TPTP Problem File: SLH0435^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Number_Theoretic_Transform/0006_Preliminary_Lemmas/prob_00256_009467__14065170_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1343 ( 636 unt; 65 typ; 0 def)
% Number of atoms : 3559 (1377 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10435 ( 256 ~; 103 |; 173 &;8633 @)
% ( 0 <=>;1270 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 202 ( 202 >; 0 *; 0 +; 0 <<)
% Number of symbols : 64 ( 61 usr; 14 con; 0-3 aty)
% Number of variables : 3249 ( 130 ^;3012 !; 107 ?;3249 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:37:55.193
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (61)
thf(sy_c_Cong_Ounique__euclidean__semiring__class_Ocong_001t__Finite____Field__Omod____ring_Itf__a_J,type,
unique9076693328225066129ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a > $o ).
thf(sy_c_Cong_Ounique__euclidean__semiring__class_Ocong_001t__Int__Oint,type,
unique651150874487253600ng_int: int > int > int > $o ).
thf(sy_c_Cong_Ounique__euclidean__semiring__class_Ocong_001t__Nat__Onat,type,
unique653641344996303876ng_nat: nat > nat > nat > $o ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Finite____Field__Omod____ring_Itf__a_J,type,
factor4631116012818856269ring_a: finite_mod_ring_a > $o ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Int__Oint,type,
factor1798656936486255268me_int: int > $o ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Nat__Onat,type,
factor1801147406995305544me_nat: nat > $o ).
thf(sy_c_Finite__Field_Oof__int__mod__ring_001tf__a,type,
finite8272632373135393572ring_a: int > finite_mod_ring_a ).
thf(sy_c_Finite__Field_Oto__int__mod__ring_001tf__a,type,
finite1095367895020317408ring_a: finite_mod_ring_a > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
minus_3609261664126569004ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
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__Finite____Field__Omod____ring_Itf__a_J,type,
one_on2109788427901206336ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
plus_p6165643967897163644ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
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_Otimes__class_Otimes_001t__Finite____Field__Omod____ring_Itf__a_J,type,
times_5121417576591743744ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
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_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zero_z7902377541816115708ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Finite____Field__Omod____ring_Itf__a_J,type,
if_Finite_mod_ring_a: $o > finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
semiri9180929696517417892ring_a: nat > finite_mod_ring_a ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Finite____Field__Omod____ring_Itf__a_J,type,
neg_nu5901776551076858996ring_a: finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_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_Pocklington_Oord_001t__Finite____Field__Omod____ring_Itf__a_J,type,
ord_Fi8078518587877182669ring_a: finite_mod_ring_a > finite_mod_ring_a > nat ).
thf(sy_c_Pocklington_Oord_001t__Int__Oint,type,
ord_int: int > int > nat ).
thf(sy_c_Pocklington_Oord_001t__Nat__Onat,type,
ord_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Finite____Field__Omod____ring_Itf__a_J,type,
power_6826135765519566523ring_a: finite_mod_ring_a > nat > finite_mod_ring_a ).
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_Prime__Powers_Oprimepow_001t__Finite____Field__Omod____ring_Itf__a_J,type,
prime_9181715797402265098ring_a: finite_mod_ring_a > $o ).
thf(sy_c_Prime__Powers_Oprimepow_001t__Int__Oint,type,
prime_primepow_int: int > $o ).
thf(sy_c_Prime__Powers_Oprimepow_001t__Nat__Onat,type,
prime_primepow_nat: nat > $o ).
thf(sy_c_Primes_Oprime__int,type,
prime_int: int > $o ).
thf(sy_c_Residue__Primitive__Roots_Ocyclic__moduli,type,
residu3389958895863328978moduli: set_nat ).
thf(sy_c_Residue__Primitive__Roots_Oresidue__primroot,type,
residu2993863765933214154imroot: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Finite____Field__Omod____ring_Itf__a_J,type,
dvd_dv7258769340395861407ring_a: finite_mod_ring_a > finite_mod_ring_a > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
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__Finite____Field__Omod____ring_Itf__a_J,type,
modulo8308552932176287283ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
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_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_g____,type,
g: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_thesis,type,
thesis: $o ).
% Relevant facts (1270)
thf(fact_0_calculation,axiom,
unique653641344996303876ng_nat @ ( power_power_nat @ g @ ( minus_minus_nat @ p @ one_one_nat ) ) @ one_one_nat @ p ).
% calculation
thf(fact_1__092_060open_062_091g_A_092_060noteq_062_A1_093_A_Imod_Ap_J_092_060close_062,axiom,
~ ( unique653641344996303876ng_nat @ g @ one_one_nat @ p ) ).
% \<open>[g \<noteq> 1] (mod p)\<close>
thf(fact_2_g__Def,axiom,
( ( residu2993863765933214154imroot @ p @ g )
& ( g != one_one_nat ) ) ).
% g_Def
thf(fact_3_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_4_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( power_power_nat @ M @ N ) )
= ( power_6826135765519566523ring_a @ ( semiri9180929696517417892ring_a @ M ) @ N ) ) ).
% of_nat_power
thf(fact_5_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_8_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_9_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_10_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_11_of__nat__1,axiom,
( ( semiri9180929696517417892ring_a @ one_one_nat )
= one_on2109788427901206336ring_a ) ).
% of_nat_1
thf(fact_12_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_13_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_14_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_15_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_16_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_17_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= one_one_nat )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_18_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= one_one_int )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_19_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_20_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_21_power__one__right,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_22_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_23_mod__homo,axiom,
( finite8272632373135393572ring_a
= ( ^ [X2: int] : ( finite8272632373135393572ring_a @ ( modulo_modulo_int @ X2 @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% mod_homo
thf(fact_24_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_25_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_26_power__one,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ one_on2109788427901206336ring_a @ N )
= one_on2109788427901206336ring_a ) ).
% power_one
thf(fact_27__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062g_O_Aresidue__primroot_Ap_Ag_A_092_060and_062_Ag_A_092_060noteq_062_A1_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [G: nat] :
~ ( ( residu2993863765933214154imroot @ p @ G )
& ( G != one_one_nat ) ) ).
% \<open>\<And>thesis. (\<And>g. residue_primroot p g \<and> g \<noteq> 1 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_28_exp__homo,axiom,
! [X: int,I: nat] :
( ( finite8272632373135393572ring_a @ ( power_power_int @ X @ I ) )
= ( power_6826135765519566523ring_a @ ( finite8272632373135393572ring_a @ X ) @ I ) ) ).
% exp_homo
thf(fact_29_int__exp__hom,axiom,
! [X: nat,I: nat] :
( ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ I )
= ( semiri1314217659103216013at_int @ ( power_power_nat @ X @ I ) ) ) ).
% int_exp_hom
thf(fact_30_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_31_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_32_mod__mod__trivial,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_33_mod__mod__trivial,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_34_zmod__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zmod_int
thf(fact_35_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_36_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_37_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_38_mod__diff__cong,axiom,
! [A: int,C: int,A2: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A2 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_39_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_40_power__mod,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,N: nat] :
( ( modulo8308552932176287283ring_a @ ( power_6826135765519566523ring_a @ ( modulo8308552932176287283ring_a @ A @ B ) @ N ) @ B )
= ( modulo8308552932176287283ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_41_power__mod,axiom,
! [A: int,B: int,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
= ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_42_power__mod,axiom,
! [A: nat,B: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
= ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_43_cong__mod__right,axiom,
! [B: nat,C: nat,A: nat] :
( ( unique653641344996303876ng_nat @ B @ ( modulo_modulo_nat @ C @ A ) @ A )
= ( unique653641344996303876ng_nat @ B @ C @ A ) ) ).
% cong_mod_right
thf(fact_44_cong__mod__right,axiom,
! [B: int,C: int,A: int] :
( ( unique651150874487253600ng_int @ B @ ( modulo_modulo_int @ C @ A ) @ A )
= ( unique651150874487253600ng_int @ B @ C @ A ) ) ).
% cong_mod_right
thf(fact_45_cong__mod__left,axiom,
! [B: nat,A: nat,C: nat] :
( ( unique653641344996303876ng_nat @ ( modulo_modulo_nat @ B @ A ) @ C @ A )
= ( unique653641344996303876ng_nat @ B @ C @ A ) ) ).
% cong_mod_left
thf(fact_46_cong__mod__left,axiom,
! [B: int,A: int,C: int] :
( ( unique651150874487253600ng_int @ ( modulo_modulo_int @ B @ A ) @ C @ A )
= ( unique651150874487253600ng_int @ B @ C @ A ) ) ).
% cong_mod_left
thf(fact_47_primroot__ord,axiom,
! [G2: nat] :
( ( residu2993863765933214154imroot @ p @ G2 )
=> ( ( ord_nat @ p @ G2 )
= ( minus_minus_nat @ p @ one_one_nat ) ) ) ).
% primroot_ord
thf(fact_48_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_49_residue__primroot__mod,axiom,
! [N: nat,X: nat] :
( ( residu2993863765933214154imroot @ N @ ( modulo_modulo_nat @ X @ N ) )
= ( residu2993863765933214154imroot @ N @ X ) ) ).
% residue_primroot_mod
thf(fact_50_residue__primroot__cong,axiom,
! [X: nat,X3: nat,N: nat] :
( ( unique653641344996303876ng_nat @ X @ X3 @ N )
=> ( ( residu2993863765933214154imroot @ N @ X )
= ( residu2993863765933214154imroot @ N @ X3 ) ) ) ).
% residue_primroot_cong
thf(fact_51_int__ops_I9_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(9)
thf(fact_52_cong__def,axiom,
( unique653641344996303876ng_nat
= ( ^ [B3: nat,C2: nat,A3: nat] :
( ( modulo_modulo_nat @ B3 @ A3 )
= ( modulo_modulo_nat @ C2 @ A3 ) ) ) ) ).
% cong_def
thf(fact_53_cong__def,axiom,
( unique651150874487253600ng_int
= ( ^ [B3: int,C2: int,A3: int] :
( ( modulo_modulo_int @ B3 @ A3 )
= ( modulo_modulo_int @ C2 @ A3 ) ) ) ) ).
% cong_def
thf(fact_54_cong__pow,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a,N: nat] :
( ( unique9076693328225066129ring_a @ B @ C @ A )
=> ( unique9076693328225066129ring_a @ ( power_6826135765519566523ring_a @ B @ N ) @ ( power_6826135765519566523ring_a @ C @ N ) @ A ) ) ).
% cong_pow
thf(fact_55_cong__pow,axiom,
! [B: nat,C: nat,A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ ( power_power_nat @ B @ N ) @ ( power_power_nat @ C @ N ) @ A ) ) ).
% cong_pow
thf(fact_56_cong__pow,axiom,
! [B: int,C: int,A: int,N: nat] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ ( power_power_int @ B @ N ) @ ( power_power_int @ C @ N ) @ A ) ) ).
% cong_pow
thf(fact_57_cong__diff,axiom,
! [B: int,C: int,A: int,D: int,E: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ D @ E @ A )
=> ( unique651150874487253600ng_int @ ( minus_minus_int @ B @ D ) @ ( minus_minus_int @ C @ E ) @ A ) ) ) ).
% cong_diff
thf(fact_58_cong__cong__mod__int,axiom,
( unique651150874487253600ng_int
= ( ^ [A3: int,B3: int,M2: int] : ( unique651150874487253600ng_int @ ( modulo_modulo_int @ A3 @ M2 ) @ ( modulo_modulo_int @ B3 @ M2 ) @ M2 ) ) ) ).
% cong_cong_mod_int
thf(fact_59_cong__int__iff,axiom,
! [M: nat,Q: nat,N: nat] :
( ( unique651150874487253600ng_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ Q ) @ ( semiri1314217659103216013at_int @ N ) )
= ( unique653641344996303876ng_nat @ M @ Q @ N ) ) ).
% cong_int_iff
thf(fact_60_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_61_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_62_one__reorient,axiom,
! [X: finite_mod_ring_a] :
( ( one_on2109788427901206336ring_a = X )
= ( X = one_on2109788427901206336ring_a ) ) ).
% one_reorient
thf(fact_63_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_64_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_65_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_66_nat__int__comparison_I1_J,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_67_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_69_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_70_cong__sym__eq,axiom,
( unique653641344996303876ng_nat
= ( ^ [B3: nat,C2: nat] : ( unique653641344996303876ng_nat @ C2 @ B3 ) ) ) ).
% cong_sym_eq
thf(fact_71_cong__sym__eq,axiom,
( unique651150874487253600ng_int
= ( ^ [B3: int,C2: int] : ( unique651150874487253600ng_int @ C2 @ B3 ) ) ) ).
% cong_sym_eq
thf(fact_72_cong__trans,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( unique653641344996303876ng_nat @ C @ D @ A )
=> ( unique653641344996303876ng_nat @ B @ D @ A ) ) ) ).
% cong_trans
thf(fact_73_cong__trans,axiom,
! [B: int,C: int,A: int,D: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ C @ D @ A )
=> ( unique651150874487253600ng_int @ B @ D @ A ) ) ) ).
% cong_trans
thf(fact_74_cong__refl,axiom,
! [B: nat,A: nat] : ( unique653641344996303876ng_nat @ B @ B @ A ) ).
% cong_refl
thf(fact_75_cong__refl,axiom,
! [B: int,A: int] : ( unique651150874487253600ng_int @ B @ B @ A ) ).
% cong_refl
thf(fact_76_cong__sym,axiom,
! [B: nat,C: nat,A: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ C @ B @ A ) ) ).
% cong_sym
thf(fact_77_cong__sym,axiom,
! [B: int,C: int,A: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ C @ B @ A ) ) ).
% cong_sym
thf(fact_78_cong__cong__mod__nat,axiom,
( unique653641344996303876ng_nat
= ( ^ [A3: nat,B3: nat,M2: nat] : ( unique653641344996303876ng_nat @ ( modulo_modulo_nat @ A3 @ M2 ) @ ( modulo_modulo_nat @ B3 @ M2 ) @ M2 ) ) ) ).
% cong_cong_mod_nat
thf(fact_79_cong__1,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a] : ( unique9076693328225066129ring_a @ B @ C @ one_on2109788427901206336ring_a ) ).
% cong_1
thf(fact_80_cong__1,axiom,
! [B: nat,C: nat] : ( unique653641344996303876ng_nat @ B @ C @ one_one_nat ) ).
% cong_1
thf(fact_81_cong__1,axiom,
! [B: int,C: int] : ( unique651150874487253600ng_int @ B @ C @ one_one_int ) ).
% cong_1
thf(fact_82_ord__1,axiom,
! [N: nat] :
( ( ord_nat @ one_one_nat @ N )
= one_one_nat ) ).
% ord_1
thf(fact_83_ord__1,axiom,
! [N: int] :
( ( ord_int @ one_one_int @ N )
= one_one_nat ) ).
% ord_1
thf(fact_84_ord__1,axiom,
! [N: finite_mod_ring_a] :
( ( ord_Fi8078518587877182669ring_a @ one_on2109788427901206336ring_a @ N )
= one_one_nat ) ).
% ord_1
thf(fact_85_ord__1__right,axiom,
! [N: nat] :
( ( ord_nat @ N @ one_one_nat )
= one_one_nat ) ).
% ord_1_right
thf(fact_86_ord__mod,axiom,
! [N: int,K: int] :
( ( ord_int @ N @ ( modulo_modulo_int @ K @ N ) )
= ( ord_int @ N @ K ) ) ).
% ord_mod
thf(fact_87_ord__mod,axiom,
! [N: nat,K: nat] :
( ( ord_nat @ N @ ( modulo_modulo_nat @ K @ N ) )
= ( ord_nat @ N @ K ) ) ).
% ord_mod
thf(fact_88_ord,axiom,
! [A: nat,N: nat] : ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ ( ord_nat @ N @ A ) ) @ one_one_nat @ N ) ).
% ord
thf(fact_89_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mod
thf(fact_90_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mod
thf(fact_91_ord__cong,axiom,
! [K1: nat,K2: nat,N: nat] :
( ( unique653641344996303876ng_nat @ K1 @ K2 @ N )
=> ( ( ord_nat @ N @ K1 )
= ( ord_nat @ N @ K2 ) ) ) ).
% ord_cong
thf(fact_92_ord__cong,axiom,
! [K1: int,K2: int,N: int] :
( ( unique651150874487253600ng_int @ K1 @ K2 @ N )
=> ( ( ord_int @ N @ K1 )
= ( ord_int @ N @ K2 ) ) ) ).
% ord_cong
thf(fact_93_nat__int__mod,axiom,
! [Pr: finite_mod_ring_a,D: nat] :
( ( unique653641344996303876ng_nat @ ( power_power_nat @ ( nat2 @ ( finite1095367895020317408ring_a @ Pr ) ) @ D ) @ one_one_nat @ p )
= ( unique651150874487253600ng_int @ ( power_power_int @ ( finite1095367895020317408ring_a @ Pr ) @ D ) @ one_one_int @ ( semiri1314217659103216013at_int @ p ) ) ) ).
% nat_int_mod
thf(fact_94_exp__rule,axiom,
! [C: finite_mod_ring_a,D: finite_mod_ring_a,E: nat] :
( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ C @ D ) @ E )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ C @ E ) @ ( power_6826135765519566523ring_a @ D @ E ) ) ) ).
% exp_rule
thf(fact_95_test,axiom,
factor1801147406995305544me_nat @ p ).
% test
thf(fact_96_residue__primroot__iff__in__cyclic__moduli,axiom,
! [M: nat] :
( ( ? [X4: nat] : ( residu2993863765933214154imroot @ M @ X4 ) )
= ( member_nat @ M @ residu3389958895863328978moduli ) ) ).
% residue_primroot_iff_in_cyclic_moduli
thf(fact_97_that,axiom,
! [Primroot: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Primroot @ ( minus_minus_nat @ p @ one_one_nat ) )
= one_on2109788427901206336ring_a )
=> ( ( Primroot != one_on2109788427901206336ring_a )
=> ( ( residu2993863765933214154imroot @ p @ ( nat2 @ ( finite1095367895020317408ring_a @ Primroot ) ) )
=> thesis ) ) ) ).
% that
thf(fact_98_ord__lift,axiom,
! [Pr: finite_mod_ring_a] :
( ( ord_int @ ( semiri1314217659103216013at_int @ p ) @ ( finite1095367895020317408ring_a @ Pr ) )
= ( ord_nat @ p @ ( nat2 @ ( finite1095367895020317408ring_a @ Pr ) ) ) ) ).
% ord_lift
thf(fact_99_mult__1,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A )
= A ) ).
% mult_1
thf(fact_100_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_101_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_102_mult_Oright__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ one_on2109788427901206336ring_a )
= A ) ).
% mult.right_neutral
thf(fact_103_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_104_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_105_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_106_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( times_times_nat @ M @ N ) )
= ( times_5121417576591743744ring_a @ ( semiri9180929696517417892ring_a @ M ) @ ( semiri9180929696517417892ring_a @ N ) ) ) ).
% of_nat_mult
thf(fact_107_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_108_mult_Oleft__commute,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ B @ ( times_5121417576591743744ring_a @ A @ C ) )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_109_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_110_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_111_mult_Ocommute,axiom,
( times_5121417576591743744ring_a
= ( ^ [A3: finite_mod_ring_a,B3: finite_mod_ring_a] : ( times_5121417576591743744ring_a @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_112_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_113_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_114_mult_Oassoc,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_115_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_116_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_117_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_118_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_119_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_120_mult_Ocomm__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ one_on2109788427901206336ring_a )
= A ) ).
% mult.comm_neutral
thf(fact_121_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_122_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_123_comm__monoid__mult__class_Omult__1,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_124_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_125_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_126_power__commuting__commutes,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat] :
( ( ( times_5121417576591743744ring_a @ X @ Y2 )
= ( times_5121417576591743744ring_a @ Y2 @ X ) )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ X @ N ) @ Y2 )
= ( times_5121417576591743744ring_a @ Y2 @ ( power_6826135765519566523ring_a @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_127_power__commuting__commutes,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= ( times_times_int @ Y2 @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y2 )
= ( times_times_int @ Y2 @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_128_power__commuting__commutes,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= ( times_times_nat @ Y2 @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y2 )
= ( times_times_nat @ Y2 @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_129_power__mult__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ N )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_130_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_131_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_132_power__commutes,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ A )
= ( times_5121417576591743744ring_a @ A @ ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_commutes
thf(fact_133_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_134_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_135_mult__of__nat__commute,axiom,
! [X: nat,Y2: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y2 )
= ( times_times_int @ Y2 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_136_mult__of__nat__commute,axiom,
! [X: nat,Y2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( semiri9180929696517417892ring_a @ X ) @ Y2 )
= ( times_5121417576591743744ring_a @ Y2 @ ( semiri9180929696517417892ring_a @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_137_mult__of__nat__commute,axiom,
! [X: nat,Y2: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y2 )
= ( times_times_nat @ Y2 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_138_mod__mult__right__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A @ ( modulo8308552932176287283ring_a @ B @ C ) ) @ C )
= ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_139_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_140_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_141_mod__mult__left__eq,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ ( modulo8308552932176287283ring_a @ A @ C ) @ B ) @ C )
= ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_142_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_143_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_144_mult__mod__right,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ C @ ( modulo8308552932176287283ring_a @ A @ B ) )
= ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_145_mult__mod__right,axiom,
! [C: int,A: int,B: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_146_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_147_mod__mult__mult2,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) )
= ( times_5121417576591743744ring_a @ ( modulo8308552932176287283ring_a @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_148_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_149_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_150_mod__mult__cong,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( ( modulo8308552932176287283ring_a @ A @ C )
= ( modulo8308552932176287283ring_a @ A2 @ C ) )
=> ( ( ( modulo8308552932176287283ring_a @ B @ C )
= ( modulo8308552932176287283ring_a @ B2 @ C ) )
=> ( ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C )
= ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A2 @ B2 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_151_mod__mult__cong,axiom,
! [A: int,C: int,A2: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A2 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_152_mod__mult__cong,axiom,
! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A2 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B2 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_153_mod__mult__eq,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ ( modulo8308552932176287283ring_a @ A @ C ) @ ( modulo8308552932176287283ring_a @ B @ C ) ) @ C )
= ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_154_mod__mult__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_155_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_156_cong__mult,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a,D: finite_mod_ring_a,E: finite_mod_ring_a] :
( ( unique9076693328225066129ring_a @ B @ C @ A )
=> ( ( unique9076693328225066129ring_a @ D @ E @ A )
=> ( unique9076693328225066129ring_a @ ( times_5121417576591743744ring_a @ B @ D ) @ ( times_5121417576591743744ring_a @ C @ E ) @ A ) ) ) ).
% cong_mult
thf(fact_157_cong__mult,axiom,
! [B: nat,C: nat,A: nat,D: nat,E: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( unique653641344996303876ng_nat @ D @ E @ A )
=> ( unique653641344996303876ng_nat @ ( times_times_nat @ B @ D ) @ ( times_times_nat @ C @ E ) @ A ) ) ) ).
% cong_mult
thf(fact_158_cong__mult,axiom,
! [B: int,C: int,A: int,D: int,E: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ D @ E @ A )
=> ( unique651150874487253600ng_int @ ( times_times_int @ B @ D ) @ ( times_times_int @ C @ E ) @ A ) ) ) ).
% cong_mult
thf(fact_159_cong__modulus__mult,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a,M: finite_mod_ring_a,N: finite_mod_ring_a] :
( ( unique9076693328225066129ring_a @ X @ Y2 @ ( times_5121417576591743744ring_a @ M @ N ) )
=> ( unique9076693328225066129ring_a @ X @ Y2 @ M ) ) ).
% cong_modulus_mult
thf(fact_160_cong__modulus__mult,axiom,
! [X: int,Y2: int,M: int,N: int] :
( ( unique651150874487253600ng_int @ X @ Y2 @ ( times_times_int @ M @ N ) )
=> ( unique651150874487253600ng_int @ X @ Y2 @ M ) ) ).
% cong_modulus_mult
thf(fact_161_cong__scalar__left,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( unique9076693328225066129ring_a @ B @ C @ A )
=> ( unique9076693328225066129ring_a @ ( times_5121417576591743744ring_a @ D @ B ) @ ( times_5121417576591743744ring_a @ D @ C ) @ A ) ) ).
% cong_scalar_left
thf(fact_162_cong__scalar__left,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ ( times_times_nat @ D @ B ) @ ( times_times_nat @ D @ C ) @ A ) ) ).
% cong_scalar_left
thf(fact_163_cong__scalar__left,axiom,
! [B: int,C: int,A: int,D: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ ( times_times_int @ D @ B ) @ ( times_times_int @ D @ C ) @ A ) ) ).
% cong_scalar_left
thf(fact_164_cong__scalar__right,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( unique9076693328225066129ring_a @ B @ C @ A )
=> ( unique9076693328225066129ring_a @ ( times_5121417576591743744ring_a @ B @ D ) @ ( times_5121417576591743744ring_a @ C @ D ) @ A ) ) ).
% cong_scalar_right
thf(fact_165_cong__scalar__right,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ ( times_times_nat @ B @ D ) @ ( times_times_nat @ C @ D ) @ A ) ) ).
% cong_scalar_right
thf(fact_166_cong__scalar__right,axiom,
! [B: int,C: int,A: int,D: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ ( times_times_int @ B @ D ) @ ( times_times_int @ C @ D ) @ A ) ) ).
% cong_scalar_right
thf(fact_167_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_168_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_169_left__right__inverse__power,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat] :
( ( ( times_5121417576591743744ring_a @ X @ Y2 )
= one_on2109788427901206336ring_a )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ X @ N ) @ ( power_6826135765519566523ring_a @ Y2 @ N ) )
= one_on2109788427901206336ring_a ) ) ).
% left_right_inverse_power
thf(fact_170_left__right__inverse__power,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_171_left__right__inverse__power,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_172_mod__mult__cong__right,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( unique9076693328225066129ring_a @ ( modulo8308552932176287283ring_a @ C @ ( times_5121417576591743744ring_a @ A @ B ) ) @ D @ A )
= ( unique9076693328225066129ring_a @ C @ D @ A ) ) ).
% mod_mult_cong_right
thf(fact_173_mod__mult__cong__right,axiom,
! [C: nat,A: nat,B: nat,D: nat] :
( ( unique653641344996303876ng_nat @ ( modulo_modulo_nat @ C @ ( times_times_nat @ A @ B ) ) @ D @ A )
= ( unique653641344996303876ng_nat @ C @ D @ A ) ) ).
% mod_mult_cong_right
thf(fact_174_mod__mult__cong__right,axiom,
! [C: int,A: int,B: int,D: int] :
( ( unique651150874487253600ng_int @ ( modulo_modulo_int @ C @ ( times_times_int @ A @ B ) ) @ D @ A )
= ( unique651150874487253600ng_int @ C @ D @ A ) ) ).
% mod_mult_cong_right
thf(fact_175_mod__mult__cong__left,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a,A: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( unique9076693328225066129ring_a @ ( modulo8308552932176287283ring_a @ C @ ( times_5121417576591743744ring_a @ B @ A ) ) @ D @ A )
= ( unique9076693328225066129ring_a @ C @ D @ A ) ) ).
% mod_mult_cong_left
thf(fact_176_mod__mult__cong__left,axiom,
! [C: nat,B: nat,A: nat,D: nat] :
( ( unique653641344996303876ng_nat @ ( modulo_modulo_nat @ C @ ( times_times_nat @ B @ A ) ) @ D @ A )
= ( unique653641344996303876ng_nat @ C @ D @ A ) ) ).
% mod_mult_cong_left
thf(fact_177_mod__mult__cong__left,axiom,
! [C: int,B: int,A: int,D: int] :
( ( unique651150874487253600ng_int @ ( modulo_modulo_int @ C @ ( times_times_int @ B @ A ) ) @ D @ A )
= ( unique651150874487253600ng_int @ C @ D @ A ) ) ).
% mod_mult_cong_left
thf(fact_178_ord__int,axiom,
! [P2: nat,X: nat] :
( ( ord_int @ ( semiri1314217659103216013at_int @ P2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_nat @ P2 @ X ) ) ).
% ord_int
thf(fact_179_to__int__mod__ring__hom_Ohom__one,axiom,
( ( finite1095367895020317408ring_a @ one_on2109788427901206336ring_a )
= one_one_int ) ).
% to_int_mod_ring_hom.hom_one
thf(fact_180_to__int__mod__ring__hom_Ohom__1__iff,axiom,
! [X: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X )
= one_one_int )
= ( X = one_on2109788427901206336ring_a ) ) ).
% to_int_mod_ring_hom.hom_1_iff
thf(fact_181_of__int__mod__ring__to__int__mod__ring,axiom,
! [X: finite_mod_ring_a] :
( ( finite8272632373135393572ring_a @ ( finite1095367895020317408ring_a @ X ) )
= X ) ).
% of_int_mod_ring_to_int_mod_ring
thf(fact_182_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_183_to__int__mod__ring__hom_Ohom__1,axiom,
! [X: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X )
= one_one_int )
=> ( X = one_on2109788427901206336ring_a ) ) ).
% to_int_mod_ring_hom.hom_1
thf(fact_184_prime__power__iff,axiom,
! [P2: finite_mod_ring_a,N: nat] :
( ( factor4631116012818856269ring_a @ ( power_6826135765519566523ring_a @ P2 @ N ) )
= ( ( factor4631116012818856269ring_a @ P2 )
& ( N = one_one_nat ) ) ) ).
% prime_power_iff
thf(fact_185_prime__power__iff,axiom,
! [P2: nat,N: nat] :
( ( factor1801147406995305544me_nat @ ( power_power_nat @ P2 @ N ) )
= ( ( factor1801147406995305544me_nat @ P2 )
& ( N = one_one_nat ) ) ) ).
% prime_power_iff
thf(fact_186_prime__power__iff,axiom,
! [P2: int,N: nat] :
( ( factor1798656936486255268me_int @ ( power_power_int @ P2 @ N ) )
= ( ( factor1798656936486255268me_int @ P2 )
& ( N = one_one_nat ) ) ) ).
% prime_power_iff
thf(fact_187_to__int__mod__ring__hom_Oeq__iff,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X )
= ( finite1095367895020317408ring_a @ Y2 ) )
= ( X = Y2 ) ) ).
% to_int_mod_ring_hom.eq_iff
thf(fact_188_homomorphism__mul__on__ring,axiom,
! [X: int,Y2: int] :
( ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ X ) @ ( finite8272632373135393572ring_a @ Y2 ) )
= ( finite8272632373135393572ring_a @ ( times_times_int @ X @ Y2 ) ) ) ).
% homomorphism_mul_on_ring
thf(fact_189_prime__power__inj,axiom,
! [A: finite_mod_ring_a,M: nat,N: nat] :
( ( factor4631116012818856269ring_a @ A )
=> ( ( ( power_6826135765519566523ring_a @ A @ M )
= ( power_6826135765519566523ring_a @ A @ N ) )
=> ( M = N ) ) ) ).
% prime_power_inj
thf(fact_190_prime__power__inj,axiom,
! [A: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
=> ( M = N ) ) ) ).
% prime_power_inj
thf(fact_191_prime__power__inj,axiom,
! [A: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
=> ( M = N ) ) ) ).
% prime_power_inj
thf(fact_192_not__prime__1,axiom,
~ ( factor4631116012818856269ring_a @ one_on2109788427901206336ring_a ) ).
% not_prime_1
thf(fact_193_not__prime__1,axiom,
~ ( factor1801147406995305544me_nat @ one_one_nat ) ).
% not_prime_1
thf(fact_194_not__prime__1,axiom,
~ ( factor1798656936486255268me_int @ one_one_int ) ).
% not_prime_1
thf(fact_195_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_196_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_197_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(4)
thf(fact_198_int__distrib_I3_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(3)
thf(fact_199_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_200_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_201_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_202_power__mult,axiom,
! [A: finite_mod_ring_a,M: nat,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( times_times_nat @ M @ N ) )
= ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_203_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_204_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_205_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_206_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_207_cong__modulus__mult__nat,axiom,
! [X: nat,Y2: nat,M: nat,N: nat] :
( ( unique653641344996303876ng_nat @ X @ Y2 @ ( times_times_nat @ M @ N ) )
=> ( unique653641344996303876ng_nat @ X @ Y2 @ M ) ) ).
% cong_modulus_mult_nat
thf(fact_208_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_209_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M3: nat,N2: nat] :
( Z3
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_210_to__int__mod__ring__hom_Oinjectivity,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X )
= ( finite1095367895020317408ring_a @ Y2 ) )
=> ( X = Y2 ) ) ).
% to_int_mod_ring_hom.injectivity
thf(fact_211_prime__nat__iff__prime,axiom,
! [K: int] :
( ( factor1801147406995305544me_nat @ ( nat2 @ K ) )
= ( factor1798656936486255268me_int @ K ) ) ).
% prime_nat_iff_prime
thf(fact_212_prime__nat__int__transfer,axiom,
! [N: nat] :
( ( factor1798656936486255268me_int @ ( semiri1314217659103216013at_int @ N ) )
= ( factor1801147406995305544me_nat @ N ) ) ).
% prime_nat_int_transfer
thf(fact_213_prime__power__mult__nat,axiom,
! [P2: nat,X: nat,Y2: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ( times_times_nat @ X @ Y2 )
= ( power_power_nat @ P2 @ K ) )
=> ? [I2: nat,J2: nat] :
( ( X
= ( power_power_nat @ P2 @ I2 ) )
& ( Y2
= ( power_power_nat @ P2 @ J2 ) ) ) ) ) ).
% prime_power_mult_nat
thf(fact_214_prime__product,axiom,
! [P2: nat,Q: nat] :
( ( factor1801147406995305544me_nat @ ( times_times_nat @ P2 @ Q ) )
=> ( ( P2 = one_one_nat )
| ( Q = one_one_nat ) ) ) ).
% prime_product
thf(fact_215_ord__max,axiom,
! [L: nat,X: finite_mod_ring_a] :
( ( L != zero_zero_nat )
=> ( ( ( power_6826135765519566523ring_a @ X @ L )
= one_on2109788427901206336ring_a )
=> ( ord_less_eq_nat @ ( ord_int @ ( semiri1314217659103216013at_int @ p ) @ ( finite1095367895020317408ring_a @ X ) ) @ L ) ) ) ).
% ord_max
thf(fact_216_inf__period_I2_J,axiom,
! [P: finite_mod_ring_a > $o,D2: finite_mod_ring_a,Q2: finite_mod_ring_a > $o] :
( ! [X5: finite_mod_ring_a,K3: finite_mod_ring_a] :
( ( P @ X5 )
= ( P @ ( minus_3609261664126569004ring_a @ X5 @ ( times_5121417576591743744ring_a @ K3 @ D2 ) ) ) )
=> ( ! [X5: finite_mod_ring_a,K3: finite_mod_ring_a] :
( ( Q2 @ X5 )
= ( Q2 @ ( minus_3609261664126569004ring_a @ X5 @ ( times_5121417576591743744ring_a @ K3 @ D2 ) ) ) )
=> ! [X6: finite_mod_ring_a,K4: finite_mod_ring_a] :
( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P @ ( minus_3609261664126569004ring_a @ X6 @ ( times_5121417576591743744ring_a @ K4 @ D2 ) ) )
| ( Q2 @ ( minus_3609261664126569004ring_a @ X6 @ ( times_5121417576591743744ring_a @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_217_inf__period_I2_J,axiom,
! [P: int > $o,D2: int,Q2: int > $o] :
( ! [X5: int,K3: int] :
( ( P @ X5 )
= ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ! [X5: int,K3: int] :
( ( Q2 @ X5 )
= ( Q2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ! [X6: int,K4: int] :
( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D2 ) ) )
| ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_218_inf__period_I1_J,axiom,
! [P: finite_mod_ring_a > $o,D2: finite_mod_ring_a,Q2: finite_mod_ring_a > $o] :
( ! [X5: finite_mod_ring_a,K3: finite_mod_ring_a] :
( ( P @ X5 )
= ( P @ ( minus_3609261664126569004ring_a @ X5 @ ( times_5121417576591743744ring_a @ K3 @ D2 ) ) ) )
=> ( ! [X5: finite_mod_ring_a,K3: finite_mod_ring_a] :
( ( Q2 @ X5 )
= ( Q2 @ ( minus_3609261664126569004ring_a @ X5 @ ( times_5121417576591743744ring_a @ K3 @ D2 ) ) ) )
=> ! [X6: finite_mod_ring_a,K4: finite_mod_ring_a] :
( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P @ ( minus_3609261664126569004ring_a @ X6 @ ( times_5121417576591743744ring_a @ K4 @ D2 ) ) )
& ( Q2 @ ( minus_3609261664126569004ring_a @ X6 @ ( times_5121417576591743744ring_a @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_219_inf__period_I1_J,axiom,
! [P: int > $o,D2: int,Q2: int > $o] :
( ! [X5: int,K3: int] :
( ( P @ X5 )
= ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ! [X5: int,K3: int] :
( ( Q2 @ X5 )
= ( Q2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ! [X6: int,K4: int] :
( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D2 ) ) )
& ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_220_left__diff__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ C )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_221_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_222_right__diff__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_223_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_224_left__diff__distrib_H,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ B @ C ) @ A )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ B @ A ) @ ( times_5121417576591743744ring_a @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_225_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_226_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_227_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_228_mult__zero__left,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a @ A )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_left
thf(fact_229_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_230_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_231_mult__zero__right,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_right
thf(fact_232_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_233_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_234_mult__eq__0__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B )
= zero_z7902377541816115708ring_a )
= ( ( A = zero_z7902377541816115708ring_a )
| ( B = zero_z7902377541816115708ring_a ) ) ) ).
% mult_eq_0_iff
thf(fact_235_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_236_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_237_mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A )
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_238_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_239_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_240_mult__cancel__right,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ C )
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_241_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_242_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_243_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_244_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_245_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_246_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_247_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_248_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_249_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_250_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_251_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_252_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_253_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_254_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_255_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_256_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_257_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_258_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_259_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_260_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_261_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_262_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_263_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_264_cong__0,axiom,
! [B: nat,C: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ zero_zero_nat )
= ( B = C ) ) ).
% cong_0
thf(fact_265_cong__0,axiom,
! [B: int,C: int] :
( ( unique651150874487253600ng_int @ B @ C @ zero_zero_int )
= ( B = C ) ) ).
% cong_0
thf(fact_266_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_267_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_268_mult__cancel__right2,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ C )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right2
thf(fact_269_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_270_mult__cancel__right1,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right1
thf(fact_271_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_272_mult__cancel__left2,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left2
thf(fact_273_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_274_mult__cancel__left1,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left1
thf(fact_275_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_276_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_277_of__nat__0,axiom,
( ( semiri9180929696517417892ring_a @ zero_zero_nat )
= zero_z7902377541816115708ring_a ) ).
% of_nat_0
thf(fact_278_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_279_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_280_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_281_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_282_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_283_mod__mult__self1__is__0,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ B @ A ) @ B )
= zero_z7902377541816115708ring_a ) ).
% mod_mult_self1_is_0
thf(fact_284_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_285_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_286_mod__mult__self2__is__0,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ B )
= zero_z7902377541816115708ring_a ) ).
% mod_mult_self2_is_0
thf(fact_287_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_288_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_289_mod__by__1,axiom,
! [A: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ A @ one_on2109788427901206336ring_a )
= zero_z7902377541816115708ring_a ) ).
% mod_by_1
thf(fact_290_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_291_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_292_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_293_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_294_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_295_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_296_ord__0__right__nat,axiom,
! [N: nat] :
( ( ( N = one_one_nat )
=> ( ( ord_nat @ N @ zero_zero_nat )
= one_one_nat ) )
& ( ( N != one_one_nat )
=> ( ( ord_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ) ) ).
% ord_0_right_nat
thf(fact_297_ord__0__nat,axiom,
! [N: nat] :
( ( ( N = one_one_nat )
=> ( ( ord_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != one_one_nat )
=> ( ( ord_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% ord_0_nat
thf(fact_298_not__residue__primroot__0__right,axiom,
! [N: nat] :
( ( residu2993863765933214154imroot @ N @ zero_zero_nat )
= ( N = one_one_nat ) ) ).
% not_residue_primroot_0_right
thf(fact_299_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_300_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_301_prime__power__eq__one__iff,axiom,
! [P2: finite_mod_ring_a,N: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( ( power_6826135765519566523ring_a @ P2 @ N )
= one_on2109788427901206336ring_a )
= ( N = zero_zero_nat ) ) ) ).
% prime_power_eq_one_iff
thf(fact_302_prime__power__eq__one__iff,axiom,
! [P2: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ( power_power_nat @ P2 @ N )
= one_one_nat )
= ( N = zero_zero_nat ) ) ) ).
% prime_power_eq_one_iff
thf(fact_303_prime__power__eq__one__iff,axiom,
! [P2: int,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( ( power_power_int @ P2 @ N )
= one_one_int )
= ( N = zero_zero_nat ) ) ) ).
% prime_power_eq_one_iff
thf(fact_304_one__eq__prime__power__iff,axiom,
! [P2: finite_mod_ring_a,N: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( one_on2109788427901206336ring_a
= ( power_6826135765519566523ring_a @ P2 @ N ) )
= ( N = zero_zero_nat ) ) ) ).
% one_eq_prime_power_iff
thf(fact_305_one__eq__prime__power__iff,axiom,
! [P2: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( one_one_nat
= ( power_power_nat @ P2 @ N ) )
= ( N = zero_zero_nat ) ) ) ).
% one_eq_prime_power_iff
thf(fact_306_one__eq__prime__power__iff,axiom,
! [P2: int,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( one_one_int
= ( power_power_int @ P2 @ N ) )
= ( N = zero_zero_nat ) ) ) ).
% one_eq_prime_power_iff
thf(fact_307_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_308_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_309_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_310_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_311_mult__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 @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_312_mult__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 @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_313_mult__mono_H,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 @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_314_mult__mono_H,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 @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_315_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_316_split__mult__pos__le,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 @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_317_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_318_mult__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 @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_319_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_320_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_321_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_322_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_323_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_324_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_325_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_326_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_327_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_328_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_329_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_330_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_331_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_332_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_333_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_334_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_335_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_336_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_337_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_338_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_339_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_340_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_341_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_342_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_343_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_344_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_345_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_346_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_347_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_348_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_349_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_350_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_351_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_352_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_353_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_354_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_355_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_356_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_357_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_358_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_359_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_360_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_361_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_362_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_363_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_364_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_365_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_366_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_367_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_368_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_369_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_370_mult__not__zero,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B )
!= zero_z7902377541816115708ring_a )
=> ( ( A != zero_z7902377541816115708ring_a )
& ( B != zero_z7902377541816115708ring_a ) ) ) ).
% mult_not_zero
thf(fact_371_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_372_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_373_divisors__zero,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B )
= zero_z7902377541816115708ring_a )
=> ( ( A = zero_z7902377541816115708ring_a )
| ( B = zero_z7902377541816115708ring_a ) ) ) ).
% divisors_zero
thf(fact_374_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_375_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_376_no__zero__divisors,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( B != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A @ B )
!= zero_z7902377541816115708ring_a ) ) ) ).
% no_zero_divisors
thf(fact_377_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_378_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_379_mult__left__cancel,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ C @ A )
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_380_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_381_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_382_mult__right__cancel,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ A @ C )
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_383_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_384_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_385_zero__neq__one,axiom,
zero_z7902377541816115708ring_a != one_on2109788427901206336ring_a ).
% zero_neq_one
thf(fact_386_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_387_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_388_power__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_389_power__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_390_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_391_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_392_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_393_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_394_mult__right__le__one__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_395_mult__left__le__one__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_396_power__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_397_power__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_398_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_399_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_400_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_401_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_402_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= one_on2109788427901206336ring_a ) )
& ( ( N != zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ) ).
% power_0_left
thf(fact_403_cong__diff__iff__cong__0__nat,axiom,
! [B: nat,A: nat,M: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( unique653641344996303876ng_nat @ ( minus_minus_nat @ A @ B ) @ zero_zero_nat @ M )
= ( unique653641344996303876ng_nat @ A @ B @ M ) ) ) ).
% cong_diff_iff_cong_0_nat
thf(fact_404_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_405_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_406_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_407_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_408_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_409_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_410_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_411_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_412_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_413_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_414_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_415_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_416_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_417_mult__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 @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_418_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_419_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_420_eq__iff__diff__eq__0,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_421_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_422_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_423_power__not__zero,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( power_6826135765519566523ring_a @ A @ N )
!= zero_z7902377541816115708ring_a ) ) ).
% power_not_zero
thf(fact_424_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_425_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_426_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_427_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_428_not__prime__0,axiom,
~ ( factor1801147406995305544me_nat @ zero_zero_nat ) ).
% not_prime_0
thf(fact_429_not__prime__0,axiom,
~ ( factor1798656936486255268me_int @ zero_zero_int ) ).
% not_prime_0
thf(fact_430_prime__ge__1__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ P2 ) ) ).
% prime_ge_1_nat
thf(fact_431_not__residue__primroot__0,axiom,
! [X: nat] :
~ ( residu2993863765933214154imroot @ zero_zero_nat @ X ) ).
% not_residue_primroot_0
thf(fact_432_prime__power__exp__nat,axiom,
! [P2: nat,N: nat,X: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( N != zero_zero_nat )
=> ( ( ( power_power_nat @ X @ N )
= ( power_power_nat @ P2 @ K ) )
=> ? [I2: nat] :
( X
= ( power_power_nat @ P2 @ I2 ) ) ) ) ) ).
% prime_power_exp_nat
thf(fact_433_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_434_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_435_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_436_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_437_power__0,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ zero_zero_nat )
= one_on2109788427901206336ring_a ) ).
% power_0
thf(fact_438_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_439_cong__diff__nat,axiom,
! [A: nat,B: nat,M: nat,C: nat,D: nat] :
( ( unique653641344996303876ng_nat @ A @ B @ M )
=> ( ( unique653641344996303876ng_nat @ C @ D @ M )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( unique653641344996303876ng_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ D ) @ M ) ) ) ) ) ).
% cong_diff_nat
thf(fact_440_cong__mult__self__right,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] : ( unique9076693328225066129ring_a @ ( times_5121417576591743744ring_a @ B @ A ) @ zero_z7902377541816115708ring_a @ A ) ).
% cong_mult_self_right
thf(fact_441_cong__mult__self__right,axiom,
! [B: nat,A: nat] : ( unique653641344996303876ng_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat @ A ) ).
% cong_mult_self_right
thf(fact_442_cong__mult__self__right,axiom,
! [B: int,A: int] : ( unique651150874487253600ng_int @ ( times_times_int @ B @ A ) @ zero_zero_int @ A ) ).
% cong_mult_self_right
thf(fact_443_cong__mult__self__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] : ( unique9076693328225066129ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ zero_z7902377541816115708ring_a @ A ) ).
% cong_mult_self_left
thf(fact_444_cong__mult__self__left,axiom,
! [A: nat,B: nat] : ( unique653641344996303876ng_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat @ A ) ).
% cong_mult_self_left
thf(fact_445_cong__mult__self__left,axiom,
! [A: int,B: int] : ( unique651150874487253600ng_int @ ( times_times_int @ A @ B ) @ zero_zero_int @ A ) ).
% cong_mult_self_left
thf(fact_446_cong__diff__iff__cong__0,axiom,
! [B: int,C: int,A: int] :
( ( unique651150874487253600ng_int @ ( minus_minus_int @ B @ C ) @ zero_zero_int @ A )
= ( unique651150874487253600ng_int @ B @ C @ A ) ) ).
% cong_diff_iff_cong_0
thf(fact_447_prime__power__inj_H_H,axiom,
! [P2: finite_mod_ring_a,Q: finite_mod_ring_a,M: nat,N: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( factor4631116012818856269ring_a @ Q )
=> ( ( ( power_6826135765519566523ring_a @ P2 @ M )
= ( power_6826135765519566523ring_a @ Q @ N ) )
= ( ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) )
| ( ( P2 = Q )
& ( M = N ) ) ) ) ) ) ).
% prime_power_inj''
thf(fact_448_prime__power__inj_H_H,axiom,
! [P2: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ( power_power_nat @ P2 @ M )
= ( power_power_nat @ Q @ N ) )
= ( ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) )
| ( ( P2 = Q )
& ( M = N ) ) ) ) ) ) ).
% prime_power_inj''
thf(fact_449_prime__power__inj_H_H,axiom,
! [P2: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ( power_power_int @ P2 @ M )
= ( power_power_int @ Q @ N ) )
= ( ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) )
| ( ( P2 = Q )
& ( M = N ) ) ) ) ) ) ).
% prime_power_inj''
thf(fact_450_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_451_cong__0__1__nat,axiom,
! [N: nat] :
( ( unique653641344996303876ng_nat @ zero_zero_nat @ one_one_nat @ N )
= ( N = one_one_nat ) ) ).
% cong_0_1_nat
thf(fact_452_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_453_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri9180929696517417892ring_a @ ( minus_minus_nat @ M @ N ) )
= ( minus_3609261664126569004ring_a @ ( semiri9180929696517417892ring_a @ M ) @ ( semiri9180929696517417892ring_a @ N ) ) ) ) ).
% of_nat_diff
thf(fact_454_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_455_power__eq__if,axiom,
( power_6826135765519566523ring_a
= ( ^ [P3: finite_mod_ring_a,M2: nat] : ( if_Finite_mod_ring_a @ ( M2 = zero_zero_nat ) @ one_on2109788427901206336ring_a @ ( times_5121417576591743744ring_a @ P3 @ ( power_6826135765519566523ring_a @ P3 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_456_power__eq__if,axiom,
( power_power_int
= ( ^ [P3: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_457_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_458_right__diff__distrib_H,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_459_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_460_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_461_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_462_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_463_diff__numeral__special_I9_J,axiom,
( ( minus_3609261664126569004ring_a @ one_on2109788427901206336ring_a @ one_on2109788427901206336ring_a )
= zero_z7902377541816115708ring_a ) ).
% diff_numeral_special(9)
thf(fact_464_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_465_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_466_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_467_mult__hom_Ohom__zero,axiom,
! [C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ C @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_hom.hom_zero
thf(fact_468_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_469_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_470_cong__prime__prod__zero__nat,axiom,
! [A: nat,B: nat,P2: nat] :
( ( unique653641344996303876ng_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat @ P2 )
=> ( ( factor1801147406995305544me_nat @ P2 )
=> ( ( unique653641344996303876ng_nat @ A @ zero_zero_nat @ P2 )
| ( unique653641344996303876ng_nat @ B @ zero_zero_nat @ P2 ) ) ) ) ).
% cong_prime_prod_zero_nat
thf(fact_471_less__eq__fract__respect,axiom,
! [B: int,B2: int,D: int,D3: int,A: int,A2: int,C: int,C3: int] :
( ( B != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( D3 != zero_zero_int )
=> ( ( ( times_times_int @ A @ B2 )
= ( times_times_int @ A2 @ B ) )
=> ( ( ( times_times_int @ C @ D3 )
= ( times_times_int @ C3 @ D ) )
=> ( ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A2 @ D3 ) @ ( times_times_int @ B2 @ D3 ) ) @ ( times_times_int @ ( times_times_int @ C3 @ B2 ) @ ( times_times_int @ B2 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_472_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_473_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_474_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_475_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_476_to__int__mod__ring__hom_Ohom__zero,axiom,
( ( finite1095367895020317408ring_a @ zero_z7902377541816115708ring_a )
= zero_zero_int ) ).
% to_int_mod_ring_hom.hom_zero
thf(fact_477_to__int__mod__ring__hom_Ohom__0__iff,axiom,
! [X: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X )
= zero_zero_int )
= ( X = zero_z7902377541816115708ring_a ) ) ).
% to_int_mod_ring_hom.hom_0_iff
thf(fact_478_of__int__mod__ring__hom_Ohom__zero,axiom,
( ( finite8272632373135393572ring_a @ zero_zero_int )
= zero_z7902377541816115708ring_a ) ).
% of_int_mod_ring_hom.hom_zero
thf(fact_479_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_480_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_481_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_482_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_483_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_484_eq__nat__nat__iff,axiom,
! [Z3: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z4 ) )
= ( Z3 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_485_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_486_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_487_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_488_to__int__mod__ring__hom_Ohom__0,axiom,
! [X: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X )
= zero_zero_int )
=> ( X = zero_z7902377541816115708ring_a ) ) ).
% to_int_mod_ring_hom.hom_0
thf(fact_489_prime__ge__0__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_eq_int @ zero_zero_int @ P2 ) ) ).
% prime_ge_0_int
thf(fact_490_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_491_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_492_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_493_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_494_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_495_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_496_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_497_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_498_nat__mult__distrib,axiom,
! [Z3: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z4 ) )
= ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z4 ) ) ) ) ).
% nat_mult_distrib
thf(fact_499_nat__diff__distrib_H,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y2 ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_500_nat__diff__distrib,axiom,
! [Z4: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ Z4 @ Z3 )
=> ( ( nat2 @ ( minus_minus_int @ Z3 @ Z4 ) )
= ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_501_nat__mod__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( modulo_modulo_int @ X @ Y2 ) )
= ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_502_nat__power__eq,axiom,
! [Z3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( power_power_int @ Z3 @ N ) )
= ( power_power_nat @ ( nat2 @ Z3 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_503_prime__int__nat__transfer,axiom,
( factor1798656936486255268me_int
= ( ^ [K5: int] :
( ( ord_less_eq_int @ zero_zero_int @ K5 )
& ( factor1801147406995305544me_nat @ ( nat2 @ K5 ) ) ) ) ) ).
% prime_int_nat_transfer
thf(fact_504_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_505_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_506_nat__mono,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_507_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_508_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_509_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_510_prime__ge__1__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_eq_int @ one_one_int @ P2 ) ) ).
% prime_ge_1_int
thf(fact_511_cong__prime__prod__zero__int,axiom,
! [A: int,B: int,P2: int] :
( ( unique651150874487253600ng_int @ ( times_times_int @ A @ B ) @ zero_zero_int @ P2 )
=> ( ( factor1798656936486255268me_int @ P2 )
=> ( ( unique651150874487253600ng_int @ A @ zero_zero_int @ P2 )
| ( unique651150874487253600ng_int @ B @ zero_zero_int @ P2 ) ) ) ) ).
% cong_prime_prod_zero_int
thf(fact_512_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_513_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_514_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_515_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_516_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_517_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_518_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_519_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_520_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_521_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_522_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_523_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_524_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_525_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_526_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_527_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_528_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_529_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_530_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_531_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_532_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_533_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_534_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_535_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_536_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_537_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_538_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_539_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_540_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_541_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_542_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_543_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_544_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_545_dual__order_Oeq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_546_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_547_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: int,B4: int] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_548_order__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_549_order__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_550_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_551_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_552_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_553_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_554_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_555_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_556_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_557_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_558_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_559_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_560_le__cases3,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_561_le__cases3,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_562_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_563_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_564_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_565_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_566_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_567_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_568_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_569_mod__nat__int__pow__eq,axiom,
! [A: int,P2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ P2 )
=> ( ( modulo_modulo_nat @ ( power_power_nat @ ( nat2 @ A ) @ N ) @ ( nat2 @ P2 ) )
= ( nat2 @ ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ P2 ) ) ) ) ) ).
% mod_nat_int_pow_eq
thf(fact_570_zmod__eq__0D,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
=> ? [Q3: int] :
( M
= ( times_times_int @ D @ Q3 ) ) ) ).
% zmod_eq_0D
thf(fact_571_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_572_eucl__induct,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [B4: int] : ( P @ B4 @ zero_zero_int )
=> ( ! [A5: int,B4: int] :
( ( B4 != zero_zero_int )
=> ( ( P @ B4 @ ( modulo_modulo_int @ A5 @ B4 ) )
=> ( P @ A5 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ).
% eucl_induct
thf(fact_573_eucl__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [B4: nat] : ( P @ B4 @ zero_zero_nat )
=> ( ! [A5: nat,B4: nat] :
( ( B4 != zero_zero_nat )
=> ( ( P @ B4 @ ( modulo_modulo_nat @ A5 @ B4 ) )
=> ( P @ A5 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ).
% eucl_induct
thf(fact_574_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_575_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_576_imp__le__cong,axiom,
! [X: int,X3: int,P: $o,P6: $o] :
( ( X = X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_577_conj__le__cong,axiom,
! [X: int,X3: int,P: $o,P6: $o] :
( ( X = X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_578_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_579_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_580_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5901776551076858996ring_a @ zero_z7902377541816115708ring_a )
= one_on2109788427901206336ring_a ) ).
% dbl_inc_simps(2)
thf(fact_581_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_582_Primes_Oprime__int__def,axiom,
prime_int = factor1798656936486255268me_int ).
% Primes.prime_int_def
thf(fact_583_ord__minimal,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ ( ord_nat @ N @ A ) )
=> ~ ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ M ) @ one_one_nat @ N ) ) ) ).
% ord_minimal
thf(fact_584_ord__works,axiom,
! [A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ ( ord_nat @ N @ A ) ) @ one_one_nat @ N )
& ! [M4: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ M4 @ ( ord_nat @ N @ A ) ) )
=> ~ ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ M4 ) @ one_one_nat @ N ) ) ) ).
% ord_works
thf(fact_585_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_586_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_587_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_588_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_589_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_590_k__bound,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% k_bound
thf(fact_591_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_592_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_593_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_594_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_595_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_596_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_597_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_598_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_599_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_600_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_601_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_602_zless__nat__conj,axiom,
! [W: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_603_zle__diff1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_604_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_605_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_606_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_607_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_608_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_609_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_610_power__eq__0__iff,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( ( power_6826135765519566523ring_a @ A @ N )
= zero_z7902377541816115708ring_a )
= ( ( A = zero_z7902377541816115708ring_a )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_611_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_612_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_613_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_614_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_615_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_616_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_617_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_618_power__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_619_power__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_620_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_621_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_622_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_623_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_624_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_625_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_626_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_627_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_628_nat__mono__iff,axiom,
! [Z3: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_629_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_630_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_631_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_632_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_633_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_634_minf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_635_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_636_minf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_637_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_638_minf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_639_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_640_minf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_641_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z6 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z6 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_642_minf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z6 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z6 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_643_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z6 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z6 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_644_minf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z6 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z6 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_645_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_646_pinf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_647_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_648_pinf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_649_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_650_pinf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_651_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_652_pinf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_653_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z6 @ X5 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z6 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_654_pinf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ Z6 @ X5 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ Z6 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_655_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z6 @ X5 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z6 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_656_pinf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ Z6 @ X5 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ Z6 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_657_linorder__neqE__linordered__idom,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_658_nat__exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ~ ( P5 @ M2 ) ) ) ) ) ).
% nat_exists_least_iff
thf(fact_659_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_660_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_661_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_662_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_663_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_664_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_665_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_666_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_667_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_668_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_669_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_670_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_671_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_672_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_673_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_674_zless__nat__eq__int__zless,axiom,
! [M: nat,Z3: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_675_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_676_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_677_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_678_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_679_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_680_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_681_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_682_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_683_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_684_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_685_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_686_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_687_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_688_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_689_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_690_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_691_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_692_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_693_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_694_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_695_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_696_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_697_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_698_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_699_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_700_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_701_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_702_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_703_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_704_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_705_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_706_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_707_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_708_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_709_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_710_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_711_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_712_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_713_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_714_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_715_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_716_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_717_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_718_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_719_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_720_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_721_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_722_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_723_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_724_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_725_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_726_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_727_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_728_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_729_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_730_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_731_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_732_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_733_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_734_order__le__less__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_735_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_736_order__less__le__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_737_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_738_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_739_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_740_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_741_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_742_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_743_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_744_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_745_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_746_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_747_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_748_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_749_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_750_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_751_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 )
=> ( ! [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_752_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 )
=> ( ! [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_753_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_754_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_755_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_756_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_757_pinf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_758_pinf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_759_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_760_pinf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ Z5 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_761_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_762_minf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_763_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_764_minf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z5 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_765_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_766_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_767_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_768_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_769_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_770_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_771_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_772_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_773_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_774_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_775_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_776_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_777_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_778_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_779_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_780_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_781_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_782_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_783_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_784_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_785_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_786_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_787_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_788_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_789_bigger__prime,axiom,
! [N: nat] :
? [P7: nat] :
( ( factor1801147406995305544me_nat @ P7 )
& ( ord_less_nat @ N @ P7 ) ) ).
% bigger_prime
thf(fact_790_cong__less__modulus__unique__nat,axiom,
! [X: nat,Y2: nat,M: nat] :
( ( unique653641344996303876ng_nat @ X @ Y2 @ M )
=> ( ( ord_less_nat @ X @ M )
=> ( ( ord_less_nat @ Y2 @ M )
=> ( X = Y2 ) ) ) ) ).
% cong_less_modulus_unique_nat
thf(fact_791_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_792_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_793_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_794_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_795_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_796_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_797_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_798_nat__less__eq__zless,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_799_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_800_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_801_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_802_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_803_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_804_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_805_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_806_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_807_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_808_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_809_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_810_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_811_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_812_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_813_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_814_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_815_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_816_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_817_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_818_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_819_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_820_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_821_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_822_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_823_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_824_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_825_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_826_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_827_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_828_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_829_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_830_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_831_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_832_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_833_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_834_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_835_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_836_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_837_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_838_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_839_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_840_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_841_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_842_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_843_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_844_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_845_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_846_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_847_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_848_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_849_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_850_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_851_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_852_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_853_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_854_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_855_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_856_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_857_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo_nat @ M3 @ N2 ) )
=> ( P @ M3 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_858_prime__gt__0__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_nat @ zero_zero_nat @ P2 ) ) ).
% prime_gt_0_nat
thf(fact_859_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N4 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_860_prime__gt__1__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_nat @ one_one_nat @ P2 ) ) ).
% prime_gt_1_nat
thf(fact_861_prime__gt__0__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_int @ zero_zero_int @ P2 ) ) ).
% prime_gt_0_int
thf(fact_862_prime__gt__1__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_int @ one_one_int @ P2 ) ) ).
% prime_gt_1_int
thf(fact_863_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_864_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_865_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_866_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_867_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_868_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_869_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_870_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_871_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_872_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_873_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_874_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_875_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_876_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_877_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_878_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_879_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_880_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_881_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_882_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_883_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_884_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_885_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_886_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_887_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_888_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_889_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_890_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_891_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_892_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_893_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ).
% zero_power
thf(fact_894_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_895_prime__power__inj_H_I1_J,axiom,
! [P2: finite_mod_ring_a,Q: finite_mod_ring_a,M: nat,N: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( factor4631116012818856269ring_a @ Q )
=> ( ( ( power_6826135765519566523ring_a @ P2 @ M )
= ( power_6826135765519566523ring_a @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( P2 = Q ) ) ) ) ) ) ).
% prime_power_inj'(1)
thf(fact_896_prime__power__inj_H_I1_J,axiom,
! [P2: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ( power_power_nat @ P2 @ M )
= ( power_power_nat @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( P2 = Q ) ) ) ) ) ) ).
% prime_power_inj'(1)
thf(fact_897_prime__power__inj_H_I1_J,axiom,
! [P2: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ( power_power_int @ P2 @ M )
= ( power_power_int @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( P2 = Q ) ) ) ) ) ) ).
% prime_power_inj'(1)
thf(fact_898_prime__power__inj_H_I2_J,axiom,
! [P2: finite_mod_ring_a,Q: finite_mod_ring_a,M: nat,N: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( factor4631116012818856269ring_a @ Q )
=> ( ( ( power_6826135765519566523ring_a @ P2 @ M )
= ( power_6826135765519566523ring_a @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( M = N ) ) ) ) ) ) ).
% prime_power_inj'(2)
thf(fact_899_prime__power__inj_H_I2_J,axiom,
! [P2: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ( power_power_nat @ P2 @ M )
= ( power_power_nat @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( M = N ) ) ) ) ) ) ).
% prime_power_inj'(2)
thf(fact_900_prime__power__inj_H_I2_J,axiom,
! [P2: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ( power_power_int @ P2 @ M )
= ( power_power_int @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( M = N ) ) ) ) ) ) ).
% prime_power_inj'(2)
thf(fact_901_prime__power__eq__imp__eq,axiom,
! [P2: finite_mod_ring_a,Q: finite_mod_ring_a,M: nat,N: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( factor4631116012818856269ring_a @ Q )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( power_6826135765519566523ring_a @ P2 @ M )
= ( power_6826135765519566523ring_a @ Q @ N ) )
=> ( P2 = Q ) ) ) ) ) ).
% prime_power_eq_imp_eq
thf(fact_902_prime__power__eq__imp__eq,axiom,
! [P2: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( power_power_nat @ P2 @ M )
= ( power_power_nat @ Q @ N ) )
=> ( P2 = Q ) ) ) ) ) ).
% prime_power_eq_imp_eq
thf(fact_903_prime__power__eq__imp__eq,axiom,
! [P2: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( power_power_int @ P2 @ M )
= ( power_power_int @ Q @ N ) )
=> ( P2 = Q ) ) ) ) ) ).
% prime_power_eq_imp_eq
thf(fact_904_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_905_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_906_plusinfinity,axiom,
! [D: int,P6: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int,K3: int] :
( ( P6 @ X5 )
= ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ Z6 @ X5 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [X_1: int] : ( P6 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_907_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int,K3: int] :
( ( P1 @ X5 )
= ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z6 )
=> ( ( P @ X5 )
= ( P1 @ X5 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_908_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_909_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_910_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_911_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo_int @ I @ K )
= I )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_912_cong__less__imp__eq__nat,axiom,
! [A: nat,M: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ M )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ M )
=> ( ( unique653641344996303876ng_nat @ A @ B @ M )
=> ( A = B ) ) ) ) ) ) ).
% cong_less_imp_eq_nat
thf(fact_913_cong__less__unique__nat,axiom,
! [M: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ? [X5: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X5 )
& ( ord_less_nat @ X5 @ M )
& ( unique653641344996303876ng_nat @ A @ X5 @ M )
& ! [Y4: nat] :
( ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
& ( ord_less_nat @ Y4 @ M )
& ( unique653641344996303876ng_nat @ A @ Y4 @ M ) )
=> ( Y4 = X5 ) ) ) ) ).
% cong_less_unique_nat
thf(fact_914_not__prime__eq__prod__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ~ ( factor1801147406995305544me_nat @ M )
=> ? [N2: nat,K3: nat] :
( ( N2
= ( times_times_nat @ M @ K3 ) )
& ( ord_less_nat @ one_one_nat @ M )
& ( ord_less_nat @ M @ N2 )
& ( ord_less_nat @ one_one_nat @ K3 )
& ( ord_less_nat @ K3 @ N2 ) ) ) ) ).
% not_prime_eq_prod_nat
thf(fact_915_cong__less__unique__int,axiom,
! [M: int,A: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ? [X5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
& ( ord_less_int @ X5 @ M )
& ( unique651150874487253600ng_int @ A @ X5 @ M )
& ! [Y4: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
& ( ord_less_int @ Y4 @ M )
& ( unique651150874487253600ng_int @ A @ Y4 @ M ) )
=> ( Y4 = X5 ) ) ) ) ).
% cong_less_unique_int
thf(fact_916_cong__less__imp__eq__int,axiom,
! [A: int,M: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ M )
=> ( ( unique651150874487253600ng_int @ A @ B @ M )
=> ( A = B ) ) ) ) ) ) ).
% cong_less_imp_eq_int
thf(fact_917_prime__primitive__root__exists,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( factor1801147406995305544me_nat @ N )
=> ? [X_12: nat] : ( residu2993863765933214154imroot @ N @ X_12 ) ) ) ).
% prime_primitive_root_exists
thf(fact_918_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_919_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_920_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_921_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_922_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_923_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_924_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_925_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_926_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_927_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_937_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_938_nat__le__eq__zle,axiom,
! [W: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_939_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_940_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_941_power__minus__mult,axiom,
! [N: nat,A: finite_mod_ring_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_942_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_943_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_944_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_945_cong__unique__inverse__prime,axiom,
! [P2: nat,X: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ord_less_nat @ X @ P2 )
=> ? [X5: nat] :
( ( ord_less_nat @ zero_zero_nat @ X5 )
& ( ord_less_nat @ X5 @ P2 )
& ( unique653641344996303876ng_nat @ ( times_times_nat @ X @ X5 ) @ one_one_nat @ P2 )
& ! [Y4: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ Y4 )
& ( ord_less_nat @ Y4 @ P2 )
& ( unique653641344996303876ng_nat @ ( times_times_nat @ X @ Y4 ) @ one_one_nat @ P2 ) )
=> ( Y4 = X5 ) ) ) ) ) ) ).
% cong_unique_inverse_prime
thf(fact_946_prime__power__not__one,axiom,
! [P2: finite_mod_ring_a,K: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( power_6826135765519566523ring_a @ P2 @ K )
!= one_on2109788427901206336ring_a ) ) ) ).
% prime_power_not_one
thf(fact_947_prime__power__not__one,axiom,
! [P2: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( power_power_nat @ P2 @ K )
!= one_one_nat ) ) ) ).
% prime_power_not_one
thf(fact_948_prime__power__not__one,axiom,
! [P2: int,K: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( power_power_int @ P2 @ K )
!= one_one_int ) ) ) ).
% prime_power_not_one
thf(fact_949_mult__le__cancel__iff1,axiom,
! [Z3: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y2 @ Z3 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_950_mult__le__cancel__iff2,axiom,
! [Z3: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_951_mult__less__iff1,axiom,
! [Z3: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y2 @ Z3 ) )
= ( ord_less_int @ X @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_952_p__fact,axiom,
( p
= ( plus_plus_nat @ ( times_times_nat @ k @ n ) @ one_one_nat ) ) ).
% p_fact
thf(fact_953_primepow__prime__power,axiom,
! [P2: finite_mod_ring_a,N: nat] :
( ( factor4631116012818856269ring_a @ P2 )
=> ( ( prime_9181715797402265098ring_a @ ( power_6826135765519566523ring_a @ P2 @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% primepow_prime_power
thf(fact_954_primepow__prime__power,axiom,
! [P2: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( prime_primepow_nat @ ( power_power_nat @ P2 @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% primepow_prime_power
thf(fact_955_primepow__prime__power,axiom,
! [P2: int,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( prime_primepow_int @ ( power_power_int @ P2 @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% primepow_prime_power
thf(fact_956_divide__out__primepow,axiom,
! [N: finite_mod_ring_a] :
( ( N != zero_z7902377541816115708ring_a )
=> ( ~ ( dvd_dv7258769340395861407ring_a @ N @ one_on2109788427901206336ring_a )
=> ~ ! [P7: finite_mod_ring_a] :
( ( factor4631116012818856269ring_a @ P7 )
=> ( ( dvd_dv7258769340395861407ring_a @ P7 @ N )
=> ! [K3: nat,N5: finite_mod_ring_a] :
( ~ ( dvd_dv7258769340395861407ring_a @ P7 @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( N
!= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ P7 @ K3 ) @ N5 ) ) ) ) ) ) ) ) ).
% divide_out_primepow
thf(fact_957_divide__out__primepow,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ N @ one_one_nat )
=> ~ ! [P7: nat] :
( ( factor1801147406995305544me_nat @ P7 )
=> ( ( dvd_dvd_nat @ P7 @ N )
=> ! [K3: nat,N5: nat] :
( ~ ( dvd_dvd_nat @ P7 @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( N
!= ( times_times_nat @ ( power_power_nat @ P7 @ K3 ) @ N5 ) ) ) ) ) ) ) ) ).
% divide_out_primepow
thf(fact_958_divide__out__primepow,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ N @ one_one_int )
=> ~ ! [P7: int] :
( ( factor1798656936486255268me_int @ P7 )
=> ( ( dvd_dvd_int @ P7 @ N )
=> ! [K3: nat,N5: int] :
( ~ ( dvd_dvd_int @ P7 @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( N
!= ( times_times_int @ ( power_power_int @ P7 @ K3 ) @ N5 ) ) ) ) ) ) ) ) ).
% divide_out_primepow
thf(fact_959_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_960_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_961_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_962_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_963_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_964_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_965_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_975_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_976_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_977_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_988_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_989_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_990_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_991_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_992_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1002_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1003_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1004_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1005_mod__add__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self1
thf(fact_1006_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_1007_mod__add__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self2
thf(fact_1008_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_1009_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_1010_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_sum__squares__eq__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1036_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_1037_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_1038_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_1039_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_1040_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_1041_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) )
= ( dvd_dv7258769340395861407ring_a @ B @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_1042_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_1043_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ B @ A ) @ ( times_5121417576591743744ring_a @ C @ A ) )
= ( dvd_dv7258769340395861407ring_a @ B @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_1044_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_1045_dvd__mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( dvd_dv7258769340395861407ring_a @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1046_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1047_dvd__mult__cancel__right,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( dvd_dv7258769340395861407ring_a @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1048_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1049_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) )
= ( dvd_dv7258769340395861407ring_a @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_1050_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_1051_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_1052_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ B @ A ) @ ( times_5121417576591743744ring_a @ C @ A ) )
= ( dvd_dv7258769340395861407ring_a @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_1053_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_1054_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_1055_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A @ one_on2109788427901206336ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C )
= ( dvd_dv7258769340395861407ring_a @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_1056_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_1057_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_1058_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ B @ one_on2109788427901206336ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C )
= ( dvd_dv7258769340395861407ring_a @ A @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_1059_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_1060_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_1061_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ one_on2109788427901206336ring_a )
= ( ( dvd_dv7258769340395861407ring_a @ A @ one_on2109788427901206336ring_a )
& ( dvd_dv7258769340395861407ring_a @ B @ one_on2109788427901206336ring_a ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_1062_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_1063_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_1064_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A @ one_on2109788427901206336ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ B @ one_on2109788427901206336ring_a )
=> ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_1065_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_1066_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_1067_algebraic__semidom__class_Ounit__prod,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A @ one_on2109788427901206336ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ B @ one_on2109788427901206336ring_a )
=> ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_1068_algebraic__semidom__class_Ounit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_1069_algebraic__semidom__class_Ounit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_1070_dvd__add__times__triv__right__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ ( times_5121417576591743744ring_a @ C @ A ) ) )
= ( dvd_dv7258769340395861407ring_a @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1071_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1072_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1073_dvd__add__times__triv__left__iff,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ A ) @ B ) )
= ( dvd_dv7258769340395861407ring_a @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1074_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1075_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1076_mod__mult__self1,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( plus_p6165643967897163644ring_a @ A @ ( times_5121417576591743744ring_a @ C @ B ) ) @ B )
= ( modulo8308552932176287283ring_a @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1077_mod__mult__self1,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1078_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1079_mod__mult__self2,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( plus_p6165643967897163644ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) @ B )
= ( modulo8308552932176287283ring_a @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1080_mod__mult__self2,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1081_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1082_mod__mult__self3,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ B ) @ A ) @ B )
= ( modulo8308552932176287283ring_a @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1083_mod__mult__self3,axiom,
! [C: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1084_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1085_mod__mult__self4,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( modulo8308552932176287283ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ B @ C ) @ A ) @ B )
= ( modulo8308552932176287283ring_a @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1086_mod__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1087_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1088_dvd__imp__mod__0,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( modulo_modulo_int @ B @ A )
= zero_zero_int ) ) ).
% dvd_imp_mod_0
thf(fact_1089_dvd__imp__mod__0,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( modulo_modulo_nat @ B @ A )
= zero_zero_nat ) ) ).
% dvd_imp_mod_0
thf(fact_1090_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_1091_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( plus_plus_nat @ M @ N ) )
= ( plus_p6165643967897163644ring_a @ ( semiri9180929696517417892ring_a @ M ) @ ( semiri9180929696517417892ring_a @ N ) ) ) ).
% of_nat_add
thf(fact_1092_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_1093_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_1094_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_1095_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_1096_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_1097_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_1098_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_1099_pow__divides__pow__iff,axiom,
! [N: nat,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) )
= ( dvd_dv7258769340395861407ring_a @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_1100_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_1101_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_1102_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_1103_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_1104_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_1105_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_1106_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_1107_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1108_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_1109_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_1110_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_1111_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1112_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1113_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1114_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1115_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_1116_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_1117_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_1118_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_1119_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1120_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1121_inf__period_I3_J,axiom,
! [D: finite_mod_ring_a,D2: finite_mod_ring_a,T: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ D @ D2 )
=> ! [X6: finite_mod_ring_a,K4: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ D @ ( plus_p6165643967897163644ring_a @ X6 @ T ) )
= ( dvd_dv7258769340395861407ring_a @ D @ ( plus_p6165643967897163644ring_a @ ( minus_3609261664126569004ring_a @ X6 @ ( times_5121417576591743744ring_a @ K4 @ D2 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_1122_inf__period_I3_J,axiom,
! [D: int,D2: int,T: int] :
( ( dvd_dvd_int @ D @ D2 )
=> ! [X6: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_1123_inf__period_I4_J,axiom,
! [D: finite_mod_ring_a,D2: finite_mod_ring_a,T: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ D @ D2 )
=> ! [X6: finite_mod_ring_a,K4: finite_mod_ring_a] :
( ( ~ ( dvd_dv7258769340395861407ring_a @ D @ ( plus_p6165643967897163644ring_a @ X6 @ T ) ) )
= ( ~ ( dvd_dv7258769340395861407ring_a @ D @ ( plus_p6165643967897163644ring_a @ ( minus_3609261664126569004ring_a @ X6 @ ( times_5121417576591743744ring_a @ K4 @ D2 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_1124_inf__period_I4_J,axiom,
! [D: int,D2: int,T: int] :
( ( dvd_dvd_int @ D @ D2 )
=> ! [X6: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_1125_unity__coeff__ex,axiom,
! [P: finite_mod_ring_a > $o,L: finite_mod_ring_a] :
( ( ? [X2: finite_mod_ring_a] : ( P @ ( times_5121417576591743744ring_a @ L @ X2 ) ) )
= ( ? [X2: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ L @ ( plus_p6165643967897163644ring_a @ X2 @ zero_z7902377541816115708ring_a ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1126_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
= ( ? [X2: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1127_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
= ( ? [X2: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1128_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D ) ) )
=> ? [X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1129_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( times_times_nat @ A @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D4 ) )
| ( ( times_times_nat @ B @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_1130_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( times_times_nat @ A @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1131_dvd__minus__add,axiom,
! [Q: nat,N: nat,R: nat,M: nat] :
( ( ord_less_eq_nat @ Q @ N )
=> ( ( ord_less_eq_nat @ Q @ ( times_times_nat @ R @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1132_cong__add__rcancel__nat,axiom,
! [X: nat,A: nat,Y2: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ X @ A ) @ ( plus_plus_nat @ Y2 @ A ) @ N )
= ( unique653641344996303876ng_nat @ X @ Y2 @ N ) ) ).
% cong_add_rcancel_nat
thf(fact_1133_cong__add__lcancel__nat,axiom,
! [A: nat,X: nat,Y2: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ A @ X ) @ ( plus_plus_nat @ A @ Y2 ) @ N )
= ( unique653641344996303876ng_nat @ X @ Y2 @ N ) ) ).
% cong_add_lcancel_nat
thf(fact_1134_cong__dvd__modulus__nat,axiom,
! [X: nat,Y2: nat,M: nat,N: nat] :
( ( unique653641344996303876ng_nat @ X @ Y2 @ M )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( unique653641344996303876ng_nat @ X @ Y2 @ N ) ) ) ).
% cong_dvd_modulus_nat
thf(fact_1135_cong__add__rcancel,axiom,
! [X: int,A: int,Y2: int,N: int] :
( ( unique651150874487253600ng_int @ ( plus_plus_int @ X @ A ) @ ( plus_plus_int @ Y2 @ A ) @ N )
= ( unique651150874487253600ng_int @ X @ Y2 @ N ) ) ).
% cong_add_rcancel
thf(fact_1136_cong__add__lcancel,axiom,
! [A: int,X: int,Y2: int,N: int] :
( ( unique651150874487253600ng_int @ ( plus_plus_int @ A @ X ) @ ( plus_plus_int @ A @ Y2 ) @ N )
= ( unique651150874487253600ng_int @ X @ Y2 @ N ) ) ).
% cong_add_lcancel
thf(fact_1137_cong__add,axiom,
! [B: nat,C: nat,A: nat,D: nat,E: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( unique653641344996303876ng_nat @ D @ E @ A )
=> ( unique653641344996303876ng_nat @ ( plus_plus_nat @ B @ D ) @ ( plus_plus_nat @ C @ E ) @ A ) ) ) ).
% cong_add
thf(fact_1138_cong__add,axiom,
! [B: int,C: int,A: int,D: int,E: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ D @ E @ A )
=> ( unique651150874487253600ng_int @ ( plus_plus_int @ B @ D ) @ ( plus_plus_int @ C @ E ) @ A ) ) ) ).
% cong_add
thf(fact_1139_cong__dvd__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( dvd_dvd_nat @ A @ B )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% cong_dvd_iff
thf(fact_1140_cong__dvd__iff,axiom,
! [B: int,C: int,A: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( dvd_dvd_int @ A @ B )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% cong_dvd_iff
thf(fact_1141_cong__dvd__modulus,axiom,
! [X: int,Y2: int,M: int,N: int] :
( ( unique651150874487253600ng_int @ X @ Y2 @ M )
=> ( ( dvd_dvd_int @ N @ M )
=> ( unique651150874487253600ng_int @ X @ Y2 @ N ) ) ) ).
% cong_dvd_modulus
thf(fact_1142_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1143_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1144_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1145_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1146_primes__dvd__imp__eq,axiom,
! [P2: nat,Q: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( dvd_dvd_nat @ P2 @ Q )
=> ( P2 = Q ) ) ) ) ).
% primes_dvd_imp_eq
thf(fact_1147_primes__dvd__imp__eq,axiom,
! [P2: int,Q: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( dvd_dvd_int @ P2 @ Q )
=> ( P2 = Q ) ) ) ) ).
% primes_dvd_imp_eq
thf(fact_1148_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_1149_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_1150_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1151_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1152_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_1153_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1154_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1155_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_1156_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_1157_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1158_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_1159_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_1160_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_1161_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_1162_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K5: nat] :
( N4
= ( plus_plus_nat @ M2 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1163_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1164_mod__add__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1165_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1166_mod__add__cong,axiom,
! [A: int,C: int,A2: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A2 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1167_mod__add__cong,axiom,
! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A2 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B2 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1168_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1169_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1170_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1171_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1172_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_1173_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_1174_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1175_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1176_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_1177_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_1178_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_1179_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_1180_dvd__mod,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ M )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_1181_dvd__mod,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_1182_mod__mod__cancel,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
= ( modulo_modulo_int @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_1183_mod__mod__cancel,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_1184_dvd__mod__iff,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_1185_dvd__mod__iff,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_1186_dvd__mod__imp__dvd,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
=> ( ( dvd_dvd_int @ C @ B )
=> ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_1187_dvd__mod__imp__dvd,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_1188_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1189_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_1190_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1191_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1192_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1193_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1194_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1195_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1196_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1197_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1198_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_1199_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1200_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1201_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1202_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1203_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1204_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_1205_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1206_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1207_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_1208_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1209_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_1210_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_1211_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X5: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y3 ) )
= D4 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y3 ) )
= D4 ) ) ) ).
% bezout1_nat
thf(fact_1212_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_1213_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_1214_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_1215_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_1216_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_1217_prime__factor__nat,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ? [P7: nat] :
( ( factor1801147406995305544me_nat @ P7 )
& ( dvd_dvd_nat @ P7 @ N ) ) ) ).
% prime_factor_nat
thf(fact_1218_prime__prime__factor,axiom,
( factor1801147406995305544me_nat
= ( ^ [N4: nat] :
( ( N4 != one_one_nat )
& ! [P3: nat] :
( ( ( factor1801147406995305544me_nat @ P3 )
& ( dvd_dvd_nat @ P3 @ N4 ) )
=> ( P3 = N4 ) ) ) ) ) ).
% prime_prime_factor
thf(fact_1219_cong__add__rcancel__0__nat,axiom,
! [X: nat,A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ X @ A ) @ A @ N )
= ( unique653641344996303876ng_nat @ X @ zero_zero_nat @ N ) ) ).
% cong_add_rcancel_0_nat
thf(fact_1220_cong__add__lcancel__0__nat,axiom,
! [A: nat,X: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ A @ X ) @ A @ N )
= ( unique653641344996303876ng_nat @ X @ zero_zero_nat @ N ) ) ).
% cong_add_lcancel_0_nat
thf(fact_1221_prime__dvd__mult__eq__nat,axiom,
! [P2: nat,A: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ P2 @ A )
| ( dvd_dvd_nat @ P2 @ B ) ) ) ) ).
% prime_dvd_mult_eq_nat
thf(fact_1222_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y2 @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y2 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1223_prime__dvd__power__nat,axiom,
! [P2: nat,X: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( dvd_dvd_nat @ P2 @ ( power_power_nat @ X @ N ) )
=> ( dvd_dvd_nat @ P2 @ X ) ) ) ).
% prime_dvd_power_nat
thf(fact_1224_cong__iff__lin__nat,axiom,
( unique653641344996303876ng_nat
= ( ^ [A3: nat,B3: nat,M2: nat] :
? [K12: nat,K22: nat] :
( ( plus_plus_nat @ B3 @ ( times_times_nat @ K12 @ M2 ) )
= ( plus_plus_nat @ A3 @ ( times_times_nat @ K22 @ M2 ) ) ) ) ) ).
% cong_iff_lin_nat
thf(fact_1225_prime__dvd__mult__eq__int,axiom,
! [P2: int,A: int,B: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ P2 @ A )
| ( dvd_dvd_int @ P2 @ B ) ) ) ) ).
% prime_dvd_mult_eq_int
thf(fact_1226_prime__dvd__power__int,axiom,
! [P2: int,X: int,N: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( dvd_dvd_int @ P2 @ ( power_power_int @ X @ N ) )
=> ( dvd_dvd_int @ P2 @ X ) ) ) ).
% prime_dvd_power_int
thf(fact_1227_nat__dvd__iff,axiom,
! [Z3: int,M: nat] :
( ( dvd_dvd_nat @ ( nat2 @ Z3 ) @ M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( dvd_dvd_int @ Z3 @ ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
thf(fact_1228_primepow__gt__0__nat,axiom,
! [N: nat] :
( ( prime_primepow_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% primepow_gt_0_nat
thf(fact_1229_primepow__multD_I2_J,axiom,
! [A: nat,B: nat] :
( ( prime_primepow_nat @ ( times_times_nat @ A @ B ) )
=> ( ( B = one_one_nat )
| ( prime_primepow_nat @ B ) ) ) ).
% primepow_multD(2)
thf(fact_1230_primepow__multD_I1_J,axiom,
! [A: nat,B: nat] :
( ( prime_primepow_nat @ ( times_times_nat @ A @ B ) )
=> ( ( A = one_one_nat )
| ( prime_primepow_nat @ A ) ) ) ).
% primepow_multD(1)
thf(fact_1231_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1232_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1233_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1234_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 ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1235_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 ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1236_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_1237_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_1238_zdvd__imp__le,axiom,
! [Z3: int,N: int] :
( ( dvd_dvd_int @ Z3 @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z3 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1239_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1240_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1241_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1242_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1243_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1244_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1245_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( ( modulo_modulo_nat @ M @ Q )
= ( modulo_modulo_nat @ N @ Q ) )
= ( dvd_dvd_nat @ Q @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1246_prime__nat__not__dvd,axiom,
! [P2: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ N @ P2 )
=> ( ( N != one_one_nat )
=> ~ ( dvd_dvd_nat @ N @ P2 ) ) ) ) ).
% prime_nat_not_dvd
thf(fact_1247_prime__nat__iff,axiom,
( factor1801147406995305544me_nat
= ( ^ [N4: nat] :
( ( ord_less_nat @ one_one_nat @ N4 )
& ! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ N4 )
=> ( ( M2 = one_one_nat )
| ( M2 = N4 ) ) ) ) ) ) ).
% prime_nat_iff
thf(fact_1248_cong__altdef__nat,axiom,
! [B: nat,A: nat,M: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( unique653641344996303876ng_nat @ A @ B @ M )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ A @ B ) ) ) ) ).
% cong_altdef_nat
thf(fact_1249_cong__to__1__nat,axiom,
! [A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ A @ one_one_nat @ N )
=> ( dvd_dvd_nat @ N @ ( minus_minus_nat @ A @ one_one_nat ) ) ) ).
% cong_to_1_nat
thf(fact_1250_mod__eq__nat2E,axiom,
! [M: nat,Q: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q )
= ( modulo_modulo_nat @ N @ Q ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1251_mod__eq__nat1E,axiom,
! [M: nat,Q: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q )
= ( modulo_modulo_nat @ N @ Q ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1252_divides__primepow__nat,axiom,
! [P2: nat,D: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( dvd_dvd_nat @ D @ ( power_power_nat @ P2 @ K ) )
= ( ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K )
& ( D
= ( power_power_nat @ P2 @ I4 ) ) ) ) ) ) ).
% divides_primepow_nat
thf(fact_1253_cong__le__nat,axiom,
! [Y2: nat,X: nat,N: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( unique653641344996303876ng_nat @ X @ Y2 @ N )
= ( ? [Q5: nat] :
( X
= ( plus_plus_nat @ ( times_times_nat @ Q5 @ N ) @ Y2 ) ) ) ) ) ).
% cong_le_nat
thf(fact_1254_prime__int__not__dvd,axiom,
! [P2: int,N: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( ord_less_int @ N @ P2 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ~ ( dvd_dvd_int @ N @ P2 ) ) ) ) ).
% prime_int_not_dvd
thf(fact_1255_primepow__power__iff__nat,axiom,
! [P2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ( prime_primepow_nat @ ( power_power_nat @ P2 @ N ) )
= ( ( prime_primepow_nat @ P2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% primepow_power_iff_nat
thf(fact_1256_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1257_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1258_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1259_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1260_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1261_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1262_mod__nat__eqI,axiom,
! [R: nat,N: nat,M: nat] :
( ( ord_less_nat @ R @ N )
=> ( ( ord_less_eq_nat @ R @ M )
=> ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
=> ( ( modulo_modulo_nat @ M @ N )
= R ) ) ) ) ).
% mod_nat_eqI
thf(fact_1263_split__mod,axiom,
! [Q2: nat > $o,M: nat,N: nat] :
( ( Q2 @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( Q2 @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) ) )
=> ( Q2 @ J3 ) ) ) ) ) ).
% split_mod
thf(fact_1264_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
= ( ( dvd_dvd_int @ L @ K )
| ( ( L = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K ) )
| ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_1265_prime__dvd__power__nat__iff,axiom,
! [P2: nat,N: nat,X: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ P2 @ ( power_power_nat @ X @ N ) )
= ( dvd_dvd_nat @ P2 @ X ) ) ) ) ).
% prime_dvd_power_nat_iff
thf(fact_1266_cong__to__1_H__nat,axiom,
! [A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ A @ one_one_nat @ N )
= ( ( ( A = zero_zero_nat )
& ( N = one_one_nat ) )
| ? [M2: nat] :
( A
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ) ).
% cong_to_1'_nat
thf(fact_1267_prime__dvd__power__int__iff,axiom,
! [P2: int,N: nat,X: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ P2 @ ( power_power_int @ X @ N ) )
= ( dvd_dvd_int @ P2 @ X ) ) ) ) ).
% prime_dvd_power_int_iff
thf(fact_1268_ord__divides,axiom,
! [A: nat,D: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ D ) @ one_one_nat @ N )
= ( dvd_dvd_nat @ ( ord_nat @ N @ A ) @ D ) ) ).
% ord_divides
thf(fact_1269_prime__power__canonical,axiom,
! [P2: nat,M: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ? [K3: nat,N2: nat] :
( ~ ( dvd_dvd_nat @ P2 @ N2 )
& ( M
= ( times_times_nat @ N2 @ ( power_power_nat @ P2 @ K3 ) ) ) ) ) ) ).
% prime_power_canonical
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ ( power_power_nat @ g @ ( minus_minus_nat @ p @ one_one_nat ) ) ) @ ( semiri1314217659103216013at_int @ p ) )
= one_one_int ) ).
%------------------------------------------------------------------------------