TPTP Problem File: SLH0346^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Median_Method/0000_Median/prob_00072_002478__14633902_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1378 ( 712 unt; 97 typ; 0 def)
% Number of atoms : 3296 (1159 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9907 ( 280 ~; 73 |; 128 &;8169 @)
% ( 0 <=>;1257 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 263 ( 263 >; 0 *; 0 +; 0 <<)
% Number of symbols : 92 ( 89 usr; 10 con; 0-3 aty)
% Number of variables : 3042 ( 147 ^;2869 !; 26 ?;3042 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:10.283
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (89)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ogroup_001t__Int__Oint,type,
group_int: ( int > int > int ) > int > ( int > int ) > $o ).
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_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
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_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Int__Oint_J,type,
plus_plus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Num__Onum_J,type,
plus_plus_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Num__Onum_J,type,
times_times_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Num__Onum_J,type,
uminus6418336269838312217et_num: set_num > set_num ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
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_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Int__Oint_J,type,
zero_zero_set_int: set_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_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_Median_Odown__ray_001t__Int__Oint,type,
down_ray_int: set_int > $o ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001t__Num__Onum,type,
down_ray_num: set_num > $o ).
thf(sy_c_Median_Odown__ray_001tf__a,type,
down_ray_a: set_a > $o ).
thf(sy_c_Median_Ointerval_001t__Int__Oint,type,
interval_int: set_int > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001t__Num__Onum,type,
interval_num: set_num > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001t__Int__Oint,type,
up_ray_int: set_int > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001t__Num__Onum,type,
up_ray_num: set_num > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
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_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
neg_numeral_sub_int: num > num > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
ring_1_iszero_int: int > $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_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Num__Onum,type,
order_Greatest_num: ( num > $o ) > num ).
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_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Int__Oint,type,
set_elt_set_plus_int: int > set_int > set_int ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Nat__Onat,type,
set_elt_set_plus_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__plus_001t__Num__Onum,type,
set_elt_set_plus_num: num > set_num > set_num ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Int__Oint,type,
set_el2930815339941905629es_int: int > set_int > set_int ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Nat__Onat,type,
set_el2933305810450955905es_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Algebras_Oelt__set__times_001t__Num__Onum,type,
set_el8714009633461510347es_num: num > set_num > set_num ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
% Relevant facts (1275)
thf(fact_0_assms,axiom,
down_ray_a @ i ).
% assms
thf(fact_1_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_2_ComplI,axiom,
! [C: int,A: set_int] :
( ~ ( member_int @ C @ A )
=> ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) ) ) ).
% ComplI
thf(fact_3_ComplI,axiom,
! [C: num,A: set_num] :
( ~ ( member_num @ C @ A )
=> ( member_num @ C @ ( uminus6418336269838312217et_num @ A ) ) ) ).
% ComplI
thf(fact_4_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_5_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_6_Compl__iff,axiom,
! [C: int,A: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
= ( ~ ( member_int @ C @ A ) ) ) ).
% Compl_iff
thf(fact_7_Compl__iff,axiom,
! [C: num,A: set_num] :
( ( member_num @ C @ ( uminus6418336269838312217et_num @ A ) )
= ( ~ ( member_num @ C @ A ) ) ) ).
% Compl_iff
thf(fact_8_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_9_Compl__eq__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( uminus_uminus_set_a @ A )
= ( uminus_uminus_set_a @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_10_verit__minus__simplify_I4_J,axiom,
! [B2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_11_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_12_neg__equal__iff__equal,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B2 ) )
= ( A2 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_13_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_14_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( uminus_uminus_set_a @ X )
= ( uminus_uminus_set_a @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_15_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_16_ComplD,axiom,
! [C: int,A: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
=> ~ ( member_int @ C @ A ) ) ).
% ComplD
thf(fact_17_ComplD,axiom,
! [C: num,A: set_num] :
( ( member_num @ C @ ( uminus6418336269838312217et_num @ A ) )
=> ~ ( member_num @ C @ A ) ) ).
% ComplD
thf(fact_18_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_19_exists__diff,axiom,
! [P: set_a > $o] :
( ( ? [S: set_a] : ( P @ ( uminus_uminus_set_a @ S ) ) )
= ( ? [X2: set_a] : ( P @ X2 ) ) ) ).
% exists_diff
thf(fact_20_double__complement,axiom,
! [A: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A ) )
= A ) ).
% double_complement
thf(fact_21_minus__equation__iff,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A2 ) ) ).
% minus_equation_iff
thf(fact_22_equation__minus__iff,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_23_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_24_down__ray__def,axiom,
( down_ray_a
= ( ^ [I: set_a] :
! [X3: a,Y2: a] :
( ( member_a @ Y2 @ I )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
=> ( member_a @ X3 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_25_down__ray__def,axiom,
( down_ray_num
= ( ^ [I: set_num] :
! [X3: num,Y2: num] :
( ( member_num @ Y2 @ I )
=> ( ( ord_less_eq_num @ X3 @ Y2 )
=> ( member_num @ X3 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_26_down__ray__def,axiom,
( down_ray_int
= ( ^ [I: set_int] :
! [X3: int,Y2: int] :
( ( member_int @ Y2 @ I )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
=> ( member_int @ X3 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_27_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I: set_nat] :
! [X3: nat,Y2: nat] :
( ( member_nat @ Y2 @ I )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( member_nat @ X3 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_28_up__ray__def,axiom,
( up_ray_a
= ( ^ [I: set_a] :
! [X3: a,Y2: a] :
( ( member_a @ X3 @ I )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
=> ( member_a @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_29_up__ray__def,axiom,
( up_ray_num
= ( ^ [I: set_num] :
! [X3: num,Y2: num] :
( ( member_num @ X3 @ I )
=> ( ( ord_less_eq_num @ X3 @ Y2 )
=> ( member_num @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_30_up__ray__def,axiom,
( up_ray_int
= ( ^ [I: set_int] :
! [X3: int,Y2: int] :
( ( member_int @ X3 @ I )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
=> ( member_int @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_31_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I: set_nat] :
! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ I )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( member_nat @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_32_Compl__subset__Compl__iff,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A ) @ ( uminus1532241313380277803et_int @ B ) )
= ( ord_less_eq_set_int @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_33_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_34_Compl__subset__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_35_Compl__anti__mono,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B ) @ ( uminus1532241313380277803et_int @ A ) ) ) ).
% Compl_anti_mono
thf(fact_36_Compl__anti__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_37_Compl__anti__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_38_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_39_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_40_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_41_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_42_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_43_compl__le__compl__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
= ( ord_less_eq_set_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_44_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_45_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_46_neg__le__iff__le,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_47_minus__add__distrib,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_48_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_49_add__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_50_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_51_add__right__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_52_subsetI,axiom,
! [A: set_num,B: set_num] :
( ! [X4: num] :
( ( member_num @ X4 @ A )
=> ( member_num @ X4 @ B ) )
=> ( ord_less_eq_set_num @ A @ B ) ) ).
% subsetI
thf(fact_53_subsetI,axiom,
! [A: set_int,B: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( member_int @ X4 @ B ) )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% subsetI
thf(fact_54_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_55_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_56_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_57_subset__antisym,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_58_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_59_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_60_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_61_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_62_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_63_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_64_add_Oright__neutral,axiom,
! [A2: set_int] :
( ( plus_plus_set_int @ A2 @ zero_zero_set_int )
= A2 ) ).
% add.right_neutral
thf(fact_65_add_Oright__neutral,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ A2 @ zero_zero_set_nat )
= A2 ) ).
% add.right_neutral
thf(fact_66_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_67_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_68_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_69_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_70_add__cancel__left__left,axiom,
! [B2: int,A2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_71_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_72_add__cancel__left__right,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_73_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_74_add__cancel__right__left,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ B2 @ A2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_75_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_76_add__cancel__right__right,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ A2 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_77_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_78_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_79_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A2: num,P: num > $o] :
( ( member_num @ A2 @ ( collect_num @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A: set_num] :
( ( collect_num
@ ^ [X3: num] : ( member_num @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_87_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_88_add__0,axiom,
! [A2: set_int] :
( ( plus_plus_set_int @ zero_zero_set_int @ A2 )
= A2 ) ).
% add_0
thf(fact_89_add__0,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_90_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_91_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_92_add__minus__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_93_minus__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_94_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_95_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_96_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_97_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_98_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_99_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_100_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_101_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_102_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_103_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_104_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_105_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_106_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_107_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_108_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_109_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_110_verit__comp__simplify1_I2_J,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_111_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_112_verit__comp__simplify1_I2_J,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_113_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_114_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_115_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_116_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_117_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_118_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_119_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_120_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_121_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_122_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_123_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_124_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_125_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_126_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_127_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_128_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_129_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_130_group__cancel_Oadd2,axiom,
! [B: int,K: int,B2: int,A2: int] :
( ( B
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_131_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_132_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_133_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_134_add_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_135_add_Oleft__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_136_add_Oright__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_137_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_138_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_139_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_140_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_141_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_142_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_143_add_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_144_add__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_145_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_146_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_147_add__left__imp__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_148_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_149_add__right__imp__eq,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_150_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_151_add__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_152_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_153_add__decreasing,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_154_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_155_add__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_156_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_157_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_158_add__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_159_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_160_add__decreasing2,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_161_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_162_add__increasing2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_163_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_164_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_165_add__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_166_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_167_add__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_168_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_169_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_170_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_171_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_172_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_173_add__le__imp__le__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_174_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_175_add__le__imp__le__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_176_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_177_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_178_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_179_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_180_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_181_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_182_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_183_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_184_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_185_verit__la__disequality,axiom,
! [A2: num,B2: num] :
( ( A2 = B2 )
| ~ ( ord_less_eq_num @ A2 @ B2 )
| ~ ( ord_less_eq_num @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_186_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_187_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_188_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_189_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_190_add_Oinverse__unique,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B2 ) ) ).
% add.inverse_unique
thf(fact_191_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_192_add__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_193_group__cancel_Oneg1,axiom,
! [A: int,K: int,A2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_194_add_Oinverse__distrib__swap,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_195_le__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_196_minus__le__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_197_le__imp__neg__le,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_198_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_199_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_200_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_201_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_202_set__plus__intro,axiom,
! [A2: num,C4: set_num,B2: num,D2: set_num] :
( ( member_num @ A2 @ C4 )
=> ( ( member_num @ B2 @ D2 )
=> ( member_num @ ( plus_plus_num @ A2 @ B2 ) @ ( plus_plus_set_num @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_203_set__plus__intro,axiom,
! [A2: nat,C4: set_nat,B2: nat,D2: set_nat] :
( ( member_nat @ A2 @ C4 )
=> ( ( member_nat @ B2 @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( plus_plus_set_nat @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_204_set__plus__intro,axiom,
! [A2: int,C4: set_int,B2: int,D2: set_int] :
( ( member_int @ A2 @ C4 )
=> ( ( member_int @ B2 @ D2 )
=> ( member_int @ ( plus_plus_int @ A2 @ B2 ) @ ( plus_plus_set_int @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_205_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_206_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_207_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_208_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_209_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_210_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_211_interval__def,axiom,
( interval_a
= ( ^ [I: set_a] :
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ I )
=> ( ( member_a @ Z @ I )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( member_a @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_212_interval__def,axiom,
( interval_num
= ( ^ [I: set_num] :
! [X3: num,Y2: num,Z: num] :
( ( member_num @ X3 @ I )
=> ( ( member_num @ Z @ I )
=> ( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z )
=> ( member_num @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_213_interval__def,axiom,
( interval_int
= ( ^ [I: set_int] :
! [X3: int,Y2: int,Z: int] :
( ( member_int @ X3 @ I )
=> ( ( member_int @ Z @ I )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( member_int @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_214_interval__def,axiom,
( interval_nat
= ( ^ [I: set_nat] :
! [X3: nat,Y2: nat,Z: nat] :
( ( member_nat @ X3 @ I )
=> ( ( member_nat @ Z @ I )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( member_nat @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_215_is__num__normalize_I8_J,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_216_set__zero__plus2,axiom,
! [A: set_nat,B: set_nat] :
( ( member_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_set_nat @ B @ ( plus_plus_set_nat @ A @ B ) ) ) ).
% set_zero_plus2
thf(fact_217_set__zero__plus2,axiom,
! [A: set_int,B: set_int] :
( ( member_int @ zero_zero_int @ A )
=> ( ord_less_eq_set_int @ B @ ( plus_plus_set_int @ A @ B ) ) ) ).
% set_zero_plus2
thf(fact_218_add__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_219_add__0__iff,axiom,
! [B2: int,A2: int] :
( ( B2
= ( plus_plus_int @ B2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_220_eq__add__iff,axiom,
! [X: int,Y: int] :
( ( X
= ( plus_plus_int @ X @ Y ) )
= ( Y = zero_zero_int ) ) ).
% eq_add_iff
thf(fact_221_nle__le,axiom,
! [A2: num,B2: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B2 ) )
= ( ( ord_less_eq_num @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_222_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_223_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_224_le__cases3,axiom,
! [X: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_225_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_226_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_227_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z3: num] : ( Y3 = Z3 ) )
= ( ^ [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
& ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_228_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_229_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_230_ord__eq__le__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( A2 = B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_231_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_232_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_233_ord__le__eq__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_234_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_235_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_236_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_237_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_238_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_239_order_Otrans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_240_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_241_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_242_order__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_243_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_244_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_245_linorder__wlog,axiom,
! [P: num > num > $o,A2: num,B2: num] :
( ! [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: num,B5: num] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_246_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_247_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_248_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: num,Z3: num] : ( Y3 = Z3 ) )
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_249_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_250_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_251_dual__order_Oantisym,axiom,
! [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_252_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_253_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_254_dual__order_Otrans,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_255_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_256_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_257_antisym,axiom,
! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_258_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_259_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_260_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z3: num] : ( Y3 = Z3 ) )
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_261_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_262_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_263_order__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_264_order__subst1,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_265_order__subst1,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_266_order__subst1,axiom,
! [A2: int,F: num > int,B2: num,C: num] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_267_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_268_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_269_order__subst1,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_270_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_271_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_272_order__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_273_order__subst2,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_274_order__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_275_order__subst2,axiom,
! [A2: int,B2: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_276_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_277_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_278_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_279_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_280_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_281_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_282_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_283_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_284_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_285_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_286_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_287_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_288_ord__eq__le__subst,axiom,
! [A2: int,F: num > int,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_289_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_290_ord__eq__le__subst,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_291_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_292_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_293_ord__eq__le__subst,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_294_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_295_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_296_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_297_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_298_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_299_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_300_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_301_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_302_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_303_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_304_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_305_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_306_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_307_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_308_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_309_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_310_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_311_is__num__normalize_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_312_set__plus__elim,axiom,
! [X: num,A: set_num,B: set_num] :
( ( member_num @ X @ ( plus_plus_set_num @ A @ B ) )
=> ~ ! [A5: num,B5: num] :
( ( X
= ( plus_plus_num @ A5 @ B5 ) )
=> ( ( member_num @ A5 @ A )
=> ~ ( member_num @ B5 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_313_set__plus__elim,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A @ B ) )
=> ~ ! [A5: nat,B5: nat] :
( ( X
= ( plus_plus_nat @ A5 @ B5 ) )
=> ( ( member_nat @ A5 @ A )
=> ~ ( member_nat @ B5 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_314_set__plus__elim,axiom,
! [X: int,A: set_int,B: set_int] :
( ( member_int @ X @ ( plus_plus_set_int @ A @ B ) )
=> ~ ! [A5: int,B5: int] :
( ( X
= ( plus_plus_int @ A5 @ B5 ) )
=> ( ( member_int @ A5 @ A )
=> ~ ( member_int @ B5 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_315_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_316_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_317_add_Ogroup__axioms,axiom,
group_int @ plus_plus_int @ zero_zero_int @ uminus_uminus_int ).
% add.group_axioms
thf(fact_318_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_319_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_320_GreatestI2__order,axiom,
! [P: num > $o,X: num,Q: num > $o] :
( ( P @ X )
=> ( ! [Y4: num] :
( ( P @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X ) )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( ! [Y5: num] :
( ( P @ Y5 )
=> ( ord_less_eq_num @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_num @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_321_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y4: int] :
( ( P @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X ) )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_322_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_323_Greatest__equality,axiom,
! [P: num > $o,X: num] :
( ( P @ X )
=> ( ! [Y4: num] :
( ( P @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X ) )
=> ( ( order_Greatest_num @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_324_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y4: int] :
( ( P @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_325_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_326_abs__of__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( abs_abs_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ) ).
% abs_of_nonpos
thf(fact_327_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_328_abs__idempotent,axiom,
! [A2: int] :
( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_idempotent
thf(fact_329_abs__0__eq,axiom,
! [A2: int] :
( ( zero_zero_int
= ( abs_abs_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_330_abs__eq__0,axiom,
! [A2: int] :
( ( ( abs_abs_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_331_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_332_abs__add__abs,axiom,
! [A2: int,B2: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_add_abs
thf(fact_333_abs__minus__cancel,axiom,
! [A2: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_minus_cancel
thf(fact_334_set__zero__plus,axiom,
! [C4: set_nat] :
( ( set_elt_set_plus_nat @ zero_zero_nat @ C4 )
= C4 ) ).
% set_zero_plus
thf(fact_335_set__zero__plus,axiom,
! [C4: set_int] :
( ( set_elt_set_plus_int @ zero_zero_int @ C4 )
= C4 ) ).
% set_zero_plus
thf(fact_336_set__plus__intro2,axiom,
! [B2: num,C4: set_num,A2: num] :
( ( member_num @ B2 @ C4 )
=> ( member_num @ ( plus_plus_num @ A2 @ B2 ) @ ( set_elt_set_plus_num @ A2 @ C4 ) ) ) ).
% set_plus_intro2
thf(fact_337_set__plus__intro2,axiom,
! [B2: nat,C4: set_nat,A2: nat] :
( ( member_nat @ B2 @ C4 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( set_elt_set_plus_nat @ A2 @ C4 ) ) ) ).
% set_plus_intro2
thf(fact_338_set__plus__intro2,axiom,
! [B2: int,C4: set_int,A2: int] :
( ( member_int @ B2 @ C4 )
=> ( member_int @ ( plus_plus_int @ A2 @ B2 ) @ ( set_elt_set_plus_int @ A2 @ C4 ) ) ) ).
% set_plus_intro2
thf(fact_339_abs__of__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_nonneg
thf(fact_340_abs__le__self__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% abs_le_self_iff
thf(fact_341_abs__le__zero__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_342_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_343_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_344_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_345_abs__le__D1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% abs_le_D1
thf(fact_346_abs__ge__self,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_self
thf(fact_347_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_348_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_349_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_350_set__plus__rearrange2,axiom,
! [A2: nat,B2: nat,C4: set_nat] :
( ( set_elt_set_plus_nat @ A2 @ ( set_elt_set_plus_nat @ B2 @ C4 ) )
= ( set_elt_set_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C4 ) ) ).
% set_plus_rearrange2
thf(fact_351_set__plus__rearrange2,axiom,
! [A2: int,B2: int,C4: set_int] :
( ( set_elt_set_plus_int @ A2 @ ( set_elt_set_plus_int @ B2 @ C4 ) )
= ( set_elt_set_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C4 ) ) ).
% set_plus_rearrange2
thf(fact_352_abs__ge__zero,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_zero
thf(fact_353_abs__triangle__ineq,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_triangle_ineq
thf(fact_354_abs__ge__minus__self,axiom,
! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_minus_self
thf(fact_355_abs__le__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
= ( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).
% abs_le_iff
thf(fact_356_abs__le__D2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).
% abs_le_D2
thf(fact_357_abs__leI,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 ) ) ) ).
% abs_leI
thf(fact_358_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_359_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_360_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_361_abs__minus__le__zero,axiom,
! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A2 ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_362_set__plus__rearrange,axiom,
! [A2: nat,C4: set_nat,B2: nat,D2: set_nat] :
( ( plus_plus_set_nat @ ( set_elt_set_plus_nat @ A2 @ C4 ) @ ( set_elt_set_plus_nat @ B2 @ D2 ) )
= ( set_elt_set_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( plus_plus_set_nat @ C4 @ D2 ) ) ) ).
% set_plus_rearrange
thf(fact_363_set__plus__rearrange,axiom,
! [A2: int,C4: set_int,B2: int,D2: set_int] :
( ( plus_plus_set_int @ ( set_elt_set_plus_int @ A2 @ C4 ) @ ( set_elt_set_plus_int @ B2 @ D2 ) )
= ( set_elt_set_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ ( plus_plus_set_int @ C4 @ D2 ) ) ) ).
% set_plus_rearrange
thf(fact_364_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_365_abs__minus,axiom,
! [A2: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_minus
thf(fact_366_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_367_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_368_abs__eq__iff_H,axiom,
! [A2: int,B2: int] :
( ( ( abs_abs_int @ A2 )
= B2 )
= ( ( ord_less_eq_int @ zero_zero_int @ B2 )
& ( ( A2 = B2 )
| ( A2
= ( uminus_uminus_int @ B2 ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_369_eq__abs__iff_H,axiom,
! [A2: int,B2: int] :
( ( A2
= ( abs_abs_int @ B2 ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ( B2 = A2 )
| ( B2
= ( uminus_uminus_int @ A2 ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_370_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_371_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_372_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_373_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_374_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_375_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_376_abs__abs,axiom,
! [A2: int] :
( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_abs
thf(fact_377_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_378_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_379_abs__eq__0__iff,axiom,
! [A2: int] :
( ( ( abs_abs_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_380_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_381_abs__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( abs_abs_int @ X )
= ( abs_abs_int @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_382_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_383_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_384_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_385_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_386_convex__bound__le,axiom,
! [X: int,A2: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A2 )
=> ( ( ord_less_eq_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_387_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_388_set__times__intro,axiom,
! [A2: num,C4: set_num,B2: num,D2: set_num] :
( ( member_num @ A2 @ C4 )
=> ( ( member_num @ B2 @ D2 )
=> ( member_num @ ( times_times_num @ A2 @ B2 ) @ ( times_times_set_num @ C4 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_389_set__times__intro,axiom,
! [A2: int,C4: set_int,B2: int,D2: set_int] :
( ( member_int @ A2 @ C4 )
=> ( ( member_int @ B2 @ D2 )
=> ( member_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_set_int @ C4 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_390_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_391_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_392_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_393_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_394_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_395_mult__eq__0__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_396_mult__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_397_mult__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_398_mult__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_399_mult__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_400_mult__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_401_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_402_add__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_403_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_404_add__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_405_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_406_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_407_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_408_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_409_neg__less__iff__less,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_410_compl__less__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_set_a @ Y @ X ) ) ).
% compl_less_compl_iff
thf(fact_411_mult__minus__left,axiom,
! [A2: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B2 ) ) ) ).
% mult_minus_left
thf(fact_412_minus__mult__minus,axiom,
! [A2: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) )
= ( times_times_int @ A2 @ B2 ) ) ).
% minus_mult_minus
thf(fact_413_mult__minus__right,axiom,
! [A2: int,B2: int] :
( ( times_times_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B2 ) ) ) ).
% mult_minus_right
thf(fact_414_abs__mult__self__eq,axiom,
! [A2: int] :
( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ A2 ) )
= ( times_times_int @ A2 @ A2 ) ) ).
% abs_mult_self_eq
thf(fact_415_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_416_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_417_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_418_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_419_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_420_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_421_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_422_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_423_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_424_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_425_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_426_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_427_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_428_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_429_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_430_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_431_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_432_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_433_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_434_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_435_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_436_zero__less__abs__iff,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
= ( A2 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_437_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_438_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_439_abs__mult__less,axiom,
! [A2: int,C: int,B2: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_440_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_441_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_442_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_443_mult__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_444_mult__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_445_mult__strict__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_446_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_447_mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_448_mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_449_mult__strict__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_450_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_451_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_452_zero__less__mult__pos2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_453_zero__less__mult__pos2,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_454_zero__less__mult__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_455_zero__less__mult__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_456_zero__less__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_457_mult__pos__neg2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_458_mult__pos__neg2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_459_mult__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_460_mult__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_461_mult__pos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_462_mult__pos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_463_mult__neg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_464_mult__neg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_465_mult__less__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_466_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_467_mult__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_468_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_469_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_470_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_471_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_472_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_473_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_474_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_475_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_476_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_477_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_478_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_479_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_480_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_481_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_482_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_483_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_484_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_485_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_486_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_487_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_488_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_489_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_490_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_491_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_492_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_493_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_494_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_495_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_496_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_497_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_498_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_499_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_500_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_501_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_502_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_503_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_504_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_505_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_506_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_507_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_508_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_509_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_510_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_511_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_512_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_513_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_514_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_515_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_516_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_517_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_518_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_519_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_520_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_521_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_522_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_523_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_524_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_525_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_526_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_527_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_528_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_529_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_530_set__times__elim,axiom,
! [X: num,A: set_num,B: set_num] :
( ( member_num @ X @ ( times_times_set_num @ A @ B ) )
=> ~ ! [A5: num,B5: num] :
( ( X
= ( times_times_num @ A5 @ B5 ) )
=> ( ( member_num @ A5 @ A )
=> ~ ( member_num @ B5 @ B ) ) ) ) ).
% set_times_elim
thf(fact_531_set__times__elim,axiom,
! [X: int,A: set_int,B: set_int] :
( ( member_int @ X @ ( times_times_set_int @ A @ B ) )
=> ~ ! [A5: int,B5: int] :
( ( X
= ( times_times_int @ A5 @ B5 ) )
=> ( ( member_int @ A5 @ A )
=> ~ ( member_int @ B5 @ B ) ) ) ) ).
% set_times_elim
thf(fact_532_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_533_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_534_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_535_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_536_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_537_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_538_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_539_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_540_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_541_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_542_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_543_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_544_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_545_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_546_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_547_mult_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_548_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_549_mult_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_550_mult__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_551_mult__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_552_mult__left__less__imp__less,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_553_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_554_mult__strict__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_555_mult__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_556_mult__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_557_mult__right__less__imp__less,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_558_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_559_mult__strict__mono_H,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_560_mult__strict__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_561_mult__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_562_mult__le__cancel__left__neg,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_563_mult__le__cancel__left__pos,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_564_mult__left__le__imp__le,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_565_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_566_mult__right__le__imp__le,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_567_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_568_mult__le__less__imp__less,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_569_mult__le__less__imp__less,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_570_mult__less__le__imp__less,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_571_mult__less__le__imp__less,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_572_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_573_verit__comp__simplify1_I3_J,axiom,
! [B6: num,A6: num] :
( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
= ( ord_less_num @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_574_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_575_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_576_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_577_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_578_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_579_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_580_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_581_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_582_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_583_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > num,C: num] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_584_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_585_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_586_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_587_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_588_order__less__le__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_589_order__less__le__subst1,axiom,
! [A2: int,F: num > int,B2: num,C: num] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_590_order__less__le__subst1,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_591_order__less__le__subst1,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_592_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_593_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_594_order__less__le__subst1,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_595_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_596_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_597_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_598_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_599_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_600_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_601_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_602_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_603_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_604_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_605_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_606_order__le__less__subst1,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_607_order__le__less__subst1,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_608_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_609_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_610_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_611_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_612_order__less__le__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_613_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_614_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_615_order__le__less__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_616_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_617_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_618_order__neq__le__trans,axiom,
! [A2: num,B2: num] :
( ( A2 != B2 )
=> ( ( ord_less_eq_num @ A2 @ B2 )
=> ( ord_less_num @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_619_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_620_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_621_order__le__neq__trans,axiom,
! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_num @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_622_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_623_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_624_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_625_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_626_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_627_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_628_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_629_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_630_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_631_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_632_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_633_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_634_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_635_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_636_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_637_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_638_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_639_dual__order_Ostrict__implies__order,axiom,
! [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ord_less_eq_num @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_640_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_641_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_642_order_Ostrict__implies__order,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ord_less_eq_num @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_643_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_644_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_645_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_646_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_647_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_648_dual__order_Ostrict__trans2,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_649_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_650_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_651_dual__order_Ostrict__trans1,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_652_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_653_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_654_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_655_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_656_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_657_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_658_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_659_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_660_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_661_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_662_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_663_order_Ostrict__trans2,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_664_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_665_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_666_order_Ostrict__trans1,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_667_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_668_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_669_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_670_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_671_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_672_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_673_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_674_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_675_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_676_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_677_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_678_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
& ~ ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_679_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_680_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_681_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_682_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_683_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_684_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_685_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_686_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_687_nless__le,axiom,
! [A2: num,B2: num] :
( ( ~ ( ord_less_num @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_688_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_689_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_690_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_691_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_692_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_693_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_694_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_695_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_696_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_697_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_698_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_699_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_700_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_701_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_702_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_703_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_704_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_705_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_706_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_707_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_708_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_709_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_710_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_711_add__strict__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_712_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_713_add__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_714_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_715_add__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_716_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_717_add__less__imp__less__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_718_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_719_add__less__imp__less__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_720_mult__not__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_721_mult__not__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_722_divisors__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_723_divisors__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_724_no__zero__divisors,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_725_no__zero__divisors,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_726_mult__left__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_727_mult__left__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_728_mult__right__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_729_mult__right__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_730_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_731_less__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_732_minus__less__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_less_iff
thf(fact_733_compl__less__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_less_swap1
thf(fact_734_compl__less__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_less_swap2
thf(fact_735_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_736_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_737_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_738_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_739_crossproduct__noteq,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_740_crossproduct__noteq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_741_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_742_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_743_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_744_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_745_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_746_comm__semiring__class_Odistrib,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_747_distrib__left,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_748_distrib__left,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_749_distrib__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_750_distrib__right,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_751_combine__common__factor,axiom,
! [A2: nat,E: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_752_combine__common__factor,axiom,
! [A2: int,E: int,B2: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_753_minus__mult__commute,axiom,
! [A2: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( times_times_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_mult_commute
thf(fact_754_square__eq__iff,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ A2 )
= ( times_times_int @ B2 @ B2 ) )
= ( ( A2 = B2 )
| ( A2
= ( uminus_uminus_int @ B2 ) ) ) ) ).
% square_eq_iff
thf(fact_755_abs__mult,axiom,
! [A2: int,B2: int] :
( ( abs_abs_int @ ( times_times_int @ A2 @ B2 ) )
= ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_mult
thf(fact_756_mult__le__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_757_mult__le__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_758_mult__le__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_759_mult__le__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_760_mult__less__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_761_mult__less__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_762_mult__less__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_763_mult__less__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_764_convex__bound__lt,axiom,
! [X: int,A2: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A2 )
=> ( ( ord_less_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_765_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_766_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_767_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_768_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_769_add__le__less__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_770_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_771_add__less__le__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_772_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_773_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_774_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_775_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_776_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_777_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_778_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_779_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_780_add__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_781_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_782_add__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_783_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_784_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_785_pos__add__strict,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_786_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_787_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_788_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_789_zero__le__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_790_mult__nonneg__nonpos2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_791_mult__nonneg__nonpos2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_792_mult__nonpos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_793_mult__nonpos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_794_mult__nonneg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_795_mult__nonneg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_796_mult__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_797_mult__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_798_split__mult__neg__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_799_split__mult__neg__le,axiom,
! [A2: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_800_mult__le__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_801_mult__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_802_mult__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_803_mult__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_804_mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_805_mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_806_mult__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_807_mult__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_808_split__mult__pos__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_809_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_810_mult__mono_H,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_811_mult__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_812_mult__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_813_mult__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_814_add__mono1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_815_add__mono1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_816_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_817_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_818_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_819_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_820_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B2: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B2 @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_821_add__scale__eq__noteq,axiom,
! [R: int,A2: int,B2: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus_int @ A2 @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B2 @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_822_abs__not__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_823_abs__of__pos,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_pos
thf(fact_824_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_825_abs__less__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( abs_abs_int @ A2 ) @ B2 )
= ( ( ord_less_int @ A2 @ B2 )
& ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).
% abs_less_iff
thf(fact_826_add__neg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_827_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_828_add__nonneg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_829_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_830_add__nonpos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_831_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_832_add__pos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_833_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_834_add__strict__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_835_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_836_add__strict__increasing2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_837_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_838_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_839_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_840_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_841_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_842_mult__left__le,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_843_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_844_mult__le__one,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_845_mult__le__one,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_846_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_847_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_848_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_849_abs__of__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( abs_abs_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ) ).
% abs_of_neg
thf(fact_850_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_851_abs__if,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_852_abs__mult__pos_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ X @ ( abs_abs_int @ Y ) )
= ( abs_abs_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_853_abs__eq__mult,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
| ( ord_less_eq_int @ A2 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B2 )
| ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A2 @ B2 ) )
= ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% abs_eq_mult
thf(fact_854_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_855_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_856_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_857_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_858_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_859_affine__ineq,axiom,
! [X: int,V: int,U: int] :
( ( ord_less_eq_int @ X @ one_one_int )
=> ( ( ord_less_eq_int @ V @ U )
=> ( ord_less_eq_int @ ( plus_plus_int @ V @ ( times_times_int @ X @ U ) ) @ ( plus_plus_int @ U @ ( times_times_int @ X @ V ) ) ) ) ) ).
% affine_ineq
thf(fact_860_mult__eq__1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ B2 @ one_one_int )
=> ( ( ( times_times_int @ A2 @ B2 )
= one_one_int )
= ( ( A2 = one_one_int )
& ( B2 = one_one_int ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_861_mult__eq__1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B2 )
= one_one_nat )
= ( ( A2 = one_one_nat )
& ( B2 = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_862_mult__le__cancel__iff2,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_863_mult__le__cancel__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_864_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_865_mult__if__delta,axiom,
! [P: $o,Q2: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q2 )
= Q2 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q2 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_866_mult__if__delta,axiom,
! [P: $o,Q2: int] :
( ( P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q2 )
= Q2 ) )
& ( ~ P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q2 )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_867_mult__delta__left,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_868_mult__delta__left,axiom,
! [B2: $o,X: int,Y: int] :
( ( B2
=> ( ( times_times_int @ ( if_int @ B2 @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_int @ ( if_int @ B2 @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_869_mult__delta__right,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_870_mult__delta__right,axiom,
! [B2: $o,X: int,Y: int] :
( ( B2
=> ( ( times_times_int @ X @ ( if_int @ B2 @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_int @ X @ ( if_int @ B2 @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_871_minf_I8_J,axiom,
! [T2: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ~ ( ord_less_eq_num @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_872_minf_I8_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ~ ( ord_less_eq_int @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_873_minf_I8_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_874_minf_I6_J,axiom,
! [T2: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ( ord_less_eq_num @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_875_minf_I6_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ord_less_eq_int @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_876_minf_I6_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_877_pinf_I8_J,axiom,
! [T2: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ( ord_less_eq_num @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_878_pinf_I8_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ord_less_eq_int @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_879_pinf_I8_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_880_pinf_I6_J,axiom,
! [T2: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ~ ( ord_less_eq_num @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_881_pinf_I6_J,axiom,
! [T2: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_882_pinf_I6_J,axiom,
! [T2: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_883_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A2 @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A2 @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A2 @ X4 )
& ( ord_less_int @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_884_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_885_set__times__intro2,axiom,
! [B2: num,C4: set_num,A2: num] :
( ( member_num @ B2 @ C4 )
=> ( member_num @ ( times_times_num @ A2 @ B2 ) @ ( set_el8714009633461510347es_num @ A2 @ C4 ) ) ) ).
% set_times_intro2
thf(fact_886_set__times__intro2,axiom,
! [B2: int,C4: set_int,A2: int] :
( ( member_int @ B2 @ C4 )
=> ( member_int @ ( times_times_int @ A2 @ B2 ) @ ( set_el2930815339941905629es_int @ A2 @ C4 ) ) ) ).
% set_times_intro2
thf(fact_887_set__one__times,axiom,
! [C4: set_nat] :
( ( set_el2933305810450955905es_nat @ one_one_nat @ C4 )
= C4 ) ).
% set_one_times
thf(fact_888_set__one__times,axiom,
! [C4: set_int] :
( ( set_el2930815339941905629es_int @ one_one_int @ C4 )
= C4 ) ).
% set_one_times
thf(fact_889_set__times__rearrange2,axiom,
! [A2: int,B2: int,C4: set_int] :
( ( set_el2930815339941905629es_int @ A2 @ ( set_el2930815339941905629es_int @ B2 @ C4 ) )
= ( set_el2930815339941905629es_int @ ( times_times_int @ A2 @ B2 ) @ C4 ) ) ).
% set_times_rearrange2
thf(fact_890_set__neg__intro,axiom,
! [A2: int,C4: set_int] :
( ( member_int @ A2 @ ( set_el2930815339941905629es_int @ ( uminus_uminus_int @ one_one_int ) @ C4 ) )
=> ( member_int @ ( uminus_uminus_int @ A2 ) @ C4 ) ) ).
% set_neg_intro
thf(fact_891_set__neg__intro2,axiom,
! [A2: int,C4: set_int] :
( ( member_int @ A2 @ C4 )
=> ( member_int @ ( uminus_uminus_int @ A2 ) @ ( set_el2930815339941905629es_int @ ( uminus_uminus_int @ one_one_int ) @ C4 ) ) ) ).
% set_neg_intro2
thf(fact_892_set__times__rearrange,axiom,
! [A2: int,C4: set_int,B2: int,D2: set_int] :
( ( times_times_set_int @ ( set_el2930815339941905629es_int @ A2 @ C4 ) @ ( set_el2930815339941905629es_int @ B2 @ D2 ) )
= ( set_el2930815339941905629es_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_set_int @ C4 @ D2 ) ) ) ).
% set_times_rearrange
thf(fact_893_set__times__plus__distrib,axiom,
! [A2: int,B2: int,C4: set_int] :
( ( set_el2930815339941905629es_int @ A2 @ ( set_elt_set_plus_int @ B2 @ C4 ) )
= ( set_elt_set_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( set_el2930815339941905629es_int @ A2 @ C4 ) ) ) ).
% set_times_plus_distrib
thf(fact_894_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_895_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_896_linepath__le__1,axiom,
! [A2: int,B2: int,U: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ B2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ U @ one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ one_one_int @ U ) @ A2 ) @ ( times_times_int @ U @ B2 ) ) @ one_one_int ) ) ) ) ) ).
% linepath_le_1
thf(fact_897_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_898_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_899_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_900_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_901_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_902_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_903_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_904_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_905_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_906_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_907_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_908_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_909_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_910_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_911_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_912_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_913_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_914_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_915_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_916_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_917_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_918_add__diff__cancel__right_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_919_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_920_add__diff__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_921_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_922_add__diff__cancel__left_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_923_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_924_add__diff__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_925_diff__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% diff_add_cancel
thf(fact_926_add__diff__cancel,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel
thf(fact_927_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_928_minus__diff__eq,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( minus_minus_int @ B2 @ A2 ) ) ).
% minus_diff_eq
thf(fact_929_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_numeral
thf(fact_930_diff__ge__0__iff__ge,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_931_diff__gt__0__iff__gt,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_932_le__add__diff__inverse2,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_933_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_934_le__add__diff__inverse,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_935_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_936_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_937_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_938_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_939_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_940_verit__minus__simplify_I3_J,axiom,
! [B2: int] :
( ( minus_minus_int @ zero_zero_int @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_941_distrib__right__numeral,axiom,
! [A2: int,B2: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_942_distrib__right__numeral,axiom,
! [A2: nat,B2: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_943_distrib__left__numeral,axiom,
! [V: num,B2: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_944_distrib__left__numeral,axiom,
! [V: num,B2: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_945_left__diff__distrib__numeral,axiom,
! [A2: int,B2: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_946_right__diff__distrib__numeral,axiom,
! [V: num,B2: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_947_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_948_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_949_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_950_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_951_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_952_uminus__add__conv__diff,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( minus_minus_int @ B2 @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_953_diff__minus__eq__add,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( plus_plus_int @ A2 @ B2 ) ) ).
% diff_minus_eq_add
thf(fact_954_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_955_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_956_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_neg_numeral
thf(fact_957_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_958_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_959_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_960_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_961_diff__eq__diff__less__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_962_diff__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_963_diff__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_964_diff__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_965_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_966_diff__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_967_diff__strict__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_968_diff__eq__diff__less,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_969_diff__strict__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_970_right__diff__distrib_H,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_971_left__diff__distrib_H,axiom,
! [B2: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_972_right__diff__distrib,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_973_left__diff__distrib,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% left_diff_distrib
thf(fact_974_diff__diff__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_975_diff__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_976_add__implies__diff,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_977_add__implies__diff,axiom,
! [C: int,B2: int,A2: int] :
( ( ( plus_plus_int @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_978_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_979_diff__add__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_980_diff__diff__eq2,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_981_add__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_982_eq__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( A2
= ( minus_minus_int @ C @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_983_diff__eq__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= C )
= ( A2
= ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_984_group__cancel_Osub1,axiom,
! [A: int,K: int,A2: int,B2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_985_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_986_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_987_minus__diff__commute,axiom,
! [B2: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_988_abs__minus__commute,axiom,
! [A2: int,B2: int] :
( ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).
% abs_minus_commute
thf(fact_989_set__plus__imp__minus,axiom,
! [A2: int,B2: int,C4: set_int] :
( ( member_int @ A2 @ ( set_elt_set_plus_int @ B2 @ C4 ) )
=> ( member_int @ ( minus_minus_int @ A2 @ B2 ) @ C4 ) ) ).
% set_plus_imp_minus
thf(fact_990_set__minus__imp__plus,axiom,
! [A2: int,B2: int,C4: set_int] :
( ( member_int @ ( minus_minus_int @ A2 @ B2 ) @ C4 )
=> ( member_int @ A2 @ ( set_elt_set_plus_int @ B2 @ C4 ) ) ) ).
% set_minus_imp_plus
thf(fact_991_set__minus__plus,axiom,
! [A2: int,B2: int,C4: set_int] :
( ( member_int @ ( minus_minus_int @ A2 @ B2 ) @ C4 )
= ( member_int @ A2 @ ( set_elt_set_plus_int @ B2 @ C4 ) ) ) ).
% set_minus_plus
thf(fact_992_diff__eq__diff__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_993_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_994_minus__diff__minus,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).
% minus_diff_minus
thf(fact_995_add__diff__add,axiom,
! [A2: int,C: int,B2: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_996_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_997_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_998_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_999_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1000_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1001_ge__iff__diff__ge__0,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_1002_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_1003_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1004_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_1005_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1006_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1007_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_1008_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1009_diff__le__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_1010_le__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_1011_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1012_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1013_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1014_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1017_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1018_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1019_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1020_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1021_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ 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_1022_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ 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_1023_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1024_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1025_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_1026_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_1027_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_1028_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1029_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1030_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ~ ( ord_less_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1031_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: int,B2: int] :
( ~ ( ord_less_int @ A2 @ B2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1032_less__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% less_diff_eq
thf(fact_1033_diff__less__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_less_eq
thf(fact_1034_mult__diff__mult,axiom,
! [X: int,Y: int,A2: int,B2: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A2 ) @ B2 ) ) ) ).
% mult_diff_mult
thf(fact_1035_eq__add__iff1,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1036_eq__add__iff2,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1037_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_1038_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_1039_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_1040_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_1041_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_1042_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_1043_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_1044_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1045_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1046_group__cancel_Osub2,axiom,
! [B: int,K: int,B2: int,A2: int] :
( ( B
= ( plus_plus_int @ K @ B2 ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.sub2
thf(fact_1047_abs__triangle__ineq2__sym,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1048_abs__triangle__ineq3,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).
% abs_triangle_ineq3
thf(fact_1049_abs__triangle__ineq2,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).
% abs_triangle_ineq2
thf(fact_1050_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_1051_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1052_ordered__ring__class_Ole__add__iff1,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1053_ordered__ring__class_Ole__add__iff2,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1054_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_1055_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1056_less__add__iff1,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1057_less__add__iff2,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1058_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_1059_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_1060_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_1061_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_1062_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1063_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_1064_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1065_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1066_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_1067_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_1068_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_1069_abs__triangle__ineq4,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_triangle_ineq4
thf(fact_1070_abs__diff__triangle__ineq,axiom,
! [A2: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1071_abs__diff__le__iff,axiom,
! [X: int,A2: int,R: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ R ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A2 @ R ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1072_abs__diff__less__iff,axiom,
! [X: int,A2: int,R: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R )
= ( ( ord_less_int @ ( minus_minus_int @ A2 @ R ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A2 @ R ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1073_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_1074_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_1075_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(169)
thf(fact_1076_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1077_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1078_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1079_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1080_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1081_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_1082_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_1083_numeral__times__minus__swap,axiom,
! [W: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_1084_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_1085_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_1086_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_1087_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1088_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1089_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1090_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1091_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_1092_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_1093_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1094_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1095_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1096_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1097_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1098_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1099_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_1100_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_1101_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_1102_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1103_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1104_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1105_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1106_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1107_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_1108_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus_num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% add_inc
thf(fact_1109_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1110_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_1111_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1112_mult__numeral__1,axiom,
! [A2: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_1113_mult__numeral__1,axiom,
! [A2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_1114_mult__numeral__1__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_1115_mult__numeral__1__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_1116_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_1117_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1118_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times_num @ X @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_1119_mult__1s__ring__1_I2_J,axiom,
! [B2: int] :
( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B2 ) ) ).
% mult_1s_ring_1(2)
thf(fact_1120_mult__1s__ring__1_I1_J,axiom,
! [B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% mult_1s_ring_1(1)
thf(fact_1121_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_1122_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_1123_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_1124_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_1125_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_1126_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_1127_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_1128_diff__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_1129_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_1130_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_1131_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_1132_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_1133_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_1134_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_1135_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_1136_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_1137_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_1138_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_1139_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_1140_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_1141_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K2: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_1142_sub__non__negative,axiom,
! [N: num,M: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% sub_non_negative
thf(fact_1143_sub__non__positive,axiom,
! [N: num,M: num] :
( ( ord_less_eq_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
= ( ord_less_eq_num @ N @ M ) ) ).
% sub_non_positive
thf(fact_1144_sub__negative,axiom,
! [N: num,M: num] :
( ( ord_less_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
= ( ord_less_num @ N @ M ) ) ).
% sub_negative
thf(fact_1145_sub__positive,axiom,
! [N: num,M: num] :
( ( ord_less_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
= ( ord_less_num @ M @ N ) ) ).
% sub_positive
thf(fact_1146_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_1147_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= one_one_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_1148_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1149_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1150_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1151_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_1152_not__iszero__1,axiom,
~ ( ring_1_iszero_int @ one_one_int ) ).
% not_iszero_1
thf(fact_1153_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1154_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1155_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1156_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1157_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1158_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1159_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1160_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1161_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1162_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1163_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1164_int__le__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1165_int__induct,axiom,
! [P: int > $o,K: int,I2: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_induct
thf(fact_1166_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1167_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1168_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1169_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_1170_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1171_eq__iff__iszero__diff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [X3: int,Y2: int] : ( ring_1_iszero_int @ ( minus_minus_int @ X3 @ Y2 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_1172_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_1173_iszero__0,axiom,
ring_1_iszero_int @ zero_zero_int ).
% iszero_0
thf(fact_1174_iszero__def,axiom,
( ring_1_iszero_int
= ( ^ [Z: int] : ( Z = zero_zero_int ) ) ) ).
% iszero_def
thf(fact_1175_eq__numeral__iff__iszero_I9_J,axiom,
! [X: num] :
( ( ( numeral_numeral_int @ X )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_1176_eq__numeral__iff__iszero_I10_J,axiom,
! [Y: num] :
( ( zero_zero_int
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_1177_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_iszero_neg_1
thf(fact_1178_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1179_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_zero_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1180_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_1181_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_1182_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y: num] :
( ( ( numeral_numeral_int @ X )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_1183_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_1184_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numeral_numeral_int @ X )
= one_one_int )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_1185_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_int
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_1186_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_1187_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_1188_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1189_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1190_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_1191_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_1192_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1193_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1194_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1195_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1196_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1197_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1198_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1199_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1200_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_1201_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_1202_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1203_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1204_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M @ 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 @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1205_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1206_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1207_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1208_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1209_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1210_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1211_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_1212_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_1213_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1214_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_1215_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_1216_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_1217_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1218_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_1219_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1220_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1221_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1222_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1223_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1224_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1225_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_1226_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_1227_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_1228_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_1229_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1230_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_1231_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_1232_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_1233_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1234_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1235_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1236_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1237_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_1238_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_1239_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_1240_sum__sqs__eq,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ X @ ( times_times_int @ Y @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( Y = X ) ) ).
% sum_sqs_eq
thf(fact_1241_left__add__twice,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_1242_left__add__twice,axiom,
! [A2: nat,B2: nat] :
( ( plus_plus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).
% left_add_twice
thf(fact_1243_mult__2__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1244_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1245_mult__2,axiom,
! [Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1246_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1247_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_1248_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1249_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1250_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_1251_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_1252_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1253_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1254_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_1255_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_1256_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_1257_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_1258_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_1259_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_1260_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_1261_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1262_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1263_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_1264_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_1265_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_1266_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_1267_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1268_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1269_nonzero__mult__div__cancel__right,axiom,
! [B2: nat,A2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1270_nonzero__mult__div__cancel__right,axiom,
! [B2: int,A2: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1271_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_1272_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_1273_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_1274_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
% 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,
! [P: $o] :
( ( P = $true )
| ( P = $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,
up_ray_a @ ( uminus_uminus_set_a @ i ) ).
%------------------------------------------------------------------------------