact_85_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_123: nat,B_93: nat,C_58: nat,D_16: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_123 @ B_93 ) @ ( plus_plus_nat @ C_58 @ D_16 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_123 @ C_58 ) @ ( plus_plus_nat @ B_93 @ D_16 ) ) ) ).
thf(fact_86_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_122: int,B_92: int,C_57: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_122 @ B_92 ) @ C_57 )
= ( plus_plus_int @ ( plus_plus_int @ A_122 @ C_57 ) @ B_92 ) ) ).
thf(fact_87_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_122: real,B_92: real,C_57: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_122 @ B_92 ) @ C_57 )
= ( plus_plus_real @ ( plus_plus_real @ A_122 @ C_57 ) @ B_92 ) ) ).
thf(fact_88_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_122: nat,B_92: nat,C_57: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_122 @ B_92 ) @ C_57 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_122 @ C_57 ) @ B_92 ) ) ).
thf(fact_89_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_121: int,B_91: int] :
( ( times_times_int @ A_121 @ B_91 )
= ( times_times_int @ B_91 @ A_121 ) ) ).
thf(fact_90_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_121: real,B_91: real] :
( ( times_times_real @ A_121 @ B_91 )
= ( times_times_real @ B_91 @ A_121 ) ) ).
thf(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_121: nat,B_91: nat] :
( ( times_times_nat @ A_121 @ B_91 )
= ( times_times_nat @ B_91 @ A_121 ) ) ).
thf(fact_92_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_93_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: real,Rx_6: real,Ry_4: real] :
( ( times_times_real @ Lx_6 @ ( times_times_real @ Rx_6 @ Ry_4 ) )
= ( times_times_real @ Rx_6 @ ( times_times_real @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_94_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_95_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_96_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: real,Rx_5: real,Ry_3: real] :
( ( times_times_real @ Lx_5 @ ( times_times_real @ Rx_5 @ Ry_3 ) )
= ( times_times_real @ ( times_times_real @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_97_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_98_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_99_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: real,Ly_4: real,Rx_4: real] :
( ( times_times_real @ ( times_times_real @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_real @ Lx_4 @ ( times_times_real @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_100_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_101_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_102_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_3: real,Rx_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_real @ ( times_times_real @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_103_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_104_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_105_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: real,Ly_2: real,Rx_2: real,Ry_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_2 @ Ly_2 ) @ ( times_times_real @ Rx_2 @ Ry_2 ) )
= ( times_times_real @ Lx_2 @ ( times_times_real @ Ly_2 @ ( times_times_real @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_106_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_107_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_108_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: real,Ly_1: real,Rx_1: real,Ry_1: real] :
( ( times_times_real @ ( times_times_real @ Lx_1 @ Ly_1 ) @ ( times_times_real @ Rx_1 @ Ry_1 ) )
= ( times_times_real @ Rx_1 @ ( times_times_real @ ( times_times_real @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_109_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_110_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_111_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: real,Ly: real,Rx: real,Ry: real] :
( ( times_times_real @ ( times_times_real @ Lx @ Ly ) @ ( times_times_real @ Rx @ Ry ) )
= ( times_times_real @ ( times_times_real @ Lx @ Rx ) @ ( times_times_real @ Ly @ Ry ) ) ) ).
thf(fact_112_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_113_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_120: int,C_56: int] :
( ( plus_plus_int @ A_120 @ C_56 )
= ( plus_plus_int @ C_56 @ A_120 ) ) ).
thf(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_120: real,C_56: real] :
( ( plus_plus_real @ A_120 @ C_56 )
= ( plus_plus_real @ C_56 @ A_120 ) ) ).
thf(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_120: nat,C_56: nat] :
( ( plus_plus_nat @ A_120 @ C_56 )
= ( plus_plus_nat @ C_56 @ A_120 ) ) ).
thf(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_119: int,C_55: int,D_15: int] :
( ( plus_plus_int @ A_119 @ ( plus_plus_int @ C_55 @ D_15 ) )
= ( plus_plus_int @ C_55 @ ( plus_plus_int @ A_119 @ D_15 ) ) ) ).
thf(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_119: real,C_55: real,D_15: real] :
( ( plus_plus_real @ A_119 @ ( plus_plus_real @ C_55 @ D_15 ) )
= ( plus_plus_real @ C_55 @ ( plus_plus_real @ A_119 @ D_15 ) ) ) ).
thf(fact_118_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_119: nat,C_55: nat,D_15: nat] :
( ( plus_plus_nat @ A_119 @ ( plus_plus_nat @ C_55 @ D_15 ) )
= ( plus_plus_nat @ C_55 @ ( plus_plus_nat @ A_119 @ D_15 ) ) ) ).
thf(fact_119_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_118: int,C_54: int,D_14: int] :
( ( plus_plus_int @ A_118 @ ( plus_plus_int @ C_54 @ D_14 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_118 @ C_54 ) @ D_14 ) ) ).
thf(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_118: real,C_54: real,D_14: real] :
( ( plus_plus_real @ A_118 @ ( plus_plus_real @ C_54 @ D_14 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_118 @ C_54 ) @ D_14 ) ) ).
thf(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_118: nat,C_54: nat,D_14: nat] :
( ( plus_plus_nat @ A_118 @ ( plus_plus_nat @ C_54 @ D_14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_118 @ C_54 ) @ D_14 ) ) ).
thf(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_117: int,B_90: int,C_53: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_117 @ B_90 ) @ C_53 )
= ( plus_plus_int @ A_117 @ ( plus_plus_int @ B_90 @ C_53 ) ) ) ).
thf(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_117: real,B_90: real,C_53: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_117 @ B_90 ) @ C_53 )
= ( plus_plus_real @ A_117 @ ( plus_plus_real @ B_90 @ C_53 ) ) ) ).
thf(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_117: nat,B_90: nat,C_53: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_117 @ B_90 ) @ C_53 )
= ( plus_plus_nat @ A_117 @ ( plus_plus_nat @ B_90 @ C_53 ) ) ) ).
thf(fact_125_split__paired__Ex,axiom,
! [P_5: product_prod_int_int > $o] :
( ( ?? @ product_prod_int_int @ P_5 )
<=> ? [A_64: int,B_89: int] : ( P_5 @ ( product_Pair_int_int @ A_64 @ B_89 ) ) ) ).
thf(fact_126_split__paired__Ex,axiom,
! [P_5: produc393999548nt_int > $o] :
( ( ?? @ produc393999548nt_int @ P_5 )
<=> ? [A_64: int,B_89: product_prod_int_int] : ( P_5 @ ( produc282740534nt_int @ A_64 @ B_89 ) ) ) ).
thf(fact_127_prod__induct3,axiom,
! [X_67: produc393999548nt_int,P_4: produc393999548nt_int > $o] :
( ! [A_64: int,B_89: int,C_52: int] : ( P_4 @ ( produc282740534nt_int @ A_64 @ ( product_Pair_int_int @ B_89 @ C_52 ) ) )
=> ( P_4 @ X_67 ) ) ).
thf(fact_128_prod__cases3,axiom,
! [Y_56: produc393999548nt_int] :
~ ! [A_64: int,B_89: int,C_52: int] :
( Y_56
!= ( produc282740534nt_int @ A_64 @ ( product_Pair_int_int @ B_89 @ C_52 ) ) ) ).
thf(fact_129_PairE,axiom,
! [P_3: product_prod_int_int] :
~ ! [X_1: int,Y_1: int] :
( P_3
!= ( product_Pair_int_int @ X_1 @ Y_1 ) ) ).
thf(fact_130_PairE,axiom,
! [P_3: produc393999548nt_int] :
~ ! [X_1: int,Y_1: product_prod_int_int] :
( P_3
!= ( produc282740534nt_int @ X_1 @ Y_1 ) ) ).
thf(fact_131_mem__def,axiom,
! [X_66: real,A_116: real > $o] :
( ( member_real @ X_66 @ A_116 )
<=> ( A_116 @ X_66 ) ) ).
thf(fact_132_prod_Oexhaust,axiom,
! [Y_55: product_prod_int_int] :
~ ! [A_64: int,B_89: int] :
( Y_55
!= ( product_Pair_int_int @ A_64 @ B_89 ) ) ).
thf(fact_133_prod_Oexhaust,axiom,
! [Y_55: produc393999548nt_int] :
~ ! [A_64: int,B_89: product_prod_int_int] :
( Y_55
!= ( produc282740534nt_int @ A_64 @ B_89 ) ) ).
thf(fact_134_curryI,axiom,
! [F_4: product_prod_int_int > $o,A_115: int,B_88: int] :
( ( F_4 @ ( product_Pair_int_int @ A_115 @ B_88 ) )
=> ( produc176579150_int_o @ F_4 @ A_115 @ B_88 ) ) ).
thf(fact_135_curryI,axiom,
! [F_4: produc393999548nt_int > $o,A_115: int,B_88: product_prod_int_int] :
( ( F_4 @ ( produc282740534nt_int @ A_115 @ B_88 ) )
=> ( produc122891093_int_o @ F_4 @ A_115 @ B_88 ) ) ).
thf(fact_136_le__add__iff1,axiom,
! [A_114: int,E_4: int,C_51: int,B_87: int,D_13: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_114 @ E_4 ) @ C_51 ) @ ( plus_plus_int @ ( times_times_int @ B_87 @ E_4 ) @ D_13 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_114 @ B_87 ) @ E_4 ) @ C_51 ) @ D_13 ) ) ).
thf(fact_137_le__add__iff1,axiom,
! [A_114: real,E_4: real,C_51: real,B_87: real,D_13: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_114 @ E_4 ) @ C_51 ) @ ( plus_plus_real @ ( times_times_real @ B_87 @ E_4 ) @ D_13 ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_114 @ B_87 ) @ E_4 ) @ C_51 ) @ D_13 ) ) ).
thf(fact_138_le__add__iff2,axiom,
! [A_113: int,E_3: int,C_50: int,B_86: int,D_12: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_113 @ E_3 ) @ C_50 ) @ ( plus_plus_int @ ( times_times_int @ B_86 @ E_3 ) @ D_12 ) )
<=> ( ord_less_eq_int @ C_50 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_86 @ A_113 ) @ E_3 ) @ D_12 ) ) ) ).
thf(fact_139_le__add__iff2,axiom,
! [A_113: real,E_3: real,C_50: real,B_86: real,D_12: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_113 @ E_3 ) @ C_50 ) @ ( plus_plus_real @ ( times_times_real @ B_86 @ E_3 ) @ D_12 ) )
<=> ( ord_less_eq_real @ C_50 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_86 @ A_113 ) @ E_3 ) @ D_12 ) ) ) ).
thf(fact_140_real__squared__diff__one__factored,axiom,
! [X_65: int] :
( ( minus_minus_int @ ( times_times_int @ X_65 @ X_65 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X_65 @ one_one_int ) @ ( minus_minus_int @ X_65 @ one_one_int ) ) ) ).
thf(fact_141_real__squared__diff__one__factored,axiom,
! [X_65: real] :
( ( minus_minus_real @ ( times_times_real @ X_65 @ X_65 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X_65 @ one_one_real ) @ ( minus_minus_real @ X_65 @ one_one_real ) ) ) ).
thf(fact_142_less__add__iff1,axiom,
! [A_112: int,E_2: int,C_49: int,B_85: int,D_11: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_112 @ E_2 ) @ C_49 ) @ ( plus_plus_int @ ( times_times_int @ B_85 @ E_2 ) @ D_11 ) )
<=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_112 @ B_85 ) @ E_2 ) @ C_49 ) @ D_11 ) ) ).
thf(fact_143_less__add__iff1,axiom,
! [A_112: real,E_2: real,C_49: real,B_85: real,D_11: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_112 @ E_2 ) @ C_49 ) @ ( plus_plus_real @ ( times_times_real @ B_85 @ E_2 ) @ D_11 ) )
<=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_112 @ B_85 ) @ E_2 ) @ C_49 ) @ D_11 ) ) ).
thf(fact_144_less__add__iff2,axiom,
! [A_111: int,E_1: int,C_48: int,B_84: int,D_10: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_111 @ E_1 ) @ C_48 ) @ ( plus_plus_int @ ( times_times_int @ B_84 @ E_1 ) @ D_10 ) )
<=> ( ord_less_int @ C_48 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_84 @ A_111 ) @ E_1 ) @ D_10 ) ) ) ).
thf(fact_145_less__add__iff2,axiom,
! [A_111: real,E_1: real,C_48: real,B_84: real,D_10: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_111 @ E_1 ) @ C_48 ) @ ( plus_plus_real @ ( times_times_real @ B_84 @ E_1 ) @ D_10 ) )
<=> ( ord_less_real @ C_48 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_84 @ A_111 ) @ E_1 ) @ D_10 ) ) ) ).
thf(fact_146_add__number__of__diff1,axiom,
! [V_18: int,W_12: int,C_47: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_18 ) @ ( minus_minus_int @ ( number_number_of_int @ W_12 ) @ C_47 ) )
= ( minus_minus_int @ ( number_number_of_int @ ( plus_plus_int @ V_18 @ W_12 ) ) @ C_47 ) ) ).
thf(fact_147_add__number__of__diff1,axiom,
! [V_18: int,W_12: int,C_47: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_18 ) @ ( minus_minus_real @ ( number267125858f_real @ W_12 ) @ C_47 ) )
= ( minus_minus_real @ ( number267125858f_real @ ( plus_plus_int @ V_18 @ W_12 ) ) @ C_47 ) ) ).
thf(fact_148_curry__conv,axiom,
! [F_3: product_prod_int_int > $o,A_110: int,B_83: int] :
( ( produc176579150_int_o @ F_3 @ A_110 @ B_83 )
<=> ( F_3 @ ( product_Pair_int_int @ A_110 @ B_83 ) ) ) ).
thf(fact_149_curry__conv,axiom,
! [F_3: produc393999548nt_int > $o,A_110: int,B_83: product_prod_int_int] :
( ( produc122891093_int_o @ F_3 @ A_110 @ B_83 )
<=> ( F_3 @ ( produc282740534nt_int @ A_110 @ B_83 ) ) ) ).
thf(fact_150_less__1__mult,axiom,
! [N_2: int,M_5: int] :
( ( ord_less_int @ one_one_int @ M_5 )
=> ( ( ord_less_int @ one_one_int @ N_2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M_5 @ N_2 ) ) ) ) ).
thf(fact_151_less__1__mult,axiom,
! [N_2: real,M_5: real] :
( ( ord_less_real @ one_one_real @ M_5 )
=> ( ( ord_less_real @ one_one_real @ N_2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M_5 @ N_2 ) ) ) ) ).
thf(fact_152_less__1__mult,axiom,
! [N_2: nat,M_5: nat] :
( ( ord_less_nat @ one_one_nat @ M_5 )
=> ( ( ord_less_nat @ one_one_nat @ N_2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M_5 @ N_2 ) ) ) ) ).
thf(fact_153_less__add__one,axiom,
! [A_109: int] : ( ord_less_int @ A_109 @ ( plus_plus_int @ A_109 @ one_one_int ) ) ).
thf(fact_154_less__add__one,axiom,
! [A_109: real] : ( ord_less_real @ A_109 @ ( plus_plus_real @ A_109 @ one_one_real ) ) ).
thf(fact_155_less__add__one,axiom,
! [A_109: nat] : ( ord_less_nat @ A_109 @ ( plus_plus_nat @ A_109 @ one_one_nat ) ) ).
thf(fact_156_add__less__le__mono,axiom,
! [C_46: int,D_9: int,A_108: int,B_82: int] :
( ( ord_less_int @ A_108 @ B_82 )
=> ( ( ord_less_eq_int @ C_46 @ D_9 )
=> ( ord_less_int @ ( plus_plus_int @ A_108 @ C_46 ) @ ( plus_plus_int @ B_82 @ D_9 ) ) ) ) ).
thf(fact_157_add__less__le__mono,axiom,
! [C_46: real,D_9: real,A_108: real,B_82: real] :
( ( ord_less_real @ A_108 @ B_82 )
=> ( ( ord_less_eq_real @ C_46 @ D_9 )
=> ( ord_less_real @ ( plus_plus_real @ A_108 @ C_46 ) @ ( plus_plus_real @ B_82 @ D_9 ) ) ) ) ).
thf(fact_158_add__less__le__mono,axiom,
! [C_46: nat,D_9: nat,A_108: nat,B_82: nat] :
( ( ord_less_nat @ A_108 @ B_82 )
=> ( ( ord_less_eq_nat @ C_46 @ D_9 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_108 @ C_46 ) @ ( plus_plus_nat @ B_82 @ D_9 ) ) ) ) ).
thf(fact_159_add__le__less__mono,axiom,
! [C_45: int,D_8: int,A_107: int,B_81: int] :
( ( ord_less_eq_int @ A_107 @ B_81 )
=> ( ( ord_less_int @ C_45 @ D_8 )
=> ( ord_less_int @ ( plus_plus_int @ A_107 @ C_45 ) @ ( plus_plus_int @ B_81 @ D_8 ) ) ) ) ).
thf(fact_160_add__le__less__mono,axiom,
! [C_45: real,D_8: real,A_107: real,B_81: real] :
( ( ord_less_eq_real @ A_107 @ B_81 )
=> ( ( ord_less_real @ C_45 @ D_8 )
=> ( ord_less_real @ ( plus_plus_real @ A_107 @ C_45 ) @ ( plus_plus_real @ B_81 @ D_8 ) ) ) ) ).
thf(fact_161_add__le__less__mono,axiom,
! [C_45: nat,D_8: nat,A_107: nat,B_81: nat] :
( ( ord_less_eq_nat @ A_107 @ B_81 )
=> ( ( ord_less_nat @ C_45 @ D_8 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_107 @ C_45 ) @ ( plus_plus_nat @ B_81 @ D_8 ) ) ) ) ).
thf(fact_162_one__reorient,axiom,
! [X_64: int] :
( ( one_one_int = X_64 )
<=> ( X_64 = one_one_int ) ) ).
thf(fact_163_one__reorient,axiom,
! [X_64: real] :
( ( one_one_real = X_64 )
<=> ( X_64 = one_one_real ) ) ).
thf(fact_164_one__reorient,axiom,
! [X_64: nat] :
( ( one_one_nat = X_64 )
<=> ( X_64 = one_one_nat ) ) ).
thf(fact_165_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_166_number__of__reorient,axiom,
! [W_11: int,X_63: int] :
( ( ( number_number_of_int @ W_11 )
= X_63 )
<=> ( X_63
= ( number_number_of_int @ W_11 ) ) ) ).
thf(fact_167_number__of__reorient,axiom,
! [W_11: int,X_63: real] :
( ( ( number267125858f_real @ W_11 )
= X_63 )
<=> ( X_63
= ( number267125858f_real @ W_11 ) ) ) ).
thf(fact_168_number__of__reorient,axiom,
! [W_11: int,X_63: nat] :
( ( ( number_number_of_nat @ W_11 )
= X_63 )
<=> ( X_63
= ( number_number_of_nat @ W_11 ) ) ) ).
thf(fact_169_le__number__of__eq__not__less,axiom,
! [V_17: int,W_10: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_17 ) @ ( number_number_of_int @ W_10 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_10 ) @ ( number_number_of_int @ V_17 ) ) ) ).
thf(fact_170_le__number__of__eq__not__less,axiom,
! [V_17: int,W_10: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ V_17 ) @ ( number267125858f_real @ W_10 ) )
<=> ~ ( ord_less_real @ ( number267125858f_real @ W_10 ) @ ( number267125858f_real @ V_17 ) ) ) ).
thf(fact_171_le__number__of__eq__not__less,axiom,
! [V_17: int,W_10: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_17 ) @ ( number_number_of_nat @ W_10 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_10 ) @ ( number_number_of_nat @ V_17 ) ) ) ).
thf(fact_172_eq__number__of,axiom,
! [X_62: int,Y_54: int] :
( ( ( number_number_of_int @ X_62 )
= ( number_number_of_int @ Y_54 ) )
<=> ( X_62 = Y_54 ) ) ).
thf(fact_173_eq__number__of,axiom,
! [X_62: int,Y_54: int] :
( ( ( number267125858f_real @ X_62 )
= ( number267125858f_real @ Y_54 ) )
<=> ( X_62 = Y_54 ) ) ).
thf(fact_174_less__number__of,axiom,
! [X_61: int,Y_53: int] :
( ( ord_less_int @ ( number_number_of_int @ X_61 ) @ ( number_number_of_int @ Y_53 ) )
<=> ( ord_less_int @ X_61 @ Y_53 ) ) ).
thf(fact_175_less__number__of,axiom,
! [X_61: int,Y_53: int] :
( ( ord_less_real @ ( number267125858f_real @ X_61 ) @ ( number267125858f_real @ Y_53 ) )
<=> ( ord_less_int @ X_61 @ Y_53 ) ) ).
thf(fact_176_le__number__of,axiom,
! [X_60: int,Y_52: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_60 ) @ ( number_number_of_int @ Y_52 ) )
<=> ( ord_less_eq_int @ X_60 @ Y_52 ) ) ).
thf(fact_177_le__number__of,axiom,
! [X_60: int,Y_52: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_60 ) @ ( number267125858f_real @ Y_52 ) )
<=> ( ord_less_eq_int @ X_60 @ Y_52 ) ) ).
thf(fact_178_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_179_linorder__neqE__linordered__idom,axiom,
! [X_59: int,Y_51: int] :
( ( X_59 != Y_51 )
=> ( ~ ( ord_less_int @ X_59 @ Y_51 )
=> ( ord_less_int @ Y_51 @ X_59 ) ) ) ).
thf(fact_180_linorder__neqE__linordered__idom,axiom,
! [X_59: real,Y_51: real] :
( ( X_59 != Y_51 )
=> ( ~ ( ord_less_real @ X_59 @ Y_51 )
=> ( ord_less_real @ Y_51 @ X_59 ) ) ) ).
thf(fact_181_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_182_order__le__neq__implies__less,axiom,
! [X_58: int,Y_50: int] :
( ( ord_less_eq_int @ X_58 @ Y_50 )
=> ( ( X_58 != Y_50 )
=> ( ord_less_int @ X_58 @ Y_50 ) ) ) ).
thf(fact_183_order__le__neq__implies__less,axiom,
! [X_58: real,Y_50: real] :
( ( ord_less_eq_real @ X_58 @ Y_50 )
=> ( ( X_58 != Y_50 )
=> ( ord_less_real @ X_58 @ Y_50 ) ) ) ).
thf(fact_184_order__le__neq__implies__less,axiom,
! [X_58: nat,Y_50: nat] :
( ( ord_less_eq_nat @ X_58 @ Y_50 )
=> ( ( X_58 != Y_50 )
=> ( ord_less_nat @ X_58 @ Y_50 ) ) ) ).
thf(fact_185_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_186_zadd__zless__mono,axiom,
! [Z_15: int,Z: int,W_9: int,W: int] :
( ( ord_less_int @ W_9 @ W )
=> ( ( ord_less_eq_int @ Z_15 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_9 @ Z_15 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_187_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_188_add__less__cancel__right,axiom,
! [A_106: int,C_44: int,B_80: int] :
( ( ord_less_int @ ( plus_plus_int @ A_106 @ C_44 ) @ ( plus_plus_int @ B_80 @ C_44 ) )
<=> ( ord_less_int @ A_106 @ B_80 ) ) ).
thf(fact_189_add__less__cancel__right,axiom,
! [A_106: real,C_44: real,B_80: real] :
( ( ord_less_real @ ( plus_plus_real @ A_106 @ C_44 ) @ ( plus_plus_real @ B_80 @ C_44 ) )
<=> ( ord_less_real @ A_106 @ B_80 ) ) ).
thf(fact_190_add__less__cancel__right,axiom,
! [A_106: nat,C_44: nat,B_80: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_106 @ C_44 ) @ ( plus_plus_nat @ B_80 @ C_44 ) )
<=> ( ord_less_nat @ A_106 @ B_80 ) ) ).
thf(fact_191_add__less__cancel__left,axiom,
! [C_43: int,A_105: int,B_79: int] :
( ( ord_less_int @ ( plus_plus_int @ C_43 @ A_105 ) @ ( plus_plus_int @ C_43 @ B_79 ) )
<=> ( ord_less_int @ A_105 @ B_79 ) ) ).
thf(fact_192_add__less__cancel__left,axiom,
! [C_43: real,A_105: real,B_79: real] :
( ( ord_less_real @ ( plus_plus_real @ C_43 @ A_105 ) @ ( plus_plus_real @ C_43 @ B_79 ) )
<=> ( ord_less_real @ A_105 @ B_79 ) ) ).
thf(fact_193_add__less__cancel__left,axiom,
! [C_43: nat,A_105: nat,B_79: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_43 @ A_105 ) @ ( plus_plus_nat @ C_43 @ B_79 ) )
<=> ( ord_less_nat @ A_105 @ B_79 ) ) ).
thf(fact_194_add__strict__right__mono,axiom,
! [C_42: int,A_104: int,B_78: int] :
( ( ord_less_int @ A_104 @ B_78 )
=> ( ord_less_int @ ( plus_plus_int @ A_104 @ C_42 ) @ ( plus_plus_int @ B_78 @ C_42 ) ) ) ).
thf(fact_195_add__strict__right__mono,axiom,
! [C_42: real,A_104: real,B_78: real] :
( ( ord_less_real @ A_104 @ B_78 )
=> ( ord_less_real @ ( plus_plus_real @ A_104 @ C_42 ) @ ( plus_plus_real @ B_78 @ C_42 ) ) ) ).
thf(fact_196_add__strict__right__mono,axiom,
! [C_42: nat,A_104: nat,B_78: nat] :
( ( ord_less_nat @ A_104 @ B_78 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_104 @ C_42 ) @ ( plus_plus_nat @ B_78 @ C_42 ) ) ) ).
thf(fact_197_add__strict__left__mono,axiom,
! [C_41: int,A_103: int,B_77: int] :
( ( ord_less_int @ A_103 @ B_77 )
=> ( ord_less_int @ ( plus_plus_int @ C_41 @ A_103 ) @ ( plus_plus_int @ C_41 @ B_77 ) ) ) ).
thf(fact_198_add__strict__left__mono,axiom,
! [C_41: real,A_103: real,B_77: real] :
( ( ord_less_real @ A_103 @ B_77 )
=> ( ord_less_real @ ( plus_plus_real @ C_41 @ A_103 ) @ ( plus_plus_real @ C_41 @ B_77 ) ) ) ).
thf(fact_199_add__strict__left__mono,axiom,
! [C_41: nat,A_103: nat,B_77: nat] :
( ( ord_less_nat @ A_103 @ B_77 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_41 @ A_103 ) @ ( plus_plus_nat @ C_41 @ B_77 ) ) ) ).
thf(fact_200_add__strict__mono,axiom,
! [C_40: int,D_7: int,A_102: int,B_76: int] :
( ( ord_less_int @ A_102 @ B_76 )
=> ( ( ord_less_int @ C_40 @ D_7 )
=> ( ord_less_int @ ( plus_plus_int @ A_102 @ C_40 ) @ ( plus_plus_int @ B_76 @ D_7 ) ) ) ) ).
thf(fact_201_add__strict__mono,axiom,
! [C_40: real,D_7: real,A_102: real,B_76: real] :
( ( ord_less_real @ A_102 @ B_76 )
=> ( ( ord_less_real @ C_40 @ D_7 )
=> ( ord_less_real @ ( plus_plus_real @ A_102 @ C_40 ) @ ( plus_plus_real @ B_76 @ D_7 ) ) ) ) ).
thf(fact_202_add__strict__mono,axiom,
! [C_40: nat,D_7: nat,A_102: nat,B_76: nat] :
( ( ord_less_nat @ A_102 @ B_76 )
=> ( ( ord_less_nat @ C_40 @ D_7 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_102 @ C_40 ) @ ( plus_plus_nat @ B_76 @ D_7 ) ) ) ) ).
thf(fact_203_add__less__imp__less__right,axiom,
! [A_101: int,C_39: int,B_75: int] :
( ( ord_less_int @ ( plus_plus_int @ A_101 @ C_39 ) @ ( plus_plus_int @ B_75 @ C_39 ) )
=> ( ord_less_int @ A_101 @ B_75 ) ) ).
thf(fact_204_add__less__imp__less__right,axiom,
! [A_101: real,C_39: real,B_75: real] :
( ( ord_less_real @ ( plus_plus_real @ A_101 @ C_39 ) @ ( plus_plus_real @ B_75 @ C_39 ) )
=> ( ord_less_real @ A_101 @ B_75 ) ) ).
thf(fact_205_add__less__imp__less__right,axiom,
! [A_101: nat,C_39: nat,B_75: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_101 @ C_39 ) @ ( plus_plus_nat @ B_75 @ C_39 ) )
=> ( ord_less_nat @ A_101 @ B_75 ) ) ).
thf(fact_206_add__less__imp__less__left,axiom,
! [C_38: int,A_100: int,B_74: int] :
( ( ord_less_int @ ( plus_plus_int @ C_38 @ A_100 ) @ ( plus_plus_int @ C_38 @ B_74 ) )
=> ( ord_less_int @ A_100 @ B_74 ) ) ).
thf(fact_207_add__less__imp__less__left,axiom,
! [C_38: real,A_100: real,B_74: real] :
( ( ord_less_real @ ( plus_plus_real @ C_38 @ A_100 ) @ ( plus_plus_real @ C_38 @ B_74 ) )
=> ( ord_less_real @ A_100 @ B_74 ) ) ).
thf(fact_208_add__less__imp__less__left,axiom,
! [C_38: nat,A_100: nat,B_74: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_38 @ A_100 ) @ ( plus_plus_nat @ C_38 @ B_74 ) )
=> ( ord_less_nat @ A_100 @ B_74 ) ) ).
thf(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_99: int] :
( ( times_times_int @ one_one_int @ A_99 )
= A_99 ) ).
thf(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_99: real] :
( ( times_times_real @ one_one_real @ A_99 )
= A_99 ) ).
thf(fact_211_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_99: nat] :
( ( times_times_nat @ one_one_nat @ A_99 )
= A_99 ) ).
thf(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_98: int] :
( ( times_times_int @ A_98 @ one_one_int )
= A_98 ) ).
thf(fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_98: real] :
( ( times_times_real @ A_98 @ one_one_real )
= A_98 ) ).
thf(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_98: nat] :
( ( times_times_nat @ A_98 @ one_one_nat )
= A_98 ) ).
thf(fact_215_mult__1__left,axiom,
! [A_97: int] :
( ( times_times_int @ one_one_int @ A_97 )
= A_97 ) ).
thf(fact_216_mult__1__left,axiom,
! [A_97: real] :
( ( times_times_real @ one_one_real @ A_97 )
= A_97 ) ).
thf(fact_217_mult__1__left,axiom,
! [A_97: nat] :
( ( times_times_nat @ one_one_nat @ A_97 )
= A_97 ) ).
thf(fact_218_mult__1,axiom,
! [A_96: int] :
( ( times_times_int @ one_one_int @ A_96 )
= A_96 ) ).
thf(fact_219_mult__1,axiom,
! [A_96: real] :
( ( times_times_real @ one_one_real @ A_96 )
= A_96 ) ).
thf(fact_220_mult__1,axiom,
! [A_96: nat] :
( ( times_times_nat @ one_one_nat @ A_96 )
= A_96 ) ).
thf(fact_221_mult__1__right,axiom,
! [A_95: int] :
( ( times_times_int @ A_95 @ one_one_int )
= A_95 ) ).
thf(fact_222_mult__1__right,axiom,
! [A_95: real] :
( ( times_times_real @ A_95 @ one_one_real )
= A_95 ) ).
thf(fact_223_mult__1__right,axiom,
! [A_95: nat] :
( ( times_times_nat @ A_95 @ one_one_nat )
= A_95 ) ).
thf(fact_224_mult_Ocomm__neutral,axiom,
! [A_94: int] :
( ( times_times_int @ A_94 @ one_one_int )
= A_94 ) ).
thf(fact_225_mult_Ocomm__neutral,axiom,
! [A_94: real] :
( ( times_times_real @ A_94 @ one_one_real )
= A_94 ) ).
thf(fact_226_mult_Ocomm__neutral,axiom,
! [A_94: nat] :
( ( times_times_nat @ A_94 @ one_one_nat )
= A_94 ) ).
thf(fact_227_diff__eq__diff__less,axiom,
! [A_93: int,B_73: int,C_37: int,D_6: int] :
( ( ( minus_minus_int @ A_93 @ B_73 )
= ( minus_minus_int @ C_37 @ D_6 ) )
=> ( ( ord_less_int @ A_93 @ B_73 )
<=> ( ord_less_int @ C_37 @ D_6 ) ) ) ).
thf(fact_228_diff__eq__diff__less,axiom,
! [A_93: real,B_73: real,C_37: real,D_6: real] :
( ( ( minus_minus_real @ A_93 @ B_73 )
= ( minus_minus_real @ C_37 @ D_6 ) )
=> ( ( ord_less_real @ A_93 @ B_73 )
<=> ( ord_less_real @ C_37 @ D_6 ) ) ) ).
thf(fact_229_add__le__cancel__right,axiom,
! [A_92: int,C_36: int,B_72: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_92 @ C_36 ) @ ( plus_plus_int @ B_72 @ C_36 ) )
<=> ( ord_less_eq_int @ A_92 @ B_72 ) ) ).
thf(fact_230_add__le__cancel__right,axiom,
! [A_92: real,C_36: real,B_72: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_92 @ C_36 ) @ ( plus_plus_real @ B_72 @ C_36 ) )
<=> ( ord_less_eq_real @ A_92 @ B_72 ) ) ).
thf(fact_231_add__le__cancel__right,axiom,
! [A_92: nat,C_36: nat,B_72: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_92 @ C_36 ) @ ( plus_plus_nat @ B_72 @ C_36 ) )
<=> ( ord_less_eq_nat @ A_92 @ B_72 ) ) ).
thf(fact_232_add__le__cancel__left,axiom,
! [C_35: int,A_91: int,B_71: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_35 @ A_91 ) @ ( plus_plus_int @ C_35 @ B_71 ) )
<=> ( ord_less_eq_int @ A_91 @ B_71 ) ) ).
thf(fact_233_add__le__cancel__left,axiom,
! [C_35: real,A_91: real,B_71: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_35 @ A_91 ) @ ( plus_plus_real @ C_35 @ B_71 ) )
<=> ( ord_less_eq_real @ A_91 @ B_71 ) ) ).
thf(fact_234_add__le__cancel__left,axiom,
! [C_35: nat,A_91: nat,B_71: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_35 @ A_91 ) @ ( plus_plus_nat @ C_35 @ B_71 ) )
<=> ( ord_less_eq_nat @ A_91 @ B_71 ) ) ).
thf(fact_235_add__right__mono,axiom,
! [C_34: int,A_90: int,B_70: int] :
( ( ord_less_eq_int @ A_90 @ B_70 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_90 @ C_34 ) @ ( plus_plus_int @ B_70 @ C_34 ) ) ) ).
thf(fact_236_add__right__mono,axiom,
! [C_34: real,A_90: real,B_70: real] :
( ( ord_less_eq_real @ A_90 @ B_70 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_90 @ C_34 ) @ ( plus_plus_real @ B_70 @ C_34 ) ) ) ).
thf(fact_237_add__right__mono,axiom,
! [C_34: nat,A_90: nat,B_70: nat] :
( ( ord_less_eq_nat @ A_90 @ B_70 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_90 @ C_34 ) @ ( plus_plus_nat @ B_70 @ C_34 ) ) ) ).
thf(fact_238_add__left__mono,axiom,
! [C_33: int,A_89: int,B_69: int] :
( ( ord_less_eq_int @ A_89 @ B_69 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_33 @ A_89 ) @ ( plus_plus_int @ C_33 @ B_69 ) ) ) ).
thf(fact_239_add__left__mono,axiom,
! [C_33: real,A_89: real,B_69: real] :
( ( ord_less_eq_real @ A_89 @ B_69 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C_33 @ A_89 ) @ ( plus_plus_real @ C_33 @ B_69 ) ) ) ).
thf(fact_240_add__left__mono,axiom,
! [C_33: nat,A_89: nat,B_69: nat] :
( ( ord_less_eq_nat @ A_89 @ B_69 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_33 @ A_89 ) @ ( plus_plus_nat @ C_33 @ B_69 ) ) ) ).
thf(fact_241_add__mono,axiom,
! [C_32: int,D_5: int,A_88: int,B_68: int] :
( ( ord_less_eq_int @ A_88 @ B_68 )
=> ( ( ord_less_eq_int @ C_32 @ D_5 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_88 @ C_32 ) @ ( plus_plus_int @ B_68 @ D_5 ) ) ) ) ).
thf(fact_242_add__mono,axiom,
! [C_32: real,D_5: real,A_88: real,B_68: real] :
( ( ord_less_eq_real @ A_88 @ B_68 )
=> ( ( ord_less_eq_real @ C_32 @ D_5 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_88 @ C_32 ) @ ( plus_plus_real @ B_68 @ D_5 ) ) ) ) ).
thf(fact_243_add__mono,axiom,
! [C_32: nat,D_5: nat,A_88: nat,B_68: nat] :
( ( ord_less_eq_nat @ A_88 @ B_68 )
=> ( ( ord_less_eq_nat @ C_32 @ D_5 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_88 @ C_32 ) @ ( plus_plus_nat @ B_68 @ D_5 ) ) ) ) ).
thf(fact_244_add__le__imp__le__right,axiom,
! [A_87: int,C_31: int,B_67: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_87 @ C_31 ) @ ( plus_plus_int @ B_67 @ C_31 ) )
=> ( ord_less_eq_int @ A_87 @ B_67 ) ) ).
thf(fact_245_add__le__imp__le__right,axiom,
! [A_87: real,C_31: real,B_67: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_87 @ C_31 ) @ ( plus_plus_real @ B_67 @ C_31 ) )
=> ( ord_less_eq_real @ A_87 @ B_67 ) ) ).
thf(fact_246_add__le__imp__le__right,axiom,
! [A_87: nat,C_31: nat,B_67: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_87 @ C_31 ) @ ( plus_plus_nat @ B_67 @ C_31 ) )
=> ( ord_less_eq_nat @ A_87 @ B_67 ) ) ).
thf(fact_247_add__le__imp__le__left,axiom,
! [C_30: int,A_86: int,B_66: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_30 @ A_86 ) @ ( plus_plus_int @ C_30 @ B_66 ) )
=> ( ord_less_eq_int @ A_86 @ B_66 ) ) ).
thf(fact_248_add__le__imp__le__left,axiom,
! [C_30: real,A_86: real,B_66: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_30 @ A_86 ) @ ( plus_plus_real @ C_30 @ B_66 ) )
=> ( ord_less_eq_real @ A_86 @ B_66 ) ) ).
thf(fact_249_add__le__imp__le__left,axiom,
! [C_30: nat,A_86: nat,B_66: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_30 @ A_86 ) @ ( plus_plus_nat @ C_30 @ B_66 ) )
=> ( ord_less_eq_nat @ A_86 @ B_66 ) ) ).
thf(fact_250_diff__eq__diff__less__eq,axiom,
! [A_85: int,B_65: int,C_29: int,D_4: int] :
( ( ( minus_minus_int @ A_85 @ B_65 )
= ( minus_minus_int @ C_29 @ D_4 ) )
=> ( ( ord_less_eq_int @ A_85 @ B_65 )
<=> ( ord_less_eq_int @ C_29 @ D_4 ) ) ) ).
thf(fact_251_diff__eq__diff__less__eq,axiom,
! [A_85: real,B_65: real,C_29: real,D_4: real] :
( ( ( minus_minus_real @ A_85 @ B_65 )
= ( minus_minus_real @ C_29 @ D_4 ) )
=> ( ( ord_less_eq_real @ A_85 @ B_65 )
<=> ( ord_less_eq_real @ C_29 @ D_4 ) ) ) ).
thf(fact_252_times__numeral__code_I5_J,axiom,
! [V: int,W: int] :
( ( times_times_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( times_times_int @ V @ W ) ) ) ).
thf(fact_253_plus__numeral__code_I9_J,axiom,
! [V: int,W: int] :
( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) ) ).
thf(fact_254_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_255_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_256_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_257_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_258_right__distrib__number__of,axiom,
! [V_16: int,B_64: int,C_28: int] :
( ( times_times_int @ ( number_number_of_int @ V_16 ) @ ( plus_plus_int @ B_64 @ C_28 ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ V_16 ) @ B_64 ) @ ( times_times_int @ ( number_number_of_int @ V_16 ) @ C_28 ) ) ) ).
thf(fact_259_right__distrib__number__of,axiom,
! [V_16: int,B_64: real,C_28: real] :
( ( times_times_real @ ( number267125858f_real @ V_16 ) @ ( plus_plus_real @ B_64 @ C_28 ) )
= ( plus_plus_real @ ( times_times_real @ ( number267125858f_real @ V_16 ) @ B_64 ) @ ( times_times_real @ ( number267125858f_real @ V_16 ) @ C_28 ) ) ) ).
thf(fact_260_right__distrib__number__of,axiom,
! [V_16: int,B_64: nat,C_28: nat] :
( ( times_times_nat @ ( number_number_of_nat @ V_16 ) @ ( plus_plus_nat @ B_64 @ C_28 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ V_16 ) @ B_64 ) @ ( times_times_nat @ ( number_number_of_nat @ V_16 ) @ C_28 ) ) ) ).
thf(fact_261_left__distrib__number__of,axiom,
! [A_84: int,B_63: int,V_15: int] :
( ( times_times_int @ ( plus_plus_int @ A_84 @ B_63 ) @ ( number_number_of_int @ V_15 ) )
= ( plus_plus_int @ ( times_times_int @ A_84 @ ( number_number_of_int @ V_15 ) ) @ ( times_times_int @ B_63 @ ( number_number_of_int @ V_15 ) ) ) ) ).
thf(fact_262_left__distrib__number__of,axiom,
! [A_84: real,B_63: real,V_15: int] :
( ( times_times_real @ ( plus_plus_real @ A_84 @ B_63 ) @ ( number267125858f_real @ V_15 ) )
= ( plus_plus_real @ ( times_times_real @ A_84 @ ( number267125858f_real @ V_15 ) ) @ ( times_times_real @ B_63 @ ( number267125858f_real @ V_15 ) ) ) ) ).
thf(fact_263_left__distrib__number__of,axiom,
! [A_84: nat,B_63: nat,V_15: int] :
( ( times_times_nat @ ( plus_plus_nat @ A_84 @ B_63 ) @ ( number_number_of_nat @ V_15 ) )
= ( plus_plus_nat @ ( times_times_nat @ A_84 @ ( number_number_of_nat @ V_15 ) ) @ ( times_times_nat @ B_63 @ ( number_number_of_nat @ V_15 ) ) ) ) ).
thf(fact_264_right__diff__distrib__number__of,axiom,
! [V_14: int,B_62: int,C_27: int] :
( ( times_times_int @ ( number_number_of_int @ V_14 ) @ ( minus_minus_int @ B_62 @ C_27 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ V_14 ) @ B_62 ) @ ( times_times_int @ ( number_number_of_int @ V_14 ) @ C_27 ) ) ) ).
thf(fact_265_right__diff__distrib__number__of,axiom,
! [V_14: int,B_62: real,C_27: real] :
( ( times_times_real @ ( number267125858f_real @ V_14 ) @ ( minus_minus_real @ B_62 @ C_27 ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ V_14 ) @ B_62 ) @ ( times_times_real @ ( number267125858f_real @ V_14 ) @ C_27 ) ) ) ).
thf(fact_266_left__diff__distrib__number__of,axiom,
! [A_83: int,B_61: int,V_13: int] :
( ( times_times_int @ ( minus_minus_int @ A_83 @ B_61 ) @ ( number_number_of_int @ V_13 ) )
= ( minus_minus_int @ ( times_times_int @ A_83 @ ( number_number_of_int @ V_13 ) ) @ ( times_times_int @ B_61 @ ( number_number_of_int @ V_13 ) ) ) ) ).
thf(fact_267_left__diff__distrib__number__of,axiom,
! [A_83: real,B_61: real,V_13: int] :
( ( times_times_real @ ( minus_minus_real @ A_83 @ B_61 ) @ ( number267125858f_real @ V_13 ) )
= ( minus_minus_real @ ( times_times_real @ A_83 @ ( number267125858f_real @ V_13 ) ) @ ( times_times_real @ B_61 @ ( number267125858f_real @ V_13 ) ) ) ) ).
thf(fact_268_number__of__mult,axiom,
! [V_12: int,W_8: int] :
( ( number_number_of_int @ ( times_times_int @ V_12 @ W_8 ) )
= ( times_times_int @ ( number_number_of_int @ V_12 ) @ ( number_number_of_int @ W_8 ) ) ) ).
thf(fact_269_number__of__mult,axiom,
! [V_12: int,W_8: int] :
( ( number267125858f_real @ ( times_times_int @ V_12 @ W_8 ) )
= ( times_times_real @ ( number267125858f_real @ V_12 ) @ ( number267125858f_real @ W_8 ) ) ) ).
thf(fact_270_arith__simps_I32_J,axiom,
! [V_11: int,W_7: int] :
( ( times_times_int @ ( number_number_of_int @ V_11 ) @ ( number_number_of_int @ W_7 ) )
= ( number_number_of_int @ ( times_times_int @ V_11 @ W_7 ) ) ) ).
thf(fact_271_arith__simps_I32_J,axiom,
! [V_11: int,W_7: int] :
( ( times_times_real @ ( number267125858f_real @ V_11 ) @ ( number267125858f_real @ W_7 ) )
= ( number267125858f_real @ ( times_times_int @ V_11 @ W_7 ) ) ) ).
thf(fact_272_mult__number__of__left,axiom,
! [V_10: int,W_6: int,Z_14: int] :
( ( times_times_int @ ( number_number_of_int @ V_10 ) @ ( times_times_int @ ( number_number_of_int @ W_6 ) @ Z_14 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_10 @ W_6 ) ) @ Z_14 ) ) ).
thf(fact_273_mult__number__of__left,axiom,
! [V_10: int,W_6: int,Z_14: real] :
( ( times_times_real @ ( number267125858f_real @ V_10 ) @ ( times_times_real @ ( number267125858f_real @ W_6 ) @ Z_14 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_10 @ W_6 ) ) @ Z_14 ) ) ).
thf(fact_274_number__of__add,axiom,
! [V_9: int,W_5: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_9 @ W_5 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_9 ) @ ( number_number_of_int @ W_5 ) ) ) ).
thf(fact_275_number__of__add,axiom,
! [V_9: int,W_5: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_9 @ W_5 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_9 ) @ ( number267125858f_real @ W_5 ) ) ) ).
thf(fact_276_add__number__of__eq,axiom,
! [V_8: int,W_4: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_8 ) @ ( number_number_of_int @ W_4 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_8 @ W_4 ) ) ) ).
thf(fact_277_add__number__of__eq,axiom,
! [V_8: int,W_4: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_8 ) @ ( number267125858f_real @ W_4 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_8 @ W_4 ) ) ) ).
thf(fact_278_add__number__of__left,axiom,
! [V_7: int,W_3: int,Z_13: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_7 ) @ ( plus_plus_int @ ( number_number_of_int @ W_3 ) @ Z_13 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_7 @ W_3 ) ) @ Z_13 ) ) ).
thf(fact_279_add__number__of__left,axiom,
! [V_7: int,W_3: int,Z_13: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_7 ) @ ( plus_plus_real @ ( number267125858f_real @ W_3 ) @ Z_13 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_7 @ W_3 ) ) @ Z_13 ) ) ).
thf(fact_280_number__of__diff,axiom,
! [V_6: int,W_2: int] :
( ( number_number_of_int @ ( minus_minus_int @ V_6 @ W_2 ) )
= ( minus_minus_int @ ( number_number_of_int @ V_6 ) @ ( number_number_of_int @ W_2 ) ) ) ).
thf(fact_281_number__of__diff,axiom,
! [V_6: int,W_2: int] :
( ( number267125858f_real @ ( minus_minus_int @ V_6 @ W_2 ) )
= ( minus_minus_real @ ( number267125858f_real @ V_6 ) @ ( number267125858f_real @ W_2 ) ) ) ).
thf(fact_282_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4: int] :
( ( plus_plus_int @ M_4 @ M_4 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_4 ) ) ).
thf(fact_283_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4: real] :
( ( plus_plus_real @ M_4 @ M_4 )
= ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ M_4 ) ) ).
thf(fact_284_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4: nat] :
( ( plus_plus_nat @ M_4 @ M_4 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_4 ) ) ).
thf(fact_285_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_3: int,A_82: int] :
( ( plus_plus_int @ M_3 @ ( times_times_int @ A_82 @ M_3 ) )
= ( times_times_int @ ( plus_plus_int @ A_82 @ one_one_int ) @ M_3 ) ) ).
thf(fact_286_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_3: real,A_82: real] :
( ( plus_plus_real @ M_3 @ ( times_times_real @ A_82 @ M_3 ) )
= ( times_times_real @ ( plus_plus_real @ A_82 @ one_one_real ) @ M_3 ) ) ).
thf(fact_287_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_3: nat,A_82: nat] :
( ( plus_plus_nat @ M_3 @ ( times_times_nat @ A_82 @ M_3 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_82 @ one_one_nat ) @ M_3 ) ) ).
thf(fact_288_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_81: int,M_2: int] :
( ( plus_plus_int @ ( times_times_int @ A_81 @ M_2 ) @ M_2 )
= ( times_times_int @ ( plus_plus_int @ A_81 @ one_one_int ) @ M_2 ) ) ).
thf(fact_289_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_81: real,M_2: real] :
( ( plus_plus_real @ ( times_times_real @ A_81 @ M_2 ) @ M_2 )
= ( times_times_real @ ( plus_plus_real @ A_81 @ one_one_real ) @ M_2 ) ) ).
thf(fact_290_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_81: nat,M_2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_81 @ M_2 ) @ M_2 )
= ( times_times_nat @ ( plus_plus_nat @ A_81 @ one_one_nat ) @ M_2 ) ) ).
thf(fact_291_curryD,axiom,
! [F_2: product_prod_int_int > $o,A_80: int,B_60: int] :
( ( produc176579150_int_o @ F_2 @ A_80 @ B_60 )
=> ( F_2 @ ( product_Pair_int_int @ A_80 @ B_60 ) ) ) ).
thf(fact_292_curryD,axiom,
! [F_2: produc393999548nt_int > $o,A_80: int,B_60: product_prod_int_int] :
( ( produc122891093_int_o @ F_2 @ A_80 @ B_60 )
=> ( F_2 @ ( produc282740534nt_int @ A_80 @ B_60 ) ) ) ).
thf(fact_293_curryE,axiom,
! [F_1: product_prod_int_int > $o,A_79: int,B_59: int] :
( ( produc176579150_int_o @ F_1 @ A_79 @ B_59 )
=> ( F_1 @ ( product_Pair_int_int @ A_79 @ B_59 ) ) ) ).
thf(fact_294_curryE,axiom,
! [F_1: produc393999548nt_int > $o,A_79: int,B_59: product_prod_int_int] :
( ( produc122891093_int_o @ F_1 @ A_79 @ B_59 )
=> ( F_1 @ ( produc282740534nt_int @ A_79 @ B_59 ) ) ) ).
thf(fact_295_order__refl,axiom,
! [X_57: int] : ( ord_less_eq_int @ X_57 @ X_57 ) ).
thf(fact_296_order__refl,axiom,
! [X_57: real] : ( ord_less_eq_real @ X_57 @ X_57 ) ).
thf(fact_297_order__refl,axiom,
! [X_57: nat] : ( ord_less_eq_nat @ X_57 @ X_57 ) ).
thf(fact_298_xt1_I8_J,axiom,
! [Z_12: int,Y_49: int,X_56: int] :
( ( ord_less_eq_int @ Y_49 @ X_56 )
=> ( ( ord_less_int @ Z_12 @ Y_49 )
=> ( ord_less_int @ Z_12 @ X_56 ) ) ) ).
thf(fact_299_xt1_I8_J,axiom,
! [Z_12: real,Y_49: real,X_56: real] :
( ( ord_less_eq_real @ Y_49 @ X_56 )
=> ( ( ord_less_real @ Z_12 @ Y_49 )
=> ( ord_less_real @ Z_12 @ X_56 ) ) ) ).
thf(fact_300_xt1_I8_J,axiom,
! [Z_12: nat,Y_49: nat,X_56: nat] :
( ( ord_less_eq_nat @ Y_49 @ X_56 )
=> ( ( ord_less_nat @ Z_12 @ Y_49 )
=> ( ord_less_nat @ Z_12 @ X_56 ) ) ) ).
thf(fact_301_order__le__less__trans,axiom,
! [Z_11: int,X_55: int,Y_48: int] :
( ( ord_less_eq_int @ X_55 @ Y_48 )
=> ( ( ord_less_int @ Y_48 @ Z_11 )
=> ( ord_less_int @ X_55 @ Z_11 ) ) ) ).
thf(fact_302_order__le__less__trans,axiom,
! [Z_11: real,X_55: real,Y_48: real] :
( ( ord_less_eq_real @ X_55 @ Y_48 )
=> ( ( ord_less_real @ Y_48 @ Z_11 )
=> ( ord_less_real @ X_55 @ Z_11 ) ) ) ).
thf(fact_303_order__le__less__trans,axiom,
! [Z_11: nat,X_55: nat,Y_48: nat] :
( ( ord_less_eq_nat @ X_55 @ Y_48 )
=> ( ( ord_less_nat @ Y_48 @ Z_11 )
=> ( ord_less_nat @ X_55 @ Z_11 ) ) ) ).
thf(fact_304_xt1_I7_J,axiom,
! [Z_10: int,Y_47: int,X_54: int] :
( ( ord_less_int @ Y_47 @ X_54 )
=> ( ( ord_less_eq_int @ Z_10 @ Y_47 )
=> ( ord_less_int @ Z_10 @ X_54 ) ) ) ).
thf(fact_305_xt1_I7_J,axiom,
! [Z_10: real,Y_47: real,X_54: real] :
( ( ord_less_real @ Y_47 @ X_54 )
=> ( ( ord_less_eq_real @ Z_10 @ Y_47 )
=> ( ord_less_real @ Z_10 @ X_54 ) ) ) ).
thf(fact_306_xt1_I7_J,axiom,
! [Z_10: nat,Y_47: nat,X_54: nat] :
( ( ord_less_nat @ Y_47 @ X_54 )
=> ( ( ord_less_eq_nat @ Z_10 @ Y_47 )
=> ( ord_less_nat @ Z_10 @ X_54 ) ) ) ).
thf(fact_307_order__less__le__trans,axiom,
! [Z_9: int,X_53: int,Y_46: int] :
( ( ord_less_int @ X_53 @ Y_46 )
=> ( ( ord_less_eq_int @ Y_46 @ Z_9 )
=> ( ord_less_int @ X_53 @ Z_9 ) ) ) ).
thf(fact_308_order__less__le__trans,axiom,
! [Z_9: real,X_53: real,Y_46: real] :
( ( ord_less_real @ X_53 @ Y_46 )
=> ( ( ord_less_eq_real @ Y_46 @ Z_9 )
=> ( ord_less_real @ X_53 @ Z_9 ) ) ) ).
thf(fact_309_order__less__le__trans,axiom,
! [Z_9: nat,X_53: nat,Y_46: nat] :
( ( ord_less_nat @ X_53 @ Y_46 )
=> ( ( ord_less_eq_nat @ Y_46 @ Z_9 )
=> ( ord_less_nat @ X_53 @ Z_9 ) ) ) ).
thf(fact_310_xt1_I11_J,axiom,
! [B_58: int,A_78: int] :
( ( ord_less_eq_int @ B_58 @ A_78 )
=> ( ( A_78 != B_58 )
=> ( ord_less_int @ B_58 @ A_78 ) ) ) ).
thf(fact_311_xt1_I11_J,axiom,
! [B_58: real,A_78: real] :
( ( ord_less_eq_real @ B_58 @ A_78 )
=> ( ( A_78 != B_58 )
=> ( ord_less_real @ B_58 @ A_78 ) ) ) ).
thf(fact_312_xt1_I11_J,axiom,
! [B_58: nat,A_78: nat] :
( ( ord_less_eq_nat @ B_58 @ A_78 )
=> ( ( A_78 != B_58 )
=> ( ord_less_nat @ B_58 @ A_78 ) ) ) ).
thf(fact_313_order__le__neq__trans,axiom,
! [A_77: int,B_57: int] :
( ( ord_less_eq_int @ A_77 @ B_57 )
=> ( ( A_77 != B_57 )
=> ( ord_less_int @ A_77 @ B_57 ) ) ) ).
thf(fact_314_order__le__neq__trans,axiom,
! [A_77: real,B_57: real] :
( ( ord_less_eq_real @ A_77 @ B_57 )
=> ( ( A_77 != B_57 )
=> ( ord_less_real @ A_77 @ B_57 ) ) ) ).
thf(fact_315_order__le__neq__trans,axiom,
! [A_77: nat,B_57: nat] :
( ( ord_less_eq_nat @ A_77 @ B_57 )
=> ( ( A_77 != B_57 )
=> ( ord_less_nat @ A_77 @ B_57 ) ) ) ).
thf(fact_316_order__le__imp__less__or__eq,axiom,
! [X_52: int,Y_45: int] :
( ( ord_less_eq_int @ X_52 @ Y_45 )
=> ( ( ord_less_int @ X_52 @ Y_45 )
| ( X_52 = Y_45 ) ) ) ).
thf(fact_317_order__le__imp__less__or__eq,axiom,
! [X_52: real,Y_45: real] :
( ( ord_less_eq_real @ X_52 @ Y_45 )
=> ( ( ord_less_real @ X_52 @ Y_45 )
| ( X_52 = Y_45 ) ) ) ).
thf(fact_318_order__le__imp__less__or__eq,axiom,
! [X_52: nat,Y_45: nat] :
( ( ord_less_eq_nat @ X_52 @ Y_45 )
=> ( ( ord_less_nat @ X_52 @ Y_45 )
| ( X_52 = Y_45 ) ) ) ).
thf(fact_319_linorder__antisym__conv2,axiom,
! [X_51: int,Y_44: int] :
( ( ord_less_eq_int @ X_51 @ Y_44 )
=> ( ~ ( ord_less_int @ X_51 @ Y_44 )
<=> ( X_51 = Y_44 ) ) ) ).
thf(fact_320_linorder__antisym__conv2,axiom,
! [X_51: real,Y_44: real] :
( ( ord_less_eq_real @ X_51 @ Y_44 )
=> ( ~ ( ord_less_real @ X_51 @ Y_44 )
<=> ( X_51 = Y_44 ) ) ) ).
thf(fact_321_linorder__antisym__conv2,axiom,
! [X_51: nat,Y_44: nat] :
( ( ord_less_eq_nat @ X_51 @ Y_44 )
=> ( ~ ( ord_less_nat @ X_51 @ Y_44 )
<=> ( X_51 = Y_44 ) ) ) ).
thf(fact_322_order__less__imp__le,axiom,
! [X_50: int,Y_43: int] :
( ( ord_less_int @ X_50 @ Y_43 )
=> ( ord_less_eq_int @ X_50 @ Y_43 ) ) ).
thf(fact_323_order__less__imp__le,axiom,
! [X_50: real,Y_43: real] :
( ( ord_less_real @ X_50 @ Y_43 )
=> ( ord_less_eq_real @ X_50 @ Y_43 ) ) ).
thf(fact_324_order__less__imp__le,axiom,
! [X_50: nat,Y_43: nat] :
( ( ord_less_nat @ X_50 @ Y_43 )
=> ( ord_less_eq_nat @ X_50 @ Y_43 ) ) ).
thf(fact_325_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_326_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_327_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_328_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_329_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_330_less__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_331_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_332_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_333_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_334_linorder__linear,axiom,
! [X_49: int,Y_42: int] :
( ( ord_less_eq_int @ X_49 @ Y_42 )
| ( ord_less_eq_int @ Y_42 @ X_49 ) ) ).
thf(fact_335_linorder__linear,axiom,
! [X_49: real,Y_42: real] :
( ( ord_less_eq_real @ X_49 @ Y_42 )
| ( ord_less_eq_real @ Y_42 @ X_49 ) ) ).
thf(fact_336_linorder__linear,axiom,
! [X_49: nat,Y_42: nat] :
( ( ord_less_eq_nat @ X_49 @ Y_42 )
| ( ord_less_eq_nat @ Y_42 @ X_49 ) ) ).
thf(fact_337_order__eq__iff,axiom,
! [X_48: int,Y_41: int] :
( ( X_48 = Y_41 )
<=> ( ( ord_less_eq_int @ X_48 @ Y_41 )
& ( ord_less_eq_int @ Y_41 @ X_48 ) ) ) ).
thf(fact_338_order__eq__iff,axiom,
! [X_48: real,Y_41: real] :
( ( X_48 = Y_41 )
<=> ( ( ord_less_eq_real @ X_48 @ Y_41 )
& ( ord_less_eq_real @ Y_41 @ X_48 ) ) ) ).
thf(fact_339_order__eq__iff,axiom,
! [X_48: nat,Y_41: nat] :
( ( X_48 = Y_41 )
<=> ( ( ord_less_eq_nat @ X_48 @ Y_41 )
& ( ord_less_eq_nat @ Y_41 @ X_48 ) ) ) ).
thf(fact_340_order__eq__refl,axiom,
! [X_47: int,Y_40: int] :
( ( X_47 = Y_40 )
=> ( ord_less_eq_int @ X_47 @ Y_40 ) ) ).
thf(fact_341_order__eq__refl,axiom,
! [X_47: real,Y_40: real] :
( ( X_47 = Y_40 )
=> ( ord_less_eq_real @ X_47 @ Y_40 ) ) ).
thf(fact_342_order__eq__refl,axiom,
! [X_47: nat,Y_40: nat] :
( ( X_47 = Y_40 )
=> ( ord_less_eq_nat @ X_47 @ Y_40 ) ) ).
thf(fact_343_order__antisym__conv,axiom,
! [Y_39: int,X_46: int] :
( ( ord_less_eq_int @ Y_39 @ X_46 )
=> ( ( ord_less_eq_int @ X_46 @ Y_39 )
<=> ( X_46 = Y_39 ) ) ) ).
thf(fact_344_order__antisym__conv,axiom,
! [Y_39: real,X_46: real] :
( ( ord_less_eq_real @ Y_39 @ X_46 )
=> ( ( ord_less_eq_real @ X_46 @ Y_39 )
<=> ( X_46 = Y_39 ) ) ) ).
thf(fact_345_order__antisym__conv,axiom,
! [Y_39: nat,X_46: nat] :
( ( ord_less_eq_nat @ Y_39 @ X_46 )
=> ( ( ord_less_eq_nat @ X_46 @ Y_39 )
<=> ( X_46 = Y_39 ) ) ) ).
thf(fact_346_ord__eq__le__trans,axiom,
! [C_26: int,A_76: int,B_56: int] :
( ( A_76 = B_56 )
=> ( ( ord_less_eq_int @ B_56 @ C_26 )
=> ( ord_less_eq_int @ A_76 @ C_26 ) ) ) ).
thf(fact_347_ord__eq__le__trans,axiom,
! [C_26: real,A_76: real,B_56: real] :
( ( A_76 = B_56 )
=> ( ( ord_less_eq_real @ B_56 @ C_26 )
=> ( ord_less_eq_real @ A_76 @ C_26 ) ) ) ).
thf(fact_348_ord__eq__le__trans,axiom,
! [C_26: nat,A_76: nat,B_56: nat] :
( ( A_76 = B_56 )
=> ( ( ord_less_eq_nat @ B_56 @ C_26 )
=> ( ord_less_eq_nat @ A_76 @ C_26 ) ) ) ).
thf(fact_349_xt1_I3_J,axiom,
! [C_25: int,A_75: int,B_55: int] :
( ( A_75 = B_55 )
=> ( ( ord_less_eq_int @ C_25 @ B_55 )
=> ( ord_less_eq_int @ C_25 @ A_75 ) ) ) ).
thf(fact_350_xt1_I3_J,axiom,
! [C_25: real,A_75: real,B_55: real] :
( ( A_75 = B_55 )
=> ( ( ord_less_eq_real @ C_25 @ B_55 )
=> ( ord_less_eq_real @ C_25 @ A_75 ) ) ) ).
thf(fact_351_xt1_I3_J,axiom,
! [C_25: nat,A_75: nat,B_55: nat] :
( ( A_75 = B_55 )
=> ( ( ord_less_eq_nat @ C_25 @ B_55 )
=> ( ord_less_eq_nat @ C_25 @ A_75 ) ) ) ).
thf(fact_352_ord__le__eq__trans,axiom,
! [C_24: int,A_74: int,B_54: int] :
( ( ord_less_eq_int @ A_74 @ B_54 )
=> ( ( B_54 = C_24 )
=> ( ord_less_eq_int @ A_74 @ C_24 ) ) ) ).
thf(fact_353_ord__le__eq__trans,axiom,
! [C_24: real,A_74: real,B_54: real] :
( ( ord_less_eq_real @ A_74 @ B_54 )
=> ( ( B_54 = C_24 )
=> ( ord_less_eq_real @ A_74 @ C_24 ) ) ) ).
thf(fact_354_ord__le__eq__trans,axiom,
! [C_24: nat,A_74: nat,B_54: nat] :
( ( ord_less_eq_nat @ A_74 @ B_54 )
=> ( ( B_54 = C_24 )
=> ( ord_less_eq_nat @ A_74 @ C_24 ) ) ) ).
thf(fact_355_xt1_I4_J,axiom,
! [C_23: int,B_53: int,A_73: int] :
( ( ord_less_eq_int @ B_53 @ A_73 )
=> ( ( B_53 = C_23 )
=> ( ord_less_eq_int @ C_23 @ A_73 ) ) ) ).
thf(fact_356_xt1_I4_J,axiom,
! [C_23: real,B_53: real,A_73: real] :
( ( ord_less_eq_real @ B_53 @ A_73 )
=> ( ( B_53 = C_23 )
=> ( ord_less_eq_real @ C_23 @ A_73 ) ) ) ).
thf(fact_357_xt1_I4_J,axiom,
! [C_23: nat,B_53: nat,A_73: nat] :
( ( ord_less_eq_nat @ B_53 @ A_73 )
=> ( ( B_53 = C_23 )
=> ( ord_less_eq_nat @ C_23 @ A_73 ) ) ) ).
thf(fact_358_order__antisym,axiom,
! [X_45: int,Y_38: int] :
( ( ord_less_eq_int @ X_45 @ Y_38 )
=> ( ( ord_less_eq_int @ Y_38 @ X_45 )
=> ( X_45 = Y_38 ) ) ) ).
thf(fact_359_order__antisym,axiom,
! [X_45: real,Y_38: real] :
( ( ord_less_eq_real @ X_45 @ Y_38 )
=> ( ( ord_less_eq_real @ Y_38 @ X_45 )
=> ( X_45 = Y_38 ) ) ) ).
thf(fact_360_order__antisym,axiom,
! [X_45: nat,Y_38: nat] :
( ( ord_less_eq_nat @ X_45 @ Y_38 )
=> ( ( ord_less_eq_nat @ Y_38 @ X_45 )
=> ( X_45 = Y_38 ) ) ) ).
thf(fact_361_order__trans,axiom,
! [Z_8: int,X_44: int,Y_37: int] :
( ( ord_less_eq_int @ X_44 @ Y_37 )
=> ( ( ord_less_eq_int @ Y_37 @ Z_8 )
=> ( ord_less_eq_int @ X_44 @ Z_8 ) ) ) ).
thf(fact_362_order__trans,axiom,
! [Z_8: real,X_44: real,Y_37: real] :
( ( ord_less_eq_real @ X_44 @ Y_37 )
=> ( ( ord_less_eq_real @ Y_37 @ Z_8 )
=> ( ord_less_eq_real @ X_44 @ Z_8 ) ) ) ).
thf(fact_363_order__trans,axiom,
! [Z_8: nat,X_44: nat,Y_37: nat] :
( ( ord_less_eq_nat @ X_44 @ Y_37 )
=> ( ( ord_less_eq_nat @ Y_37 @ Z_8 )
=> ( ord_less_eq_nat @ X_44 @ Z_8 ) ) ) ).
thf(fact_364_xt1_I5_J,axiom,
! [Y_36: int,X_43: int] :
( ( ord_less_eq_int @ Y_36 @ X_43 )
=> ( ( ord_less_eq_int @ X_43 @ Y_36 )
=> ( X_43 = Y_36 ) ) ) ).
thf(fact_365_xt1_I5_J,axiom,
! [Y_36: real,X_43: real] :
( ( ord_less_eq_real @ Y_36 @ X_43 )
=> ( ( ord_less_eq_real @ X_43 @ Y_36 )
=> ( X_43 = Y_36 ) ) ) ).
thf(fact_366_xt1_I5_J,axiom,
! [Y_36: nat,X_43: nat] :
( ( ord_less_eq_nat @ Y_36 @ X_43 )
=> ( ( ord_less_eq_nat @ X_43 @ Y_36 )
=> ( X_43 = Y_36 ) ) ) ).
thf(fact_367_xt1_I6_J,axiom,
! [Z_7: int,Y_35: int,X_42: int] :
( ( ord_less_eq_int @ Y_35 @ X_42 )
=> ( ( ord_less_eq_int @ Z_7 @ Y_35 )
=> ( ord_less_eq_int @ Z_7 @ X_42 ) ) ) ).
thf(fact_368_xt1_I6_J,axiom,
! [Z_7: real,Y_35: real,X_42: real] :
( ( ord_less_eq_real @ Y_35 @ X_42 )
=> ( ( ord_less_eq_real @ Z_7 @ Y_35 )
=> ( ord_less_eq_real @ Z_7 @ X_42 ) ) ) ).
thf(fact_369_xt1_I6_J,axiom,
! [Z_7: nat,Y_35: nat,X_42: nat] :
( ( ord_less_eq_nat @ Y_35 @ X_42 )
=> ( ( ord_less_eq_nat @ Z_7 @ Y_35 )
=> ( ord_less_eq_nat @ Z_7 @ X_42 ) ) ) ).
thf(fact_370_linorder__le__cases,axiom,
! [X_41: int,Y_34: int] :
( ~ ( ord_less_eq_int @ X_41 @ Y_34 )
=> ( ord_less_eq_int @ Y_34 @ X_41 ) ) ).
thf(fact_371_linorder__le__cases,axiom,
! [X_41: real,Y_34: real] :
( ~ ( ord_less_eq_real @ X_41 @ Y_34 )
=> ( ord_less_eq_real @ Y_34 @ X_41 ) ) ).
thf(fact_372_linorder__le__cases,axiom,
! [X_41: nat,Y_34: nat] :
( ~ ( ord_less_eq_nat @ X_41 @ Y_34 )
=> ( ord_less_eq_nat @ Y_34 @ X_41 ) ) ).
thf(fact_373_order__less__irrefl,axiom,
! [X_40: int] :
~ ( ord_less_int @ X_40 @ X_40 ) ).
thf(fact_374_order__less__irrefl,axiom,
! [X_40: real] :
~ ( ord_less_real @ X_40 @ X_40 ) ).
thf(fact_375_order__less__irrefl,axiom,
! [X_40: nat] :
~ ( ord_less_nat @ X_40 @ X_40 ) ).
thf(fact_376_linorder__neq__iff,axiom,
! [X_39: int,Y_33: int] :
( ( X_39 != Y_33 )
<=> ( ( ord_less_int @ X_39 @ Y_33 )
| ( ord_less_int @ Y_33 @ X_39 ) ) ) ).
thf(fact_377_linorder__neq__iff,axiom,
! [X_39: real,Y_33: real] :
( ( X_39 != Y_33 )
<=> ( ( ord_less_real @ X_39 @ Y_33 )
| ( ord_less_real @ Y_33 @ X_39 ) ) ) ).
thf(fact_378_linorder__neq__iff,axiom,
! [X_39: nat,Y_33: nat] :
( ( X_39 != Y_33 )
<=> ( ( ord_less_nat @ X_39 @ Y_33 )
| ( ord_less_nat @ Y_33 @ X_39 ) ) ) ).
thf(fact_379_not__less__iff__gr__or__eq,axiom,
! [X_38: int,Y_32: int] :
( ~ ( ord_less_int @ X_38 @ Y_32 )
<=> ( ( ord_less_int @ Y_32 @ X_38 )
| ( X_38 = Y_32 ) ) ) ).
thf(fact_380_not__less__iff__gr__or__eq,axiom,
! [X_38: real,Y_32: real] :
( ~ ( ord_less_real @ X_38 @ Y_32 )
<=> ( ( ord_less_real @ Y_32 @ X_38 )
| ( X_38 = Y_32 ) ) ) ).
thf(fact_381_not__less__iff__gr__or__eq,axiom,
! [X_38: nat,Y_32: nat] :
( ~ ( ord_less_nat @ X_38 @ Y_32 )
<=> ( ( ord_less_nat @ Y_32 @ X_38 )
| ( X_38 = Y_32 ) ) ) ).
thf(fact_382_linorder__less__linear,axiom,
! [X_37: int,Y_31: int] :
( ( ord_less_int @ X_37 @ Y_31 )
| ( X_37 = Y_31 )
| ( ord_less_int @ Y_31 @ X_37 ) ) ).
thf(fact_383_linorder__less__linear,axiom,
! [X_37: real,Y_31: real] :
( ( ord_less_real @ X_37 @ Y_31 )
| ( X_37 = Y_31 )
| ( ord_less_real @ Y_31 @ X_37 ) ) ).
thf(fact_384_linorder__less__linear,axiom,
! [X_37: nat,Y_31: nat] :
( ( ord_less_nat @ X_37 @ Y_31 )
| ( X_37 = Y_31 )
| ( ord_less_nat @ Y_31 @ X_37 ) ) ).
thf(fact_385_linorder__antisym__conv3,axiom,
! [Y_30: int,X_36: int] :
( ~ ( ord_less_int @ Y_30 @ X_36 )
=> ( ~ ( ord_less_int @ X_36 @ Y_30 )
<=> ( X_36 = Y_30 ) ) ) ).
thf(fact_386_linorder__antisym__conv3,axiom,
! [Y_30: real,X_36: real] :
( ~ ( ord_less_real @ Y_30 @ X_36 )
=> ( ~ ( ord_less_real @ X_36 @ Y_30 )
<=> ( X_36 = Y_30 ) ) ) ).
thf(fact_387_linorder__antisym__conv3,axiom,
! [Y_30: nat,X_36: nat] :
( ~ ( ord_less_nat @ Y_30 @ X_36 )
=> ( ~ ( ord_less_nat @ X_36 @ Y_30 )
<=> ( X_36 = Y_30 ) ) ) ).
thf(fact_388_linorder__neqE,axiom,
! [X_35: int,Y_29: int] :
( ( X_35 != Y_29 )
=> ( ~ ( ord_less_int @ X_35 @ Y_29 )
=> ( ord_less_int @ Y_29 @ X_35 ) ) ) ).
thf(fact_389_linorder__neqE,axiom,
! [X_35: real,Y_29: real] :
( ( X_35 != Y_29 )
=> ( ~ ( ord_less_real @ X_35 @ Y_29 )
=> ( ord_less_real @ Y_29 @ X_35 ) ) ) ).
thf(fact_390_linorder__neqE,axiom,
! [X_35: nat,Y_29: nat] :
( ( X_35 != Y_29 )
=> ( ~ ( ord_less_nat @ X_35 @ Y_29 )
=> ( ord_less_nat @ Y_29 @ X_35 ) ) ) ).
thf(fact_391_less__imp__neq,axiom,
! [X_34: int,Y_28: int] :
( ( ord_less_int @ X_34 @ Y_28 )
=> ( X_34 != Y_28 ) ) ).
thf(fact_392_less__imp__neq,axiom,
! [X_34: real,Y_28: real] :
( ( ord_less_real @ X_34 @ Y_28 )
=> ( X_34 != Y_28 ) ) ).
thf(fact_393_less__imp__neq,axiom,
! [X_34: nat,Y_28: nat] :
( ( ord_less_nat @ X_34 @ Y_28 )
=> ( X_34 != Y_28 ) ) ).
thf(fact_394_order__less__not__sym,axiom,
! [X_33: int,Y_27: int] :
( ( ord_less_int @ X_33 @ Y_27 )
=> ~ ( ord_less_int @ Y_27 @ X_33 ) ) ).
thf(fact_395_order__less__not__sym,axiom,
! [X_33: real,Y_27: real] :
( ( ord_less_real @ X_33 @ Y_27 )
=> ~ ( ord_less_real @ Y_27 @ X_33 ) ) ).
thf(fact_396_order__less__not__sym,axiom,
! [X_33: nat,Y_27: nat] :
( ( ord_less_nat @ X_33 @ Y_27 )
=> ~ ( ord_less_nat @ Y_27 @ X_33 ) ) ).
thf(fact_397_order__less__imp__not__less,axiom,
! [X_32: int,Y_26: int] :
( ( ord_less_int @ X_32 @ Y_26 )
=> ~ ( ord_less_int @ Y_26 @ X_32 ) ) ).
thf(fact_398_order__less__imp__not__less,axiom,
! [X_32: real,Y_26: real] :
( ( ord_less_real @ X_32 @ Y_26 )
=> ~ ( ord_less_real @ Y_26 @ X_32 ) ) ).
thf(fact_399_order__less__imp__not__less,axiom,
! [X_32: nat,Y_26: nat] :
( ( ord_less_nat @ X_32 @ Y_26 )
=> ~ ( ord_less_nat @ Y_26 @ X_32 ) ) ).
thf(fact_400_order__less__imp__not__eq,axiom,
! [X_31: int,Y_25: int] :
( ( ord_less_int @ X_31 @ Y_25 )
=> ( X_31 != Y_25 ) ) ).
thf(fact_401_order__less__imp__not__eq,axiom,
! [X_31: real,Y_25: real] :
( ( ord_less_real @ X_31 @ Y_25 )
=> ( X_31 != Y_25 ) ) ).
thf(fact_402_order__less__imp__not__eq,axiom,
! [X_31: nat,Y_25: nat] :
( ( ord_less_nat @ X_31 @ Y_25 )
=> ( X_31 != Y_25 ) ) ).
thf(fact_403_order__less__imp__not__eq2,axiom,
! [X_30: int,Y_24: int] :
( ( ord_less_int @ X_30 @ Y_24 )
=> ( Y_24 != X_30 ) ) ).
thf(fact_404_order__less__imp__not__eq2,axiom,
! [X_30: real,Y_24: real] :
( ( ord_less_real @ X_30 @ Y_24 )
=> ( Y_24 != X_30 ) ) ).
thf(fact_405_order__less__imp__not__eq2,axiom,
! [X_30: nat,Y_24: nat] :
( ( ord_less_nat @ X_30 @ Y_24 )
=> ( Y_24 != X_30 ) ) ).
thf(fact_406_order__less__imp__triv,axiom,
! [P_2: $o,X_29: int,Y_23: int] :
( ( ord_less_int @ X_29 @ Y_23 )
=> ( ( ord_less_int @ Y_23 @ X_29 )
=> P_2 ) ) ).
thf(fact_407_order__less__imp__triv,axiom,
! [P_2: $o,X_29: real,Y_23: real] :
( ( ord_less_real @ X_29 @ Y_23 )
=> ( ( ord_less_real @ Y_23 @ X_29 )
=> P_2 ) ) ).
thf(fact_408_order__less__imp__triv,axiom,
! [P_2: $o,X_29: nat,Y_23: nat] :
( ( ord_less_nat @ X_29 @ Y_23 )
=> ( ( ord_less_nat @ Y_23 @ X_29 )
=> P_2 ) ) ).
thf(fact_409_order__less__asym_H,axiom,
! [A_72: int,B_52: int] :
( ( ord_less_int @ A_72 @ B_52 )
=> ~ ( ord_less_int @ B_52 @ A_72 ) ) ).
thf(fact_410_order__less__asym_H,axiom,
! [A_72: real,B_52: real] :
( ( ord_less_real @ A_72 @ B_52 )
=> ~ ( ord_less_real @ B_52 @ A_72 ) ) ).
thf(fact_411_order__less__asym_H,axiom,
! [A_72: nat,B_52: nat] :
( ( ord_less_nat @ A_72 @ B_52 )
=> ~ ( ord_less_nat @ B_52 @ A_72 ) ) ).
thf(fact_412_xt1_I9_J,axiom,
! [B_51: int,A_71: int] :
( ( ord_less_int @ B_51 @ A_71 )
=> ~ ( ord_less_int @ A_71 @ B_51 ) ) ).
thf(fact_413_xt1_I9_J,axiom,
! [B_51: real,A_71: real] :
( ( ord_less_real @ B_51 @ A_71 )
=> ~ ( ord_less_real @ A_71 @ B_51 ) ) ).
thf(fact_414_xt1_I9_J,axiom,
! [B_51: nat,A_71: nat] :
( ( ord_less_nat @ B_51 @ A_71 )
=> ~ ( ord_less_nat @ A_71 @ B_51 ) ) ).
thf(fact_415_ord__eq__less__trans,axiom,
! [C_22: int,A_70: int,B_50: int] :
( ( A_70 = B_50 )
=> ( ( ord_less_int @ B_50 @ C_22 )
=> ( ord_less_int @ A_70 @ C_22 ) ) ) ).
thf(fact_416_ord__eq__less__trans,axiom,
! [C_22: real,A_70: real,B_50: real] :
( ( A_70 = B_50 )
=> ( ( ord_less_real @ B_50 @ C_22 )
=> ( ord_less_real @ A_70 @ C_22 ) ) ) ).
thf(fact_417_ord__eq__less__trans,axiom,
! [C_22: nat,A_70: nat,B_50: nat] :
( ( A_70 = B_50 )
=> ( ( ord_less_nat @ B_50 @ C_22 )
=> ( ord_less_nat @ A_70 @ C_22 ) ) ) ).
thf(fact_418_xt1_I1_J,axiom,
! [C_21: int,A_69: int,B_49: int] :
( ( A_69 = B_49 )
=> ( ( ord_less_int @ C_21 @ B_49 )
=> ( ord_less_int @ C_21 @ A_69 ) ) ) ).
thf(fact_419_xt1_I1_J,axiom,
! [C_21: real,A_69: real,B_49: real] :
( ( A_69 = B_49 )
=> ( ( ord_less_real @ C_21 @ B_49 )
=> ( ord_less_real @ C_21 @ A_69 ) ) ) ).
thf(fact_420_xt1_I1_J,axiom,
! [C_21: nat,A_69: nat,B_49: nat] :
( ( A_69 = B_49 )
=> ( ( ord_less_nat @ C_21 @ B_49 )
=> ( ord_less_nat @ C_21 @ A_69 ) ) ) ).
thf(fact_421_ord__less__eq__trans,axiom,
! [C_20: int,A_68: int,B_48: int] :
( ( ord_less_int @ A_68 @ B_48 )
=> ( ( B_48 = C_20 )
=> ( ord_less_int @ A_68 @ C_20 ) ) ) ).
thf(fact_422_ord__less__eq__trans,axiom,
! [C_20: real,A_68: real,B_48: real] :
( ( ord_less_real @ A_68 @ B_48 )
=> ( ( B_48 = C_20 )
=> ( ord_less_real @ A_68 @ C_20 ) ) ) ).
thf(fact_423_ord__less__eq__trans,axiom,
! [C_20: nat,A_68: nat,B_48: nat] :
( ( ord_less_nat @ A_68 @ B_48 )
=> ( ( B_48 = C_20 )
=> ( ord_less_nat @ A_68 @ C_20 ) ) ) ).
thf(fact_424_xt1_I2_J,axiom,
! [C_19: int,B_47: int,A_67: int] :
( ( ord_less_int @ B_47 @ A_67 )
=> ( ( B_47 = C_19 )
=> ( ord_less_int @ C_19 @ A_67 ) ) ) ).
thf(fact_425_xt1_I2_J,axiom,
! [C_19: real,B_47: real,A_67: real] :
( ( ord_less_real @ B_47 @ A_67 )
=> ( ( B_47 = C_19 )
=> ( ord_less_real @ C_19 @ A_67 ) ) ) ).
thf(fact_426_xt1_I2_J,axiom,
! [C_19: nat,B_47: nat,A_67: nat] :
( ( ord_less_nat @ B_47 @ A_67 )
=> ( ( B_47 = C_19 )
=> ( ord_less_nat @ C_19 @ A_67 ) ) ) ).
thf(fact_427_order__less__trans,axiom,
! [Z_6: int,X_28: int,Y_22: int] :
( ( ord_less_int @ X_28 @ Y_22 )
=> ( ( ord_less_int @ Y_22 @ Z_6 )
=> ( ord_less_int @ X_28 @ Z_6 ) ) ) ).
thf(fact_428_order__less__trans,axiom,
! [Z_6: real,X_28: real,Y_22: real] :
( ( ord_less_real @ X_28 @ Y_22 )
=> ( ( ord_less_real @ Y_22 @ Z_6 )
=> ( ord_less_real @ X_28 @ Z_6 ) ) ) ).
thf(fact_429_order__less__trans,axiom,
! [Z_6: nat,X_28: nat,Y_22: nat] :
( ( ord_less_nat @ X_28 @ Y_22 )
=> ( ( ord_less_nat @ Y_22 @ Z_6 )
=> ( ord_less_nat @ X_28 @ Z_6 ) ) ) ).
thf(fact_430_xt1_I10_J,axiom,
! [Z_5: int,Y_21: int,X_27: int] :
( ( ord_less_int @ Y_21 @ X_27 )
=> ( ( ord_less_int @ Z_5 @ Y_21 )
=> ( ord_less_int @ Z_5 @ X_27 ) ) ) ).
thf(fact_431_xt1_I10_J,axiom,
! [Z_5: real,Y_21: real,X_27: real] :
( ( ord_less_real @ Y_21 @ X_27 )
=> ( ( ord_less_real @ Z_5 @ Y_21 )
=> ( ord_less_real @ Z_5 @ X_27 ) ) ) ).
thf(fact_432_xt1_I10_J,axiom,
! [Z_5: nat,Y_21: nat,X_27: nat] :
( ( ord_less_nat @ Y_21 @ X_27 )
=> ( ( ord_less_nat @ Z_5 @ Y_21 )
=> ( ord_less_nat @ Z_5 @ X_27 ) ) ) ).
thf(fact_433_order__less__asym,axiom,
! [X_26: int,Y_20: int] :
( ( ord_less_int @ X_26 @ Y_20 )
=> ~ ( ord_less_int @ Y_20 @ X_26 ) ) ).
thf(fact_434_order__less__asym,axiom,
! [X_26: real,Y_20: real] :
( ( ord_less_real @ X_26 @ Y_20 )
=> ~ ( ord_less_real @ Y_20 @ X_26 ) ) ).
thf(fact_435_order__less__asym,axiom,
! [X_26: nat,Y_20: nat] :
( ( ord_less_nat @ X_26 @ Y_20 )
=> ~ ( ord_less_nat @ Y_20 @ X_26 ) ) ).
thf(fact_436_linorder__cases,axiom,
! [X_25: int,Y_19: int] :
( ~ ( ord_less_int @ X_25 @ Y_19 )
=> ( ( X_25 != Y_19 )
=> ( ord_less_int @ Y_19 @ X_25 ) ) ) ).
thf(fact_437_linorder__cases,axiom,
! [X_25: real,Y_19: real] :
( ~ ( ord_less_real @ X_25 @ Y_19 )
=> ( ( X_25 != Y_19 )
=> ( ord_less_real @ Y_19 @ X_25 ) ) ) ).
thf(fact_438_linorder__cases,axiom,
! [X_25: nat,Y_19: nat] :
( ~ ( ord_less_nat @ X_25 @ Y_19 )
=> ( ( X_25 != Y_19 )
=> ( ord_less_nat @ Y_19 @ X_25 ) ) ) ).
thf(fact_439_linorder__not__less,axiom,
! [X_24: int,Y_18: int] :
( ~ ( ord_less_int @ X_24 @ Y_18 )
<=> ( ord_less_eq_int @ Y_18 @ X_24 ) ) ).
thf(fact_440_linorder__not__less,axiom,
! [X_24: real,Y_18: real] :
( ~ ( ord_less_real @ X_24 @ Y_18 )
<=> ( ord_less_eq_real @ Y_18 @ X_24 ) ) ).
thf(fact_441_linorder__not__less,axiom,
! [X_24: nat,Y_18: nat] :
( ~ ( ord_less_nat @ X_24 @ Y_18 )
<=> ( ord_less_eq_nat @ Y_18 @ X_24 ) ) ).
thf(fact_442_linorder__not__le,axiom,
! [X_23: int,Y_17: int] :
( ~ ( ord_less_eq_int @ X_23 @ Y_17 )
<=> ( ord_less_int @ Y_17 @ X_23 ) ) ).
thf(fact_443_linorder__not__le,axiom,
! [X_23: real,Y_17: real] :
( ~ ( ord_less_eq_real @ X_23 @ Y_17 )
<=> ( ord_less_real @ Y_17 @ X_23 ) ) ).
thf(fact_444_linorder__not__le,axiom,
! [X_23: nat,Y_17: nat] :
( ~ ( ord_less_eq_nat @ X_23 @ Y_17 )
<=> ( ord_less_nat @ Y_17 @ X_23 ) ) ).
thf(fact_445_linorder__le__less__linear,axiom,
! [X_22: int,Y_16: int] :
( ( ord_less_eq_int @ X_22 @ Y_16 )
| ( ord_less_int @ Y_16 @ X_22 ) ) ).
thf(fact_446_linorder__le__less__linear,axiom,
! [X_22: real,Y_16: real] :
( ( ord_less_eq_real @ X_22 @ Y_16 )
| ( ord_less_real @ Y_16 @ X_22 ) ) ).
thf(fact_447_linorder__le__less__linear,axiom,
! [X_22: nat,Y_16: nat] :
( ( ord_less_eq_nat @ X_22 @ Y_16 )
| ( ord_less_nat @ Y_16 @ X_22 ) ) ).
thf(fact_448_order__less__le,axiom,
! [X_21: int,Y_15: int] :
( ( ord_less_int @ X_21 @ Y_15 )
<=> ( ( ord_less_eq_int @ X_21 @ Y_15 )
& ( X_21 != Y_15 ) ) ) ).
thf(fact_449_order__less__le,axiom,
! [X_21: real,Y_15: real] :
( ( ord_less_real @ X_21 @ Y_15 )
<=> ( ( ord_less_eq_real @ X_21 @ Y_15 )
& ( X_21 != Y_15 ) ) ) ).
thf(fact_450_order__less__le,axiom,
! [X_21: nat,Y_15: nat] :
( ( ord_less_nat @ X_21 @ Y_15 )
<=> ( ( ord_less_eq_nat @ X_21 @ Y_15 )
& ( X_21 != Y_15 ) ) ) ).
thf(fact_451_less__le__not__le,axiom,
! [X_20: int,Y_14: int] :
( ( ord_less_int @ X_20 @ Y_14 )
<=> ( ( ord_less_eq_int @ X_20 @ Y_14 )
& ~ ( ord_less_eq_int @ Y_14 @ X_20 ) ) ) ).
thf(fact_452_less__le__not__le,axiom,
! [X_20: real,Y_14: real] :
( ( ord_less_real @ X_20 @ Y_14 )
<=> ( ( ord_less_eq_real @ X_20 @ Y_14 )
& ~ ( ord_less_eq_real @ Y_14 @ X_20 ) ) ) ).
thf(fact_453_less__le__not__le,axiom,
! [X_20: nat,Y_14: nat] :
( ( ord_less_nat @ X_20 @ Y_14 )
<=> ( ( ord_less_eq_nat @ X_20 @ Y_14 )
& ~ ( ord_less_eq_nat @ Y_14 @ X_20 ) ) ) ).
thf(fact_454_order__le__less,axiom,
! [X_19: int,Y_13: int] :
( ( ord_less_eq_int @ X_19 @ Y_13 )
<=> ( ( ord_less_int @ X_19 @ Y_13 )
| ( X_19 = Y_13 ) ) ) ).
thf(fact_455_order__le__less,axiom,
! [X_19: real,Y_13: real] :
( ( ord_less_eq_real @ X_19 @ Y_13 )
<=> ( ( ord_less_real @ X_19 @ Y_13 )
| ( X_19 = Y_13 ) ) ) ).
thf(fact_456_order__le__less,axiom,
! [X_19: nat,Y_13: nat] :
( ( ord_less_eq_nat @ X_19 @ Y_13 )
<=> ( ( ord_less_nat @ X_19 @ Y_13 )
| ( X_19 = Y_13 ) ) ) ).
thf(fact_457_leI,axiom,
! [X_18: int,Y_12: int] :
( ~ ( ord_less_int @ X_18 @ Y_12 )
=> ( ord_less_eq_int @ Y_12 @ X_18 ) ) ).
thf(fact_458_leI,axiom,
! [X_18: real,Y_12: real] :
( ~ ( ord_less_real @ X_18 @ Y_12 )
=> ( ord_less_eq_real @ Y_12 @ X_18 ) ) ).
thf(fact_459_leI,axiom,
! [X_18: nat,Y_12: nat] :
( ~ ( ord_less_nat @ X_18 @ Y_12 )
=> ( ord_less_eq_nat @ Y_12 @ X_18 ) ) ).
thf(fact_460_not__leE,axiom,
! [Y_11: int,X_17: int] :
( ~ ( ord_less_eq_int @ Y_11 @ X_17 )
=> ( ord_less_int @ X_17 @ Y_11 ) ) ).
thf(fact_461_not__leE,axiom,
! [Y_11: real,X_17: real] :
( ~ ( ord_less_eq_real @ Y_11 @ X_17 )
=> ( ord_less_real @ X_17 @ Y_11 ) ) ).
thf(fact_462_not__leE,axiom,
! [Y_11: nat,X_17: nat] :
( ~ ( ord_less_eq_nat @ Y_11 @ X_17 )
=> ( ord_less_nat @ X_17 @ Y_11 ) ) ).
thf(fact_463_linorder__antisym__conv1,axiom,
! [X_16: int,Y_10: int] :
( ~ ( ord_less_int @ X_16 @ Y_10 )
=> ( ( ord_less_eq_int @ X_16 @ Y_10 )
<=> ( X_16 = Y_10 ) ) ) ).
thf(fact_464_linorder__antisym__conv1,axiom,
! [X_16: real,Y_10: real] :
( ~ ( ord_less_real @ X_16 @ Y_10 )
=> ( ( ord_less_eq_real @ X_16 @ Y_10 )
<=> ( X_16 = Y_10 ) ) ) ).
thf(fact_465_linorder__antisym__conv1,axiom,
! [X_16: nat,Y_10: nat] :
( ~ ( ord_less_nat @ X_16 @ Y_10 )
=> ( ( ord_less_eq_nat @ X_16 @ Y_10 )
<=> ( X_16 = Y_10 ) ) ) ).
thf(fact_466_order__neq__le__trans,axiom,
! [A_66: int,B_46: int] :
( ( A_66 != B_46 )
=> ( ( ord_less_eq_int @ A_66 @ B_46 )
=> ( ord_less_int @ A_66 @ B_46 ) ) ) ).
thf(fact_467_order__neq__le__trans,axiom,
! [A_66: real,B_46: real] :
( ( A_66 != B_46 )
=> ( ( ord_less_eq_real @ A_66 @ B_46 )
=> ( ord_less_real @ A_66 @ B_46 ) ) ) ).
thf(fact_468_order__neq__le__trans,axiom,
! [A_66: nat,B_46: nat] :
( ( A_66 != B_46 )
=> ( ( ord_less_eq_nat @ A_66 @ B_46 )
=> ( ord_less_nat @ A_66 @ B_46 ) ) ) ).
thf(fact_469_xt1_I12_J,axiom,
! [A_65: int,B_45: int] :
( ( A_65 != B_45 )
=> ( ( ord_less_eq_int @ B_45 @ A_65 )
=> ( ord_less_int @ B_45 @ A_65 ) ) ) ).
thf(fact_470_xt1_I12_J,axiom,
! [A_65: real,B_45: real] :
( ( A_65 != B_45 )
=> ( ( ord_less_eq_real @ B_45 @ A_65 )
=> ( ord_less_real @ B_45 @ A_65 ) ) ) ).
thf(fact_471_xt1_I12_J,axiom,
! [A_65: nat,B_45: nat] :
( ( A_65 != B_45 )
=> ( ( ord_less_eq_nat @ B_45 @ A_65 )
=> ( ord_less_nat @ B_45 @ A_65 ) ) ) ).
thf(fact_472_leD,axiom,
! [Y_9: int,X_15: int] :
( ( ord_less_eq_int @ Y_9 @ X_15 )
=> ~ ( ord_less_int @ X_15 @ Y_9 ) ) ).
thf(fact_473_leD,axiom,
! [Y_9: real,X_15: real] :
( ( ord_less_eq_real @ Y_9 @ X_15 )
=> ~ ( ord_less_real @ X_15 @ Y_9 ) ) ).
thf(fact_474_leD,axiom,
! [Y_9: nat,X_15: nat] :
( ( ord_less_eq_nat @ Y_9 @ X_15 )
=> ~ ( ord_less_nat @ X_15 @ Y_9 ) ) ).
thf(fact_475_norR__mem__unique__aux,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ B @ one_one_int ) )
=> ( ord_less_int @ A @ B ) ) ).
thf(fact_476_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_477_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_478_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_479_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_480_dense__le,axiom,
! [Z_4: real,Y_8: real] :
( ! [X_1: real] :
( ( ord_less_real @ X_1 @ Y_8 )
=> ( ord_less_eq_real @ X_1 @ Z_4 ) )
=> ( ord_less_eq_real @ Y_8 @ Z_4 ) ) ).
thf(fact_481_dense__le__bounded,axiom,
! [Z_3: real,X_14: real,Y_7: real] :
( ( ord_less_real @ X_14 @ Y_7 )
=> ( ! [W_1: real] :
( ( ord_less_real @ X_14 @ W_1 )
=> ( ( ord_less_real @ W_1 @ Y_7 )
=> ( ord_less_eq_real @ W_1 @ Z_3 ) ) )
=> ( ord_less_eq_real @ Y_7 @ Z_3 ) ) ) ).
thf(fact_482_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_483_Euler_Oaux2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ A @ C )
=> ( ( ord_less_int @ B @ C )
=> ( ( ord_less_eq_int @ A @ B )
| ( ord_less_eq_int @ B @ A ) ) ) ) ).
thf(fact_484_d22set__induct__old,axiom,
! [X: int,P: int > $o] :
( ! [A_64: int] :
( ( ( ord_less_int @ one_one_int @ A_64 )
=> ( P @ ( minus_minus_int @ A_64 @ one_one_int ) ) )
=> ( P @ A_64 ) )
=> ( P @ X ) ) ).
thf(fact_485_convex__bound__lt,axiom,
! [V_5: int,U_1: int,Y_6: int,X_13: int,A_63: int] :
( ( ord_less_int @ X_13 @ A_63 )
=> ( ( ord_less_int @ Y_6 @ A_63 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_5 )
=> ( ( ( plus_plus_int @ U_1 @ V_5 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U_1 @ X_13 ) @ ( times_times_int @ V_5 @ Y_6 ) ) @ A_63 ) ) ) ) ) ) ).
thf(fact_486_convex__bound__lt,axiom,
! [V_5: real,U_1: real,Y_6: real,X_13: real,A_63: real] :
( ( ord_less_real @ X_13 @ A_63 )
=> ( ( ord_less_real @ Y_6 @ A_63 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_1 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_5 )
=> ( ( ( plus_plus_real @ U_1 @ V_5 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U_1 @ X_13 ) @ ( times_times_real @ V_5 @ Y_6 ) ) @ A_63 ) ) ) ) ) ) ).
thf(fact_487_number__of__le__ceiling,axiom,
! [V_4: int,X_12: real] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_4 ) @ ( archim856651990g_real @ X_12 ) )
<=> ( ord_less_real @ ( minus_minus_real @ ( number267125858f_real @ V_4 ) @ one_one_real ) @ X_12 ) ) ).
thf(fact_488_ceiling__less__number__of,axiom,
! [X_11: real,V_3: int] :
( ( ord_less_int @ ( archim856651990g_real @ X_11 ) @ ( number_number_of_int @ V_3 ) )
<=> ( ord_less_eq_real @ X_11 @ ( minus_minus_real @ ( number267125858f_real @ V_3 ) @ one_one_real ) ) ) ).
thf(fact_489_zero__less__ceiling,axiom,
! [X_10: real] :
( ( ord_less_int @ zero_zero_int @ ( archim856651990g_real @ X_10 ) )
<=> ( ord_less_real @ zero_zero_real @ X_10 ) ) ).
thf(fact_490_zero__reorient,axiom,
! [X_9: int] :
( ( zero_zero_int = X_9 )
<=> ( X_9 = zero_zero_int ) ) ).
thf(fact_491_zero__reorient,axiom,
! [X_9: real] :
( ( zero_zero_real = X_9 )
<=> ( X_9 = zero_zero_real ) ) ).
thf(fact_492_zero__reorient,axiom,
! [X_9: nat] :
( ( zero_zero_nat = X_9 )
<=> ( X_9 = zero_zero_nat ) ) ).
thf(fact_493_ceiling__zero,axiom,
( ( archim856651990g_real @ zero_zero_real )
= zero_zero_int ) ).
thf(fact_494_ceiling__le__zero,axiom,
! [X_8: real] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X_8 ) @ zero_zero_int )
<=> ( ord_less_eq_real @ X_8 @ zero_zero_real ) ) ).
thf(fact_495_ceiling__less__one,axiom,
! [X_7: real] :
( ( ord_less_int @ ( archim856651990g_real @ X_7 ) @ one_one_int )
<=> ( ord_less_eq_real @ X_7 @ zero_zero_real ) ) ).
thf(fact_496_one__le__ceiling,axiom,
! [X_6: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim856651990g_real @ X_6 ) )
<=> ( ord_less_real @ zero_zero_real @ X_6 ) ) ).
thf(fact_497_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_62: nat] :
( ( times_times_nat @ zero_zero_nat @ A_62 )
= zero_zero_nat ) ).
thf(fact_498_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_62: real] :
( ( times_times_real @ zero_zero_real @ A_62 )
= zero_zero_real ) ).
thf(fact_499_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_62: int] :
( ( times_times_int @ zero_zero_int @ A_62 )
= zero_zero_int ) ).
thf(fact_500_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_61: nat] :
( ( times_times_nat @ A_61 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_501_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_61: real] :
( ( times_times_real @ A_61 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_502_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_61: int] :
( ( times_times_int @ A_61 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_503_mult__zero__left,axiom,
! [A_60: nat] :
( ( times_times_nat @ zero_zero_nat @ A_60 )
= zero_zero_nat ) ).
thf(fact_504_mult__zero__left,axiom,
! [A_60: real] :
( ( times_times_real @ zero_zero_real @ A_60 )
= zero_zero_real ) ).
thf(fact_505_mult__zero__left,axiom,
! [A_60: int] :
( ( times_times_int @ zero_zero_int @ A_60 )
= zero_zero_int ) ).
thf(fact_506_mult_Ozero__left,axiom,
! [B_44: real] :
( ( times_times_real @ zero_zero_real @ B_44 )
= zero_zero_real ) ).
thf(fact_507_mult__left_Ozero,axiom,
! [Y_5: real] :
( ( times_times_real @ zero_zero_real @ Y_5 )
= zero_zero_real ) ).
thf(fact_508_mult__zero__right,axiom,
! [A_59: nat] :
( ( times_times_nat @ A_59 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_509_mult__zero__right,axiom,
! [A_59: real] :
( ( times_times_real @ A_59 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_510_mult__zero__right,axiom,
! [A_59: int] :
( ( times_times_int @ A_59 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_511_mult_Ozero__right,axiom,
! [A_58: real] :
( ( times_times_real @ A_58 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_512_mult__right_Ozero,axiom,
! [X_5: real] :
( ( times_times_real @ X_5 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_513_mult__eq__0__iff,axiom,
! [A_57: real,B_43: real] :
( ( ( times_times_real @ A_57 @ B_43 )
= zero_zero_real )
<=> ( ( A_57 = zero_zero_real )
| ( B_43 = zero_zero_real ) ) ) ).
thf(fact_514_mult__eq__0__iff,axiom,
! [A_57: int,B_43: int] :
( ( ( times_times_int @ A_57 @ B_43 )
= zero_zero_int )
<=> ( ( A_57 = zero_zero_int )
| ( B_43 = zero_zero_int ) ) ) ).
thf(fact_515_no__zero__divisors,axiom,
! [B_42: nat,A_56: nat] :
( ( A_56 != zero_zero_nat )
=> ( ( B_42 != zero_zero_nat )
=> ( ( times_times_nat @ A_56 @ B_42 )
!= zero_zero_nat ) ) ) ).
thf(fact_516_no__zero__divisors,axiom,
! [B_42: real,A_56: real] :
( ( A_56 != zero_zero_real )
=> ( ( B_42 != zero_zero_real )
=> ( ( times_times_real @ A_56 @ B_42 )
!= zero_zero_real ) ) ) ).
thf(fact_517_no__zero__divisors,axiom,
! [B_42: int,A_56: int] :
( ( A_56 != zero_zero_int )
=> ( ( B_42 != zero_zero_int )
=> ( ( times_times_int @ A_56 @ B_42 )
!= zero_zero_int ) ) ) ).
thf(fact_518_divisors__zero,axiom,
! [A_55: nat,B_41: nat] :
( ( ( times_times_nat @ A_55 @ B_41 )
= zero_zero_nat )
=> ( ( A_55 = zero_zero_nat )
| ( B_41 = zero_zero_nat ) ) ) ).
thf(fact_519_divisors__zero,axiom,
! [A_55: real,B_41: real] :
( ( ( times_times_real @ A_55 @ B_41 )
= zero_zero_real )
=> ( ( A_55 = zero_zero_real )
| ( B_41 = zero_zero_real ) ) ) ).
thf(fact_520_divisors__zero,axiom,
! [A_55: int,B_41: int] :
( ( ( times_times_int @ A_55 @ B_41 )
= zero_zero_int )
=> ( ( A_55 = zero_zero_int )
| ( B_41 = zero_zero_int ) ) ) ).
thf(fact_521_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_54: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_54 )
= A_54 ) ).
thf(fact_522_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_54: real] :
( ( plus_plus_real @ zero_zero_real @ A_54 )
= A_54 ) ).
thf(fact_523_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_54: int] :
( ( plus_plus_int @ zero_zero_int @ A_54 )
= A_54 ) ).
thf(fact_524_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_53: nat] :
( ( plus_plus_nat @ A_53 @ zero_zero_nat )
= A_53 ) ).
thf(fact_525_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_53: real] :
( ( plus_plus_real @ A_53 @ zero_zero_real )
= A_53 ) ).
thf(fact_526_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_53: int] :
( ( plus_plus_int @ A_53 @ zero_zero_int )
= A_53 ) ).
thf(fact_527_add__0__iff,axiom,
! [B_40: nat,A_52: nat] :
( ( B_40
= ( plus_plus_nat @ B_40 @ A_52 ) )
<=> ( A_52 = zero_zero_nat ) ) ).
thf(fact_528_add__0__iff,axiom,
! [B_40: real,A_52: real] :
( ( B_40
= ( plus_plus_real @ B_40 @ A_52 ) )
<=> ( A_52 = zero_zero_real ) ) ).
thf(fact_529_add__0__iff,axiom,
! [B_40: int,A_52: int] :
( ( B_40
= ( plus_plus_int @ B_40 @ A_52 ) )
<=> ( A_52 = zero_zero_int ) ) ).
thf(fact_530_add__0__left,axiom,
! [A_51: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_51 )
= A_51 ) ).
thf(fact_531_add__0__left,axiom,
! [A_51: real] :
( ( plus_plus_real @ zero_zero_real @ A_51 )
= A_51 ) ).
thf(fact_532_add__0__left,axiom,
! [A_51: int] :
( ( plus_plus_int @ zero_zero_int @ A_51 )
= A_51 ) ).
thf(fact_533_add__0,axiom,
! [A_50: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_50 )
= A_50 ) ).
thf(fact_534_add__0,axiom,
! [A_50: real] :
( ( plus_plus_real @ zero_zero_real @ A_50 )
= A_50 ) ).
thf(fact_535_add__0,axiom,
! [A_50: int] :
( ( plus_plus_int @ zero_zero_int @ A_50 )
= A_50 ) ).
thf(fact_536_double__zero__sym,axiom,
! [A_49: real] :
( ( zero_zero_real
= ( plus_plus_real @ A_49 @ A_49 ) )
<=> ( A_49 = zero_zero_real ) ) ).
thf(fact_537_double__zero__sym,axiom,
! [A_49: int] :
( ( zero_zero_int
= ( plus_plus_int @ A_49 @ A_49 ) )
<=> ( A_49 = zero_zero_int ) ) ).
thf(fact_538_add__0__right,axiom,
! [A_48: nat] :
( ( plus_plus_nat @ A_48 @ zero_zero_nat )
= A_48 ) ).
thf(fact_539_add__0__right,axiom,
! [A_48: real] :
( ( plus_plus_real @ A_48 @ zero_zero_real )
= A_48 ) ).
thf(fact_540_add__0__right,axiom,
! [A_48: int] :
( ( plus_plus_int @ A_48 @ zero_zero_int )
= A_48 ) ).
thf(fact_541_add_Ocomm__neutral,axiom,
! [A_47: nat] :
( ( plus_plus_nat @ A_47 @ zero_zero_nat )
= A_47 ) ).
thf(fact_542_add_Ocomm__neutral,axiom,
! [A_47: real] :
( ( plus_plus_real @ A_47 @ zero_zero_real )
= A_47 ) ).
thf(fact_543_add_Ocomm__neutral,axiom,
! [A_47: int] :
( ( plus_plus_int @ A_47 @ zero_zero_int )
= A_47 ) ).
thf(fact_544_double__eq__0__iff,axiom,
! [A_46: real] :
( ( ( plus_plus_real @ A_46 @ A_46 )
= zero_zero_real )
<=> ( A_46 = zero_zero_real ) ) ).
thf(fact_545_double__eq__0__iff,axiom,
! [A_46: int] :
( ( ( plus_plus_int @ A_46 @ A_46 )
= zero_zero_int )
<=> ( A_46 = zero_zero_int ) ) ).
thf(fact_546_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_547_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
thf(fact_548_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_549_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_550_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_551_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_552_diff__0__right,axiom,
! [A_45: real] :
( ( minus_minus_real @ A_45 @ zero_zero_real )
= A_45 ) ).
thf(fact_553_diff__0__right,axiom,
! [A_45: int] :
( ( minus_minus_int @ A_45 @ zero_zero_int )
= A_45 ) ).
thf(fact_554_diff__self,axiom,
! [A_44: real] :
( ( minus_minus_real @ A_44 @ A_44 )
= zero_zero_real ) ).
thf(fact_555_diff__self,axiom,
! [A_44: int] :
( ( minus_minus_int @ A_44 @ A_44 )
= zero_zero_int ) ).
thf(fact_556_eq__iff__diff__eq__0,axiom,
! [A_43: real,B_39: real] :
( ( A_43 = B_39 )
<=> ( ( minus_minus_real @ A_43 @ B_39 )
= zero_zero_real ) ) ).
thf(fact_557_eq__iff__diff__eq__0,axiom,
! [A_43: int,B_39: int] :
( ( A_43 = B_39 )
<=> ( ( minus_minus_int @ A_43 @ B_39 )
= zero_zero_int ) ) ).
thf(fact_558_right__minus__eq,axiom,
! [A_42: real,B_38: real] :
( ( ( minus_minus_real @ A_42 @ B_38 )
= zero_zero_real )
<=> ( A_42 = B_38 ) ) ).
thf(fact_559_right__minus__eq,axiom,
! [A_42: int,B_38: int] :
( ( ( minus_minus_int @ A_42 @ B_38 )
= zero_zero_int )
<=> ( A_42 = B_38 ) ) ).
thf(fact_560_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_561_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_562_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_563_ceiling__less__cancel,axiom,
! [X_4: real,Y_4: real] :
( ( ord_less_int @ ( archim856651990g_real @ X_4 ) @ ( archim856651990g_real @ Y_4 ) )
=> ( ord_less_real @ X_4 @ Y_4 ) ) ).
thf(fact_564_ceiling__mono,axiom,
! [Y_3: real,X_3: real] :
( ( ord_less_eq_real @ Y_3 @ X_3 )
=> ( ord_less_eq_int @ ( archim856651990g_real @ Y_3 ) @ ( archim856651990g_real @ X_3 ) ) ) ).
thf(fact_565_ceiling__one,axiom,
( ( archim856651990g_real @ one_one_real )
= one_one_int ) ).
thf(fact_566_ceiling__number__of,axiom,
! [V_2: int] :
( ( archim856651990g_real @ ( number267125858f_real @ V_2 ) )
= ( number_number_of_int @ V_2 ) ) ).
thf(fact_567_zero__le__square,axiom,
! [A_41: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_41 @ A_41 ) ) ).
thf(fact_568_zero__le__square,axiom,
! [A_41: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_41 @ A_41 ) ) ).
thf(fact_569_zero__le__mult__iff,axiom,
! [A_40: real,B_37: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_40 @ B_37 ) )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_40 )
& ( ord_less_eq_real @ zero_zero_real @ B_37 ) )
| ( ( ord_less_eq_real @ A_40 @ zero_zero_real )
& ( ord_less_eq_real @ B_37 @ zero_zero_real ) ) ) ) ).
thf(fact_570_zero__le__mult__iff,axiom,
! [A_40: int,B_37: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_40 @ B_37 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_40 )
& ( ord_less_eq_int @ zero_zero_int @ B_37 ) )
| ( ( ord_less_eq_int @ A_40 @ zero_zero_int )
& ( ord_less_eq_int @ B_37 @ zero_zero_int ) ) ) ) ).
thf(fact_571_mult__le__0__iff,axiom,
! [A_39: real,B_36: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_39 @ B_36 ) @ zero_zero_real )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_39 )
& ( ord_less_eq_real @ B_36 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_39 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_36 ) ) ) ) ).
thf(fact_572_mult__le__0__iff,axiom,
! [A_39: int,B_36: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_39 @ B_36 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_39 )
& ( ord_less_eq_int @ B_36 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_39 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_36 ) ) ) ) ).
thf(fact_573_mult__nonneg__nonneg,axiom,
! [B_35: nat,A_38: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_38 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_35 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_38 @ B_35 ) ) ) ) ).
thf(fact_574_mult__nonneg__nonneg,axiom,
! [B_35: real,A_38: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_38 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_35 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_38 @ B_35 ) ) ) ) ).
thf(fact_575_mult__nonneg__nonneg,axiom,
! [B_35: int,A_38: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_38 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_35 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_38 @ B_35 ) ) ) ) ).
thf(fact_576_mult__nonneg__nonpos,axiom,
! [B_34: nat,A_37: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_37 )
=> ( ( ord_less_eq_nat @ B_34 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_37 @ B_34 ) @ zero_zero_nat ) ) ) ).
thf(fact_577_mult__nonneg__nonpos,axiom,
! [B_34: real,A_37: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_37 )
=> ( ( ord_less_eq_real @ B_34 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_37 @ B_34 ) @ zero_zero_real ) ) ) ).
thf(fact_578_mult__nonneg__nonpos,axiom,
! [B_34: int,A_37: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_37 )
=> ( ( ord_less_eq_int @ B_34 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_37 @ B_34 ) @ zero_zero_int ) ) ) ).
thf(fact_579_mult__nonneg__nonpos2,axiom,
! [B_33: nat,A_36: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_36 )
=> ( ( ord_less_eq_nat @ B_33 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_33 @ A_36 ) @ zero_zero_nat ) ) ) ).
thf(fact_580_mult__nonneg__nonpos2,axiom,
! [B_33: real,A_36: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_36 )
=> ( ( ord_less_eq_real @ B_33 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B_33 @ A_36 ) @ zero_zero_real ) ) ) ).
thf(fact_581_mult__nonneg__nonpos2,axiom,
! [B_33: int,A_36: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_36 )
=> ( ( ord_less_eq_int @ B_33 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_33 @ A_36 ) @ zero_zero_int ) ) ) ).
thf(fact_582_mult__nonpos__nonneg,axiom,
! [B_32: nat,A_35: nat] :
( ( ord_less_eq_nat @ A_35 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_32 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_35 @ B_32 ) @ zero_zero_nat ) ) ) ).
thf(fact_583_mult__nonpos__nonneg,axiom,
! [B_32: real,A_35: real] :
( ( ord_less_eq_real @ A_35 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_32 )
=> ( ord_less_eq_real @ ( times_times_real @ A_35 @ B_32 ) @ zero_zero_real ) ) ) ).
thf(fact_584_mult__nonpos__nonneg,axiom,
! [B_32: int,A_35: int] :
( ( ord_less_eq_int @ A_35 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_32 )
=> ( ord_less_eq_int @ ( times_times_int @ A_35 @ B_32 ) @ zero_zero_int ) ) ) ).
thf(fact_585_mult__nonpos__nonpos,axiom,
! [B_31: real,A_34: real] :
( ( ord_less_eq_real @ A_34 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_31 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_34 @ B_31 ) ) ) ) ).
thf(fact_586_mult__nonpos__nonpos,axiom,
! [B_31: int,A_34: int] :
( ( ord_less_eq_int @ A_34 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_31 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_34 @ B_31 ) ) ) ) ).
thf(fact_587_mult__right__mono,axiom,
! [C_18: nat,A_33: nat,B_30: nat] :
( ( ord_less_eq_nat @ A_33 @ B_30 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_18 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_33 @ C_18 ) @ ( times_times_nat @ B_30 @ C_18 ) ) ) ) ).
thf(fact_588_mult__right__mono,axiom,
! [C_18: real,A_33: real,B_30: real] :
( ( ord_less_eq_real @ A_33 @ B_30 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_18 )
=> ( ord_less_eq_real @ ( times_times_real @ A_33 @ C_18 ) @ ( times_times_real @ B_30 @ C_18 ) ) ) ) ).
thf(fact_589_mult__right__mono,axiom,
! [C_18: int,A_33: int,B_30: int] :
( ( ord_less_eq_int @ A_33 @ B_30 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_18 )
=> ( ord_less_eq_int @ ( times_times_int @ A_33 @ C_18 ) @ ( times_times_int @ B_30 @ C_18 ) ) ) ) ).
thf(fact_590_mult__left__mono,axiom,
! [C_17: nat,A_32: nat,B_29: nat] :
( ( ord_less_eq_nat @ A_32 @ B_29 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_17 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_17 @ A_32 ) @ ( times_times_nat @ C_17 @ B_29 ) ) ) ) ).
thf(fact_591_mult__left__mono,axiom,
! [C_17: real,A_32: real,B_29: real] :
( ( ord_less_eq_real @ A_32 @ B_29 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_17 )
=> ( ord_less_eq_real @ ( times_times_real @ C_17 @ A_32 ) @ ( times_times_real @ C_17 @ B_29 ) ) ) ) ).
thf(fact_592_mult__left__mono,axiom,
! [C_17: int,A_32: int,B_29: int] :
( ( ord_less_eq_int @ A_32 @ B_29 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_17 )
=> ( ord_less_eq_int @ ( times_times_int @ C_17 @ A_32 ) @ ( times_times_int @ C_17 @ B_29 ) ) ) ) ).
thf(fact_593_comm__mult__left__mono,axiom,
! [C_16: nat,A_31: nat,B_28: nat] :
( ( ord_less_eq_nat @ A_31 @ B_28 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_16 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_16 @ A_31 ) @ ( times_times_nat @ C_16 @ B_28 ) ) ) ) ).
thf(fact_594_comm__mult__left__mono,axiom,
! [C_16: real,A_31: real,B_28: real] :
( ( ord_less_eq_real @ A_31 @ B_28 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_16 )
=> ( ord_less_eq_real @ ( times_times_real @ C_16 @ A_31 ) @ ( times_times_real @ C_16 @ B_28 ) ) ) ) ).
thf(fact_595_comm__mult__left__mono,axiom,
! [C_16: int,A_31: int,B_28: int] :
( ( ord_less_eq_int @ A_31 @ B_28 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_16 )
=> ( ord_less_eq_int @ ( times_times_int @ C_16 @ A_31 ) @ ( times_times_int @ C_16 @ B_28 ) ) ) ) ).
thf(fact_596_mult__right__mono__neg,axiom,
! [C_15: real,B_27: real,A_30: real] :
( ( ord_less_eq_real @ B_27 @ A_30 )
=> ( ( ord_less_eq_real @ C_15 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_30 @ C_15 ) @ ( times_times_real @ B_27 @ C_15 ) ) ) ) ).
thf(fact_597_mult__right__mono__neg,axiom,
! [C_15: int,B_27: int,A_30: int] :
( ( ord_less_eq_int @ B_27 @ A_30 )
=> ( ( ord_less_eq_int @ C_15 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_30 @ C_15 ) @ ( times_times_int @ B_27 @ C_15 ) ) ) ) ).
thf(fact_598_mult__left__mono__neg,axiom,
! [C_14: real,B_26: real,A_29: real] :
( ( ord_less_eq_real @ B_26 @ A_29 )
=> ( ( ord_less_eq_real @ C_14 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C_14 @ A_29 ) @ ( times_times_real @ C_14 @ B_26 ) ) ) ) ).
thf(fact_599_mult__left__mono__neg,axiom,
! [C_14: int,B_26: int,A_29: int] :
( ( ord_less_eq_int @ B_26 @ A_29 )
=> ( ( ord_less_eq_int @ C_14 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_14 @ A_29 ) @ ( times_times_int @ C_14 @ B_26 ) ) ) ) ).
thf(fact_600_mult__mono_H,axiom,
! [C_13: nat,D_3: nat,A_28: nat,B_25: nat] :
( ( ord_less_eq_nat @ A_28 @ B_25 )
=> ( ( ord_less_eq_nat @ C_13 @ D_3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_28 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_13 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_28 @ C_13 ) @ ( times_times_nat @ B_25 @ D_3 ) ) ) ) ) ) ).
thf(fact_601_mult__mono_H,axiom,
! [C_13: real,D_3: real,A_28: real,B_25: real] :
( ( ord_less_eq_real @ A_28 @ B_25 )
=> ( ( ord_less_eq_real @ C_13 @ D_3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_28 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_13 )
=> ( ord_less_eq_real @ ( times_times_real @ A_28 @ C_13 ) @ ( times_times_real @ B_25 @ D_3 ) ) ) ) ) ) ).
thf(fact_602_mult__mono_H,axiom,
! [C_13: int,D_3: int,A_28: int,B_25: int] :
( ( ord_less_eq_int @ A_28 @ B_25 )
=> ( ( ord_less_eq_int @ C_13 @ D_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_28 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_13 )
=> ( ord_less_eq_int @ ( times_times_int @ A_28 @ C_13 ) @ ( times_times_int @ B_25 @ D_3 ) ) ) ) ) ) ).
thf(fact_603_mult__mono,axiom,
! [C_12: nat,D_2: nat,A_27: nat,B_24: nat] :
( ( ord_less_eq_nat @ A_27 @ B_24 )
=> ( ( ord_less_eq_nat @ C_12 @ D_2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_24 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_12 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_27 @ C_12 ) @ ( times_times_nat @ B_24 @ D_2 ) ) ) ) ) ) ).
thf(fact_604_mult__mono,axiom,
! [C_12: real,D_2: real,A_27: real,B_24: real] :
( ( ord_less_eq_real @ A_27 @ B_24 )
=> ( ( ord_less_eq_real @ C_12 @ D_2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_24 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_12 )
=> ( ord_less_eq_real @ ( times_times_real @ A_27 @ C_12 ) @ ( times_times_real @ B_24 @ D_2 ) ) ) ) ) ) ).
thf(fact_605_mult__mono,axiom,
! [C_12: int,D_2: int,A_27: int,B_24: int] :
( ( ord_less_eq_int @ A_27 @ B_24 )
=> ( ( ord_less_eq_int @ C_12 @ D_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_24 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_12 )
=> ( ord_less_eq_int @ ( times_times_int @ A_27 @ C_12 ) @ ( times_times_int @ B_24 @ D_2 ) ) ) ) ) ) ).
thf(fact_606_split__mult__pos__le,axiom,
! [B_23: real,A_26: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_26 )
& ( ord_less_eq_real @ zero_zero_real @ B_23 ) )
| ( ( ord_less_eq_real @ A_26 @ zero_zero_real )
& ( ord_less_eq_real @ B_23 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_26 @ B_23 ) ) ) ).
thf(fact_607_split__mult__pos__le,axiom,
! [B_23: int,A_26: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_26 )
& ( ord_less_eq_int @ zero_zero_int @ B_23 ) )
| ( ( ord_less_eq_int @ A_26 @ zero_zero_int )
& ( ord_less_eq_int @ B_23 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_26 @ B_23 ) ) ) ).
thf(fact_608_split__mult__neg__le,axiom,
! [B_22: nat,A_25: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_25 )
& ( ord_less_eq_nat @ B_22 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_25 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_22 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_25 @ B_22 ) @ zero_zero_nat ) ) ).
thf(fact_609_split__mult__neg__le,axiom,
! [B_22: real,A_25: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_25 )
& ( ord_less_eq_real @ B_22 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_25 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_22 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A_25 @ B_22 ) @ zero_zero_real ) ) ).
thf(fact_610_split__mult__neg__le,axiom,
! [B_22: int,A_25: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_25 )
& ( ord_less_eq_int @ B_22 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_25 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_22 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_25 @ B_22 ) @ zero_zero_int ) ) ).
thf(fact_611_not__square__less__zero,axiom,
! [A_24: real] :
~ ( ord_less_real @ ( times_times_real @ A_24 @ A_24 ) @ zero_zero_real ) ).
thf(fact_612_not__square__less__zero,axiom,
! [A_24: int] :
~ ( ord_less_int @ ( times_times_int @ A_24 @ A_24 ) @ zero_zero_int ) ).
thf(fact_613_mult__less__cancel__right__disj,axiom,
! [A_23: real,C_11: real,B_21: real] :
( ( ord_less_real @ ( times_times_real @ A_23 @ C_11 ) @ ( times_times_real @ B_21 @ C_11 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_11 )
& ( ord_less_real @ A_23 @ B_21 ) )
| ( ( ord_less_real @ C_11 @ zero_zero_real )
& ( ord_less_real @ B_21 @ A_23 ) ) ) ) ).
thf(fact_614_mult__less__cancel__right__disj,axiom,
! [A_23: int,C_11: int,B_21: int] :
( ( ord_less_int @ ( times_times_int @ A_23 @ C_11 ) @ ( times_times_int @ B_21 @ C_11 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_11 )
& ( ord_less_int @ A_23 @ B_21 ) )
| ( ( ord_less_int @ C_11 @ zero_zero_int )
& ( ord_less_int @ B_21 @ A_23 ) ) ) ) ).
thf(fact_615_mult__less__cancel__left__disj,axiom,
! [C_10: real,A_22: real,B_20: real] :
( ( ord_less_real @ ( times_times_real @ C_10 @ A_22 ) @ ( times_times_real @ C_10 @ B_20 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_10 )
& ( ord_less_real @ A_22 @ B_20 ) )
| ( ( ord_less_real @ C_10 @ zero_zero_real )
& ( ord_less_real @ B_20 @ A_22 ) ) ) ) ).
thf(fact_616_mult__less__cancel__left__disj,axiom,
! [C_10: int,A_22: int,B_20: int] :
( ( ord_less_int @ ( times_times_int @ C_10 @ A_22 ) @ ( times_times_int @ C_10 @ B_20 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_10 )
& ( ord_less_int @ A_22 @ B_20 ) )
| ( ( ord_less_int @ C_10 @ zero_zero_int )
& ( ord_less_int @ B_20 @ A_22 ) ) ) ) ).
thf(fact_617_mult__less__cancel__left__pos,axiom,
! [A_21: real,B_19: real,C_9: real] :
( ( ord_less_real @ zero_zero_real @ C_9 )
=> ( ( ord_less_real @ ( times_times_real @ C_9 @ A_21 ) @ ( times_times_real @ C_9 @ B_19 ) )
<=> ( ord_less_real @ A_21 @ B_19 ) ) ) ).
thf(fact_618_mult__less__cancel__left__pos,axiom,
! [A_21: int,B_19: int,C_9: int] :
( ( ord_less_int @ zero_zero_int @ C_9 )
=> ( ( ord_less_int @ ( times_times_int @ C_9 @ A_21 ) @ ( times_times_int @ C_9 @ B_19 ) )
<=> ( ord_less_int @ A_21 @ B_19 ) ) ) ).
thf(fact_619_mult__pos__pos,axiom,
! [B_18: nat,A_20: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_20 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_18 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_20 @ B_18 ) ) ) ) ).
thf(fact_620_mult__pos__pos,axiom,
! [B_18: real,A_20: real] :
( ( ord_less_real @ zero_zero_real @ A_20 )
=> ( ( ord_less_real @ zero_zero_real @ B_18 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_20 @ B_18 ) ) ) ) ).
thf(fact_621_mult__pos__pos,axiom,
! [B_18: int,A_20: int] :
( ( ord_less_int @ zero_zero_int @ A_20 )
=> ( ( ord_less_int @ zero_zero_int @ B_18 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_20 @ B_18 ) ) ) ) ).
thf(fact_622_mult__pos__neg,axiom,
! [B_17: nat,A_19: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_19 )
=> ( ( ord_less_nat @ B_17 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_19 @ B_17 ) @ zero_zero_nat ) ) ) ).
thf(fact_623_mult__pos__neg,axiom,
! [B_17: real,A_19: real] :
( ( ord_less_real @ zero_zero_real @ A_19 )
=> ( ( ord_less_real @ B_17 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_19 @ B_17 ) @ zero_zero_real ) ) ) ).
thf(fact_624_mult__pos__neg,axiom,
! [B_17: int,A_19: int] :
( ( ord_less_int @ zero_zero_int @ A_19 )
=> ( ( ord_less_int @ B_17 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_19 @ B_17 ) @ zero_zero_int ) ) ) ).
thf(fact_625_mult__pos__neg2,axiom,
! [B_16: nat,A_18: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_18 )
=> ( ( ord_less_nat @ B_16 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B_16 @ A_18 ) @ zero_zero_nat ) ) ) ).
thf(fact_626_mult__pos__neg2,axiom,
! [B_16: real,A_18: real] :
( ( ord_less_real @ zero_zero_real @ A_18 )
=> ( ( ord_less_real @ B_16 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B_16 @ A_18 ) @ zero_zero_real ) ) ) ).
thf(fact_627_mult__pos__neg2,axiom,
! [B_16: int,A_18: int] :
( ( ord_less_int @ zero_zero_int @ A_18 )
=> ( ( ord_less_int @ B_16 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_16 @ A_18 ) @ zero_zero_int ) ) ) ).
thf(fact_628_zero__less__mult__pos,axiom,
! [A_17: nat,B_15: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_17 @ B_15 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_17 )
=> ( ord_less_nat @ zero_zero_nat @ B_15 ) ) ) ).
thf(fact_629_zero__less__mult__pos,axiom,
! [A_17: real,B_15: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_17 @ B_15 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_17 )
=> ( ord_less_real @ zero_zero_real @ B_15 ) ) ) ).
thf(fact_630_zero__less__mult__pos,axiom,
! [A_17: int,B_15: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_17 @ B_15 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_17 )
=> ( ord_less_int @ zero_zero_int @ B_15 ) ) ) ).
thf(fact_631_zero__less__mult__pos2,axiom,
! [B_14: nat,A_16: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B_14 @ A_16 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_16 )
=> ( ord_less_nat @ zero_zero_nat @ B_14 ) ) ) ).
thf(fact_632_zero__less__mult__pos2,axiom,
! [B_14: real,A_16: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B_14 @ A_16 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_16 )
=> ( ord_less_real @ zero_zero_real @ B_14 ) ) ) ).
thf(fact_633_zero__less__mult__pos2,axiom,
! [B_14: int,A_16: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B_14 @ A_16 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_16 )
=> ( ord_less_int @ zero_zero_int @ B_14 ) ) ) ).
thf(fact_634_mult__less__cancel__left__neg,axiom,
! [A_15: real,B_13: real,C_8: real] :
( ( ord_less_real @ C_8 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C_8 @ A_15 ) @ ( times_times_real @ C_8 @ B_13 ) )
<=> ( ord_less_real @ B_13 @ A_15 ) ) ) ).
thf(fact_635_mult__less__cancel__left__neg,axiom,
! [A_15: int,B_13: int,C_8: int] :
( ( ord_less_int @ C_8 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C_8 @ A_15 ) @ ( times_times_int @ C_8 @ B_13 ) )
<=> ( ord_less_int @ B_13 @ A_15 ) ) ) ).
thf(fact_636_mult__neg__pos,axiom,
! [B_12: nat,A_14: nat] :
( ( ord_less_nat @ A_14 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B_12 )
=> ( ord_less_nat @ ( times_times_nat @ A_14 @ B_12 ) @ zero_zero_nat ) ) ) ).
thf(fact_637_mult__neg__pos,axiom,
! [B_12: real,A_14: real] :
( ( ord_less_real @ A_14 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B_12 )
=> ( ord_less_real @ ( times_times_real @ A_14 @ B_12 ) @ zero_zero_real ) ) ) ).
thf(fact_638_mult__neg__pos,axiom,
! [B_12: int,A_14: int] :
( ( ord_less_int @ A_14 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_12 )
=> ( ord_less_int @ ( times_times_int @ A_14 @ B_12 ) @ zero_zero_int ) ) ) ).
thf(fact_639_mult__neg__neg,axiom,
! [B_11: real,A_13: real] :
( ( ord_less_real @ A_13 @ zero_zero_real )
=> ( ( ord_less_real @ B_11 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_13 @ B_11 ) ) ) ) ).
thf(fact_640_mult__neg__neg,axiom,
! [B_11: int,A_13: int] :
( ( ord_less_int @ A_13 @ zero_zero_int )
=> ( ( ord_less_int @ B_11 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_13 @ B_11 ) ) ) ) ).
thf(fact_641_mult__strict__right__mono,axiom,
! [C_7: nat,A_12: nat,B_10: nat] :
( ( ord_less_nat @ A_12 @ B_10 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_7 )
=> ( ord_less_nat @ ( times_times_nat @ A_12 @ C_7 ) @ ( times_times_nat @ B_10 @ C_7 ) ) ) ) ).
thf(fact_642_mult__strict__right__mono,axiom,
! [C_7: real,A_12: real,B_10: real] :
( ( ord_less_real @ A_12 @ B_10 )
=> ( ( ord_less_real @ zero_zero_real @ C_7 )
=> ( ord_less_real @ ( times_times_real @ A_12 @ C_7 ) @ ( times_times_real @ B_10 @ C_7 ) ) ) ) ).
thf(fact_643_mult__strict__right__mono,axiom,
! [C_7: int,A_12: int,B_10: int] :
( ( ord_less_int @ A_12 @ B_10 )
=> ( ( ord_less_int @ zero_zero_int @ C_7 )
=> ( ord_less_int @ ( times_times_int @ A_12 @ C_7 ) @ ( times_times_int @ B_10 @ C_7 ) ) ) ) ).
thf(fact_644_mult__strict__left__mono,axiom,
! [C_6: nat,A_11: nat,B_9: nat] :
( ( ord_less_nat @ A_11 @ B_9 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_6 )
=> ( ord_less_nat @ ( times_times_nat @ C_6 @ A_11 ) @ ( times_times_nat @ C_6 @ B_9 ) ) ) ) ).
thf(fact_645_mult__strict__left__mono,axiom,
! [C_6: real,A_11: real,B_9: real] :
( ( ord_less_real @ A_11 @ B_9 )
=> ( ( ord_less_real @ zero_zero_real @ C_6 )
=> ( ord_less_real @ ( times_times_real @ C_6 @ A_11 ) @ ( times_times_real @ C_6 @ B_9 ) ) ) ) ).
thf(fact_646_mult__strict__left__mono,axiom,
! [C_6: int,A_11: int,B_9: int] :
( ( ord_less_int @ A_11 @ B_9 )
=> ( ( ord_less_int @ zero_zero_int @ C_6 )
=> ( ord_less_int @ ( times_times_int @ C_6 @ A_11 ) @ ( times_times_int @ C_6 @ B_9 ) ) ) ) ).
thf(fact_647_comm__mult__strict__left__mono,axiom,
! [C_5: nat,A_10: nat,B_8: nat] :
( ( ord_less_nat @ A_10 @ B_8 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_5 )
=> ( ord_less_nat @ ( times_times_nat @ C_5 @ A_10 ) @ ( times_times_nat @ C_5 @ B_8 ) ) ) ) ).
thf(fact_648_comm__mult__strict__left__mono,axiom,
! [C_5: real,A_10: real,B_8: real] :
( ( ord_less_real @ A_10 @ B_8 )
=> ( ( ord_less_real @ zero_zero_real @ C_5 )
=> ( ord_less_real @ ( times_times_real @ C_5 @ A_10 ) @ ( times_times_real @ C_5 @ B_8 ) ) ) ) ).
thf(fact_649_comm__mult__strict__left__mono,axiom,
! [C_5: int,A_10: int,B_8: int] :
( ( ord_less_int @ A_10 @ B_8 )
=> ( ( ord_less_int @ zero_zero_int @ C_5 )
=> ( ord_less_int @ ( times_times_int @ C_5 @ A_10 ) @ ( times_times_int @ C_5 @ B_8 ) ) ) ) ).
thf(fact_650_mult__strict__right__mono__neg,axiom,
! [C_4: real,B_7: real,A_9: real] :
( ( ord_less_real @ B_7 @ A_9 )
=> ( ( ord_less_real @ C_4 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_9 @ C_4 ) @ ( times_times_real @ B_7 @ C_4 ) ) ) ) ).
thf(fact_651_mult__strict__right__mono__neg,axiom,
! [C_4: int,B_7: int,A_9: int] :
( ( ord_less_int @ B_7 @ A_9 )
=> ( ( ord_less_int @ C_4 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_9 @ C_4 ) @ ( times_times_int @ B_7 @ C_4 ) ) ) ) ).
thf(fact_652_mult__strict__left__mono__neg,axiom,
! [C_3: real,B_6: real,A_8: real] :
( ( ord_less_real @ B_6 @ A_8 )
=> ( ( ord_less_real @ C_3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C_3 @ A_8 ) @ ( times_times_real @ C_3 @ B_6 ) ) ) ) ).
thf(fact_653_mult__strict__left__mono__neg,axiom,
! [C_3: int,B_6: int,A_8: int] :
( ( ord_less_int @ B_6 @ A_8 )
=> ( ( ord_less_int @ C_3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C_3 @ A_8 ) @ ( times_times_int @ C_3 @ B_6 ) ) ) ) ).
thf(fact_654_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_7: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_7 @ A_7 ) )
<=> ( ord_less_eq_real @ zero_zero_real @ A_7 ) ) ).
thf(fact_655_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_7: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_7 @ A_7 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A_7 ) ) ).
thf(fact_656_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_6: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_6 @ A_6 ) @ zero_zero_real )
<=> ( ord_less_eq_real @ A_6 @ zero_zero_real ) ) ).
thf(fact_657_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_6: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_6 @ A_6 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ A_6 @ zero_zero_int ) ) ).
thf(fact_658_add__nonneg__nonneg,axiom,
! [B_5: nat,A_5: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A_5 @ B_5 ) ) ) ) ).
thf(fact_659_add__nonneg__nonneg,axiom,
! [B_5: real,A_5: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_5 @ B_5 ) ) ) ) ).
thf(fact_660_add__nonneg__nonneg,axiom,
! [B_5: int,A_5: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_5 @ B_5 ) ) ) ) ).
thf(fact_661_add__nonneg__eq__0__iff,axiom,
! [Y_2: nat,X_2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X_2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_2 )
=> ( ( ( plus_plus_nat @ X_2 @ Y_2 )
= zero_zero_nat )
<=> ( ( X_2 = zero_zero_nat )
& ( Y_2 = zero_zero_nat ) ) ) ) ) ).
thf(fact_662_add__nonneg__eq__0__iff,axiom,
! [Y_2: real,X_2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_2 )
=> ( ( ( plus_plus_real @ X_2 @ Y_2 )
= zero_zero_real )
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ) ) ).
thf(fact_663_add__nonneg__eq__0__iff,axiom,
! [Y_2: int,X_2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_2 )
=> ( ( ( plus_plus_int @ X_2 @ Y_2 )
= zero_zero_int )
<=> ( ( X_2 = zero_zero_int )
& ( Y_2 = zero_zero_int ) ) ) ) ) ).
thf(fact_664_add__increasing,axiom,
! [B_4: nat,C_2: nat,A_4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_4 )
=> ( ( ord_less_eq_nat @ B_4 @ C_2 )
=> ( ord_less_eq_nat @ B_4 @ ( plus_plus_nat @ A_4 @ C_2 ) ) ) ) ).
thf(fact_665_add__increasing,axiom,
! [B_4: real,C_2: real,A_4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_4 )
=> ( ( ord_less_eq_real @ B_4 @ C_2 )
=> ( ord_less_eq_real @ B_4 @ ( plus_plus_real @ A_4 @ C_2 ) ) ) ) ).
thf(fact_666_add__increasing,axiom,
! [B_4: int,C_2: int,A_4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_4 )
=> ( ( ord_less_eq_int @ B_4 @ C_2 )
=> ( ord_less_eq_int @ B_4 @ ( plus_plus_int @ A_4 @ C_2 ) ) ) ) ).
thf(fact_667_add__increasing2,axiom,
! [B_3: nat,A_3: nat,C_1: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C_1 )
=> ( ( ord_less_eq_nat @ B_3 @ A_3 )
=> ( ord_less_eq_nat @ B_3 @ ( plus_plus_nat @ A_3 @ C_1 ) ) ) ) ).
thf(fact_668_add__increasing2,axiom,
! [B_3: real,A_3: real,C_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ C_1 )
=> ( ( ord_less_eq_real @ B_3 @ A_3 )
=> ( ord_less_eq_real @ B_3 @ ( plus_plus_real @ A_3 @ C_1 ) ) ) ) ).
thf(fact_669_add__increasing2,axiom,
! [B_3: int,A_3: int,C_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ C_1 )
=> ( ( ord_less_eq_int @ B_3 @ A_3 )
=> ( ord_less_eq_int @ B_3 @ ( plus_plus_int @ A_3 @ C_1 ) ) ) ) ).
thf(fact_670_add__nonpos__nonpos,axiom,
! [B_2: nat,A_2: nat] :
( ( ord_less_eq_nat @ A_2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_2 @ B_2 ) @ zero_zero_nat ) ) ) ).
thf(fact_671_add__nonpos__nonpos,axiom,
! [B_2: real,A_2: real] :
( ( ord_less_eq_real @ A_2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_2 @ B_2 ) @ zero_zero_real ) ) ) ).
thf(fact_672_add__nonpos__nonpos,axiom,
! [B_2: int,A_2: int] :
( ( ord_less_eq_int @ A_2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_2 @ B_2 ) @ zero_zero_int ) ) ) ).
thf(fact_673_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_1 @ A_1 ) )
<=> ( ord_less_real @ zero_zero_real @ A_1 ) ) ).
thf(fact_674_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_1 @ A_1 ) )
<=> ( ord_less_int @ zero_zero_int @ A_1 ) ) ).
thf(fact_675_Euler_Oaux1,axiom,
! [A: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ A )
=> ( ( X
!= ( minus_minus_int @ A @ one_one_int ) )
=> ( ord_less_int @ X @ ( minus_minus_int @ A @ one_one_int ) ) ) ) ) ).
thf(fact_676_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_677_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_678_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_679_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_680_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_681_pos__zmult__eq__1__iff,axiom,
! [N_1: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N_1 )
= one_one_int )
<=> ( ( M = one_one_int )
& ( N_1 = one_one_int ) ) ) ) ).
thf(fact_682_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_683_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_684_self__quotient__aux2,axiom,
! [R: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_685_self__quotient__aux1,axiom,
! [R: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_int @ R @ A )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_686_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R: int,B_1: int,Q_1: int,R_1: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) @ zero_zero_int )
=> ( ( ord_less_int @ R @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
thf(fact_687_unique__quotient__lemma__neg,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_1 ) @ R_1 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
thf(fact_688_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R: int,B_1: int,Q_1: int,R_1: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R )
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_int @ R_1 @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ B )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
thf(fact_689_unique__quotient__lemma,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_1 ) @ R_1 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ( ord_less_int @ R @ B )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
thf(fact_690_q__neg__lemma,axiom,
! [B_1: int,Q_1: int,R_1: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ Q_1 @ zero_zero_int ) ) ) ) ).
thf(fact_691_q__pos__lemma,axiom,
! [B_1: int,Q_1: int,R_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_int @ R_1 @ B_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ zero_zero_int @ Q_1 ) ) ) ) ).
thf(fact_692_xzgcd__linear,axiom,
! [M: int,R: int,S: int,T: int,N_1: int] :
( ( ord_less_int @ zero_zero_int @ N_1 )
=> ( ( ( xzgcd @ M @ N_1 )
= ( produc282740534nt_int @ R @ ( product_Pair_int_int @ S @ T ) ) )
=> ( R
= ( plus_plus_int @ ( times_times_int @ S @ M ) @ ( times_times_int @ T @ N_1 ) ) ) ) ) ).
thf(fact_693_xzgcda__linear,axiom,
! [M: int,N_1: int,R_1: int,S_1: int,S: int,T_1: int,T: int,Rn: int,Sn: int,Tn: int,R: int] :
( ( ord_less_int @ zero_zero_int @ R )
=> ( ( ( xzgcda @ M @ N_1 @ R_1 @ R @ S_1 @ S @ T_1 @ T )
= ( produc282740534nt_int @ Rn @ ( product_Pair_int_int @ Sn @ Tn ) ) )
=> ( ( R_1
= ( plus_plus_int @ ( times_times_int @ S_1 @ M ) @ ( times_times_int @ T_1 @ N_1 ) ) )
=> ( ( R
= ( plus_plus_int @ ( times_times_int @ S @ M ) @ ( times_times_int @ T @ N_1 ) ) )
=> ( Rn
= ( plus_plus_int @ ( times_times_int @ Sn @ M ) @ ( times_times_int @ Tn @ N_1 ) ) ) ) ) ) ) ).
thf(fact_694_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_695_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_696_pos__zmult__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
thf(fact_697_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_698_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_699_xzgcd__def,axiom,
! [M: int,N_1: int] :
( ( xzgcd @ M @ N_1 )
= ( xzgcda @ M @ N_1 @ M @ N_1 @ one_one_int @ zero_zero_int @ zero_zero_int @ one_one_int ) ) ).
thf(fact_700_decr__mult__lemma,axiom,
! [K: int,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D_1 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) ) ) ) ) ).
thf(fact_701_incr__mult__lemma,axiom,
! [K: int,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D_1 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) ) ) ) ) ).
thf(fact_702_divmod__int__relI,axiom,
! [A: int,B: int,Q: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
& ( ord_less_int @ R @ B ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ R )
& ( ord_less_eq_int @ R @ zero_zero_int ) ) ) )
=> ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R ) ) ) ) ).
thf(fact_703_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_704_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_705_self__remainder,axiom,
! [A: int,Q: int,R: int] :
( ( divmod_int_rel @ A @ A @ ( product_Pair_int_int @ Q @ R ) )
=> ( ( A != zero_zero_int )
=> ( R = zero_zero_int ) ) ) ).
thf(fact_706_unique__quotient,axiom,
! [Q_1: int,R_1: int,A: int,B: int,Q: int,R: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R ) )
=> ( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q_1 @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( Q = Q_1 ) ) ) ) ).
thf(fact_707_unique__remainder,axiom,
! [Q_1: int,R_1: int,A: int,B: int,Q: int,R: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R ) )
=> ( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q_1 @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( R = R_1 ) ) ) ) ).
thf(fact_708_self__quotient,axiom,
! [A: int,Q: int,R: int] :
( ( divmod_int_rel @ A @ A @ ( product_Pair_int_int @ Q @ R ) )
=> ( ( A != zero_zero_int )
=> ( Q = one_one_int ) ) ) ).
thf(fact_709_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_710_tsub__eq,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_711_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_712_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_713_minusinfinity,axiom,
! [P: int > $o,P1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int,K_1: int] :
( ( P1 @ X_1 )
<=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D_1 ) ) ) )
=> ( ? [Z_2: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z_2 )
=> ( ( P @ X_1 )
<=> ( P1 @ X_1 ) ) )
=> ( ( ?? @ int @ P1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_714_plusinfinity,axiom,
! [P: int > $o,P_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int,K_1: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D_1 ) ) ) )
=> ( ? [Z_2: int] :
! [X_1: int] :
( ( ord_less_int @ Z_2 @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( ?? @ int @ P_1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_715_zfact_Osimps,axiom,
! [N_1: int] :
( ( ( ord_less_eq_int @ N_1 @ zero_zero_int )
=> ( ( zfact @ N_1 )
= one_one_int ) )
& ( ~ ( ord_less_eq_int @ N_1 @ zero_zero_int )
=> ( ( zfact @ N_1 )
= ( times_times_int @ N_1 @ ( zfact @ ( minus_minus_int @ N_1 @ one_one_int ) ) ) ) ) ) ).
thf(fact_716_le__mult__floor,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim1246769320r_real @ A ) @ ( archim1246769320r_real @ B ) ) @ ( archim1246769320r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
thf(fact_717_posDivAlg__correct,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( divmod_int_rel @ A @ B @ ( posDivAlg @ A @ B ) ) ) ) ).
thf(fact_718_incr__lemma,axiom,
! [Z: int,X: int,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( 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_1 ) ) ) ) ).
thf(fact_719_decr__lemma,axiom,
! [X: int,Z: int,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( 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_1 ) ) @ Z ) ) ).
thf(fact_720_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_721_real__le__linear,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
| ( ord_less_eq_real @ W @ Z ) ) ).
thf(fact_722_real__add__left__mono,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( plus_plus_real @ Z @ X ) @ ( plus_plus_real @ Z @ Y ) ) ) ).
thf(fact_723_real__le__trans,axiom,
! [K: real,I: real,J: real] :
( ( ord_less_eq_real @ I @ J )
=> ( ( ord_less_eq_real @ J @ K )
=> ( ord_less_eq_real @ I @ K ) ) ) ).
thf(fact_724_real__le__antisym,axiom,
! [Z: real,W: real] :
( ( ord_less_eq_real @ Z @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_725_real__add__mult__distrib,axiom,
! [Z1: real,Z2: real,W: real] :
( ( times_times_real @ ( plus_plus_real @ Z1 @ Z2 ) @ W )
= ( plus_plus_real @ ( times_times_real @ Z1 @ W ) @ ( times_times_real @ Z2 @ W ) ) ) ).
thf(fact_726_real__mult__assoc,axiom,
! [Z1: real,Z2: real,Z3: real] :
( ( times_times_real @ ( times_times_real @ Z1 @ Z2 ) @ Z3 )
= ( times_times_real @ Z1 @ ( times_times_real @ Z2 @ Z3 ) ) ) ).
thf(fact_727_real__mult__commute,axiom,
! [Z: real,W: real] :
( ( times_times_real @ Z @ W )
= ( times_times_real @ W @ Z ) ) ).
thf(fact_728_real__mult__1,axiom,
! [Z: real] :
( ( times_times_real @ one_one_real @ Z )
= Z ) ).
thf(fact_729_less__eq__real__def,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
thf(fact_730_real__less__def,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
<=> ( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_731_real__le__eq__diff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ord_less_eq_real @ ( minus_minus_real @ X @ Y ) @ zero_zero_real ) ) ).
thf(fact_732_real__mult__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
<=> ( A = B ) ) ) ).
thf(fact_733_real__mult__right__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
<=> ( A = B ) ) ) ).
thf(fact_734_real__two__squares__add__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
thf(fact_735_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_736_real__mult__less__iff1,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
<=> ( ord_less_real @ X @ Y ) ) ) ).
thf(fact_737_real__mult__order,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ Y ) ) ) ) ).
thf(fact_738_real__mult__less__mono2,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) ) ) ) ).
thf(fact_739_real__mult__le__cancel__iff2,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ).
thf(fact_740_real__mult__le__cancel__iff1,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ).
thf(fact_741_abs__zmult__eq__1,axiom,
! [M: int,N_1: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N_1 ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
thf(fact_742_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
<=> ( Z = zero_zero_int ) ) ).
thf(fact_743_posDivAlg__0,axiom,
! [B: int] :
( ( posDivAlg @ zero_zero_int @ B )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ).
thf(fact_744_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_745_nat__intermed__int__val,axiom,
! [K: int,F: nat > int,N_1: nat,M: nat] :
( ! [I_1: nat] :
( ( ( ord_less_eq_nat @ M @ I_1 )
& ( ord_less_nat @ I_1 @ N_1 ) )
=> ( 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_nat @ M @ N_1 )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N_1 ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ M @ I_1 )
& ( ord_less_eq_nat @ I_1 @ N_1 )
& ( ( F @ I_1 )
= K ) ) ) ) ) ) ).
thf(fact_746_nat0__intermed__int__val,axiom,
! [K: int,F: nat > int,N_1: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N_1 )
=> ( 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_1 ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N_1 )
& ( ( F @ I_1 )
= K ) ) ) ) ) ).
thf(fact_747_int__val__lemma,axiom,
! [K: int,F: nat > int,N_1: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N_1 )
=> ( 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_1 ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N_1 )
& ( ( F @ I_1 )
= K ) ) ) ) ) ).
thf(fact_748_mult__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( times_times_nat @ M @ N_1 )
= zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( times_times_nat @ M @ N_1 )
= ( plus_plus_nat @ N_1 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ) ) ) ).
thf(fact_749_abs__add__one__not__less__self,axiom,
! [X: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( abs_abs_real @ X ) @ one_one_real ) @ X ) ).
thf(fact_750_abs__add__one__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
thf(fact_751_le0,axiom,
! [N_1: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N_1 ) ).
thf(fact_752_less__zeroE,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ zero_zero_nat ) ).
thf(fact_753_sin__bound__lemma,axiom,
! [U: real,V: real,X: real,Y: real] :
( ( X = Y )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
thf(fact_754_nat__less__add__iff2,axiom,
! [U: nat,M: nat,N_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
<=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N_1 ) ) ) ) ).
thf(fact_755_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N_1 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N_1 ) ) ) ).
thf(fact_756_diff__mult__distrib,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N_1 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N_1 @ K ) ) ) ).
thf(fact_757_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_758_nat__mult__commute,axiom,
! [M: nat,N_1: nat] :
( ( times_times_nat @ M @ N_1 )
= ( times_times_nat @ N_1 @ M ) ) ).
thf(fact_759_nat__mult__assoc,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N_1 ) @ K )
= ( times_times_nat @ M @ ( times_times_nat @ N_1 @ K ) ) ) ).
thf(fact_760_mult__0,axiom,
! [N_1: nat] :
( ( times_times_nat @ zero_zero_nat @ N_1 )
= zero_zero_nat ) ).
thf(fact_761_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_762_mult__is__0,axiom,
! [M: nat,N_1: nat] :
( ( ( times_times_nat @ M @ N_1 )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N_1 = zero_zero_nat ) ) ) ).
thf(fact_763_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N_1 ) )
<=> ( ( K = zero_zero_nat )
| ( M = N_1 ) ) ) ).
thf(fact_764_mult__cancel1,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N_1 ) )
<=> ( ( M = N_1 )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_765_mult__cancel2,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N_1 @ K ) )
<=> ( ( M = N_1 )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_766_diffs0__imp__equal,axiom,
! [M: nat,N_1: nat] :
( ( ( minus_minus_nat @ M @ N_1 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N_1 @ M )
= zero_zero_nat )
=> ( M = N_1 ) ) ) ).
thf(fact_767_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_768_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_769_diff__0__eq__0,axiom,
! [N_1: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N_1 )
= zero_zero_nat ) ).
thf(fact_770_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N_1: nat] :
( ( ( ord_less_nat @ M @ N_1 )
=> ( P @ N_1 @ M ) )
=> ( ( ( M = N_1 )
=> ( P @ N_1 @ M ) )
=> ( ( ( ord_less_nat @ N_1 @ M )
=> ( P @ N_1 @ M ) )
=> ( P @ N_1 @ M ) ) ) ) ).
thf(fact_771_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_772_less__not__refl2,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ N_1 @ M )
=> ( M != N_1 ) ) ).
thf(fact_773_less__irrefl__nat,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ N_1 ) ).
thf(fact_774_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_775_nat__neq__iff,axiom,
! [M: nat,N_1: nat] :
( ( M != N_1 )
<=> ( ( ord_less_nat @ M @ N_1 )
| ( ord_less_nat @ N_1 @ M ) ) ) ).
thf(fact_776_less__not__refl,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ N_1 ) ).
thf(fact_777_diff__less__mono2,axiom,
! [L: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N_1 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_778_less__imp__diff__less,axiom,
! [N_1: nat,J: nat,K: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N_1 ) @ K ) ) ).
thf(fact_779_diff__add__inverse2,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N_1 ) @ N_1 )
= M ) ).
thf(fact_780_diff__add__inverse,axiom,
! [N_1: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N_1 @ M ) @ N_1 )
= M ) ).
thf(fact_781_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_782_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_783_diff__cancel2,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N_1 @ K ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_784_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N_1 @ K ) )
<=> ( M = N_1 ) ) ).
thf(fact_785_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N_1 ) )
<=> ( M = N_1 ) ) ).
thf(fact_786_nat__add__assoc,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N_1 ) @ K )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N_1 @ K ) ) ) ).
thf(fact_787_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_788_nat__add__commute,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ M @ N_1 )
= ( plus_plus_nat @ N_1 @ M ) ) ).
thf(fact_789_add__mult__distrib2,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N_1 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N_1 ) ) ) ).
thf(fact_790_add__mult__distrib,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N_1 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N_1 @ K ) ) ) ).
thf(fact_791_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
thf(fact_792_Nat_Odiff__le__self,axiom,
! [M: nat,N_1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N_1 ) @ M ) ).
thf(fact_793_diff__le__mono2,axiom,
! [L: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N_1 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_794_diff__le__mono,axiom,
! [L: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N_1 @ L ) ) ) ).
thf(fact_795_diff__diff__cancel,axiom,
! [I: nat,N_1: nat] :
( ( ord_less_eq_nat @ I @ N_1 )
=> ( ( minus_minus_nat @ N_1 @ ( minus_minus_nat @ N_1 @ I ) )
= I ) ) ).
thf(fact_796_eq__diff__iff,axiom,
! [N_1: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N_1 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N_1 @ K ) )
<=> ( M = N_1 ) ) ) ) ).
thf(fact_797_Nat_Odiff__diff__eq,axiom,
! [N_1: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N_1 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N_1 @ K ) )
= ( minus_minus_nat @ M @ N_1 ) ) ) ) ).
thf(fact_798_le__diff__iff,axiom,
! [N_1: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N_1 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N_1 @ K ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ) ) ).
thf(fact_799_le__refl,axiom,
! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ N_1 ) ).
thf(fact_800_nat__le__linear,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
| ( ord_less_eq_nat @ N_1 @ M ) ) ).
thf(fact_801_eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( M = N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_802_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_803_le__antisym,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( ord_less_eq_nat @ N_1 @ M )
=> ( M = N_1 ) ) ) ).
thf(fact_804_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_805_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_806_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_807_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_808_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_809_nat__mult__1,axiom,
! [N_1: nat] :
( ( times_times_nat @ one_one_nat @ N_1 )
= N_1 ) ).
thf(fact_810_nat__1__eq__mult__iff,axiom,
! [M: nat,N_1: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N_1 ) )
<=> ( ( M = one_one_nat )
& ( N_1 = one_one_nat ) ) ) ).
thf(fact_811_nat__mult__1__right,axiom,
! [N_1: nat] :
( ( times_times_nat @ N_1 @ one_one_nat )
= N_1 ) ).
thf(fact_812_nat__mult__eq__1__iff,axiom,
! [M: nat,N_1: nat] :
( ( ( times_times_nat @ M @ N_1 )
= one_one_nat )
<=> ( ( M = one_one_nat )
& ( N_1 = one_one_nat ) ) ) ).
thf(fact_813_zero__less__diff,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N_1 @ M ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_814_diff__less,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N_1 ) @ M ) ) ) ).
thf(fact_815_not__less0,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ zero_zero_nat ) ).
thf(fact_816_neq0__conv,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ).
thf(fact_817_less__nat__zero__code,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ zero_zero_nat ) ).
thf(fact_818_gr__implies__not0,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( N_1 != zero_zero_nat ) ) ).
thf(fact_819_gr0I,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ).
thf(fact_820_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_821_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_822_nat__mult__less__cancel1,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_823_nat__mult__eq__cancel1,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N_1 ) )
<=> ( M = N_1 ) ) ) ).
thf(fact_824_mult__less__cancel2,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N_1 @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_825_mult__less__cancel1,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_826_nat__0__less__mult__iff,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ) ).
thf(fact_827_plus__nat_Oadd__0,axiom,
! [N_1: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N_1 )
= N_1 ) ).
thf(fact_828_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_829_add__is__0,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N_1 = zero_zero_nat ) ) ) ).
thf(fact_830_add__eq__self__zero,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= M )
=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_831_diff__add__0,axiom,
! [N_1: nat,M: nat] :
( ( minus_minus_nat @ N_1 @ ( plus_plus_nat @ N_1 @ M ) )
= zero_zero_nat ) ).
thf(fact_832_diff__is__0__eq,axiom,
! [M: nat,N_1: nat] :
( ( ( minus_minus_nat @ M @ N_1 )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_833_diff__is__0__eq_H,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( minus_minus_nat @ M @ N_1 )
= zero_zero_nat ) ) ).
thf(fact_834_less__eq__nat_Osimps_I1_J,axiom,
! [N_1: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N_1 ) ).
thf(fact_835_le__0__eq,axiom,
! [N_1: nat] :
( ( ord_less_eq_nat @ N_1 @ zero_zero_nat )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_836_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
thf(fact_837_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
thf(fact_838_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_839_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_840_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_841_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_842_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_843_less__add__eq__less,axiom,
! [M: nat,N_1: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N_1 ) )
=> ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_844_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_845_add__diff__inverse,axiom,
! [M: nat,N_1: nat] :
( ~ ( ord_less_nat @ M @ N_1 )
=> ( ( plus_plus_nat @ N_1 @ ( minus_minus_nat @ M @ N_1 ) )
= M ) ) ).
thf(fact_846_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_847_mult__eq__self__implies__10,axiom,
! [M: nat,N_1: nat] :
( ( M
= ( times_times_nat @ M @ N_1 ) )
=> ( ( N_1 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
thf(fact_848_less__or__eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( ( ord_less_nat @ M @ N_1 )
| ( M = N_1 ) )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_849_le__neq__implies__less,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( M != N_1 )
=> ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_850_less__imp__le__nat,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_851_le__eq__less__or__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
<=> ( ( ord_less_nat @ M @ N_1 )
| ( M = N_1 ) ) ) ).
thf(fact_852_nat__less__le,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
<=> ( ( ord_less_eq_nat @ M @ N_1 )
& ( M != N_1 ) ) ) ).
thf(fact_853_less__diff__iff,axiom,
! [N_1: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N_1 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N_1 @ K ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ) ) ).
thf(fact_854_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
thf(fact_855_diff__diff__right,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_856_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
thf(fact_857_le__add__diff,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ord_less_eq_nat @ K @ N_1 )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N_1 @ M ) @ K ) ) ) ).
thf(fact_858_le__add__diff__inverse,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( plus_plus_nat @ N_1 @ ( minus_minus_nat @ M @ N_1 ) )
= M ) ) ).
thf(fact_859_add__diff__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
thf(fact_860_le__diff__conv2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_861_le__add__diff__inverse2,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N_1 ) @ N_1 )
= M ) ) ).
thf(fact_862_le__imp__diff__is__add,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
<=> ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
thf(fact_863_diff__add__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
thf(fact_864_add__diff__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
thf(fact_865_diff__add__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
thf(fact_866_nat__eq__add__iff2,axiom,
! [U: nat,M: nat,N_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N_1 ) ) ) ) ).
thf(fact_867_nat__diff__add__eq2,axiom,
! [U: nat,M: nat,N_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N_1 ) ) ) ) ).
thf(fact_868_nat__le__add__iff2,axiom,
! [U: nat,M: nat,N_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N_1 ) ) ) ) ).
thf(fact_869_nat__eq__add__iff1,axiom,
! [U: nat,M: nat,N_1: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N_1 ) ) ) ).
thf(fact_870_nat__diff__add__eq1,axiom,
! [U: nat,M: nat,N_1: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N_1 ) ) ) ).
thf(fact_871_nat__le__add__iff1,axiom,
! [U: nat,M: nat,N_1: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N_1 ) ) ) ).
thf(fact_872_add__leE,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N_1 )
=> ~ ( ( ord_less_eq_nat @ M @ N_1 )
=> ~ ( ord_less_eq_nat @ K @ N_1 ) ) ) ).
thf(fact_873_add__leD1,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_874_add__leD2,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N_1 )
=> ( ord_less_eq_nat @ K @ N_1 ) ) ).
thf(fact_875_add__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 @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_876_add__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_877_trans__le__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_878_trans__le__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_879_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_880_le__iff__add,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
<=> ? [K_1: nat] :
( N_1
= ( plus_plus_nat @ M @ K_1 ) ) ) ).
thf(fact_881_le__add1,axiom,
! [N_1: nat,M: nat] : ( ord_less_eq_nat @ N_1 @ ( plus_plus_nat @ N_1 @ M ) ) ).
thf(fact_882_le__add2,axiom,
! [N_1: nat,M: nat] : ( ord_less_eq_nat @ N_1 @ ( plus_plus_nat @ M @ N_1 ) ) ).
thf(fact_883_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X
= ( minus_minus_real @ Y @ Z ) )
<=> ( Y
= ( plus_plus_real @ X @ Z ) ) ) ).
thf(fact_884_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P @ D ) ) ) ) ).
thf(fact_885_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P @ D ) ) ) ) ).
thf(fact_886_add__gr__0,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ) ).
thf(fact_887_mult__le__cancel1,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ) ).
thf(fact_888_mult__le__cancel2,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N_1 @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ) ).
thf(fact_889_nat__mult__le__cancel1,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ) ).
thf(fact_890_nat__less__add__iff1,axiom,
! [U: nat,M: nat,N_1: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N_1 ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N_1 ) ) ) ).
thf(fact_891_ex__least__nat__less,axiom,
! [N_1: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N_1 )
=> ? [K_1: nat] :
( ( ord_less_nat @ K_1 @ N_1 )
& ! [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K_1 )
=> ~ ( P @ I_1 ) )
& ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
thf(fact_892_rabs__ratiotest__lemma,axiom,
! [X: real,Y: real,C: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( times_times_real @ C @ ( abs_abs_real @ Y ) ) )
=> ( X = zero_zero_real ) ) ) ).
thf(fact_893_mono__nat__linear__lb,axiom,
! [M: nat,K: nat,F: nat > nat] :
( ! [M_1: nat,N: nat] :
( ( ord_less_nat @ M_1 @ N )
=> ( ord_less_nat @ ( F @ M_1 ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_894_ex__least__nat__le,axiom,
! [N_1: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N_1 )
=> ? [K_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N_1 )
& ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ K_1 )
=> ~ ( P @ I_1 ) )
& ( P @ K_1 ) ) ) ) ).
thf(fact_895_le__Suc__ex__iff,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
<=> ? [N: nat] :
( L
= ( plus_plus_nat @ K @ N ) ) ) ).
thf(fact_896_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ( plus_plus_nat @ I @ K_1 )
= J ) ) ) ).
thf(fact_897_not__real__square__gt__zero,axiom,
! [X: real] :
( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) )
<=> ( X = zero_zero_real ) ) ).
thf(fact_898_mult__left__cancel,axiom,
! [N_1: nat,M: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ N_1 )
= ( times_times_nat @ K @ M ) )
=> ( N_1 = M ) ) ) ).
thf(fact_899_lemma__interval,axiom,
! [B: real,A: real,X: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ( ord_less_eq_real @ A @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ B ) ) ) ) ) ) ).
thf(fact_900_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N: nat] :
( L
= ( plus_plus_nat @ K @ N ) ) ) ).
thf(fact_901_less__mono__imp__le__mono,axiom,
! [I: nat,J: nat,F: nat > nat] :
( ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ ( F @ I_1 ) @ ( F @ J_1 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
thf(fact_902_lemma__interval__lt,axiom,
! [B: real,A: real,X: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ( ord_less_real @ A @ Y_1 )
& ( ord_less_real @ Y_1 @ B ) ) ) ) ) ) ).
thf(fact_903_real__lbound__gt__zero,axiom,
! [D2: real,D1: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
thf(fact_904_natceiling__add__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ one_one_nat ) ) ) ).
thf(fact_905_natceiling__number__of__eq,axiom,
! [N_1: int] :
( ( natceiling @ ( number267125858f_real @ N_1 ) )
= ( number_number_of_nat @ N_1 ) ) ).
thf(fact_906_natceiling__zero,axiom,
( ( natceiling @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_907_zero__le__natceiling,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natceiling @ X ) ) ).
thf(fact_908_natceiling__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ ( natceiling @ Y ) ) ) ).
thf(fact_909_natceiling__one,axiom,
( ( natceiling @ one_one_real )
= one_one_nat ) ).
thf(fact_910_natceiling__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natceiling @ X )
= zero_zero_nat ) ) ).
thf(fact_911_natceiling__le__eq__one,axiom,
! [X: real] :
( ( ord_less_eq_nat @ ( natceiling @ X ) @ one_one_nat )
<=> ( ord_less_eq_real @ X @ one_one_real ) ) ).
thf(fact_912_natfloor__add__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ one_one_nat ) ) ) ).
thf(fact_913_nat__mult__eq__one,axiom,
! [N_1: nat,M: nat] :
( ( ( times_times_nat @ N_1 @ M )
= one_one_nat )
<=> ( ( N_1 = one_one_nat )
& ( M = one_one_nat ) ) ) ).
thf(fact_914_natceiling__eq,axiom,
! [N_1: nat,X: real] :
( ( ord_less_real @ ( real_nat @ N_1 ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( real_nat @ N_1 ) @ one_one_real ) )
=> ( ( natceiling @ X )
= ( plus_plus_nat @ N_1 @ one_one_nat ) ) ) ) ).
thf(fact_915_real__of__nat__ge__zero,axiom,
! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( real_nat @ N_1 ) ) ).
thf(fact_916_natceiling__real__of__nat,axiom,
! [N_1: nat] :
( ( natceiling @ ( real_nat @ N_1 ) )
= N_1 ) ).
thf(fact_917_real__of__nat__less__iff,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_real @ ( real_nat @ N_1 ) @ ( real_nat @ M ) )
<=> ( ord_less_nat @ N_1 @ M ) ) ).
thf(fact_918_real__of__nat__zero,axiom,
( ( real_nat @ zero_zero_nat )
= zero_zero_real ) ).
thf(fact_919_real__of__nat__zero__iff,axiom,
! [N_1: nat] :
( ( ( real_nat @ N_1 )
= zero_zero_real )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_920_abs__real__of__nat__cancel,axiom,
! [X: nat] :
( ( abs_abs_real @ ( real_nat @ X ) )
= ( real_nat @ X ) ) ).
thf(fact_921_natfloor__real__of__nat,axiom,
! [N_1: nat] :
( ( natfloor @ ( real_nat @ N_1 ) )
= N_1 ) ).
thf(fact_922_Ints__real__of__nat,axiom,
! [N_1: nat] : ( member_real @ ( real_nat @ N_1 ) @ ring_1_Ints_real ) ).
thf(fact_923_natfloor__number__of__eq,axiom,
! [N_1: int] :
( ( natfloor @ ( number267125858f_real @ N_1 ) )
= ( number_number_of_nat @ N_1 ) ) ).
thf(fact_924_real__of__nat__inject,axiom,
! [N_1: nat,M: nat] :
( ( ( real_nat @ N_1 )
= ( real_nat @ M ) )
<=> ( N_1 = M ) ) ).
thf(fact_925_real__natfloor__le,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_nat @ ( natfloor @ X ) ) @ X ) ) ).
thf(fact_926_le__natfloor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( real_nat @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( natfloor @ A ) ) ) ).
thf(fact_927_real__natfloor__gt__diff__one,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ one_one_real ) @ ( real_nat @ ( natfloor @ X ) ) ) ).
thf(fact_928_natfloor__subtract,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A ) @ X )
=> ( ( natfloor @ ( minus_minus_real @ X @ ( real_nat @ A ) ) )
= ( minus_minus_nat @ ( natfloor @ X ) @ A ) ) ) ).
thf(fact_929_real__natfloor__add__one__gt,axiom,
! [X: real] : ( ord_less_real @ X @ ( plus_plus_real @ ( real_nat @ ( natfloor @ X ) ) @ one_one_real ) ) ).
thf(fact_930_le__natfloor__eq,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ A @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ ( real_nat @ A ) @ X ) ) ) ).
thf(fact_931_not__real__of__nat__less__zero,axiom,
! [N_1: nat] :
~ ( ord_less_real @ ( real_nat @ N_1 ) @ zero_zero_real ) ).
thf(fact_932_real__of__nat__le__iff,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_real @ ( real_nat @ N_1 ) @ ( real_nat @ M ) )
<=> ( ord_less_eq_nat @ N_1 @ M ) ) ).
thf(fact_933_less__natfloor,axiom,
! [N_1: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( real_nat @ N_1 ) )
=> ( ord_less_nat @ ( natfloor @ X ) @ N_1 ) ) ) ).
thf(fact_934_real__of__nat__mult,axiom,
! [M: nat,N_1: nat] :
( ( real_nat @ ( times_times_nat @ M @ N_1 ) )
= ( times_times_real @ ( real_nat @ M ) @ ( real_nat @ N_1 ) ) ) ).
thf(fact_935_natfloor__add,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ ( real_nat @ A ) ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ A ) ) ) ).
thf(fact_936_real__of__nat__add,axiom,
! [M: nat,N_1: nat] :
( ( real_nat @ ( plus_plus_nat @ M @ N_1 ) )
= ( plus_plus_real @ ( real_nat @ M ) @ ( real_nat @ N_1 ) ) ) ).
thf(fact_937_ge__natfloor__plus__one__imp__gt,axiom,
! [Z: real,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( natfloor @ Z ) @ one_one_nat ) @ N_1 )
=> ( ord_less_real @ Z @ ( real_nat @ N_1 ) ) ) ).
thf(fact_938_natfloor__eq,axiom,
! [N_1: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ N_1 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_nat @ N_1 ) @ one_one_real ) )
=> ( ( natfloor @ X )
= N_1 ) ) ) ).
thf(fact_939_real__of__nat__1,axiom,
( ( real_nat @ one_one_nat )
= one_one_real ) ).
thf(fact_940_real__natceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( real_nat @ ( natceiling @ X ) ) ) ).
thf(fact_941_natfloor__zero,axiom,
( ( natfloor @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_942_zero__le__natfloor,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natfloor @ X ) ) ).
thf(fact_943_natfloor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natfloor @ X ) @ ( natfloor @ Y ) ) ) ).
thf(fact_944_natfloor__one,axiom,
( ( natfloor @ one_one_real )
= one_one_nat ) ).
thf(fact_945_real__of__nat__le__zero__cancel__iff,axiom,
! [N_1: nat] :
( ( ord_less_eq_real @ ( real_nat @ N_1 ) @ zero_zero_real )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_946_real__of__nat__diff,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( real_nat @ ( minus_minus_nat @ M @ N_1 ) )
= ( minus_minus_real @ ( real_nat @ M ) @ ( real_nat @ N_1 ) ) ) ) ).
thf(fact_947_natceiling__le,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_real @ X @ ( real_nat @ A ) )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ A ) ) ).
thf(fact_948_natfloor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natfloor @ X )
= zero_zero_nat ) ) ).
thf(fact_949_real__of__nat__gt__zero__cancel__iff,axiom,
! [N_1: nat] :
( ( ord_less_real @ zero_zero_real @ ( real_nat @ N_1 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ).
thf(fact_950_nat__less__real__le,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ N_1 @ M )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_nat @ N_1 ) @ one_one_real ) @ ( real_nat @ M ) ) ) ).
thf(fact_951_nat__le__real__less,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
<=> ( ord_less_real @ ( real_nat @ N_1 ) @ ( plus_plus_real @ ( real_nat @ M ) @ one_one_real ) ) ) ).
thf(fact_952_natceiling__le__eq,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( natceiling @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( real_nat @ A ) ) ) ) ).
thf(fact_953_natceiling__subtract,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A ) @ X )
=> ( ( natceiling @ ( minus_minus_real @ X @ ( real_nat @ A ) ) )
= ( minus_minus_nat @ ( natceiling @ X ) @ A ) ) ) ).
thf(fact_954_le__natfloor__eq__one,axiom,
! [X: real] :
( ( ord_less_eq_nat @ one_one_nat @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ one_one_real @ X ) ) ).
thf(fact_955_natceiling__add,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ ( real_nat @ A ) ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ A ) ) ) ).
thf(fact_956_le__mult__natfloor,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( natfloor @ A ) @ ( natfloor @ B ) ) @ ( natfloor @ ( times_times_real @ A @ B ) ) ) ) ) ).
thf(fact_957_reals__Archimedean6,axiom,
! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ? [N: nat] :
( ( ord_less_eq_real @ ( real_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) @ R )
& ( ord_less_real @ R @ ( real_nat @ N ) ) ) ) ).
thf(fact_958_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y_1: real] :
? [N: nat] : ( ord_less_real @ Y_1 @ ( times_times_real @ ( real_nat @ N ) @ X ) ) ) ).
thf(fact_959_natceiling__add__number__of,axiom,
! [X: real,N_1: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N_1 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ ( number267125858f_real @ N_1 ) ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ ( number_number_of_nat @ N_1 ) ) ) ) ) ).
thf(fact_960_not__neg__0,axiom,
~ ( nat_neg @ zero_zero_int ) ).
thf(fact_961_not__neg__1,axiom,
~ ( nat_neg @ one_one_int ) ).
thf(fact_962_neg__def,axiom,
! [Z_1: int] :
( ( nat_neg @ Z_1 )
<=> ( ord_less_int @ Z_1 @ zero_zero_int ) ) ).
thf(fact_963_not__neg__eq__ge__0,axiom,
! [X: int] :
( ~ ( nat_neg @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_964_neg__imp__number__of__eq__0,axiom,
! [V: int] :
( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( number_number_of_nat @ V )
= zero_zero_nat ) ) ).
thf(fact_965_eq__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ( number_number_of_nat @ V )
= ( number_number_of_nat @ V_1 ) )
<=> ( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ord_less_eq_int @ ( number_number_of_int @ V_1 ) @ zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( number_number_of_int @ V )
= zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( V = V_1 ) ) ) ) ) ) ).
thf(fact_966_nat__number__of__add__left,axiom,
! [V_1: int,K: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V ) @ K ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) @ K ) ) ) ) ) ) ).
thf(fact_967_real__of__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( real_nat @ ( number_number_of_nat @ V ) )
= zero_zero_real ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( real_nat @ ( number_number_of_nat @ V ) )
= ( number267125858f_real @ V ) ) ) ) ).
thf(fact_968_le__natfloor__eq__number__of,axiom,
! [X: real,N_1: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N_1 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( number_number_of_nat @ N_1 ) @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ ( number267125858f_real @ N_1 ) @ X ) ) ) ) ).
thf(fact_969_natceiling__le__eq__number__of,axiom,
! [X: real,N_1: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N_1 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( natceiling @ X ) @ ( number_number_of_nat @ N_1 ) )
<=> ( ord_less_eq_real @ X @ ( number267125858f_real @ N_1 ) ) ) ) ) ).
thf(fact_970_natfloor__add__number__of,axiom,
! [X: real,N_1: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N_1 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ ( number267125858f_real @ N_1 ) ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ ( number_number_of_nat @ N_1 ) ) ) ) ) ).
thf(fact_971_reals__Archimedean2,axiom,
! [X: real] :
? [N: nat] : ( ord_less_real @ X @ ( real_nat @ N ) ) ).
thf(fact_972_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_973_termination__basic__simps_I3_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_974_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_975_termination__basic__simps_I1_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_976_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_977_ceiling__eq2,axiom,
! [N_1: int,X: real] :
( ( ord_less_real @ ( real_int @ N_1 ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( real_int @ N_1 ) @ one_one_real ) )
=> ( ( archim856651990g_real @ X )
= ( plus_plus_int @ N_1 @ one_one_int ) ) ) ) ).
thf(fact_978_real__of__int__add,axiom,
! [X: int,Y: int] :
( ( real_int @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_real @ ( real_int @ X ) @ ( real_int @ Y ) ) ) ).
thf(fact_979_ceiling__add,axiom,
! [X: real,A: int] :
( ( archim856651990g_real @ ( plus_plus_real @ X @ ( real_int @ A ) ) )
= ( plus_plus_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_980_real__of__int__floor__cancel,axiom,
! [X: real] :
( ( ( real_int @ ( archim1246769320r_real @ X ) )
= X )
<=> ? [N: int] :
( X
= ( real_int @ N ) ) ) ).
thf(fact_981_floor__real__of__int,axiom,
! [N_1: int] :
( ( archim1246769320r_real @ ( real_int @ N_1 ) )
= N_1 ) ).
thf(fact_982_real__of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( real_int @ ( archim856651990g_real @ X ) )
= X )
<=> ? [N: int] :
( X
= ( real_int @ N ) ) ) ).
thf(fact_983_ceiling__real__of__int,axiom,
! [N_1: int] :
( ( archim856651990g_real @ ( real_int @ N_1 ) )
= N_1 ) ).
thf(fact_984_real__number__of,axiom,
! [V: int] :
( ( real_int @ ( number_number_of_int @ V ) )
= ( number267125858f_real @ V ) ) ).
thf(fact_985_real__of__int__inject,axiom,
! [X: int,Y: int] :
( ( ( real_int @ X )
= ( real_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_986_real__of__int__ceiling__ge,axiom,
! [R: real] : ( ord_less_eq_real @ R @ ( real_int @ ( archim856651990g_real @ R ) ) ) ).
thf(fact_987_real__of__one,axiom,
( ( real_int @ one_one_int )
= one_one_real ) ).
thf(fact_988_real__of__int__mult,axiom,
! [X: int,Y: int] :
( ( real_int @ ( times_times_int @ X @ Y ) )
= ( times_times_real @ ( real_int @ X ) @ ( real_int @ Y ) ) ) ).
thf(fact_989_ceiling__le__eq,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_990_ceiling__le,axiom,
! [X: real,A: int] :
( ( ord_less_eq_real @ X @ ( real_int @ A ) )
=> ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_991_ceiling__le__real,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A )
=> ( ord_less_eq_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_992_real__of__int__le__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_real @ ( real_int @ X ) @ ( real_int @ Y ) )
<=> ( ord_less_eq_int @ X @ Y ) ) ).
thf(fact_993_real__of__int__less__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_real @ ( real_int @ X ) @ ( real_int @ Y ) )
<=> ( ord_less_int @ X @ Y ) ) ).
thf(fact_994_less__ceiling__eq,axiom,
! [A: int,X: real] :
( ( ord_less_int @ A @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_995_real__of__int__zero__cancel,axiom,
! [X: int] :
( ( ( real_int @ X )
= zero_zero_real )
<=> ( X = zero_zero_int ) ) ).
thf(fact_996_real__of__int__zero,axiom,
( ( real_int @ zero_zero_int )
= zero_zero_real ) ).
thf(fact_997_ceiling__subtract,axiom,
! [X: real,A: int] :
( ( archim856651990g_real @ ( minus_minus_real @ X @ ( real_int @ A ) ) )
= ( minus_minus_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_998_real__of__int__diff,axiom,
! [X: int,Y: int] :
( ( real_int @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_real @ ( real_int @ X ) @ ( real_int @ Y ) ) ) ).
thf(fact_999_real__of__int__floor__le,axiom,
! [R: real] : ( ord_less_eq_real @ ( real_int @ ( archim1246769320r_real @ R ) ) @ R ) ).
thf(fact_1000_real__of__int__abs,axiom,
! [X: int] :
( ( real_int @ ( abs_abs_int @ X ) )
= ( abs_abs_real @ ( real_int @ X ) ) ) ).
thf(fact_1001_Ints__real__of__int,axiom,
! [X: int] : ( member_real @ ( real_int @ X ) @ ring_1_Ints_real ) ).
thf(fact_1002_real__of__int__ceiling__le__add__one,axiom,
! [R: real] : ( ord_less_eq_real @ ( real_int @ ( archim856651990g_real @ R ) ) @ ( plus_plus_real @ R @ one_one_real ) ) ).
thf(fact_1003_real__of__int__ceiling__diff__one__le,axiom,
! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_int @ ( archim856651990g_real @ R ) ) @ one_one_real ) @ R ) ).
thf(fact_1004_floor__less__eq,axiom,
! [X: real,A: int] :
( ( ord_less_int @ ( archim1246769320r_real @ X ) @ A )
<=> ( ord_less_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_1005_le__floor,axiom,
! [A: int,X: real] :
( ( ord_less_eq_real @ ( real_int @ A ) @ X )
=> ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) ) ) ).
thf(fact_1006_le__floor__eq,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_1007_real__le__floor,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) )
=> ( ord_less_eq_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_1008_floor__add,axiom,
! [X: real,A: int] :
( ( archim1246769320r_real @ ( plus_plus_real @ X @ ( real_int @ A ) ) )
= ( plus_plus_int @ ( archim1246769320r_real @ X ) @ A ) ) ).
thf(fact_1009_floor__subtract,axiom,
! [X: real,A: int] :
( ( archim1246769320r_real @ ( minus_minus_real @ X @ ( real_int @ A ) ) )
= ( minus_minus_int @ ( archim1246769320r_real @ X ) @ A ) ) ).
thf(fact_1010_real__of__int__gt__zero__cancel__iff,axiom,
! [N_1: int] :
( ( ord_less_real @ zero_zero_real @ ( real_int @ N_1 ) )
<=> ( ord_less_int @ zero_zero_int @ N_1 ) ) ).
thf(fact_1011_real__of__int__lt__zero__cancel__iff,axiom,
! [N_1: int] :
( ( ord_less_real @ ( real_int @ N_1 ) @ zero_zero_real )
<=> ( ord_less_int @ N_1 @ zero_zero_int ) ) ).
thf(fact_1012_real__of__int__ge__zero__cancel__iff,axiom,
! [N_1: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_int @ N_1 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ N_1 ) ) ).
thf(fact_1013_real__of__int__le__zero__cancel__iff,axiom,
! [N_1: int] :
( ( ord_less_eq_real @ ( real_int @ N_1 ) @ zero_zero_real )
<=> ( ord_less_eq_int @ N_1 @ zero_zero_int ) ) ).
thf(fact_1014_number__of__less__real__of__int__iff,axiom,
! [N_1: int,M: int] :
( ( ord_less_real @ ( number267125858f_real @ N_1 ) @ ( real_int @ M ) )
<=> ( ord_less_int @ ( number_number_of_int @ N_1 ) @ M ) ) ).
thf(fact_1015_number__of__less__real__of__int__iff2,axiom,
! [M: int,N_1: int] :
( ( ord_less_real @ ( real_int @ M ) @ ( number267125858f_real @ N_1 ) )
<=> ( ord_less_int @ M @ ( number_number_of_int @ N_1 ) ) ) ).
thf(fact_1016_number__of__le__real__of__int__iff,axiom,
! [N_1: int,M: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ N_1 ) @ ( real_int @ M ) )
<=> ( ord_less_eq_int @ ( number_number_of_int @ N_1 ) @ M ) ) ).
thf(fact_1017_number__of__le__real__of__int__iff2,axiom,
! [M: int,N_1: int] :
( ( ord_less_eq_real @ ( real_int @ M ) @ ( number267125858f_real @ N_1 ) )
<=> ( ord_less_eq_int @ M @ ( number_number_of_int @ N_1 ) ) ) ).
thf(fact_1018_int__less__real__le,axiom,
! [N_1: int,M: int] :
( ( ord_less_int @ N_1 @ M )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_int @ N_1 ) @ one_one_real ) @ ( real_int @ M ) ) ) ).
thf(fact_1019_int__le__real__less,axiom,
! [N_1: int,M: int] :
( ( ord_less_eq_int @ N_1 @ M )
<=> ( ord_less_real @ ( real_int @ N_1 ) @ ( plus_plus_real @ ( real_int @ M ) @ one_one_real ) ) ) ).
thf(fact_1020_lemma__floor2,axiom,
! [N_1: int,X: int] :
( ( ord_less_real @ ( real_int @ N_1 ) @ ( plus_plus_real @ ( real_int @ X ) @ one_one_real ) )
=> ( ord_less_eq_int @ N_1 @ X ) ) ).
thf(fact_1021_ceiling__eq3,axiom,
! [N_1: int,X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( real_int @ N_1 ) @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ ( real_int @ N_1 ) )
=> ( ( archim856651990g_real @ X )
= N_1 ) ) ) ).
thf(fact_1022_ceiling__less__eq,axiom,
! [X: real,A: int] :
( ( ord_less_int @ ( archim856651990g_real @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( minus_minus_real @ ( real_int @ A ) @ one_one_real ) ) ) ).
thf(fact_1023_le__ceiling__eq,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( minus_minus_real @ ( real_int @ A ) @ one_one_real ) @ X ) ) ).
thf(fact_1024_real__of__int__floor__add__one__ge,axiom,
! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( real_int @ ( archim1246769320r_real @ R ) ) @ one_one_real ) ) ).
thf(fact_1025_real__of__int__floor__add__one__gt,axiom,
! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( real_int @ ( archim1246769320r_real @ R ) ) @ one_one_real ) ) ).
thf(fact_1026_floor__eq,axiom,
! [N_1: int,X: real] :
( ( ord_less_real @ ( real_int @ N_1 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ N_1 ) @ one_one_real ) )
=> ( ( archim1246769320r_real @ X )
= N_1 ) ) ) ).
thf(fact_1027_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( real_int @ ( archim1246769320r_real @ R ) ) ) ).
thf(fact_1028_real__of__int__floor__gt__diff__one,axiom,
! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( real_int @ ( archim1246769320r_real @ R ) ) ) ).
thf(fact_1029_lemma__floor,axiom,
! [N_1: int,M: int,R: real] :
( ( ord_less_eq_real @ ( real_int @ M ) @ R )
=> ( ( ord_less_real @ R @ ( plus_plus_real @ ( real_int @ N_1 ) @ one_one_real ) )
=> ( ord_less_eq_int @ M @ N_1 ) ) ) ).
thf(fact_1030_ceiling__eq,axiom,
! [N_1: int,X: real] :
( ( ord_less_real @ ( real_int @ N_1 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ N_1 ) @ one_one_real ) )
=> ( ( archim856651990g_real @ X )
= ( plus_plus_int @ N_1 @ one_one_int ) ) ) ) ).
thf(fact_1031_floor__eq2,axiom,
! [N_1: int,X: real] :
( ( ord_less_eq_real @ ( real_int @ N_1 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ N_1 ) @ one_one_real ) )
=> ( ( archim1246769320r_real @ X )
= N_1 ) ) ) ).
thf(fact_1032_less__floor__eq,axiom,
! [A: int,X: real] :
( ( ord_less_int @ A @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_int @ A ) @ one_one_real ) @ X ) ) ).
thf(fact_1033_floor__le__eq,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim1246769320r_real @ X ) @ A )
<=> ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ A ) @ one_one_real ) ) ) ).
thf(fact_1034_reals__Archimedean__6b__int,axiom,
! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ? [N: int] :
( ( ord_less_eq_real @ ( real_int @ N ) @ R )
& ( ord_less_real @ R @ ( real_int @ ( plus_plus_int @ N @ one_one_int ) ) ) ) ) ).
thf(fact_1035_reals__Archimedean__6c__int,axiom,
! [R: real] :
( ( ord_less_real @ R @ zero_zero_real )
=> ? [N: int] :
( ( ord_less_eq_real @ ( real_int @ N ) @ R )
& ( ord_less_real @ R @ ( real_int @ ( plus_plus_int @ N @ one_one_int ) ) ) ) ) ).
thf(fact_1036_real__lb__ub__int,axiom,
! [R: real] :
? [N: int] :
( ( ord_less_eq_real @ ( real_int @ N ) @ R )
& ( ord_less_real @ R @ ( real_int @ ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
thf(fact_1037_ceiling__real__of__nat,axiom,
! [N_1: nat] :
( ( archim856651990g_real @ ( real_nat @ N_1 ) )
= ( semiri1621563631at_int @ N_1 ) ) ).
thf(fact_1038_int__less__0__conv,axiom,
! [K: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
thf(fact_1039_zero__zle__int,axiom,
! [N_1: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N_1 ) ) ).
thf(fact_1040_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P @ X_1 ) )
<=> ! [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_1041_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( P @ X_1 ) )
<=> ? [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_1042_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z ) ) ).
thf(fact_1043_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_nat @ X @ Y ) ) ).
thf(fact_1044_zless__int,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1045_abs__int__eq,axiom,
! [M: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
= ( semiri1621563631at_int @ M ) ) ).
thf(fact_1046_int__eq__0__conv,axiom,
! [N_1: nat] :
( ( ( semiri1621563631at_int @ N_1 )
= zero_zero_int )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_1047_int__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_1048_transfer__int__nat__numerals_I1_J,axiom,
( zero_zero_int
= ( semiri1621563631at_int @ zero_zero_nat ) ) ).
thf(fact_1049_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_1050_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_1051_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ X @ Y ) ) ) ).
thf(fact_1052_zadd__int,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N_1 ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N_1 ) ) ) ).
thf(fact_1053_zadd__int__left,axiom,
! [M: nat,N_1: nat,Z: int] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N_1 ) @ Z ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N_1 ) ) @ Z ) ) ).
thf(fact_1054_zle__iff__zadd,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ Z )
<=> ? [N: nat] :
( Z
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ N ) ) ) ) ).
thf(fact_1055_zmult__int,axiom,
! [M: nat,N_1: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N_1 ) )
= ( semiri1621563631at_int @ ( times_times_nat @ M @ N_1 ) ) ) ).
thf(fact_1056_int__mult,axiom,
! [M: nat,N_1: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M @ N_1 ) )
= ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_1057_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( times_times_nat @ X @ Y ) ) ) ).
thf(fact_1058_zle__int,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1059_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_1060_not__neg__int,axiom,
! [N_1: nat] :
~ ( nat_neg @ ( semiri1621563631at_int @ N_1 ) ) ).
thf(fact_1061_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
thf(fact_1062_real__of__int__of__nat__eq,axiom,
! [N_1: nat] :
( ( real_int @ ( semiri1621563631at_int @ N_1 ) )
= ( real_nat @ N_1 ) ) ).
thf(fact_1063_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
thf(fact_1064_floor__real__of__nat,axiom,
! [N_1: nat] :
( ( archim1246769320r_real @ ( real_nat @ N_1 ) )
= ( semiri1621563631at_int @ N_1 ) ) ).
thf(fact_1065_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X: nat,Y: nat] :
( ( nat_tsub @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) ) ).
thf(fact_1066_int__le__0__conv,axiom,
! [N_1: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N_1 ) @ zero_zero_int )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_1067_zdiff__int,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( minus_minus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N_1 ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1068_zero__less__int__conv,axiom,
! [N_1: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N_1 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ).
thf(fact_1069_zmult__zless__mono2__lemma,axiom,
! [K: nat,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ J ) ) ) ) ).
thf(fact_1070_int__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
= zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
= ( number_number_of_int @ V ) ) ) ) ).
thf(fact_1071_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) )
<=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
thf(fact_1072_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
& ( K
= ( semiri1621563631at_int @ N ) ) ) ) ).
thf(fact_1073_int__int__eq,axiom,
! [M: nat,N_1: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N_1 ) )
<=> ( M = N_1 ) ) ).
thf(fact_1074_int__if__cong,axiom,
! [X: nat,Y: nat,P: $o] :
( ( P
=> ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) )
& ( ~ P
=> ( ( semiri1621563631at_int @ Y )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) ) ) ).
thf(fact_1075_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_1076_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N: nat] :
( K
= ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_1077_nonneg__eq__int,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ~ ! [M_1: nat] :
( Z
!= ( semiri1621563631at_int @ M_1 ) ) ) ).
thf(fact_1078_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N: nat] :
( K
!= ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_1079_int__diff__cases,axiom,
! [Z: int] :
~ ! [M_1: nat,N: nat] :
( Z
!= ( minus_minus_int @ ( semiri1621563631at_int @ M_1 ) @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_1080_mult__nat__number__of,axiom,
! [V_1: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ ( times_times_int @ V @ V_1 ) ) ) ) ) ).
thf(fact_1081_nat__number__of__mult__left,axiom,
! [V_1: int,K: nat,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V @ V_1 ) ) @ K ) ) ) ) ).
thf(fact_1082_diff__bin__simps_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ pls )
= K ) ).
thf(fact_1083_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_1084_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_1085_add__Pls,axiom,
! [K: int] :
( ( plus_plus_int @ pls @ K )
= K ) ).
thf(fact_1086_add__Pls__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ pls )
= K ) ).
thf(fact_1087_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_1088_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_1089_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_1090_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_1091_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_1092_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_1093_not__neg__number__of__Pls,axiom,
~ ( nat_neg @ ( number_number_of_int @ pls ) ) ).
thf(fact_1094_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V ) )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_1095_eq__number__of__0,axiom,
! [V: int] :
( ( ( number_number_of_nat @ V )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_1096_less__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ord_less_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
<=> ( ( ( ord_less_int @ V @ V_1 )
=> ( ord_less_int @ pls @ V_1 ) )
& ( ord_less_int @ V @ V_1 ) ) ) ).
thf(fact_1097_number__of2,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ pls ) ).
thf(fact_1098_le__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
<=> ( ~ ( ord_less_eq_int @ V @ V_1 )
=> ( ord_less_eq_int @ V @ pls ) ) ) ).
thf(fact_1099_less__0__number__of,axiom,
! [V: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
<=> ( ord_less_int @ pls @ V ) ) ).
thf(fact_1100_add__nat__number__of,axiom,
! [V_1: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ V_1 ) ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ V ) ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) ) ) ) ) ) ).
thf(fact_1101_nat__1__add__number__of,axiom,
! [V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
= one_one_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_1102_nat__number__of__add__1,axiom,
! [V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= one_one_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_1103_succ__def,axiom,
! [K: int] :
( ( succ @ K )
= ( plus_plus_int @ K @ one_one_int ) ) ).
thf(fact_1104_nat__number__of__diff__1,axiom,
! [V: int] :
( ( ( ord_less_eq_int @ V @ pls )
=> ( ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ord_less_eq_int @ V @ pls )
=> ( ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= ( number_number_of_nat @ ( pred @ V ) ) ) ) ) ).
thf(fact_1105_Suc__nat__number__of__add,axiom,
! [N_1: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N_1 ) )
= ( plus_plus_nat @ one_one_nat @ N_1 ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N_1 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( succ @ V ) ) @ N_1 ) ) ) ) ).
thf(fact_1106_lessI,axiom,
! [N_1: nat] : ( ord_less_nat @ N_1 @ ( suc @ N_1 ) ) ).
thf(fact_1107_Suc__mono,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_1 ) ) ) ).
thf(fact_1108_zero__less__Suc,axiom,
! [N_1: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N_1 ) ) ).
thf(fact_1109_succ__pred,axiom,
! [X: int] :
( ( succ @ ( pred @ X ) )
= X ) ).
thf(fact_1110_less__Suc__eq__0__disj,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
<=> ( ( M = zero_zero_nat )
| ? [J_1: nat] :
( ( M
= ( suc @ J_1 ) )
& ( ord_less_nat @ J_1 @ N_1 ) ) ) ) ).
thf(fact_1111_less__Suc0,axiom,
! [N_1: nat] :
( ( ord_less_nat @ N_1 @ ( suc @ zero_zero_nat ) )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_1112_gr0__conv__Suc,axiom,
! [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
<=> ? [M_1: nat] :
( N_1
= ( suc @ M_1 ) ) ) ).
thf(fact_1113_add__is__1,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_1114_one__is__add,axiom,
! [M: nat,N_1: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N_1 ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_1115_mult__eq__1__iff,axiom,
! [M: nat,N_1: nat] :
( ( ( times_times_nat @ M @ N_1 )
= ( suc @ zero_zero_nat ) )
<=> ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_1116_less__iff__Suc__add,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
<=> ? [K_1: nat] :
( N_1
= ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_1117_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
thf(fact_1118_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
thf(fact_1119_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_1120_less__eq__Suc__le,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ N_1 @ M )
<=> ( ord_less_eq_nat @ ( suc @ N_1 ) @ M ) ) ).
thf(fact_1121_less__Suc__eq__le,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1122_Suc__le__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N_1 )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1123_le__imp__less__Suc,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_1124_Suc__leI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N_1 ) ) ).
thf(fact_1125_le__less__Suc__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( ord_less_nat @ N_1 @ ( suc @ M ) )
<=> ( N_1 = M ) ) ) ).
thf(fact_1126_Suc__le__lessD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N_1 )
=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1127_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1128_diff__less__Suc,axiom,
! [M: nat,N_1: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N_1 ) @ ( suc @ M ) ) ).
thf(fact_1129_mult__Suc,axiom,
! [M: nat,N_1: nat] :
( ( times_times_nat @ ( suc @ M ) @ N_1 )
= ( plus_plus_nat @ N_1 @ ( times_times_nat @ M @ N_1 ) ) ) ).
thf(fact_1130_mult__Suc__right,axiom,
! [M: nat,N_1: nat] :
( ( times_times_nat @ M @ ( suc @ N_1 ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N_1 ) ) ) ).
thf(fact_1131_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1132_Suc__eq__plus1__left,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
= ( plus_plus_nat @ one_one_nat @ N_1 ) ) ).
thf(fact_1133_Suc__eq__plus1,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
= ( plus_plus_nat @ N_1 @ one_one_nat ) ) ).
thf(fact_1134_Suc__diff__le,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N_1 )
= ( suc @ ( minus_minus_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1135_diff__Suc__1,axiom,
! [N_1: nat] :
( ( minus_minus_nat @ ( suc @ N_1 ) @ one_one_nat )
= N_1 ) ).
thf(fact_1136_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ M @ ( suc @ N_1 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ).
thf(fact_1137_Suc__leD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1138_le__SucE,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N_1 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N_1 )
=> ( M
= ( suc @ N_1 ) ) ) ) ).
thf(fact_1139_le__SucI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_1140_Suc__le__mono,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N_1 ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N_1 @ M ) ) ).
thf(fact_1141_le__Suc__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N_1 ) )
<=> ( ( ord_less_eq_nat @ M @ N_1 )
| ( M
= ( suc @ N_1 ) ) ) ) ).
thf(fact_1142_not__less__eq__eq,axiom,
! [M: nat,N_1: nat] :
( ~ ( ord_less_eq_nat @ M @ N_1 )
<=> ( ord_less_eq_nat @ ( suc @ N_1 ) @ M ) ) ).
thf(fact_1143_Suc__n__not__le__n,axiom,
! [N_1: nat] :
~ ( ord_less_eq_nat @ ( suc @ N_1 ) @ N_1 ) ).
thf(fact_1144_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_1145_nat_Osimps_I2_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
thf(fact_1146_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_1147_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] :
( ( suc @ Nat_2 )
!= zero_zero_nat ) ).
thf(fact_1148_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_1149_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_1150_not__less__eq,axiom,
! [M: nat,N_1: nat] :
( ~ ( ord_less_nat @ M @ N_1 )
<=> ( ord_less_nat @ N_1 @ ( suc @ M ) ) ) ).
thf(fact_1151_less__Suc__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
<=> ( ( ord_less_nat @ M @ N_1 )
| ( M = N_1 ) ) ) ).
thf(fact_1152_Suc__less__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1153_not__less__less__Suc__eq,axiom,
! [N_1: nat,M: nat] :
( ~ ( ord_less_nat @ N_1 @ M )
=> ( ( ord_less_nat @ N_1 @ ( suc @ M ) )
<=> ( N_1 = M ) ) ) ).
thf(fact_1154_less__antisym,axiom,
! [N_1: nat,M: nat] :
( ~ ( ord_less_nat @ N_1 @ M )
=> ( ( ord_less_nat @ N_1 @ ( suc @ M ) )
=> ( M = N_1 ) ) ) ).
thf(fact_1155_less__SucI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_1156_Suc__lessI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ( ( suc @ M )
!= N_1 )
=> ( ord_less_nat @ ( suc @ M ) @ N_1 ) ) ) ).
thf(fact_1157_less__trans__Suc,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
thf(fact_1158_less__SucE,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
=> ( ~ ( ord_less_nat @ M @ N_1 )
=> ( M = N_1 ) ) ) ).
thf(fact_1159_Suc__lessD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N_1 )
=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1160_Suc__less__SucD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_1 ) )
=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1161_add__Suc__right,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ M @ ( suc @ N_1 ) )
= ( suc @ ( plus_plus_nat @ M @ N_1 ) ) ) ).
thf(fact_1162_add__Suc,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_1 )
= ( suc @ ( plus_plus_nat @ M @ N_1 ) ) ) ).
thf(fact_1163_add__Suc__shift,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_1 )
= ( plus_plus_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_1164_Suc__diff__diff,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N_1 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N_1 ) @ K ) ) ).
thf(fact_1165_diff__Suc__Suc,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_1166_n__not__Suc__n,axiom,
! [N_1: nat] :
( N_1
!= ( suc @ N_1 ) ) ).
thf(fact_1167_Suc__n__not__n,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
!= N_1 ) ).
thf(fact_1168_nat_Oinject,axiom,
! [Nat_1: nat,Nat: nat] :
( ( ( suc @ Nat_1 )
= ( suc @ Nat ) )
<=> ( Nat_1 = Nat ) ) ).
thf(fact_1169_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_1170_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N_1 ) )
<=> ( M = N_1 ) ) ).
thf(fact_1171_pred__def,axiom,
! [K: int] :
( ( pred @ K )
= ( minus_minus_int @ K @ one_one_int ) ) ).
thf(fact_1172_le__iff__pred__less,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ L )
<=> ( ord_less_int @ ( pred @ K ) @ L ) ) ).
thf(fact_1173_Suc__diff__number__of,axiom,
! [M: nat,V: int] :
( ( ord_less_int @ pls @ V )
=> ( ( minus_minus_nat @ ( suc @ M ) @ ( number_number_of_nat @ V ) )
= ( minus_minus_nat @ M @ ( number_number_of_nat @ ( pred @ V ) ) ) ) ) ).
thf(fact_1174_one__less__mult,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N_1 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1175_n__less__n__mult__m,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N_1 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N_1 @ ( times_times_nat @ N_1 @ M ) ) ) ) ).
thf(fact_1176_n__less__m__mult__n,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N_1 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N_1 @ ( times_times_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1177_mn__eq__m__one,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( times_times_nat @ M @ N_1 )
= M )
=> ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_1178_prod__mn__less__k,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( ( times_times_nat @ M @ N_1 )
= K )
=> ( ord_less_nat @ N_1 @ K ) ) ) ) ) ).
thf(fact_1179_one__less__k,axiom,
! [M: nat,K: nat] :
( ( M
!= ( times_times_nat @ M @ K ) )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ K ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ K ) ) ) ).
thf(fact_1180_one__less__m,axiom,
! [M: nat,K: nat] :
( ( M
!= ( times_times_nat @ M @ K ) )
=> ( ( M
!= ( suc @ zero_zero_nat ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M ) ) ) ).
thf(fact_1181_Suc__pred,axiom,
! [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ( suc @ ( minus_minus_nat @ N_1 @ ( suc @ zero_zero_nat ) ) )
= N_1 ) ) ).
thf(fact_1182_diff__Suc__less,axiom,
! [I: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ord_less_nat @ ( minus_minus_nat @ N_1 @ ( suc @ I ) ) @ N_1 ) ) ).
thf(fact_1183_one__le__mult__iff,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N_1 ) )
<=> ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N_1 ) ) ) ).
thf(fact_1184_real__of__nat__Suc__gt__zero,axiom,
! [N_1: nat] : ( ord_less_real @ zero_zero_real @ ( real_nat @ ( suc @ N_1 ) ) ) ).
thf(fact_1185_int__Suc0__eq__1,axiom,
( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
thf(fact_1186_real__of__nat__one,axiom,
( ( real_nat @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
thf(fact_1187_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J ) ) ) ) ).
thf(fact_1188_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ M )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
thf(fact_1189_zless__iff__Suc__zadd,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
<=> ? [N: nat] :
( Z
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ ( suc @ N ) ) ) ) ) ).
thf(fact_1190_int__Suc,axiom,
! [M: nat] :
( ( semiri1621563631at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_1191_real__of__nat__Suc,axiom,
! [N_1: nat] :
( ( real_nat @ ( suc @ N_1 ) )
= ( plus_plus_real @ ( real_nat @ N_1 ) @ one_one_real ) ) ).
thf(fact_1192_Suc__pred_H,axiom,
! [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( N_1
= ( suc @ ( minus_minus_nat @ N_1 @ one_one_nat ) ) ) ) ).
thf(fact_1193_Suc__diff__1,axiom,
! [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ( suc @ ( minus_minus_nat @ N_1 @ one_one_nat ) )
= N_1 ) ) ).
thf(fact_1194_add__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_1 )
= N_1 ) )
& ( ( M != zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_1 )
= ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ) ) ) ).
thf(fact_1195_expand__Suc,axiom,
! [V: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
=> ( ( number_number_of_nat @ V )
= ( suc @ ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat ) ) ) ) ).
thf(fact_1196_neg__number__of__pred__iff__0,axiom,
! [V: int] :
( ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
<=> ( ( number_number_of_nat @ V )
= zero_zero_nat ) ) ).
thf(fact_1197_Suc__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( number_number_of_nat @ V ) )
= one_one_nat ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( number_number_of_nat @ V ) )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_1198_reals__Archimedean6a,axiom,
! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ? [N: nat] :
( ( ord_less_eq_real @ ( real_nat @ N ) @ R )
& ( ord_less_real @ R @ ( real_nat @ ( suc @ N ) ) ) ) ) ).
%----Helper facts (3)
thf(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_int @ ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ a @ b ) ) @ ( twoSqu2057625106sum2sq @ ( product_Pair_int_int @ p @ q ) ) )
= ( twoSqu2057625106sum2sq @ ( 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 ) ) ) ) ) ).
%------------------------------------------------------------------------------