TPTP Problem File: SLH0305^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc3/0006_Encoding/prob_00074_002629__11868674_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1326 ( 624 unt; 55 typ; 0 def)
% Number of atoms : 3275 (1203 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9509 ( 300 ~; 69 |; 140 &;7662 @)
% ( 0 <=>;1338 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 482 ( 482 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 49 usr; 7 con; 0-8 aty)
% Number of variables : 3125 ( 136 ^;2900 !; 89 ?;3125 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:27:10.323
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__List__Olist_It__Syntax__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__Typerep__Otyperep,type,
typerep: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Syntax__Ofm,type,
fm: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (49)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Encoding_Ofm__of__nat,type,
fm_of_nat: nat > fm ).
thf(sy_c_Encoding_Ofm__of__nat__rel,type,
fm_of_nat_rel: nat > nat > $o ).
thf(sy_c_Encoding_Onat__of__fm,type,
nat_of_fm: fm > nat ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
euclid4774559944035922753ze_int: int > nat ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
euclid4777050414544973029ze_nat: nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Syntax__Ofm,type,
size_size_fm: fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
size_size_typerep: typerep > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_less_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Syntax_Ofm_OFalsity,type,
falsity: fm ).
thf(sy_c_Syntax_Ofm_OImp,type,
imp: fm > fm > fm ).
thf(sy_c_Syntax_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_Syntax_Ofm_Osize__fm,type,
size_fm: fm > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
% Relevant facts (1265)
thf(fact_0_fm__nat,axiom,
! [P: fm] :
( ( fm_of_nat @ ( nat_of_fm @ P ) )
= P ) ).
% fm_nat
thf(fact_1_fm__of__nat_Osimps_I1_J,axiom,
( ( fm_of_nat @ zero_zero_nat )
= falsity ) ).
% fm_of_nat.simps(1)
thf(fact_2_nat__of__fm_Osimps_I1_J,axiom,
( ( nat_of_fm @ falsity )
= zero_zero_nat ) ).
% nat_of_fm.simps(1)
thf(fact_3_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_4_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_5_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_6_fm_Osize__gen_I1_J,axiom,
( ( size_fm @ falsity )
= zero_zero_nat ) ).
% fm.size_gen(1)
thf(fact_7_fm_Osize_I5_J,axiom,
( ( size_size_fm @ falsity )
= zero_zero_nat ) ).
% fm.size(5)
thf(fact_8_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_9_fm__of__nat_Opsimps_I1_J,axiom,
( ( accp_nat @ fm_of_nat_rel @ zero_zero_nat )
=> ( ( fm_of_nat @ zero_zero_nat )
= falsity ) ) ).
% fm_of_nat.psimps(1)
thf(fact_10_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_11_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_12_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_13_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_14_size__0,axiom,
( ( euclid4777050414544973029ze_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% size_0
thf(fact_15_size__0,axiom,
( ( euclid4774559944035922753ze_int @ zero_zero_int )
= zero_zero_nat ) ).
% size_0
thf(fact_16_euclidean__size__eq__0__iff,axiom,
! [B: nat] :
( ( ( euclid4777050414544973029ze_nat @ B )
= zero_zero_nat )
= ( B = zero_zero_nat ) ) ).
% euclidean_size_eq_0_iff
thf(fact_17_euclidean__size__eq__0__iff,axiom,
! [B: int] :
( ( ( euclid4774559944035922753ze_int @ B )
= zero_zero_nat )
= ( B = zero_zero_int ) ) ).
% euclidean_size_eq_0_iff
thf(fact_18_size__neq__size__imp__neq,axiom,
! [X: fm,Y: fm] :
( ( ( size_size_fm @ X )
!= ( size_size_fm @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_19_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_20_size__neq__size__imp__neq,axiom,
! [X: typerep,Y: typerep] :
( ( ( size_size_typerep @ X )
!= ( size_size_typerep @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_21_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_22_euclidean__size__greater__0__iff,axiom,
! [B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4777050414544973029ze_nat @ B ) )
= ( B != zero_zero_nat ) ) ).
% euclidean_size_greater_0_iff
thf(fact_23_euclidean__size__greater__0__iff,axiom,
! [B: int] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4774559944035922753ze_int @ B ) )
= ( B != zero_zero_int ) ) ).
% euclidean_size_greater_0_iff
thf(fact_24_char_Osize__gen,axiom,
! [X1: $o,X2: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_25_fm_Osize_I6_J,axiom,
! [X21: nat,X22: list_tm] :
( ( size_size_fm @ ( pre @ X21 @ X22 ) )
= zero_zero_nat ) ).
% fm.size(6)
thf(fact_26_fm_Osize__gen_I2_J,axiom,
! [X21: nat,X22: list_tm] :
( ( size_fm @ ( pre @ X21 @ X22 ) )
= zero_zero_nat ) ).
% fm.size_gen(2)
thf(fact_27_accp__induct__rule,axiom,
! [R: nat > nat > $o,A: nat,P2: nat > $o] :
( ( accp_nat @ R @ A )
=> ( ! [X9: nat] :
( ( accp_nat @ R @ X9 )
=> ( ! [Y2: nat] :
( ( R @ Y2 @ X9 )
=> ( P2 @ Y2 ) )
=> ( P2 @ X9 ) ) )
=> ( P2 @ A ) ) ) ).
% accp_induct_rule
thf(fact_28_not__accp__down,axiom,
! [R2: nat > nat > $o,X: nat] :
( ~ ( accp_nat @ R2 @ X )
=> ~ ! [Z: nat] :
( ( R2 @ Z @ X )
=> ( accp_nat @ R2 @ Z ) ) ) ).
% not_accp_down
thf(fact_29_accp__downward,axiom,
! [R: nat > nat > $o,B: nat,A: nat] :
( ( accp_nat @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_nat @ R @ A ) ) ) ).
% accp_downward
thf(fact_30_accp__induct,axiom,
! [R: nat > nat > $o,A: nat,P2: nat > $o] :
( ( accp_nat @ R @ A )
=> ( ! [X9: nat] :
( ( accp_nat @ R @ X9 )
=> ( ! [Y2: nat] :
( ( R @ Y2 @ X9 )
=> ( P2 @ Y2 ) )
=> ( P2 @ X9 ) ) )
=> ( P2 @ A ) ) ) ).
% accp_induct
thf(fact_31_accpI,axiom,
! [R: nat > nat > $o,X: nat] :
( ! [Y3: nat] :
( ( R @ Y3 @ X )
=> ( accp_nat @ R @ Y3 ) )
=> ( accp_nat @ R @ X ) ) ).
% accpI
thf(fact_32_char_Oinject,axiom,
! [X1: $o,X2: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y22: $o,Y32: $o,Y4: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
( ( ( char2 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 )
= ( char2 @ Y1 @ Y22 @ Y32 @ Y4 @ Y5 @ Y6 @ Y7 @ Y8 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 )
& ( X3 = Y32 )
& ( X4 = Y4 )
& ( X5 = Y5 )
& ( X6 = Y6 )
& ( X7 = Y7 )
& ( X8 = Y8 ) ) ) ).
% char.inject
thf(fact_33_fm_Oinject_I1_J,axiom,
! [X21: nat,X22: list_tm,Y21: nat,Y222: list_tm] :
( ( ( pre @ X21 @ X22 )
= ( pre @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X22 = Y222 ) ) ) ).
% fm.inject(1)
thf(fact_34_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_35_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_36_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_37_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_38_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_39_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_40_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P2 @ M ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_41_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P2 @ M ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_42_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_43_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_44_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_45_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_46_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_47_char_Oexhaust,axiom,
! [Y: char] :
~ ! [X12: $o,X23: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( Y
!= ( char2 @ X12 @ X23 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) ) ).
% char.exhaust
thf(fact_48_char_Osize_I2_J,axiom,
! [X1: $o,X2: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_49_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_50_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_51_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_52_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_53_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_54_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_55_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_56_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_57_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_58_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_59_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P2 @ M ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_60_fm_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_tm] :
( falsity
!= ( pre @ X21 @ X22 ) ) ).
% fm.distinct(1)
thf(fact_61_accp_Ocases,axiom,
! [R: nat > nat > $o,A: nat] :
( ( accp_nat @ R @ A )
=> ! [Y2: nat] :
( ( R @ Y2 @ A )
=> ( accp_nat @ R @ Y2 ) ) ) ).
% accp.cases
thf(fact_62_accp_Osimps,axiom,
( accp_nat
= ( ^ [R3: nat > nat > $o,A2: nat] :
? [X10: nat] :
( ( A2 = X10 )
& ! [Y9: nat] :
( ( R3 @ Y9 @ X10 )
=> ( accp_nat @ R3 @ Y9 ) ) ) ) ) ).
% accp.simps
thf(fact_63_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_64_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_65_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_66_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_67_typerep_Osize__neq,axiom,
! [X: typerep] :
( ( size_size_typerep @ X )
!= zero_zero_nat ) ).
% typerep.size_neq
thf(fact_68_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ~ ( ord_less_nat @ T @ X11 ) ) ).
% minf(7)
thf(fact_69_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ~ ( ord_less_int @ T @ X11 ) ) ).
% minf(7)
thf(fact_70_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_71_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_72_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_73_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_74_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_75_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_76_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_77_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_78_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_79_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_80_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% of_nat_mask_eq
thf(fact_81_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_nat_mask_eq
thf(fact_82_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_83_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_84_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_85_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_86_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_87_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_88_pinf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ( ( ( P2 @ X11 )
& ( Q @ X11 ) )
= ( ( P3 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_89_pinf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ( ( ( P2 @ X11 )
& ( Q @ X11 ) )
= ( ( P3 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_90_pinf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ( ( ( P2 @ X11 )
| ( Q @ X11 ) )
= ( ( P3 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_91_pinf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ( ( ( P2 @ X11 )
| ( Q @ X11 ) )
= ( ( P3 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_92_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ( X11 != T ) ) ).
% pinf(3)
thf(fact_93_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ( X11 != T ) ) ).
% pinf(3)
thf(fact_94_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ( X11 != T ) ) ).
% pinf(4)
thf(fact_95_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ( X11 != T ) ) ).
% pinf(4)
thf(fact_96_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ~ ( ord_less_nat @ X11 @ T ) ) ).
% pinf(5)
thf(fact_97_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ~ ( ord_less_int @ X11 @ T ) ) ).
% pinf(5)
thf(fact_98_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ( ord_less_nat @ T @ X11 ) ) ).
% pinf(7)
thf(fact_99_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ( ord_less_int @ T @ X11 ) ) ).
% pinf(7)
thf(fact_100_minf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ( ( ( P2 @ X11 )
& ( Q @ X11 ) )
= ( ( P3 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(1)
thf(fact_101_minf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ( ( ( P2 @ X11 )
& ( Q @ X11 ) )
= ( ( P3 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(1)
thf(fact_102_minf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ( ( ( P2 @ X11 )
| ( Q @ X11 ) )
= ( ( P3 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(2)
thf(fact_103_minf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ( P2 @ X9 )
= ( P3 @ X9 ) ) )
=> ( ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ( Q @ X9 )
= ( Q2 @ X9 ) ) )
=> ? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ( ( ( P2 @ X11 )
| ( Q @ X11 ) )
= ( ( P3 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(2)
thf(fact_104_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ( X11 != T ) ) ).
% minf(3)
thf(fact_105_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ( X11 != T ) ) ).
% minf(3)
thf(fact_106_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ( X11 != T ) ) ).
% minf(4)
thf(fact_107_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ( X11 != T ) ) ).
% minf(4)
thf(fact_108_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ( ord_less_nat @ X11 @ T ) ) ).
% minf(5)
thf(fact_109_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ( ord_less_int @ X11 @ T ) ) ).
% minf(5)
thf(fact_110_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid4777050414544973029ze_nat @ ( semiri1316708129612266289at_nat @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_111_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid4774559944035922753ze_int @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_112_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_113_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_114_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_115_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_116_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_117_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_118_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_119_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_120_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_121_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_122_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_123_nat_Oinject,axiom,
! [X2: nat,Y22: nat] :
( ( ( suc @ X2 )
= ( suc @ Y22 ) )
= ( X2 = Y22 ) ) ).
% nat.inject
thf(fact_124_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_125_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_126_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_127_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_128_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_129_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_130_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power_nat @ X @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_131_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_132_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_133_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_134_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_135_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_136_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_137_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_138_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_139_of__nat__power,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M2 @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ N ) ) ).
% of_nat_power
thf(fact_140_of__nat__power,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M2 @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M2 ) @ N ) ) ).
% of_nat_power
thf(fact_141_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_142_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_143_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_144_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_145_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_146_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_147_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_148_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_149_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_150_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_151_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_152_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_153_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_154_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_155_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_156_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_157_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_158_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_159_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_160_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_161_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_162_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_163_int__if,axiom,
! [P2: $o,A: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_164_nat__int__comparison_I1_J,axiom,
( ( ^ [Y10: nat,Z3: nat] : ( Y10 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_165_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_166_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_167_conj__le__cong,axiom,
! [X: int,X13: int,P2: $o,P3: $o] :
( ( X = X13 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X13 )
=> ( P2 = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X13 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_168_imp__le__cong,axiom,
! [X: int,X13: int,P2: $o,P3: $o] :
( ( X = X13 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X13 )
=> ( P2 = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X13 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_169_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_170_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_171_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_172_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_173_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_174_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_175_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_176_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_177_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_178_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_179_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_180_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_181_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_182_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X9: nat] : ( R2 @ X9 @ X9 )
=> ( ! [X9: nat,Y3: nat,Z: nat] :
( ( R2 @ X9 @ Y3 )
=> ( ( R2 @ Y3 @ Z )
=> ( R2 @ X9 @ Z ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_183_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_184_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( P2 @ M ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_185_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_186_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_187_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_188_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_189_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_190_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_191_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_192_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_193_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_194_lift__Suc__antimono__le,axiom,
! [F: nat > nat > nat > $o,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le2646555220125990790_nat_o @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_195_lift__Suc__antimono__le,axiom,
! [F: nat > nat > $o,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat_o @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat_o @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_196_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_197_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_198_lift__Suc__mono__le,axiom,
! [F: nat > nat > nat > $o,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le2646555220125990790_nat_o @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_199_lift__Suc__mono__le,axiom,
! [F: nat > nat > $o,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat_o @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat_o @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_200_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_201_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_202_verit__comp__simplify1_I2_J,axiom,
! [A: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_203_verit__comp__simplify1_I2_J,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_204_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_205_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_206_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_207_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_208_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_209_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_210_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_211_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_212_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_213_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_214_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_215_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_216_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_217_accp__subset,axiom,
! [R1: nat > nat > $o,R22: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
=> ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_218_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_219_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_220_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_221_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_222_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_223_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A3: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_224_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_225_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_226_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_227_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K2 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_228_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_229_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_230_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_231_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_232_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_233_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_234_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_235_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X11 ) ) ).
% minf(8)
thf(fact_236_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ~ ( ord_less_eq_int @ T @ X11 ) ) ).
% minf(8)
thf(fact_237_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z )
=> ( ord_less_eq_nat @ X11 @ T ) ) ).
% minf(6)
thf(fact_238_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z )
=> ( ord_less_eq_int @ X11 @ T ) ) ).
% minf(6)
thf(fact_239_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ( ord_less_eq_nat @ T @ X11 ) ) ).
% pinf(8)
thf(fact_240_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ( ord_less_eq_int @ T @ X11 ) ) ).
% pinf(8)
thf(fact_241_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z @ X11 )
=> ~ ( ord_less_eq_nat @ X11 @ T ) ) ).
% pinf(6)
thf(fact_242_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X11: int] :
( ( ord_less_int @ Z @ X11 )
=> ~ ( ord_less_eq_int @ X11 @ T ) ) ).
% pinf(6)
thf(fact_243_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_244_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_245_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_246_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_247_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_248_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_249_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X9: nat] : ( P2 @ X9 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X9: nat,Y3: nat] :
( ( P2 @ X9 @ Y3 )
=> ( P2 @ ( suc @ X9 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_250_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_251_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_252_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_253_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_254_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_255_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_256_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_257_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P2 @ I3 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_258_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_259_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_260_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_261_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_262_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_263_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_264_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_265_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_266_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_267_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_268_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_269_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_270_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_271_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_272_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_273_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_274_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_275_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_276_accp__subset__induct,axiom,
! [D: nat > $o,R2: nat > nat > $o,X: nat,P2: nat > $o] :
( ( ord_less_eq_nat_o @ D @ ( accp_nat @ R2 ) )
=> ( ! [X9: nat,Z: nat] :
( ( D @ X9 )
=> ( ( R2 @ Z @ X9 )
=> ( D @ Z ) ) )
=> ( ( D @ X )
=> ( ! [X9: nat] :
( ( D @ X9 )
=> ( ! [Z2: nat] :
( ( R2 @ Z2 @ X9 )
=> ( P2 @ Z2 ) )
=> ( P2 @ X9 ) ) )
=> ( P2 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_277_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_278_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_279_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_280_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_281_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_282_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_283_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_284_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_285_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_286_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_287_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_288_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_289_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_290_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_291_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_292_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_293_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_294_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_295_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_296_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_297_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_298_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_299_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_300_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_301_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_302_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_303_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_304_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_305_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_306_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_307_order__refl,axiom,
! [X: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ X @ X ) ).
% order_refl
thf(fact_308_order__refl,axiom,
! [X: nat > $o] : ( ord_less_eq_nat_o @ X @ X ) ).
% order_refl
thf(fact_309_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_310_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_311_dual__order_Orefl,axiom,
! [A: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_312_dual__order_Orefl,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_313_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_314_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_315_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_316_leD,axiom,
! [Y: nat > nat > $o,X: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ Y @ X )
=> ~ ( ord_less_nat_nat_o @ X @ Y ) ) ).
% leD
thf(fact_317_leD,axiom,
! [Y: nat > $o,X: nat > $o] :
( ( ord_less_eq_nat_o @ Y @ X )
=> ~ ( ord_less_nat_o @ X @ Y ) ) ).
% leD
thf(fact_318_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_319_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_320_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_321_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_322_nless__le,axiom,
! [A: nat > nat > $o,B: nat > nat > $o] :
( ( ~ ( ord_less_nat_nat_o @ A @ B ) )
= ( ~ ( ord_le2646555220125990790_nat_o @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_323_nless__le,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ~ ( ord_less_nat_o @ A @ B ) )
= ( ~ ( ord_less_eq_nat_o @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_324_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_325_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_326_antisym__conv1,axiom,
! [X: nat > nat > $o,Y: nat > nat > $o] :
( ~ ( ord_less_nat_nat_o @ X @ Y )
=> ( ( ord_le2646555220125990790_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_327_antisym__conv1,axiom,
! [X: nat > $o,Y: nat > $o] :
( ~ ( ord_less_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_328_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_329_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_330_predicate2I,axiom,
! [P2: nat > nat > $o,Q: nat > nat > $o] :
( ! [X9: nat,Y3: nat] :
( ( P2 @ X9 @ Y3 )
=> ( Q @ X9 @ Y3 ) )
=> ( ord_le2646555220125990790_nat_o @ P2 @ Q ) ) ).
% predicate2I
thf(fact_331_predicate1I,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X9: nat] :
( ( P2 @ X9 )
=> ( Q @ X9 ) )
=> ( ord_less_eq_nat_o @ P2 @ Q ) ) ).
% predicate1I
thf(fact_332_less__fun__def,axiom,
( ord_less_nat_nat_o
= ( ^ [F2: nat > nat > $o,G: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ F2 @ G )
& ~ ( ord_le2646555220125990790_nat_o @ G @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_333_less__fun__def,axiom,
( ord_less_nat_o
= ( ^ [F2: nat > $o,G: nat > $o] :
( ( ord_less_eq_nat_o @ F2 @ G )
& ~ ( ord_less_eq_nat_o @ G @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_334_euclidean__size__nat__less__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ M2 ) @ ( euclid4777050414544973029ze_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% euclidean_size_nat_less_eq_iff
thf(fact_335_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X9: nat] :
( ( P2 @ X9 )
& ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X9 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_336_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_337_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_338_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_339_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_340_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_341_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_342_predicate1D,axiom,
! [P2: nat > $o,Q: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ P2 @ Q )
=> ( ( P2 @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_343_predicate2D,axiom,
! [P2: nat > nat > $o,Q: nat > nat > $o,X: nat,Y: nat] :
( ( ord_le2646555220125990790_nat_o @ P2 @ Q )
=> ( ( P2 @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_344_rev__predicate1D,axiom,
! [P2: nat > $o,X: nat,Q: nat > $o] :
( ( P2 @ X )
=> ( ( ord_less_eq_nat_o @ P2 @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_345_rev__predicate2D,axiom,
! [P2: nat > nat > $o,X: nat,Y: nat,Q: nat > nat > $o] :
( ( P2 @ X @ Y )
=> ( ( ord_le2646555220125990790_nat_o @ P2 @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_346_order__antisym__conv,axiom,
! [Y: nat > nat > $o,X: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ Y @ X )
=> ( ( ord_le2646555220125990790_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_347_order__antisym__conv,axiom,
! [Y: nat > $o,X: nat > $o] :
( ( ord_less_eq_nat_o @ Y @ X )
=> ( ( ord_less_eq_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_348_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_349_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_350_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_351_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_352_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_353_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_354_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_355_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_356_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_357_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat > $o,C2: nat > $o] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_358_ord__le__eq__subst,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > nat,C2: nat] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_359_ord__le__eq__subst,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > int,C2: int] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_360_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat > nat > $o,C2: nat > nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_le2646555220125990790_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_361_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat > nat > $o,C2: nat > nat > $o] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_le2646555220125990790_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_362_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_363_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_364_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_365_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_366_ord__eq__le__subst,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_367_ord__eq__le__subst,axiom,
! [A: nat > $o,F: int > nat > $o,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_368_ord__eq__le__subst,axiom,
! [A: nat,F: ( nat > $o ) > nat,B: nat > $o,C2: nat > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_369_ord__eq__le__subst,axiom,
! [A: int,F: ( nat > $o ) > int,B: nat > $o,C2: nat > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_370_ord__eq__le__subst,axiom,
! [A: nat > nat > $o,F: nat > nat > nat > $o,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_le2646555220125990790_nat_o @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_371_ord__eq__le__subst,axiom,
! [A: nat > nat > $o,F: int > nat > nat > $o,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_le2646555220125990790_nat_o @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_372_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_373_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_374_order__eq__refl,axiom,
! [X: nat > nat > $o,Y: nat > nat > $o] :
( ( X = Y )
=> ( ord_le2646555220125990790_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_375_order__eq__refl,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( X = Y )
=> ( ord_less_eq_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_376_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_377_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_378_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_379_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_380_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_381_order__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_382_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_383_order__subst2,axiom,
! [A: int,B: int,F: int > nat > $o,C2: nat > $o] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_384_order__subst2,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > nat,C2: nat] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_385_order__subst2,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > int,C2: int] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_386_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat > nat > $o,C2: nat > nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2646555220125990790_nat_o @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_le2646555220125990790_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_387_order__subst2,axiom,
! [A: int,B: int,F: int > nat > nat > $o,C2: nat > nat > $o] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le2646555220125990790_nat_o @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_le2646555220125990790_nat_o @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_388_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_389_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_390_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_391_order__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_392_order__subst1,axiom,
! [A: nat,F: ( nat > $o ) > nat,B: nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_393_order__subst1,axiom,
! [A: int,F: ( nat > $o ) > int,B: nat > $o,C2: nat > $o] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C2 )
=> ( ! [X9: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_394_order__subst1,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C2: nat] :
( ( ord_less_eq_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_395_order__subst1,axiom,
! [A: nat > $o,F: int > nat > $o,B: int,C2: int] :
( ( ord_less_eq_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_396_order__subst1,axiom,
! [A: nat,F: ( nat > nat > $o ) > nat,B: nat > nat > $o,C2: nat > nat > $o] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2646555220125990790_nat_o @ B @ C2 )
=> ( ! [X9: nat > nat > $o,Y3: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_397_order__subst1,axiom,
! [A: int,F: ( nat > nat > $o ) > int,B: nat > nat > $o,C2: nat > nat > $o] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le2646555220125990790_nat_o @ B @ C2 )
=> ( ! [X9: nat > nat > $o,Y3: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_398_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y10: nat > nat > $o,Z3: nat > nat > $o] : ( Y10 = Z3 ) )
= ( ^ [A2: nat > nat > $o,B2: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A2 @ B2 )
& ( ord_le2646555220125990790_nat_o @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_399_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y10: nat > $o,Z3: nat > $o] : ( Y10 = Z3 ) )
= ( ^ [A2: nat > $o,B2: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
& ( ord_less_eq_nat_o @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_400_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y10: nat,Z3: nat] : ( Y10 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_401_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_402_le__fun__def,axiom,
( ord_le2646555220125990790_nat_o
= ( ^ [F2: nat > nat > $o,G: nat > nat > $o] :
! [X10: nat] : ( ord_less_eq_nat_o @ ( F2 @ X10 ) @ ( G @ X10 ) ) ) ) ).
% le_fun_def
thf(fact_403_le__fun__def,axiom,
( ord_less_eq_nat_o
= ( ^ [F2: nat > $o,G: nat > $o] :
! [X10: nat] : ( ord_less_eq_o @ ( F2 @ X10 ) @ ( G @ X10 ) ) ) ) ).
% le_fun_def
thf(fact_404_le__funI,axiom,
! [F: nat > nat > $o,G2: nat > nat > $o] :
( ! [X9: nat] : ( ord_less_eq_nat_o @ ( F @ X9 ) @ ( G2 @ X9 ) )
=> ( ord_le2646555220125990790_nat_o @ F @ G2 ) ) ).
% le_funI
thf(fact_405_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_406_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_407_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_408_dual__order_Otrans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_409_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_410_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_411_dual__order_Oeq__iff,axiom,
( ( ^ [Y10: nat,Z3: nat] : ( Y10 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_412_dual__order_Oeq__iff,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_413_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_414_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_415_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_416_order__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z4 )
=> ( ord_less_eq_int @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_417_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_418_order_Otrans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_419_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_420_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_421_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_422_ord__le__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_423_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_424_ord__eq__le__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_425_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y10: nat,Z3: nat] : ( Y10 = Z3 ) )
= ( ^ [X10: nat,Y9: nat] :
( ( ord_less_eq_nat @ X10 @ Y9 )
& ( ord_less_eq_nat @ Y9 @ X10 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_426_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [X10: int,Y9: int] :
( ( ord_less_eq_int @ X10 @ Y9 )
& ( ord_less_eq_int @ Y9 @ X10 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_427_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_428_le__cases3,axiom,
! [X: int,Y: int,Z4: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z4 ) )
=> ( ( ( ord_less_eq_int @ X @ Z4 )
=> ~ ( ord_less_eq_int @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z4 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z4 )
=> ~ ( ord_less_eq_int @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z4 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_429_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_430_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_431_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_432_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_433_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_434_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_435_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_436_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_437_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_438_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_439_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_440_order__less__imp__triv,axiom,
! [X: int,Y: int,P2: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_441_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_442_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_443_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_444_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_445_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_446_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_447_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_448_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_449_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_450_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_451_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_452_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_453_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_454_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_455_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_456_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_457_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_458_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_459_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_460_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_461_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_462_order__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_463_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_464_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_465_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_466_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_467_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_468_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_469_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_470_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_471_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_472_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_473_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_474_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_475_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_476_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_477_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_478_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_479_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_480_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_481_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_482_linorder__less__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P2 @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_483_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X14: nat] : ( P4 @ X14 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N4 )
=> ~ ( P5 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_484_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_485_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_486_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_487_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_488_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_489_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_490_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_491_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_492_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X9: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X9 )
=> ( P2 @ Y2 ) )
=> ( P2 @ X9 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_493_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_494_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_495_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_496_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_497_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_498_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_499_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_500_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_501_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_502_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_503_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_504_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_505_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_506_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_507_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_508_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_509_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_510_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_511_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_512_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_513_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_514_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_515_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_516_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_517_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_eq_nat @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_518_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_519_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_eq_int @ X9 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_520_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_521_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_522_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X9: nat,Y3: nat] :
( ( ord_less_nat @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_523_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X9: int,Y3: int] :
( ( ord_less_int @ X9 @ Y3 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_524_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_525_order__less__le__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_526_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_527_order__le__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_528_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_529_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_530_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_531_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_532_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_533_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_534_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_535_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_536_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_537_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_538_order__less__le,axiom,
( ord_less_nat
= ( ^ [X10: nat,Y9: nat] :
( ( ord_less_eq_nat @ X10 @ Y9 )
& ( X10 != Y9 ) ) ) ) ).
% order_less_le
thf(fact_539_order__less__le,axiom,
( ord_less_int
= ( ^ [X10: int,Y9: int] :
( ( ord_less_eq_int @ X10 @ Y9 )
& ( X10 != Y9 ) ) ) ) ).
% order_less_le
thf(fact_540_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X10: nat,Y9: nat] :
( ( ord_less_nat @ X10 @ Y9 )
| ( X10 = Y9 ) ) ) ) ).
% order_le_less
thf(fact_541_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X10: int,Y9: int] :
( ( ord_less_int @ X10 @ Y9 )
| ( X10 = Y9 ) ) ) ) ).
% order_le_less
thf(fact_542_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_543_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_544_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_545_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_546_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_547_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_548_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_549_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_550_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_551_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_552_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_553_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_554_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_555_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_556_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_557_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_558_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_559_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_560_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_561_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_562_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_563_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_564_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_565_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_566_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_567_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_568_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X10: nat,Y9: nat] :
( ( ord_less_eq_nat @ X10 @ Y9 )
& ~ ( ord_less_eq_nat @ Y9 @ X10 ) ) ) ) ).
% less_le_not_le
thf(fact_569_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X10: int,Y9: int] :
( ( ord_less_eq_int @ X10 @ Y9 )
& ~ ( ord_less_eq_int @ Y9 @ X10 ) ) ) ) ).
% less_le_not_le
thf(fact_570_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_571_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_572_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P2 @ I2 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_573_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X11: nat] :
( ( ( ord_less_eq_nat @ A @ X11 )
& ( ord_less_nat @ X11 @ C3 ) )
=> ( P2 @ X11 ) )
& ! [D2: nat] :
( ! [X9: nat] :
( ( ( ord_less_eq_nat @ A @ X9 )
& ( ord_less_nat @ X9 @ D2 ) )
=> ( P2 @ X9 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_574_complete__interval,axiom,
! [A: int,B: int,P2: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X11: int] :
( ( ( ord_less_eq_int @ A @ X11 )
& ( ord_less_int @ X11 @ C3 ) )
=> ( P2 @ X11 ) )
& ! [D2: int] :
( ! [X9: int] :
( ( ( ord_less_eq_int @ A @ X9 )
& ( ord_less_int @ X9 @ D2 ) )
=> ( P2 @ X9 ) )
=> ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_575_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_576_power__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_577_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_578_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_579_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_580_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_581_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_582_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_583_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_584_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_585_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_586_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_587_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_588_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_589_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_590_euclidean__size__1,axiom,
( ( euclid4777050414544973029ze_nat @ one_one_nat )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_591_euclidean__size__1,axiom,
( ( euclid4774559944035922753ze_int @ one_one_int )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_592_power__inject__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_593_power__inject__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M2 )
= ( power_power_int @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_594_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_595_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_596_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_597_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_598_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_599_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_600_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_601_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_602_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_603_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_604_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_605_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_606_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_607_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_608_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_609_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_610_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_611_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_612_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_613_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_614_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_615_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_616_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_617_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_618_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_619_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_620_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_621_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_622_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_623_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_624_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_625_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_626_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_627_int__one__le__iff__zero__less,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ one_one_int @ Z4 )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% int_one_le_iff_zero_less
thf(fact_628_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_629_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_630_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_631_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_632_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_633_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_634_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_635_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_636_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_637_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_638_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_639_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_640_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_641_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_642_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_643_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_644_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_645_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_646_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_647_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_648_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_649_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_650_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_651_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_652_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_653_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_654_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_655_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_656_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_657_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_658_one__less__nat__eq,axiom,
! [Z4: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ one_one_int @ Z4 ) ) ).
% one_less_nat_eq
thf(fact_659_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_660_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_661_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_662_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_663_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_664_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_665_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_666_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_667_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_668_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_669_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_670_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_671_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_672_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_673_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_674_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_675_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_676_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_677_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_678_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_679_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_680_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_681_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_682_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_683_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_684_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_685_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_686_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_687_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_688_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_689_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_690_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_691_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_692_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_693_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_694_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_695_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_696_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_697_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_698_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_699_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_700_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_701_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_702_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_703_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_704_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_705_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_706_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_707_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_708_nat__le__0,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ Z4 @ zero_zero_int )
=> ( ( nat2 @ Z4 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_709_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_710_zless__nat__conj,axiom,
! [W: int,Z4: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z4 ) )
= ( ( ord_less_int @ zero_zero_int @ Z4 )
& ( ord_less_int @ W @ Z4 ) ) ) ).
% zless_nat_conj
thf(fact_711_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_712_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_713_int__nat__eq,axiom,
! [Z4: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z4 ) )
= Z4 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z4 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_714_zero__less__nat__eq,axiom,
! [Z4: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z4 ) )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% zero_less_nat_eq
thf(fact_715_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_716_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_717_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_718_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_719_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_720_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_721_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_722_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_723_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_724_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_725_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_726_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_727_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_728_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_729_int__cases2,axiom,
! [Z4: int] :
( ! [N2: nat] :
( Z4
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z4
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_730_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 ) ) ).
% diff_commute
thf(fact_731_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_732_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_733_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_734_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_735_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_736_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( A = B )
= ( C2 = D3 ) ) ) ).
% diff_eq_diff_eq
thf(fact_737_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_738_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_739_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_740_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_741_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_742_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_743_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_744_diff__strict__mono,axiom,
! [A: int,B: int,D3: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D3 @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_745_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_746_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_747_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_748_eq__iff__diff__eq__0,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_749_diff__mono,axiom,
! [A: int,B: int,D3: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D3 @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_750_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_751_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_752_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_753_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_754_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_755_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_756_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_757_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_758_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_759_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X14: nat] : ( P4 @ X14 ) )
= ( ^ [P5: nat > $o] :
? [X10: int] :
( ( ord_less_eq_int @ zero_zero_int @ X10 )
& ( P5 @ ( nat2 @ X10 ) ) ) ) ) ).
% ex_nat
thf(fact_760_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X14: nat] : ( P4 @ X14 ) )
= ( ^ [P5: nat > $o] :
! [X10: int] :
( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P5 @ ( nat2 @ X10 ) ) ) ) ) ).
% all_nat
thf(fact_761_eq__nat__nat__iff,axiom,
! [Z4: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z4 )
= ( nat2 @ Z5 ) )
= ( Z4 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_762_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_763_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_764_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_765_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_766_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_767_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_768_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_769_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_770_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_771_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_772_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_773_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_774_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_775_int__of__nat__induct,axiom,
! [P2: int > $o,Z4: int] :
( ! [N2: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P2 @ Z4 ) ) ) ).
% int_of_nat_induct
thf(fact_776_int__cases,axiom,
! [Z4: int] :
( ! [N2: nat] :
( Z4
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z4
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_777_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_778_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% nat_mask_eq
thf(fact_779_nat__mono__iff,axiom,
! [Z4: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ W @ Z4 ) ) ) ).
% nat_mono_iff
thf(fact_780_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_781_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_782_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_783_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z4: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z4 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z4 ) ) ).
% zless_nat_eq_int_zless
thf(fact_784_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_785_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_786_int__eq__iff,axiom,
! [M2: nat,Z4: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z4 )
= ( ( M2
= ( nat2 @ Z4 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z4 ) ) ) ).
% int_eq_iff
thf(fact_787_nat__0__le,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z4 ) )
= Z4 ) ) ).
% nat_0_le
thf(fact_788_nat__power__eq,axiom,
! [Z4: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( nat2 @ ( power_power_int @ Z4 @ N ) )
= ( power_power_nat @ ( nat2 @ Z4 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_789_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_790_int__cases4,axiom,
! [M2: int] :
( ! [N2: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_791_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_792_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_793_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_794_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_795_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_796_nat__less__eq__zless,axiom,
! [W: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ W @ Z4 ) ) ) ).
% nat_less_eq_zless
thf(fact_797_split__nat,axiom,
! [P2: nat > $o,I: int] :
( ( P2 @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P2 @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_798_nat__le__eq__zle,axiom,
! [W: int,Z4: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z4 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z4 ) )
= ( ord_less_eq_int @ W @ Z4 ) ) ) ).
% nat_le_eq_zle
thf(fact_799_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_800_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_801_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_802_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_803_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_804_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_805_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_806_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_807_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_808_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_809_zle__diff1__eq,axiom,
! [W: int,Z4: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z4 @ one_one_int ) )
= ( ord_less_int @ W @ Z4 ) ) ).
% zle_diff1_eq
thf(fact_810_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_811_int__diff__cases,axiom,
! [Z4: int] :
~ ! [M4: nat,N2: nat] :
( Z4
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_812_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_813_int__le__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_814_int__less__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_815_int__minus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_816_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_817_nat__diff__distrib,axiom,
! [Z5: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z4 )
=> ( ( nat2 @ ( minus_minus_int @ Z4 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z4 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_818_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_819_zdiff__int__split,axiom,
! [P2: int > $o,X: nat,Y: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_820_int__power__div__base,axiom,
! [M2: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M2 ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_821_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_822_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_823_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_824_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_825_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_826_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_827_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_828_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_829_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_830_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_831_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_832_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_833_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_834_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_835_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_836_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_837_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_838_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_839_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_840_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_841_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_842_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_843_zdiv__mono1,axiom,
! [A: int,A3: int,B: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A3 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_844_zdiv__mono2,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_845_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_846_zdiv__mono1__neg,axiom,
! [A: int,A3: int,B: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_847_zdiv__mono2__neg,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_848_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_849_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_850_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_851_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_852_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_853_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_854_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_855_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_856_power__diff,axiom,
! [A: int,N: nat,M2: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_857_power__diff,axiom,
! [A: nat,N: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_858_verit__less__mono__div__int2,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A5 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_859_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_860_power__diff__power__eq,axiom,
! [A: int,N: nat,M2: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_861_power__diff__power__eq,axiom,
! [A: nat,N: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_862_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_863_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_864_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_865_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_866_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_867_Suc__div__le__mono,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_868_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_869_zdiv__int,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_870_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_871_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_872_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_873_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_874_div__if,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M6 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_875_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_876_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_877_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_878_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_879_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_880_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_881_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_882_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_883_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_884_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_885_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_886_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_887_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_888_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_889_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_890_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_891_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_892_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_893_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_894_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_895_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_896_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_897_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_898_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_899_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_900_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_901_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_902_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_903_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_904_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_905_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_906_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_907_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_908_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_909_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_910_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_911_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_912_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_913_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_914_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_915_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_916_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_917_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_918_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_919_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_920_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_921_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_922_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_923_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_924_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_925_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_926_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_927_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_928_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_929_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_930_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_931_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_932_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_933_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_934_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_935_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_936_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_937_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_938_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_939_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_940_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_941_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_942_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_943_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_944_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_945_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_946_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_947_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_948_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_949_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_950_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_951_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_952_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_953_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_954_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_955_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_956_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_957_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_958_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_959_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_960_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_961_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_962_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_963_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_964_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_965_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_966_abs__power__minus,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_967_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_968_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_969_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_970_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_971_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_972_zle__add1__eq__le,axiom,
! [W: int,Z4: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z4 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z4 ) ) ).
% zle_add1_eq_le
thf(fact_973_zabs__less__one__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ ( abs_abs_int @ Z4 ) @ one_one_int )
= ( Z4 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_974_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
= ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_975_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_976_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_977_group__cancel_Oneg1,axiom,
! [A5: int,K: int,A: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_978_power__abs,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ A @ N ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% power_abs
thf(fact_979_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_980_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_981_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_982_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_983_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_984_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_985_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_986_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_987_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_988_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_989_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_990_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_991_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_992_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_993_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_994_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_995_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_996_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_997_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_998_group__cancel_Oadd1,axiom,
! [A5: int,K: int,A: int,B: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A5 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_999_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A: nat,B: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A5 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1000_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1001_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1002_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1003_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1004_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1005_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1006_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1007_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1008_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1009_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1010_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_1011_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_1012_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1013_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1014_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1015_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1016_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1017_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1018_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1019_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1020_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1021_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1022_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1023_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1024_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1025_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1026_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1027_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1028_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1029_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1030_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1031_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: 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 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1032_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1033_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_1034_add__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_1035_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1036_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1037_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_1038_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1039_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1040_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C: nat] :
( B2
= ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% le_iff_add
thf(fact_1041_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1042_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1043_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1044_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1045_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_1046_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_1047_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_1048_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C2: int,D3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C2 @ D3 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1049_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_1050_group__cancel_Osub1,axiom,
! [A5: int,K: int,A: int,B: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A5 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1051_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_1052_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_1053_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_1054_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1055_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_1056_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1057_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1058_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1059_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_1060_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1061_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1062_abs__diff__less__iff,axiom,
! [X: int,A: int,R: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1063_abs__diff__le__iff,axiom,
! [X: int,A: int,R: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1064_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_1065_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1066_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1067_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_1068_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_1069_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1070_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_1071_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_1072_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_1073_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_1074_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_1075_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1076_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1077_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1078_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1079_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1080_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1081_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1082_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1083_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1084_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1085_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1086_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1087_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1088_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1089_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1090_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1091_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1092_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1093_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1094_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1095_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_1096_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_1097_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_1098_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_1099_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1100_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1101_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1102_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1103_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1104_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1105_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1106_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1107_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1108_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1109_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1110_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1111_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_1112_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_1113_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1114_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1115_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1116_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1117_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1118_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1119_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1120_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1121_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1122_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1123_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1124_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1125_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1126_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1127_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1128_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1129_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_1130_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_1131_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1132_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1133_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_1134_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1135_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_1136_euclidean__size__int__less__eq__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ K ) @ ( euclid4774559944035922753ze_int @ L ) )
= ( ord_less_eq_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ).
% euclidean_size_int_less_eq_iff
thf(fact_1137_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1138_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1139_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1140_odd__nonzero,axiom,
! [Z4: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z4 ) @ Z4 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1141_int__ge__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1142_zless__add1__eq,axiom,
! [W: int,Z4: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z4 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z4 )
| ( W = Z4 ) ) ) ).
% zless_add1_eq
thf(fact_1143_int__gr__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1144_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N4: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1145_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X10: int] : ( plus_plus_int @ ( plus_plus_int @ X10 @ X10 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1146_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1147_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1148_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_1149_abs__if,axiom,
( abs_abs_int
= ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% abs_if
thf(fact_1150_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_1151_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% abs_if_raw
thf(fact_1152_zero__le__power__abs,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_1153_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1154_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1155_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1156_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1157_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1158_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1159_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1160_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1161_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1162_add__strict__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1163_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1164_add__strict__increasing2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1165_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1166_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1167_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_1168_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1169_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_1170_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1171_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1172_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1173_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1174_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N4: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1175_odd__less__0__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z4 ) @ Z4 ) @ zero_zero_int )
= ( ord_less_int @ Z4 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1176_zless__imp__add1__zle,axiom,
! [W: int,Z4: int] :
( ( ord_less_int @ W @ Z4 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z4 ) ) ).
% zless_imp_add1_zle
thf(fact_1177_add1__zle__eq,axiom,
! [W: int,Z4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z4 )
= ( ord_less_int @ W @ Z4 ) ) ).
% add1_zle_eq
thf(fact_1178_int__induct,axiom,
! [P2: int > $o,K: int,I: int] :
( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_induct
thf(fact_1179_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X10: int] : ( minus_minus_int @ ( plus_plus_int @ X10 @ X10 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_1180_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_1181_le__imp__0__less,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z4 ) ) ) ).
% le_imp_0_less
thf(fact_1182_Suc__as__int,axiom,
( suc
= ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1183_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1184_Suc__nat__eq__nat__zadd1,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( suc @ ( nat2 @ Z4 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z4 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1185_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1186_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1187_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1188_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1189_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1190_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1191_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1192_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 ) ) ) ).
% diff_diff_left
thf(fact_1193_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1194_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: 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 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1195_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: 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 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1196_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: 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 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1197_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1198_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1199_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1200_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1201_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1202_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1203_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1204_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1205_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1206_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1207_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1208_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1209_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1210_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1211_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1212_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1213_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1214_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1215_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1216_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: 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 ) ) ) ) ).
% add_le_mono
thf(fact_1217_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1218_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1219_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1220_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1221_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1222_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1223_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1224_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1225_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1226_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1227_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1228_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1229_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1230_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1231_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1232_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1233_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1234_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1235_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1236_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1237_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 ) ) ).
% less_diff_conv
thf(fact_1238_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1239_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1240_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: 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 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1241_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: 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 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1242_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: 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 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1243_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 ) ) ) ).
% le_diff_conv
thf(fact_1244_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1245_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1246_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1247_zadd__int__left,axiom,
! [M2: nat,N: nat,Z4: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z4 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z4 ) ) ).
% zadd_int_left
thf(fact_1248_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1249_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1250_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P2 @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1251_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P2 @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1252_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1253_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_1254_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1255_nat__add__distrib,axiom,
! [Z4: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z4 @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z4 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1256_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1257_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1258_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
= ( P2 @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
=> ( P2 @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1259_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M7: nat] :
( ( P2 @ X )
=> ( ! [X9: nat] :
( ( P2 @ X9 )
=> ( ord_less_eq_nat @ X9 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X11: nat] :
( ( P2 @ X11 )
=> ( ord_less_eq_nat @ X11 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1260_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1261_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1262_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1263_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_1264_fm_Osize__gen_I3_J,axiom,
! [X31: fm,X322: fm] :
( ( size_fm @ ( imp @ X31 @ X322 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X31 ) @ ( size_fm @ X322 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(3)
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [Y3: fm] :
? [X11: nat] :
( Y3
= ( fm_of_nat @ X11 ) ) ).
%------------------------------------------------------------------------------