TPTP Problem File: SLH0288^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Digit_Expansions/0001_Carries/prob_00256_010697__5676134_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1319 ( 700 unt; 51 typ; 0 def)
% Number of atoms : 3144 (1177 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9841 ( 320 ~; 55 |; 86 &;8174 @)
% ( 0 <=>;1206 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 212 ( 212 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 48 usr; 11 con; 0-3 aty)
% Number of variables : 2954 ( 142 ^;2794 !; 18 ?;2954 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:21:10.603
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
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 ).
% Explicit typings (48)
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(sy_c_Bits__Digits_Onth__bit,type,
bits_nth_bit: nat > nat > nat ).
thf(sy_c_Carries_Obin__carry,type,
bin_carry: nat > nat > nat > nat ).
thf(sy_c_Carries_Obin__narry,type,
bin_narry: nat > nat > nat > nat ).
thf(sy_c_Carries_Obin__narry2,type,
bin_narry2: nat > nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > 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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > 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_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__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_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_v_a,type,
a: nat ).
thf(sy_v_b,type,
b: nat ).
thf(sy_v_k____,type,
k: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1264)
thf(fact_0_bin__narry__equiv,axiom,
bin_narry = bin_narry2 ).
% bin_narry_equiv
thf(fact_1_carry__bounded,axiom,
! [A: nat,B: nat,K: nat] : ( ord_less_eq_nat @ ( bin_carry @ A @ B @ K ) @ one_one_nat ) ).
% carry_bounded
thf(fact_2__092_060open_062bin__carry_Aa_Ab_A_Ik_A_N_A1_J_A_061_A0_092_060close_062,axiom,
( ( bin_carry @ a @ b @ ( minus_minus_nat @ k @ one_one_nat ) )
= zero_zero_nat ) ).
% \<open>bin_carry a b (k - 1) = 0\<close>
thf(fact_3__092_060open_0620_A_061_Abin__carry_Aa_Ab_Ak_092_060close_062,axiom,
( zero_zero_nat
= ( bin_carry @ a @ b @ k ) ) ).
% \<open>0 = bin_carry a b k\<close>
thf(fact_4_c1,axiom,
( one_one_nat
= ( bin_carry @ a @ b @ k ) ) ).
% c1
thf(fact_5_Carries_Orev__induct,axiom,
! [I: nat,K: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) )
=> ( P @ I ) ) ) ) ).
% Carries.rev_induct
thf(fact_6_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_7_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_8_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_9_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_10_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_11_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_12_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_13_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_14_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_15_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_16_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_17_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_18_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_19_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_20_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_21_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_22_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_23_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_24_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_25_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_26_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_27_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_28_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_29_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_30_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_31_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_32_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_33_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_34_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_35_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_36_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_37_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_38_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_39_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_40_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_41_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_42_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_43_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_44_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_45_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_46_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_47_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_48_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_49_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_50_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_51_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_52_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_53_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_54_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_55_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_56_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_57_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_58_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_59_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_60_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_61_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_62_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_63_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_64_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_65_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_66_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_67_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_70_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_71_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_72_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_73_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_74_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_75_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_76_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_77_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_78_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_79_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_80_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_81_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_82_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_83_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_84_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_85_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_86_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_87_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_88_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_89_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_90_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_91_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_92_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_93_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_94_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_95_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_96_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_97_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_98_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_99_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_100_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_101_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_102_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_103_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_104_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_105_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_106_le__cases3,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_107_le__cases3,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ( ord_less_eq_num @ X @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X )
=> ~ ( ord_less_eq_num @ X @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X )
=> ~ ( ord_less_eq_num @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_108_le__cases3,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_109_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_110_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [X3: num,Y5: num] :
( ( ord_less_eq_num @ X3 @ Y5 )
& ( ord_less_eq_num @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_111_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_112_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_113_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_114_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_115_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_116_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_117_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_118_order__antisym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_119_order__antisym,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_120_order__antisym,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_121_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_122_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_123_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_124_order__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_125_order__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_126_order__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_127_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_128_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: num,B3: num] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_129_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_130_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_131_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_132_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_133_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_134_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_135_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_136_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_137_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_138_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_139_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_140_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_141_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_142_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_143_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_144_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_145_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_150_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_151_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_152_order__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_153_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_154_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_159_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_160_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_161_order__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_162_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_163_order__eq__refl,axiom,
! [X: nat,Y4: nat] :
( ( X = Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_164_order__eq__refl,axiom,
! [X: num,Y4: num] :
( ( X = Y4 )
=> ( ord_less_eq_num @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_165_order__eq__refl,axiom,
! [X: int,Y4: int] :
( ( X = Y4 )
=> ( ord_less_eq_int @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_166_linorder__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_167_linorder__linear,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
| ( ord_less_eq_num @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_168_linorder__linear,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
| ( ord_less_eq_int @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_169_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_170_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_171_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_172_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_173_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_174_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_175_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_176_ord__eq__le__subst,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_177_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_178_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_179_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_180_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_181_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_182_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_183_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_184_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_185_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_186_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_187_linorder__le__cases,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_188_linorder__le__cases,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_189_linorder__le__cases,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_190_order__antisym__conv,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ( ( ord_less_eq_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_191_order__antisym__conv,axiom,
! [Y4: num,X: num] :
( ( ord_less_eq_num @ Y4 @ X )
=> ( ( ord_less_eq_num @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_192_order__antisym__conv,axiom,
! [Y4: int,X: int] :
( ( ord_less_eq_int @ Y4 @ X )
=> ( ( ord_less_eq_int @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_193_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_194_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_195_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_196_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_197_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_198_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_199_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_200_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_201_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_202_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_203_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_204_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_205_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_206_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_207_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_208_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_209_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_210_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_211_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_212_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_213_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_214_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_215_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_216_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_217_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_226_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_227_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_228_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_229_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_230_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_231_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_232_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_233_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_234_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_235_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_236_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_237_eq__numeral__iff__iszero_I12_J,axiom,
! [Y4: num] :
( ( zero_zero_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y4 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y4 ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_238_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_239_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_240_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_241_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_242_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_243_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_244_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_245_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X2: num] :
( ( P @ X2 )
=> ( P @ ( inc @ X2 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_256_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_257_sub__inc__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( inc @ N ) @ one )
= ( numeral_numeral_int @ N ) ) ).
% sub_inc_One_eq
thf(fact_258_not__iszero__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ one ) ) ).
% not_iszero_Numeral1
thf(fact_259_eq__numeral__iff__iszero_I1_J,axiom,
! [X: num,Y4: num] :
( ( ( numeral_numeral_int @ X )
= ( numeral_numeral_int @ Y4 ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ Y4 ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_260_eq__numeral__iff__iszero_I6_J,axiom,
! [Y4: num] :
( ( one_one_int
= ( numeral_numeral_int @ Y4 ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y4 ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_261_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_262_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_263_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y4: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y4 ) ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y4 @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_264_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_265_iszero__0,axiom,
ring_1_iszero_int @ zero_zero_int ).
% iszero_0
thf(fact_266_iszero__def,axiom,
( ring_1_iszero_int
= ( ^ [Z3: int] : ( Z3 = zero_zero_int ) ) ) ).
% iszero_def
thf(fact_267_not__iszero__numeral,axiom,
! [W: num] :
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ).
% not_iszero_numeral
thf(fact_268_not__iszero__1,axiom,
~ ( ring_1_iszero_int @ one_one_int ) ).
% not_iszero_1
thf(fact_269_eq__iff__iszero__diff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y5: int] : ( ring_1_iszero_int @ ( minus_minus_int @ X3 @ Y5 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_270_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_271_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_272_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_273_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_274_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_275_eq__numeral__iff__iszero_I10_J,axiom,
! [Y4: num] :
( ( zero_zero_int
= ( numeral_numeral_int @ Y4 ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y4 ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_276_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_277_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_iszero_neg_1
thf(fact_278_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_279_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_288_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_289_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_290_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_291_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_292_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_293_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_294_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_295_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_296_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_297_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_298_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_299_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_300_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_301_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_302_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_303_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y4: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y4 ) )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_304_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y4: nat] :
( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_305_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_306_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_307_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_308_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_309_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_310_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_311_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_312_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_313_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_314_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_315_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_316_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_317_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_318_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_319_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_320_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_321_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_322_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_323_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_324_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_325_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_326_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_327_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_328_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_329_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_330_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_331_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_332_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_333_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_334_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_335_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_336_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_337_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_338_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_339_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_340_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_341_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_342_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_343_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_344_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_345_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_346_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_347_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_348_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_349_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_350_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_351_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_352_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_353_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_354_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_355_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_356_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_357_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_358_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_359_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_360_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y4: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% semiring_norm(167)
thf(fact_361_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_362_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_363_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_364_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_365_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_366_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_367_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_368_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_369_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_370_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_371_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_372_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_373_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_374_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_375_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_376_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_377_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_378_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_379_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_380_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_381_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_382_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y4: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y4 ) ) ).
% semiring_norm(165)
thf(fact_383_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y4: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y4 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y4 ) ) ).
% semiring_norm(166)
thf(fact_384_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_385_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_386_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_387_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_388_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_389_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_390_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_391_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_392_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_393_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_394_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_395_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_396_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_397_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_398_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_399_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_400_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_401_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_402_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_403_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_404_group__cancel_Oadd1,axiom,
! [A4: int,K: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A4 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_405_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_406_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_407_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_408_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_409_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_410_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_411_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_412_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_413_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_414_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_415_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_416_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_417_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_418_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_419_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_420_num_Oexhaust,axiom,
! [Y4: num] :
( ( Y4 != one )
=> ( ! [X23: num] :
( Y4
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y4
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_421_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_422_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_423_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_424_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_425_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_426_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_427_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_428_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_429_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_430_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_431_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_432_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_433_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_434_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_435_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_436_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_437_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_438_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_439_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_440_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_441_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_442_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_443_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_444_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_445_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_446_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_447_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_448_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_449_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_450_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_451_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_452_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_453_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_454_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_455_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_456_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_457_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_458_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_459_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_460_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_461_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_462_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_463_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_464_group__cancel_Osub1,axiom,
! [A4: int,K: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A4 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_465_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_466_group__cancel_Oneg1,axiom,
! [A4: int,K: int,A: int] :
( ( A4
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_467_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_468_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_469_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_470_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_471_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_472_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_473_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_474_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_475_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_476_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_477_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_478_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_479_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_480_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_481_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_482_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_483_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_484_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_485_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_486_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_487_add__inc,axiom,
! [X: num,Y4: num] :
( ( plus_plus_num @ X @ ( inc @ Y4 ) )
= ( inc @ ( plus_plus_num @ X @ Y4 ) ) ) ).
% add_inc
thf(fact_488_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_489_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_490_add__nonpos__eq__0__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y4 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_491_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_492_add__nonneg__eq__0__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ( plus_plus_int @ X @ Y4 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_493_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_494_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_495_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_496_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_497_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_498_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_499_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_500_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_501_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_502_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_503_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_504_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_505_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_506_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_507_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_508_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_509_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_510_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_511_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_512_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_513_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_514_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_515_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_516_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_517_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_518_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_519_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_520_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_521_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_522_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_523_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_524_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_525_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_526_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_527_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_528_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_529_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_530_group__cancel_Osub2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_531_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_532_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_533_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_534_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_535_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_536_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_537_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_538_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_539_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_540_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_541_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_542_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y4: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ Y4 ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y4 ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_543_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y4: num] :
( ( ( numeral_numeral_int @ X )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y4 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y4 ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_544_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_545_eq__numeral__iff__iszero_I8_J,axiom,
! [Y4: num] :
( ( one_one_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y4 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y4 ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_546_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_547_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_548_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_549_k1,axiom,
( ( bits_nth_bit @ ( plus_plus_nat @ a @ b ) @ k )
!= ( modulo_modulo_nat @ ( plus_plus_nat @ ( bits_nth_bit @ a @ k ) @ ( bits_nth_bit @ b @ k ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% k1
thf(fact_550_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_551_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_552_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_553__092_060open_062a_A_092_060exclamdown_062_A_Ik_A_N_A1_J_A_L_Ab_A_092_060exclamdown_062_A_Ik_A_N_A1_J_A_060_A1_092_060close_062,axiom,
ord_less_nat @ ( plus_plus_nat @ ( bits_nth_bit @ a @ ( minus_minus_nat @ k @ one_one_nat ) ) @ ( bits_nth_bit @ b @ ( minus_minus_nat @ k @ one_one_nat ) ) ) @ one_one_nat ).
% \<open>a \<exclamdown> (k - 1) + b \<exclamdown> (k - 1) < 1\<close>
thf(fact_554_k2,axiom,
! [R: nat] :
( ( ord_less_nat @ R @ k )
=> ( ( bits_nth_bit @ ( plus_plus_nat @ a @ b ) @ R )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( bits_nth_bit @ a @ R ) @ ( bits_nth_bit @ b @ R ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% k2
thf(fact_555_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_556_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_557_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_558_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_559_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_560_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_561_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_562_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_563_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_564_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_565_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_566_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_567_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_568_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_569_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_570_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_571_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_572_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_573_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_574_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_575_p,axiom,
! [R: nat] :
( ( ord_less_nat @ R @ n )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( bits_nth_bit @ a @ R ) @ ( bits_nth_bit @ b @ R ) ) @ one_one_nat ) ) ).
% p
thf(fact_576_nq,axiom,
( ( bits_nth_bit @ ( plus_plus_nat @ a @ b ) @ n )
!= ( modulo_modulo_nat @ ( plus_plus_nat @ ( bits_nth_bit @ a @ n ) @ ( bits_nth_bit @ b @ n ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nq
thf(fact_577__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062k_O_A_092_060lbrakk_062_Ia_A_L_Ab_J_A_092_060exclamdown_062_Ak_A_092_060noteq_062_A_Ia_A_092_060exclamdown_062_Ak_A_L_Ab_A_092_060exclamdown_062_Ak_J_Amod_A2_059_A_092_060forall_062r_060k_O_A_Ia_A_L_Ab_J_A_092_060exclamdown_062_Ar_A_061_A_Ia_A_092_060exclamdown_062_Ar_A_L_Ab_A_092_060exclamdown_062_Ar_J_Amod_A2_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [K3: nat] :
( ( ( bits_nth_bit @ ( plus_plus_nat @ a @ b ) @ K3 )
!= ( modulo_modulo_nat @ ( plus_plus_nat @ ( bits_nth_bit @ a @ K3 ) @ ( bits_nth_bit @ b @ K3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ~ ! [R: nat] :
( ( ord_less_nat @ R @ K3 )
=> ( ( bits_nth_bit @ ( plus_plus_nat @ a @ b ) @ R )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( bits_nth_bit @ a @ R ) @ ( bits_nth_bit @ b @ R ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>k. \<lbrakk>(a + b) \<exclamdown> k \<noteq> (a \<exclamdown> k + b \<exclamdown> k) mod 2; \<forall>r<k. (a + b) \<exclamdown> r = (a \<exclamdown> r + b \<exclamdown> r) mod 2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_578_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_579_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_580_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_581_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_582_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_583_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_584_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_585_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_586_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_587_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_588_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_589_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_590_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_591_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_592_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_593_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_594_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_595_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_596_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_597_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_598_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_599_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_600_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_601_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_602_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_603_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_604_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_605_one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_mod_two_eq_one
thf(fact_606_one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_mod_two_eq_one
thf(fact_607_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% bits_one_mod_two_eq_one
thf(fact_608_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_one_mod_two_eq_one
thf(fact_609_sub__num__simps_I4_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% sub_num_simps(4)
thf(fact_610_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_611_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_612_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_613_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_614_minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% minus_1_mod_2_eq
thf(fact_615_bits__minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_minus_1_mod_2_eq
thf(fact_616_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_617_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_618_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_619_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_620_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_621_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_622_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_623_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_624_linorder__neqE__nat,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_625_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_626_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_627_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_628_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_629_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_630_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_631_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_632_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_633_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_634_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_635_order__less__imp__not__less,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_636_order__less__imp__not__less,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_637_order__less__imp__not__less,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_638_order__less__imp__not__eq2,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_639_order__less__imp__not__eq2,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_640_order__less__imp__not__eq2,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_641_order__less__imp__not__eq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_642_order__less__imp__not__eq,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_643_order__less__imp__not__eq,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_644_linorder__less__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_645_linorder__less__linear,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_num @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_646_linorder__less__linear,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_int @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_647_order__less__imp__triv,axiom,
! [X: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_648_order__less__imp__triv,axiom,
! [X: num,Y4: num,P: $o] :
( ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_num @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_649_order__less__imp__triv,axiom,
! [X: int,Y4: int,P: $o] :
( ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_int @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_650_order__less__not__sym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_651_order__less__not__sym,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_652_order__less__not__sym,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_653_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_654_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_655_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_656_order__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_657_order__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_658_order__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_659_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_660_order__less__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_661_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_662_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_663_order__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_664_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_665_order__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_666_order__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_667_order__less__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_668_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_669_order__less__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_670_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_671_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_672_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_673_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_674_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_675_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_676_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_677_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_678_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_679_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_680_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_681_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_682_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_683_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_684_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_685_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_686_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_687_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_688_ord__eq__less__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_689_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_690_ord__eq__less__subst,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_691_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_692_order__less__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_693_order__less__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_694_order__less__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_695_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_696_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_697_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_698_linorder__neq__iff,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
= ( ( ord_less_nat @ X @ Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_699_linorder__neq__iff,axiom,
! [X: num,Y4: num] :
( ( X != Y4 )
= ( ( ord_less_num @ X @ Y4 )
| ( ord_less_num @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_700_linorder__neq__iff,axiom,
! [X: int,Y4: int] :
( ( X != Y4 )
= ( ( ord_less_int @ X @ Y4 )
| ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_701_order__less__asym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_702_order__less__asym,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_703_order__less__asym,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_704_linorder__neqE,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_705_linorder__neqE,axiom,
! [X: num,Y4: num] :
( ( X != Y4 )
=> ( ~ ( ord_less_num @ X @ Y4 )
=> ( ord_less_num @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_706_linorder__neqE,axiom,
! [X: int,Y4: int] :
( ( X != Y4 )
=> ( ~ ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_707_linorder__neqE__linordered__idom,axiom,
! [X: int,Y4: int] :
( ( X != Y4 )
=> ( ~ ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_708_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_709_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_710_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_711_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_712_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_713_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_714_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_715_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_716_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_717_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_718_not__less__iff__gr__or__eq,axiom,
! [X: num,Y4: num] :
( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( ( ord_less_num @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_719_not__less__iff__gr__or__eq,axiom,
! [X: int,Y4: int] :
( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( ( ord_less_int @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_720_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_721_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_722_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_723_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_724_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: num] : ( P @ A3 @ A3 )
=> ( ! [A3: num,B3: num] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_725_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_726_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_727_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_728_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_729_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_730_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_731_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_732_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_733_linorder__cases,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_734_linorder__cases,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_num @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_num @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_735_linorder__cases,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_int @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_736_antisym__conv3,axiom,
! [Y4: nat,X: nat] :
( ~ ( ord_less_nat @ Y4 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_737_antisym__conv3,axiom,
! [Y4: num,X: num] :
( ~ ( ord_less_num @ Y4 @ X )
=> ( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_738_antisym__conv3,axiom,
! [Y4: int,X: int] :
( ~ ( ord_less_int @ Y4 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_739_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X2 )
=> ( P @ Y2 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_740_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_741_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_742_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_743_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_744_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_745_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_746_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_747_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_748_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_749_less__imp__neq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_750_less__imp__neq,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_751_less__imp__neq,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_752_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_753_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_754_lt__ex,axiom,
! [X: int] :
? [Y: int] : ( ord_less_int @ Y @ X ) ).
% lt_ex
thf(fact_755_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_756_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_757_leD,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ~ ( ord_less_nat @ X @ Y4 ) ) ).
% leD
thf(fact_758_leD,axiom,
! [Y4: num,X: num] :
( ( ord_less_eq_num @ Y4 @ X )
=> ~ ( ord_less_num @ X @ Y4 ) ) ).
% leD
thf(fact_759_leD,axiom,
! [Y4: int,X: int] :
( ( ord_less_eq_int @ Y4 @ X )
=> ~ ( ord_less_int @ X @ Y4 ) ) ).
% leD
thf(fact_760_leI,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ).
% leI
thf(fact_761_leI,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_num @ X @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X ) ) ).
% leI
thf(fact_762_leI,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_int @ X @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X ) ) ).
% leI
thf(fact_763_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_764_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_765_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_766_antisym__conv1,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_767_antisym__conv1,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_768_antisym__conv1,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_769_antisym__conv2,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_770_antisym__conv2,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_771_antisym__conv2,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_772_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_773_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y5: num] :
( ( ord_less_eq_num @ X3 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_774_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_775_not__le__imp__less,axiom,
! [Y4: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X )
=> ( ord_less_nat @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_776_not__le__imp__less,axiom,
! [Y4: num,X: num] :
( ~ ( ord_less_eq_num @ Y4 @ X )
=> ( ord_less_num @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_777_not__le__imp__less,axiom,
! [Y4: int,X: int] :
( ~ ( ord_less_eq_int @ Y4 @ X )
=> ( ord_less_int @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_778_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_779_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_780_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_781_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_782_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_783_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_784_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_785_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_786_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_787_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_788_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_789_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_790_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_791_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_792_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_793_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_794_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_795_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_796_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_797_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_798_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_799_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_800_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_801_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_802_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_803_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_804_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_805_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_806_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_807_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_808_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_809_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_810_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_811_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_812_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_813_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_814_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_815_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y5: num] :
( ( ord_less_num @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_816_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_817_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_818_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y5: num] :
( ( ord_less_eq_num @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_819_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_820_linorder__not__le,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y4 ) )
= ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_821_linorder__not__le,axiom,
! [X: num,Y4: num] :
( ( ~ ( ord_less_eq_num @ X @ Y4 ) )
= ( ord_less_num @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_822_linorder__not__le,axiom,
! [X: int,Y4: int] :
( ( ~ ( ord_less_eq_int @ X @ Y4 ) )
= ( ord_less_int @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_823_linorder__not__less,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_824_linorder__not__less,axiom,
! [X: num,Y4: num] :
( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( ord_less_eq_num @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_825_linorder__not__less,axiom,
! [X: int,Y4: int] :
( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( ord_less_eq_int @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_826_order__less__imp__le,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_827_order__less__imp__le,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_eq_num @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_828_order__less__imp__le,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_eq_int @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_829_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_830_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_831_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_832_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_833_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_834_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_835_order__le__less__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_836_order__le__less__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_837_order__le__less__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_838_order__less__le__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_839_order__less__le__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_840_order__less__le__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_841_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_842_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_843_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_844_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_845_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_846_order__le__less__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_847_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_848_order__le__less__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_849_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_850_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_851_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_852_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_853_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_854_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_855_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_856_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_857_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_858_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_859_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_860_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_861_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_862_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_863_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_864_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_865_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_866_order__less__le__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_867_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_868_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_869_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_870_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_871_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_872_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_873_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_874_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_875_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_876_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_877_linorder__le__less__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_878_linorder__le__less__linear,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
| ( ord_less_num @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_879_linorder__le__less__linear,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
| ( ord_less_int @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_880_order__le__imp__less__or__eq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_881_order__le__imp__less__or__eq,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_num @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_882_order__le__imp__less__or__eq,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_int @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_883_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_884_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_885_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_886_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_887_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_888_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_889_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_890_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_891_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_892_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_893_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_894_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_895_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_896_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_897_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_898_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_899_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_900_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_901_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_902_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_903_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_904_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_905_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_906_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_907_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_908_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_909_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_910_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_911_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_912_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_913_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_914_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_915_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_916_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_917_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_918_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_919_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_920_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_921_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_922_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_923_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_924_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_925_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_926_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_927_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_928_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_929_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_930_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_931_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_932_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_933_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_934_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_935_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_936_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_937_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_938_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_939_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_940_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_941_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_942_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_943_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_944_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_945_cong__exp__iff__simps_I6_J,axiom,
! [Q: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_946_cong__exp__iff__simps_I6_J,axiom,
! [Q: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_947_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_948_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_949_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_950_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_951_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_952_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_953_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_954_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_955_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_956_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_957_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_958_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_959_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_960_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_961_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_962_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_963_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_964_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_965_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_966_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_967_add__less__zeroD,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y4 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y4 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_968_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_969_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_970_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_971_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_972_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_973_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_974_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_975_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_976_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_977_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_978_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_979_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_980_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_981_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_982_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_983_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_984_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_985_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_986_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_987_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_988_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_989_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_990_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_991_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_992_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_993_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_994_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_995_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_996_sum__digit__formula,axiom,
! [A: nat,B: nat,K: nat] :
( ( bits_nth_bit @ ( plus_plus_nat @ A @ B ) @ K )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( bits_nth_bit @ A @ K ) @ ( bits_nth_bit @ B @ K ) ) @ ( bin_carry @ A @ B @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% sum_digit_formula
thf(fact_997_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_998_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_999_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1000_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1001_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1002_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1003_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1004_dif__narry__formula,axiom,
! [B: nat,A: nat,K: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ( ord_less_nat @ ( bits_nth_bit @ A @ K ) @ ( plus_plus_nat @ ( bits_nth_bit @ B @ K ) @ ( bin_narry @ A @ B @ K ) ) )
=> ( ( bin_narry @ A @ B @ ( plus_plus_nat @ K @ one_one_nat ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( bits_nth_bit @ A @ K ) @ ( plus_plus_nat @ ( bits_nth_bit @ B @ K ) @ ( bin_narry @ A @ B @ K ) ) )
=> ( ( bin_narry @ A @ B @ ( plus_plus_nat @ K @ one_one_nat ) )
= zero_zero_nat ) ) ) ) ).
% dif_narry_formula
thf(fact_1005_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_1006_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_1007_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_1008_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_1009_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_1010_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_1011_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1012_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1013_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1014_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1015_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1016_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1017_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1018_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1019_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1020_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1021_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1022_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1023_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1024_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_dif__digit__formula,axiom,
! [B: nat,A: nat,K: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( bits_nth_bit @ ( minus_minus_nat @ A @ B ) @ K )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( bits_nth_bit @ A @ K ) @ ( bits_nth_bit @ B @ K ) ) @ ( bin_narry @ A @ B @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% dif_digit_formula
thf(fact_1035_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_1036_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_1037_cong__exp__iff__simps_I7_J,axiom,
! [Q: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_1038_cong__exp__iff__simps_I7_J,axiom,
! [Q: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_1039_bin__carry__bounded,axiom,
( bin_carry
= ( ^ [A2: nat,B2: nat,K2: nat] : ( modulo_modulo_nat @ ( bin_carry @ A2 @ B2 @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% bin_carry_bounded
thf(fact_1040_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_1041_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_1042_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_1043_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_1044_minus__mod__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_1045_mod__minus__minus,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_1046_minus__mod__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mod_self2
thf(fact_1047_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1048_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1049_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1050_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1051_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1052_mod__diff__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_1053_mod__diff__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_1054_mod__diff__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_1055_mod__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_eq
thf(fact_1056_mod__minus__right,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_1057_mod__minus__cong,axiom,
! [A: int,B: int,A5: int] :
( ( ( modulo_modulo_int @ A @ B )
= ( modulo_modulo_int @ A5 @ B ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_1058_mod__minus__eq,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_1059_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1060_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1061_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_1062_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_1063_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1064_aux0__digit__wise__equiv,axiom,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ? [K3: nat] :
( ( bits_nth_bit @ A @ K3 )
= one_one_nat ) ) ).
% aux0_digit_wise_equiv
thf(fact_1065_nth__bit__bounded,axiom,
! [A: nat,K: nat] : ( ord_less_eq_nat @ ( bits_nth_bit @ A @ K ) @ one_one_nat ) ).
% nth_bit_bounded
thf(fact_1066_obtain__smallest,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ( P @ K3 )
& ! [A6: nat] :
( ( ord_less_nat @ A6 @ K3 )
=> ~ ( P @ A6 ) ) ) ) ).
% obtain_smallest
thf(fact_1067_aux1__digit__wise__equiv,axiom,
! [A: nat] :
( ( ! [K2: nat] :
( ( bits_nth_bit @ A @ K2 )
= zero_zero_nat ) )
= ( A = zero_zero_nat ) ) ).
% aux1_digit_wise_equiv
thf(fact_1068_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
=> ( P @ M4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1069_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1070_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1071_signed__take__bit__0,axiom,
! [A: int] :
( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_1072_bin__narry__def,axiom,
( bin_narry
= ( ^ [A2: nat,B2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( modulo_modulo_nat @ B2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).
% bin_narry_def
thf(fact_1073_sum__carry__formula,axiom,
! [A: nat,B: nat,K: nat] :
( ( bin_carry @ A @ B @ ( plus_plus_nat @ K @ one_one_nat ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( bits_nth_bit @ A @ K ) @ ( bits_nth_bit @ B @ K ) ) @ ( bin_carry @ A @ B @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% sum_carry_formula
thf(fact_1074_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1075_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1076_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1077_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1078_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1079_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1080_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1081_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1082_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1083_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_1084_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1085_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1086_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_1087_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1088_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_1089_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1090_signed__take__bit__of__0,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% signed_take_bit_of_0
thf(fact_1091_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1092_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1093_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1094_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1095_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_1096_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_1097_bits__mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_1098_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_1099_mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_1100_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_1101_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_1102_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1103_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1104_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% signed_take_bit_of_minus_1
thf(fact_1105_signed__take__bit__numeral__of__1,axiom,
! [K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_numeral_of_1
thf(fact_1106_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y4: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y4 ) )
= ( ord_less_nat @ X @ Y4 ) ) ) ).
% power_strict_increasing_iff
thf(fact_1107_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y4: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y4 ) )
= ( ord_less_nat @ X @ Y4 ) ) ) ).
% power_strict_increasing_iff
thf(fact_1108_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1109_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1110_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1111_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1112_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1113_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1114_power__increasing__iff,axiom,
! [B: nat,X: nat,Y4: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power_increasing_iff
thf(fact_1115_power__increasing__iff,axiom,
! [B: int,X: nat,Y4: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power_increasing_iff
thf(fact_1116_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_1117_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_1118_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_1119_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1120_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1121_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_1122_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_1123_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_1124_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_1125_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_1126_power2__eq__iff__nonneg,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y4 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1127_power2__eq__iff__nonneg,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y4 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1128_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_1129_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_1130_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_1131_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1132_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1133_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_1134_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_1135_power__diff,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_1136_power__diff,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_1137_div__exp__eq,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_1138_div__exp__eq,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_1139_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1140_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1141_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1142_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1143_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1144_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1145_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1146_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1147_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1148_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1149_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1150_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1151_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1152_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= A )
= ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1153_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1154_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1155_div__exp__mod__exp__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_1156_div__exp__mod__exp__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_1157_nth__bit__def,axiom,
( bits_nth_bit
= ( ^ [Num: nat,K2: nat] : ( modulo_modulo_nat @ ( divide_divide_nat @ Num @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% nth_bit_def
thf(fact_1158_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1159_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1160_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1161_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1162_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1163_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1164_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1165_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_1166_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1167_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_1168_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1169_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1170_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1171_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1172_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1173_power__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1174_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1175_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1176_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_1177_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_1178_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1179_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1180_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1181_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1182_bin__carry__def,axiom,
( bin_carry
= ( ^ [A2: nat,B2: nat,K2: nat] : ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( modulo_modulo_nat @ B2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% bin_carry_def
thf(fact_1183_div__less__mono,axiom,
! [A4: nat,B4: nat,N: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A4 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B4 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1184_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1185_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1186_power__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1187_power__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1188_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1189_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1190_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1191_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1192_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1193_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1194_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1195_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1196_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1197_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1198_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_1199_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_1200_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_1201_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_1202_power2__commute,axiom,
! [X: int,Y4: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1203_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_1204_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_1205_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_1206_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_1207_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_1208_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_1209_div__sub,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) )
=> ( ( divide_divide_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( minus_minus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ one_one_nat ) ) )
& ( ~ ( ord_less_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) )
=> ( ( divide_divide_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_sub
thf(fact_1210_bin__narry2__def,axiom,
( bin_narry2
= ( ^ [A2: nat,B2: nat,K2: nat] : ( modulo_modulo_nat @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) ) @ ( modulo_modulo_nat @ B2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% bin_narry2_def
thf(fact_1211_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1212_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1213_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_1214_power2__eq__imp__eq,axiom,
! [X: nat,Y4: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( X = Y4 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1215_power2__eq__imp__eq,axiom,
! [X: int,Y4: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( X = Y4 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1216_power2__le__imp__le,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power2_le_imp_le
thf(fact_1217_power2__le__imp__le,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% power2_le_imp_le
thf(fact_1218_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_1219_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_1220_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_1221_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1222_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1223_sum__power2__eq__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1224_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1225_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1226_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_1227_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1228_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1229_power2__less__imp__less,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ord_less_nat @ X @ Y4 ) ) ) ).
% power2_less_imp_less
thf(fact_1230_power2__less__imp__less,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ord_less_int @ X @ Y4 ) ) ) ).
% power2_less_imp_less
thf(fact_1231_sum__power2__ge__zero,axiom,
! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1232_sum__power2__le__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1233_not__sum__power2__lt__zero,axiom,
! [X: int,Y4: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_1234_sum__power2__gt__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y4 != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1235_square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
=> ( ( ord_less_eq_int @ X @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_1236_bits__stable__imp__add__self,axiom,
! [A: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_nat ) ) ).
% bits_stable_imp_add_self
thf(fact_1237_bits__stable__imp__add__self,axiom,
! [A: int] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= zero_zero_int ) ) ).
% bits_stable_imp_add_self
thf(fact_1238_verit__le__mono__div,axiom,
! [A4: nat,B4: nat,N: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B4 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B4 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1239_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1240_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1241_aux2__digit__wise__equiv,axiom,
! [K: nat,A: nat] :
( ! [R2: nat] :
( ( ord_less_nat @ R2 @ K )
=> ( ( bits_nth_bit @ A @ R2 )
= zero_zero_nat ) )
=> ( ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
= zero_zero_nat ) ) ).
% aux2_digit_wise_equiv
thf(fact_1242_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1243_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1244_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1245_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1246_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1247_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1248_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1249_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1250_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1251_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1252_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1253_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1254_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1255_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_1256_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1257_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1258_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1259_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1260_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1261_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_1262_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1263_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
% Helper facts (3)
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,Y4: nat] :
( ( if_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y4: nat] :
( ( if_nat @ $true @ X @ Y4 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------