TPTP Problem File: SLH0454^1.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 : Fishers_Inequality/0039_Vector_Matrix_Mod/prob_00392_015174__28231838_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1375 ( 717 unt; 93 typ; 0 def)
% Number of atoms : 3143 (1384 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 10003 ( 117 ~; 65 |; 177 &;8611 @)
% ( 0 <=>;1033 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 253 ( 253 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 82 usr; 12 con; 0-3 aty)
% Number of variables : 2954 ( 62 ^;2849 !; 43 ?;2954 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:50:15.517
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Matrix__Ovec_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
vec_Fi4296597256376197294ring_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
mat_Fi443519135453058082ring_a: $tType ).
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Real__Oreal_J,type,
vec_real: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Int__Oint_J,type,
vec_int: $tType ).
thf(ty_n_t__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__itself_Itf__a_J,type,
itself_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (82)
thf(sy_c_Finite__Field_Oto__int__mod__ring_001tf__a,type,
finite1095367895020317408ring_a: finite_mod_ring_a > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
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_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
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_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
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_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
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_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
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_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
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_Oring__1__class_Oof__int_001t__Finite____Field__Omod____ring_Itf__a_J,type,
ring_18169885480643366966ring_a: int > finite_mod_ring_a ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Matrix_Omap__vec_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_ve1378714242357356997ring_a: ( finite_mod_ring_a > finite_mod_ring_a ) > vec_Fi4296597256376197294ring_a > vec_Fi4296597256376197294ring_a ).
thf(sy_c_Matrix_Omap__vec_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Int__Oint,type,
map_ve50164598268879916_a_int: ( finite_mod_ring_a > int ) > vec_Fi4296597256376197294ring_a > vec_int ).
thf(sy_c_Matrix_Omap__vec_001t__Int__Oint_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_ve1626356591056799954ring_a: ( int > finite_mod_ring_a ) > vec_int > vec_Fi4296597256376197294ring_a ).
thf(sy_c_Matrix_Omap__vec_001t__Int__Oint_001t__Int__Oint,type,
map_vec_int_int: ( int > int ) > vec_int > vec_int ).
thf(sy_c_Matrix_Omap__vec_001t__Int__Oint_001t__Real__Oreal,type,
map_vec_int_real: ( int > real ) > vec_int > vec_real ).
thf(sy_c_Matrix_Ounit__vec_001t__Finite____Field__Omod____ring_Itf__a_J,type,
unit_v6648808783362777473ring_a: nat > nat > vec_Fi4296597256376197294ring_a ).
thf(sy_c_Matrix_Ounit__vec_001t__Int__Oint,type,
unit_vec_int: nat > nat > vec_int ).
thf(sy_c_Matrix_Ozero__vec_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zero_v1122566495420860901ring_a: nat > vec_Fi4296597256376197294ring_a ).
thf(sy_c_Matrix_Ozero__vec_001t__Int__Oint,type,
zero_vec_int: nat > vec_int ).
thf(sy_c_Matrix_Ozero__vec_001t__Real__Oreal,type,
zero_vec_real: nat > vec_real ).
thf(sy_c_Nat_OSuc,type,
suc: 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_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Pure_Otype_001tf__a,type,
type_a: itself_a ).
thf(sy_c_Ring__Hom_Oinj__zero__hom_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
ring_i7582611058579704515ring_a: ( finite_mod_ring_a > finite_mod_ring_a ) > $o ).
thf(sy_c_Ring__Hom_Oinj__zero__hom_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Int__Oint,type,
ring_i3475985505635201454_a_int: ( finite_mod_ring_a > int ) > $o ).
thf(sy_c_Ring__Hom_Oinj__zero__hom_001t__Int__Oint_001t__Finite____Field__Omod____ring_Itf__a_J,type,
ring_i5052177498423121492ring_a: ( int > finite_mod_ring_a ) > $o ).
thf(sy_c_Ring__Hom_Oinj__zero__hom_001t__Int__Oint_001t__Int__Oint,type,
ring_i7091729277519141597nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Oinj__zero__hom_001t__Int__Oint_001t__Real__Oreal,type,
ring_i2058719125409607389t_real: ( int > real ) > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
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_Transfer_Obi__total_001t__Int__Oint_001t__Finite____Field__Omod____ring_Itf__a_J,type,
bi_tot623839499179674737ring_a: ( int > finite_mod_ring_a > $o ) > $o ).
thf(sy_c_Transfer_Obi__total_001t__Matrix__Omat_It__Int__Oint_J_001t__Matrix__Omat_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
bi_tot8466079176331800225ring_a: ( mat_int > mat_Fi443519135453058082ring_a > $o ) > $o ).
thf(sy_c_Transfer_Obi__total_001t__Matrix__Ovec_It__Int__Oint_J_001t__Matrix__Ovec_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
bi_tot310863827317176225ring_a: ( vec_int > vec_Fi4296597256376197294ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oleft__total_001t__Int__Oint_001t__Finite____Field__Omod____ring_Itf__a_J,type,
left_t8941605147251654769ring_a: ( int > finite_mod_ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oleft__total_001t__Matrix__Omat_It__Int__Oint_J_001t__Matrix__Omat_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
left_t6774590509652353697ring_a: ( mat_int > mat_Fi443519135453058082ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oleft__total_001t__Matrix__Ovec_It__Int__Oint_J_001t__Matrix__Ovec_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
left_t7842747197492505505ring_a: ( vec_int > vec_Fi4296597256376197294ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oright__total_001t__Int__Oint_001t__Finite____Field__Omod____ring_Itf__a_J,type,
right_2909008500229168952ring_a: ( int > finite_mod_ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oright__total_001t__Matrix__Omat_It__Int__Oint_J_001t__Matrix__Omat_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
right_6272992325622558440ring_a: ( mat_int > mat_Fi443519135453058082ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oright__total_001t__Matrix__Ovec_It__Int__Oint_J_001t__Matrix__Ovec_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
right_7341149013462710248ring_a: ( vec_int > vec_Fi4296597256376197294ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oright__unique_001t__Int__Oint_001t__Finite____Field__Omod____ring_Itf__a_J,type,
right_7581627888863641665ring_a: ( int > finite_mod_ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oright__unique_001t__Matrix__Omat_It__Int__Oint_J_001t__Matrix__Omat_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
right_1339649294516398705ring_a: ( mat_int > mat_Fi443519135453058082ring_a > $o ) > $o ).
thf(sy_c_Transfer_Oright__unique_001t__Matrix__Ovec_It__Int__Oint_J_001t__Matrix__Ovec_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
right_2407805982356550513ring_a: ( vec_int > vec_Fi4296597256376197294ring_a > $o ) > $o ).
thf(sy_c_Vector__Matrix__Mod_Omat__mod,type,
vector2311527483241216340at_mod: int > $o ).
thf(sy_c_Vector__Matrix__Mod_Omat__mod_Ovec__mod,type,
vector1199560857299316530ec_mod: int > vec_int > vec_int ).
thf(sy_c_Vector__Matrix__Mod_Omat__mod__type_001tf__a,type,
vector2974885785920725647type_a: itself_a > int > $o ).
thf(sy_c_Vector__Matrix__Mod_Omat__mod__type_OMM__Rel_001tf__a,type,
vector2798151085771547010_Rel_a: int > mat_int > mat_Fi443519135453058082ring_a > $o ).
thf(sy_c_Vector__Matrix__Mod_Omat__mod__type_OMV__Rel_001tf__a,type,
vector5230790432342831627_Rel_a: int > vec_int > vec_Fi4296597256376197294ring_a > $o ).
thf(sy_c_Vector__Matrix__Mod_Omod__type_001tf__a,type,
vector4745807456731380595type_a: itself_a > int > $o ).
thf(sy_c_Vector__Matrix__Mod_Omod__type_OM,type,
vector_Matrix_mod_M: int > int > int ).
thf(sy_c_Vector__Matrix__Mod_Omod__type_OM__Rel_001tf__a,type,
vector3024231343486012919_Rel_a: int > int > finite_mod_ring_a > $o ).
thf(sy_c_Vector__Matrix__Mod_Omod__type_Oinv__M,type,
vector2349909385534816566_inv_M: int > int > int ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_v_m,type,
m: int ).
% Relevant facts (1276)
thf(fact_0_mat__mod__type_OMV__Rel_Ocong,axiom,
vector5230790432342831627_Rel_a = vector5230790432342831627_Rel_a ).
% mat_mod_type.MV_Rel.cong
thf(fact_1_right__unique__MV__Rel,axiom,
right_2407805982356550513ring_a @ ( vector5230790432342831627_Rel_a @ m ) ).
% right_unique_MV_Rel
thf(fact_2_mat__mod__axioms,axiom,
vector2311527483241216340at_mod @ m ).
% mat_mod_axioms
thf(fact_3_right__totalE,axiom,
! [A: vec_int > vec_Fi4296597256376197294ring_a > $o,Y: vec_Fi4296597256376197294ring_a] :
( ( right_7341149013462710248ring_a @ A )
=> ~ ! [X: vec_int] :
~ ( A @ X @ Y ) ) ).
% right_totalE
thf(fact_4_right__totalE,axiom,
! [A: int > finite_mod_ring_a > $o,Y: finite_mod_ring_a] :
( ( right_2909008500229168952ring_a @ A )
=> ~ ! [X: int] :
~ ( A @ X @ Y ) ) ).
% right_totalE
thf(fact_5_right__totalE,axiom,
! [A: mat_int > mat_Fi443519135453058082ring_a > $o,Y: mat_Fi443519135453058082ring_a] :
( ( right_6272992325622558440ring_a @ A )
=> ~ ! [X: mat_int] :
~ ( A @ X @ Y ) ) ).
% right_totalE
thf(fact_6_right__totalI,axiom,
! [A: vec_int > vec_Fi4296597256376197294ring_a > $o] :
( ! [Y2: vec_Fi4296597256376197294ring_a] :
? [X2: vec_int] : ( A @ X2 @ Y2 )
=> ( right_7341149013462710248ring_a @ A ) ) ).
% right_totalI
thf(fact_7_right__totalI,axiom,
! [A: int > finite_mod_ring_a > $o] :
( ! [Y2: finite_mod_ring_a] :
? [X2: int] : ( A @ X2 @ Y2 )
=> ( right_2909008500229168952ring_a @ A ) ) ).
% right_totalI
thf(fact_8_right__totalI,axiom,
! [A: mat_int > mat_Fi443519135453058082ring_a > $o] :
( ! [Y2: mat_Fi443519135453058082ring_a] :
? [X2: mat_int] : ( A @ X2 @ Y2 )
=> ( right_6272992325622558440ring_a @ A ) ) ).
% right_totalI
thf(fact_9_right__total__def,axiom,
( right_7341149013462710248ring_a
= ( ^ [R: vec_int > vec_Fi4296597256376197294ring_a > $o] :
! [Y3: vec_Fi4296597256376197294ring_a] :
? [X3: vec_int] : ( R @ X3 @ Y3 ) ) ) ).
% right_total_def
thf(fact_10_right__total__def,axiom,
( right_2909008500229168952ring_a
= ( ^ [R: int > finite_mod_ring_a > $o] :
! [Y3: finite_mod_ring_a] :
? [X3: int] : ( R @ X3 @ Y3 ) ) ) ).
% right_total_def
thf(fact_11_right__total__def,axiom,
( right_6272992325622558440ring_a
= ( ^ [R: mat_int > mat_Fi443519135453058082ring_a > $o] :
! [Y3: mat_Fi443519135453058082ring_a] :
? [X3: mat_int] : ( R @ X3 @ Y3 ) ) ) ).
% right_total_def
thf(fact_12_one__MV__Rel,axiom,
! [N: nat,I: nat] : ( vector5230790432342831627_Rel_a @ m @ ( unit_vec_int @ N @ I ) @ ( unit_v6648808783362777473ring_a @ N @ I ) ) ).
% one_MV_Rel
thf(fact_13_mat__mod__type__axioms,axiom,
vector2974885785920725647type_a @ type_a @ m ).
% mat_mod_type_axioms
thf(fact_14_M__M,axiom,
! [X4: int] :
( ( vector_Matrix_mod_M @ m @ ( vector_Matrix_mod_M @ m @ X4 ) )
= ( vector_Matrix_mod_M @ m @ X4 ) ) ).
% M_M
thf(fact_15_zero__MV__Rel,axiom,
! [N: nat] : ( vector5230790432342831627_Rel_a @ m @ ( zero_vec_int @ N ) @ ( zero_v1122566495420860901ring_a @ N ) ) ).
% zero_MV_Rel
thf(fact_16_left__total__M__Rel,axiom,
left_t8941605147251654769ring_a @ ( vector3024231343486012919_Rel_a @ m ) ).
% left_total_M_Rel
thf(fact_17_vec__mod__unit,axiom,
! [N: nat,I: nat] :
( ( vector1199560857299316530ec_mod @ m @ ( unit_vec_int @ N @ I ) )
= ( unit_vec_int @ N @ I ) ) ).
% vec_mod_unit
thf(fact_18_right__total__M__Rel,axiom,
right_2909008500229168952ring_a @ ( vector3024231343486012919_Rel_a @ m ) ).
% right_total_M_Rel
thf(fact_19_vec__mod__zero,axiom,
! [N: nat] :
( ( vector1199560857299316530ec_mod @ m @ ( zero_vec_int @ N ) )
= ( zero_vec_int @ N ) ) ).
% vec_mod_zero
thf(fact_20_mat__mod__type_Oaxioms_I1_J,axiom,
! [M: int] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( vector2311527483241216340at_mod @ M ) ) ).
% mat_mod_type.axioms(1)
thf(fact_21_mod__type_OM_Ocong,axiom,
vector_Matrix_mod_M = vector_Matrix_mod_M ).
% mod_type.M.cong
thf(fact_22_mat__mod_Ovec__mod_Ocong,axiom,
vector1199560857299316530ec_mod = vector1199560857299316530ec_mod ).
% mat_mod.vec_mod.cong
thf(fact_23_mat__mod_Ovec__mod__unit,axiom,
! [M: int,N: nat,I: nat] :
( ( vector2311527483241216340at_mod @ M )
=> ( ( vector1199560857299316530ec_mod @ M @ ( unit_vec_int @ N @ I ) )
= ( unit_vec_int @ N @ I ) ) ) ).
% mat_mod.vec_mod_unit
thf(fact_24_mat__mod_Ovec__mod__zero,axiom,
! [M: int,N: nat] :
( ( vector2311527483241216340at_mod @ M )
=> ( ( vector1199560857299316530ec_mod @ M @ ( zero_vec_int @ N ) )
= ( zero_vec_int @ N ) ) ) ).
% mat_mod.vec_mod_zero
thf(fact_25_mat__mod__type_Oone__MV__Rel,axiom,
! [M: int,N: nat,I: nat] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( vector5230790432342831627_Rel_a @ M @ ( unit_vec_int @ N @ I ) @ ( unit_v6648808783362777473ring_a @ N @ I ) ) ) ).
% mat_mod_type.one_MV_Rel
thf(fact_26_mat__mod__type_Ozero__MV__Rel,axiom,
! [M: int,N: nat] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( vector5230790432342831627_Rel_a @ M @ ( zero_vec_int @ N ) @ ( zero_v1122566495420860901ring_a @ N ) ) ) ).
% mat_mod_type.zero_MV_Rel
thf(fact_27_left__totalE,axiom,
! [R2: int > finite_mod_ring_a > $o] :
( ( left_t8941605147251654769ring_a @ R2 )
=> ! [X2: int] :
? [X_1: finite_mod_ring_a] : ( R2 @ X2 @ X_1 ) ) ).
% left_totalE
thf(fact_28_left__totalI,axiom,
! [R2: int > finite_mod_ring_a > $o] :
( ! [X: int] :
? [X_12: finite_mod_ring_a] : ( R2 @ X @ X_12 )
=> ( left_t8941605147251654769ring_a @ R2 ) ) ).
% left_totalI
thf(fact_29_right__uniqueD,axiom,
! [A: vec_int > vec_Fi4296597256376197294ring_a > $o,X4: vec_int,Y: vec_Fi4296597256376197294ring_a,Z: vec_Fi4296597256376197294ring_a] :
( ( right_2407805982356550513ring_a @ A )
=> ( ( A @ X4 @ Y )
=> ( ( A @ X4 @ Z )
=> ( Y = Z ) ) ) ) ).
% right_uniqueD
thf(fact_30_right__uniqueD,axiom,
! [A: mat_int > mat_Fi443519135453058082ring_a > $o,X4: mat_int,Y: mat_Fi443519135453058082ring_a,Z: mat_Fi443519135453058082ring_a] :
( ( right_1339649294516398705ring_a @ A )
=> ( ( A @ X4 @ Y )
=> ( ( A @ X4 @ Z )
=> ( Y = Z ) ) ) ) ).
% right_uniqueD
thf(fact_31_right__uniqueI,axiom,
! [A: vec_int > vec_Fi4296597256376197294ring_a > $o] :
( ! [X: vec_int,Y2: vec_Fi4296597256376197294ring_a,Z2: vec_Fi4296597256376197294ring_a] :
( ( A @ X @ Y2 )
=> ( ( A @ X @ Z2 )
=> ( Y2 = Z2 ) ) )
=> ( right_2407805982356550513ring_a @ A ) ) ).
% right_uniqueI
thf(fact_32_right__uniqueI,axiom,
! [A: mat_int > mat_Fi443519135453058082ring_a > $o] :
( ! [X: mat_int,Y2: mat_Fi443519135453058082ring_a,Z2: mat_Fi443519135453058082ring_a] :
( ( A @ X @ Y2 )
=> ( ( A @ X @ Z2 )
=> ( Y2 = Z2 ) ) )
=> ( right_1339649294516398705ring_a @ A ) ) ).
% right_uniqueI
thf(fact_33_left__total__def,axiom,
( left_t8941605147251654769ring_a
= ( ^ [R: int > finite_mod_ring_a > $o] :
! [X3: int] :
? [X5: finite_mod_ring_a] : ( R @ X3 @ X5 ) ) ) ).
% left_total_def
thf(fact_34_right__unique__def,axiom,
( right_2407805982356550513ring_a
= ( ^ [R: vec_int > vec_Fi4296597256376197294ring_a > $o] :
! [X3: vec_int,Y3: vec_Fi4296597256376197294ring_a,Z3: vec_Fi4296597256376197294ring_a] :
( ( R @ X3 @ Y3 )
=> ( ( R @ X3 @ Z3 )
=> ( Y3 = Z3 ) ) ) ) ) ).
% right_unique_def
thf(fact_35_right__unique__def,axiom,
( right_1339649294516398705ring_a
= ( ^ [R: mat_int > mat_Fi443519135453058082ring_a > $o] :
! [X3: mat_int,Y3: mat_Fi443519135453058082ring_a,Z3: mat_Fi443519135453058082ring_a] :
( ( R @ X3 @ Y3 )
=> ( ( R @ X3 @ Z3 )
=> ( Y3 = Z3 ) ) ) ) ) ).
% right_unique_def
thf(fact_36_mat__mod__type_Oright__unique__MV__Rel,axiom,
! [M: int] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( right_2407805982356550513ring_a @ ( vector5230790432342831627_Rel_a @ M ) ) ) ).
% mat_mod_type.right_unique_MV_Rel
thf(fact_37_mod__type_OM__Rel_Ocong,axiom,
vector3024231343486012919_Rel_a = vector3024231343486012919_Rel_a ).
% mod_type.M_Rel.cong
thf(fact_38_mod__type__axioms,axiom,
vector4745807456731380595type_a @ type_a @ m ).
% mod_type_axioms
thf(fact_39_M__inv__M__id,axiom,
! [X4: int] :
( ( vector_Matrix_mod_M @ m @ ( vector2349909385534816566_inv_M @ m @ X4 ) )
= ( vector_Matrix_mod_M @ m @ X4 ) ) ).
% M_inv_M_id
thf(fact_40_right__total__MM__Rel,axiom,
right_6272992325622558440ring_a @ ( vector2798151085771547010_Rel_a @ m ) ).
% right_total_MM_Rel
thf(fact_41_M__Rel__def,axiom,
! [X4: int,X6: finite_mod_ring_a] :
( ( vector3024231343486012919_Rel_a @ m @ X4 @ X6 )
= ( ( vector_Matrix_mod_M @ m @ X4 )
= ( finite1095367895020317408ring_a @ X6 ) ) ) ).
% M_Rel_def
thf(fact_42_functional__relation,axiom,
! [R2: int > finite_mod_ring_a > $o] :
( ( right_7581627888863641665ring_a @ R2 )
=> ( ( left_t8941605147251654769ring_a @ R2 )
=> ! [X2: int] :
? [Xa: finite_mod_ring_a] :
( ( R2 @ X2 @ Xa )
& ! [Y4: finite_mod_ring_a] :
( ( R2 @ X2 @ Y4 )
=> ( Y4 = Xa ) ) ) ) ) ).
% functional_relation
thf(fact_43_functional__relation,axiom,
! [R2: vec_int > vec_Fi4296597256376197294ring_a > $o] :
( ( right_2407805982356550513ring_a @ R2 )
=> ( ( left_t7842747197492505505ring_a @ R2 )
=> ! [X2: vec_int] :
? [Xa: vec_Fi4296597256376197294ring_a] :
( ( R2 @ X2 @ Xa )
& ! [Y4: vec_Fi4296597256376197294ring_a] :
( ( R2 @ X2 @ Y4 )
=> ( Y4 = Xa ) ) ) ) ) ).
% functional_relation
thf(fact_44_functional__relation,axiom,
! [R2: mat_int > mat_Fi443519135453058082ring_a > $o] :
( ( right_1339649294516398705ring_a @ R2 )
=> ( ( left_t6774590509652353697ring_a @ R2 )
=> ! [X2: mat_int] :
? [Xa: mat_Fi443519135453058082ring_a] :
( ( R2 @ X2 @ Xa )
& ! [Y4: mat_Fi443519135453058082ring_a] :
( ( R2 @ X2 @ Y4 )
=> ( Y4 = Xa ) ) ) ) ) ).
% functional_relation
thf(fact_45_bi__total__M__Rel,axiom,
bi_tot623839499179674737ring_a @ ( vector3024231343486012919_Rel_a @ m ) ).
% bi_total_M_Rel
thf(fact_46_M__def,axiom,
! [X4: int] :
( ( vector_Matrix_mod_M @ m @ X4 )
= ( modulo_modulo_int @ X4 @ m ) ) ).
% M_def
thf(fact_47_M__to__int__mod__ring,axiom,
! [X4: finite_mod_ring_a] :
( ( vector_Matrix_mod_M @ m @ ( finite1095367895020317408ring_a @ X4 ) )
= ( finite1095367895020317408ring_a @ X4 ) ) ).
% M_to_int_mod_ring
thf(fact_48_mod__type_Oleft__total__M__Rel,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( left_t8941605147251654769ring_a @ ( vector3024231343486012919_Rel_a @ M ) ) ) ).
% mod_type.left_total_M_Rel
thf(fact_49_mat__mod__type_Ointro,axiom,
! [M: int] :
( ( vector2311527483241216340at_mod @ M )
=> ( ( vector4745807456731380595type_a @ type_a @ M )
=> ( vector2974885785920725647type_a @ type_a @ M ) ) ) ).
% mat_mod_type.intro
thf(fact_50_mat__mod__type__def,axiom,
! [M: int] :
( ( vector2974885785920725647type_a @ type_a @ M )
= ( ( vector2311527483241216340at_mod @ M )
& ( vector4745807456731380595type_a @ type_a @ M ) ) ) ).
% mat_mod_type_def
thf(fact_51_M__0,axiom,
( ( vector_Matrix_mod_M @ m @ zero_zero_int )
= zero_zero_int ) ).
% M_0
thf(fact_52_right__unique__MM__Rel,axiom,
right_1339649294516398705ring_a @ ( vector2798151085771547010_Rel_a @ m ) ).
% right_unique_MM_Rel
thf(fact_53_mod__to__int__mod__ring,axiom,
! [X4: finite_mod_ring_a] :
( ( modulo_modulo_int @ ( finite1095367895020317408ring_a @ X4 ) @ m )
= ( finite1095367895020317408ring_a @ X4 ) ) ).
% mod_to_int_mod_ring
thf(fact_54_bi__total__def,axiom,
( bi_tot623839499179674737ring_a
= ( ^ [R: int > finite_mod_ring_a > $o] :
( ! [X3: int] :
? [X5: finite_mod_ring_a] : ( R @ X3 @ X5 )
& ! [Y3: finite_mod_ring_a] :
? [X3: int] : ( R @ X3 @ Y3 ) ) ) ) ).
% bi_total_def
thf(fact_55_mat__mod__type_OMM__Rel_Ocong,axiom,
vector2798151085771547010_Rel_a = vector2798151085771547010_Rel_a ).
% mat_mod_type.MM_Rel.cong
thf(fact_56_mod__type_Oinv__M_Ocong,axiom,
vector2349909385534816566_inv_M = vector2349909385534816566_inv_M ).
% mod_type.inv_M.cong
thf(fact_57_mat__mod__type_Omod__to__int__mod__ring,axiom,
! [M: int,X4: finite_mod_ring_a] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( ( modulo_modulo_int @ ( finite1095367895020317408ring_a @ X4 ) @ M )
= ( finite1095367895020317408ring_a @ X4 ) ) ) ).
% mat_mod_type.mod_to_int_mod_ring
thf(fact_58_mod__type_OM__to__int__mod__ring,axiom,
! [M: int,X4: finite_mod_ring_a] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ ( finite1095367895020317408ring_a @ X4 ) )
= ( finite1095367895020317408ring_a @ X4 ) ) ) ).
% mod_type.M_to_int_mod_ring
thf(fact_59_mod__type_OM__def,axiom,
! [M: int,X4: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ X4 )
= ( modulo_modulo_int @ X4 @ M ) ) ) ).
% mod_type.M_def
thf(fact_60_mod__type_Obi__total__M__Rel,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( bi_tot623839499179674737ring_a @ ( vector3024231343486012919_Rel_a @ M ) ) ) ).
% mod_type.bi_total_M_Rel
thf(fact_61_mod__type_OM__0,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ zero_zero_int )
= zero_zero_int ) ) ).
% mod_type.M_0
thf(fact_62_mat__mod__type_Oright__unique__MM__Rel,axiom,
! [M: int] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( right_1339649294516398705ring_a @ ( vector2798151085771547010_Rel_a @ M ) ) ) ).
% mat_mod_type.right_unique_MM_Rel
thf(fact_63_mod__type_OM__inv__M__id,axiom,
! [M: int,X4: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ ( vector2349909385534816566_inv_M @ M @ X4 ) )
= ( vector_Matrix_mod_M @ M @ X4 ) ) ) ).
% mod_type.M_inv_M_id
thf(fact_64_mod__type_OM__Rel__def,axiom,
! [M: int,X4: int,X6: finite_mod_ring_a] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector3024231343486012919_Rel_a @ M @ X4 @ X6 )
= ( ( vector_Matrix_mod_M @ M @ X4 )
= ( finite1095367895020317408ring_a @ X6 ) ) ) ) ).
% mod_type.M_Rel_def
thf(fact_65_bi__total__alt__def,axiom,
( bi_tot310863827317176225ring_a
= ( ^ [A2: vec_int > vec_Fi4296597256376197294ring_a > $o] :
( ( left_t7842747197492505505ring_a @ A2 )
& ( right_7341149013462710248ring_a @ A2 ) ) ) ) ).
% bi_total_alt_def
thf(fact_66_bi__total__alt__def,axiom,
( bi_tot623839499179674737ring_a
= ( ^ [A2: int > finite_mod_ring_a > $o] :
( ( left_t8941605147251654769ring_a @ A2 )
& ( right_2909008500229168952ring_a @ A2 ) ) ) ) ).
% bi_total_alt_def
thf(fact_67_bi__total__alt__def,axiom,
( bi_tot8466079176331800225ring_a
= ( ^ [A2: mat_int > mat_Fi443519135453058082ring_a > $o] :
( ( left_t6774590509652353697ring_a @ A2 )
& ( right_6272992325622558440ring_a @ A2 ) ) ) ) ).
% bi_total_alt_def
thf(fact_68_bi__totalI,axiom,
! [R2: vec_int > vec_Fi4296597256376197294ring_a > $o] :
( ( left_t7842747197492505505ring_a @ R2 )
=> ( ( right_7341149013462710248ring_a @ R2 )
=> ( bi_tot310863827317176225ring_a @ R2 ) ) ) ).
% bi_totalI
thf(fact_69_bi__totalI,axiom,
! [R2: int > finite_mod_ring_a > $o] :
( ( left_t8941605147251654769ring_a @ R2 )
=> ( ( right_2909008500229168952ring_a @ R2 )
=> ( bi_tot623839499179674737ring_a @ R2 ) ) ) ).
% bi_totalI
thf(fact_70_bi__totalI,axiom,
! [R2: mat_int > mat_Fi443519135453058082ring_a > $o] :
( ( left_t6774590509652353697ring_a @ R2 )
=> ( ( right_6272992325622558440ring_a @ R2 )
=> ( bi_tot8466079176331800225ring_a @ R2 ) ) ) ).
% bi_totalI
thf(fact_71_mod__type_OM__M,axiom,
! [M: int,X4: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ ( vector_Matrix_mod_M @ M @ X4 ) )
= ( vector_Matrix_mod_M @ M @ X4 ) ) ) ).
% mod_type.M_M
thf(fact_72_mat__mod__type_Oaxioms_I2_J,axiom,
! [M: int] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( vector4745807456731380595type_a @ type_a @ M ) ) ).
% mat_mod_type.axioms(2)
thf(fact_73_mod__type_Oright__total__M__Rel,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( right_2909008500229168952ring_a @ ( vector3024231343486012919_Rel_a @ M ) ) ) ).
% mod_type.right_total_M_Rel
thf(fact_74_mat__mod__type_Oright__total__MM__Rel,axiom,
! [M: int] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( right_6272992325622558440ring_a @ ( vector2798151085771547010_Rel_a @ M ) ) ) ).
% mat_mod_type.right_total_MM_Rel
thf(fact_75_bits__mod__0,axiom,
! [A3: int] :
( ( modulo_modulo_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_76_bits__mod__0,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_77_mod__self,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ A3 )
= zero_zero_int ) ).
% mod_self
thf(fact_78_mod__self,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% mod_self
thf(fact_79_mod__by__0,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ zero_zero_int )
= A3 ) ).
% mod_by_0
thf(fact_80_mod__by__0,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% mod_by_0
thf(fact_81_mod__0,axiom,
! [A3: int] :
( ( modulo_modulo_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mod_0
thf(fact_82_mod__0,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mod_0
thf(fact_83_zero__M__Rel,axiom,
vector3024231343486012919_Rel_a @ m @ zero_zero_int @ zero_z7902377541816115708ring_a ).
% zero_M_Rel
thf(fact_84_MV__Rel__def,axiom,
! [V: vec_int,V2: vec_Fi4296597256376197294ring_a] :
( ( vector5230790432342831627_Rel_a @ m @ V @ V2 )
= ( ( vector1199560857299316530ec_mod @ m @ V )
= ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ V2 ) ) ) ).
% MV_Rel_def
thf(fact_85_to__int__mod__ring__of__int__M,axiom,
! [X4: int] :
( ( finite1095367895020317408ring_a @ ( ring_18169885480643366966ring_a @ X4 ) )
= ( vector_Matrix_mod_M @ m @ X4 ) ) ).
% to_int_mod_ring_of_int_M
thf(fact_86_vec__mod__to__int__vec,axiom,
! [V: vec_Fi4296597256376197294ring_a] :
( ( vector1199560857299316530ec_mod @ m @ ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ V ) )
= ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ V ) ) ).
% vec_mod_to_int_vec
thf(fact_87_to__int__mod__ring__hom_Oeq__iff,axiom,
! [X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X4 )
= ( finite1095367895020317408ring_a @ Y ) )
= ( X4 = Y ) ) ).
% to_int_mod_ring_hom.eq_iff
thf(fact_88_mod__mod__trivial,axiom,
! [A3: int,B: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A3 @ B ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% mod_mod_trivial
thf(fact_89_mod__mod__trivial,axiom,
! [A3: nat,B: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A3 @ B ) @ B )
= ( modulo_modulo_nat @ A3 @ B ) ) ).
% mod_mod_trivial
thf(fact_90_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_91_to__int__mod__ring__hom_Ohom__0__iff,axiom,
! [X4: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X4 )
= zero_zero_int )
= ( X4 = zero_z7902377541816115708ring_a ) ) ).
% to_int_mod_ring_hom.hom_0_iff
thf(fact_92_to__int__mod__ring__hom_Ovec__hom__zero__iff,axiom,
! [X4: vec_Fi4296597256376197294ring_a,N: nat] :
( ( ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ X4 )
= ( zero_vec_int @ N ) )
= ( X4
= ( zero_v1122566495420860901ring_a @ N ) ) ) ).
% to_int_mod_ring_hom.vec_hom_zero_iff
thf(fact_93_to__int__mod__ring__hom_Ovec__hom__inj,axiom,
! [V: vec_Fi4296597256376197294ring_a,W: vec_Fi4296597256376197294ring_a] :
( ( ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ V )
= ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ W ) )
=> ( V = W ) ) ).
% to_int_mod_ring_hom.vec_hom_inj
thf(fact_94_to__int__mod__ring__hom_Ohom__0,axiom,
! [X4: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X4 )
= zero_zero_int )
=> ( X4 = zero_z7902377541816115708ring_a ) ) ).
% to_int_mod_ring_hom.hom_0
thf(fact_95_to__int__mod__ring__hom_Oinjectivity,axiom,
! [X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X4 )
= ( finite1095367895020317408ring_a @ Y ) )
=> ( X4 = Y ) ) ).
% to_int_mod_ring_hom.injectivity
thf(fact_96_mat__mod__type_Ovec__mod__to__int__vec,axiom,
! [M: int,V: vec_Fi4296597256376197294ring_a] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( ( vector1199560857299316530ec_mod @ M @ ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ V ) )
= ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ V ) ) ) ).
% mat_mod_type.vec_mod_to_int_vec
thf(fact_97_mod__type_Oto__int__mod__ring__of__int__M,axiom,
! [M: int,X4: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( finite1095367895020317408ring_a @ ( ring_18169885480643366966ring_a @ X4 ) )
= ( vector_Matrix_mod_M @ M @ X4 ) ) ) ).
% mod_type.to_int_mod_ring_of_int_M
thf(fact_98_mod__type_Ozero__M__Rel,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( vector3024231343486012919_Rel_a @ M @ zero_zero_int @ zero_z7902377541816115708ring_a ) ) ).
% mod_type.zero_M_Rel
thf(fact_99_mat__mod__type_OMV__Rel__def,axiom,
! [M: int,V: vec_int,V2: vec_Fi4296597256376197294ring_a] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( ( vector5230790432342831627_Rel_a @ M @ V @ V2 )
= ( ( vector1199560857299316530ec_mod @ M @ V )
= ( map_ve50164598268879916_a_int @ finite1095367895020317408ring_a @ V2 ) ) ) ) ).
% mat_mod_type.MV_Rel_def
thf(fact_100_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_hom.hom_zero
thf(fact_101_of__int__hom_Ohom__zero,axiom,
( ( ring_18169885480643366966ring_a @ zero_zero_int )
= zero_z7902377541816115708ring_a ) ).
% of_int_hom.hom_zero
thf(fact_102_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_hom.hom_zero
thf(fact_103_of__int__hom_Ohom__0__iff,axiom,
! [X4: int] :
( ( ( ring_1_of_int_int @ X4 )
= zero_zero_int )
= ( X4 = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_104_of__int__hom_Ohom__0__iff,axiom,
! [X4: int] :
( ( ( ring_1_of_int_real @ X4 )
= zero_zero_real )
= ( X4 = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_105_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_106_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_107_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_108_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_109_of__int__hom_Ohom__0,axiom,
! [X4: int] :
( ( ( ring_1_of_int_int @ X4 )
= zero_zero_int )
=> ( X4 = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_110_of__int__hom_Ohom__0,axiom,
! [X4: int] :
( ( ( ring_1_of_int_real @ X4 )
= zero_zero_real )
=> ( X4 = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_111_Ring__Hom_Oof__int__hom_Oeq__iff,axiom,
! [X4: int,Y: int] :
( ( ( ring_1_of_int_real @ X4 )
= ( ring_1_of_int_real @ Y ) )
= ( X4 = Y ) ) ).
% Ring_Hom.of_int_hom.eq_iff
thf(fact_112_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_113_Matrix__Vector__Extras_Oof__int__hom_Oeq__iff,axiom,
! [X4: int,Univ: set_int,Y: int] :
( ( member_int @ X4 @ Univ )
=> ( ( member_int @ Y @ Univ )
=> ( ( ( ring_1_of_int_real @ X4 )
= ( ring_1_of_int_real @ Y ) )
= ( X4 = Y ) ) ) ) ).
% Matrix_Vector_Extras.of_int_hom.eq_iff
thf(fact_114_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_115_zero__reorient,axiom,
! [X4: finite_mod_ring_a] :
( ( zero_z7902377541816115708ring_a = X4 )
= ( X4 = zero_z7902377541816115708ring_a ) ) ).
% zero_reorient
thf(fact_116_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_117_zero__reorient,axiom,
! [X4: real] :
( ( zero_zero_real = X4 )
= ( X4 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_118_of__int__hom_Oinjectivity__lim,axiom,
! [X4: int,Univ: set_int,Y: int] :
( ( member_int @ X4 @ Univ )
=> ( ( member_int @ Y @ Univ )
=> ( ( ( ring_1_of_int_real @ X4 )
= ( ring_1_of_int_real @ Y ) )
=> ( X4 = Y ) ) ) ) ).
% of_int_hom.injectivity_lim
thf(fact_119_of__int__hom_Oinjectivity,axiom,
! [X4: int,Y: int] :
( ( ( ring_1_of_int_real @ X4 )
= ( ring_1_of_int_real @ Y ) )
=> ( X4 = Y ) ) ).
% of_int_hom.injectivity
thf(fact_120_eucl__induct,axiom,
! [P: int > int > $o,A3: int,B: int] :
( ! [B2: int] : ( P @ B2 @ zero_zero_int )
=> ( ! [A4: int,B2: int] :
( ( B2 != zero_zero_int )
=> ( ( P @ B2 @ ( modulo_modulo_int @ A4 @ B2 ) )
=> ( P @ A4 @ B2 ) ) )
=> ( P @ A3 @ B ) ) ) ).
% eucl_induct
thf(fact_121_eucl__induct,axiom,
! [P: nat > nat > $o,A3: nat,B: nat] :
( ! [B2: nat] : ( P @ B2 @ zero_zero_nat )
=> ( ! [A4: nat,B2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( P @ B2 @ ( modulo_modulo_nat @ A4 @ B2 ) )
=> ( P @ A4 @ B2 ) ) )
=> ( P @ A3 @ B ) ) ) ).
% eucl_induct
thf(fact_122_Matrix__Vector__Extras_Oof__int__hom_Ovec__hom__zero__iff,axiom,
! [X4: vec_int,N: nat] :
( ( ( map_vec_int_int @ ring_1_of_int_int @ X4 )
= ( zero_vec_int @ N ) )
= ( X4
= ( zero_vec_int @ N ) ) ) ).
% Matrix_Vector_Extras.of_int_hom.vec_hom_zero_iff
thf(fact_123_Matrix__Vector__Extras_Oof__int__hom_Ovec__hom__zero__iff,axiom,
! [X4: vec_int,N: nat] :
( ( ( map_vec_int_real @ ring_1_of_int_real @ X4 )
= ( zero_vec_real @ N ) )
= ( X4
= ( zero_vec_int @ N ) ) ) ).
% Matrix_Vector_Extras.of_int_hom.vec_hom_zero_iff
thf(fact_124_M__1,axiom,
( ( vector_Matrix_mod_M @ m @ one_one_int )
= one_one_int ) ).
% M_1
thf(fact_125_mod__type_OM__1,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ one_one_int )
= one_one_int ) ) ).
% mod_type.M_1
thf(fact_126_M__times_I2_J,axiom,
! [X4: int,Y: int] :
( ( vector_Matrix_mod_M @ m @ ( times_times_int @ X4 @ ( vector_Matrix_mod_M @ m @ Y ) ) )
= ( vector_Matrix_mod_M @ m @ ( times_times_int @ X4 @ Y ) ) ) ).
% M_times(2)
thf(fact_127_M__times_I1_J,axiom,
! [X4: int,Y: int] :
( ( vector_Matrix_mod_M @ m @ ( times_times_int @ ( vector_Matrix_mod_M @ m @ X4 ) @ Y ) )
= ( vector_Matrix_mod_M @ m @ ( times_times_int @ X4 @ Y ) ) ) ).
% M_times(1)
thf(fact_128_mod__type_OM__times_I1_J,axiom,
! [M: int,X4: int,Y: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ ( times_times_int @ ( vector_Matrix_mod_M @ M @ X4 ) @ Y ) )
= ( vector_Matrix_mod_M @ M @ ( times_times_int @ X4 @ Y ) ) ) ) ).
% mod_type.M_times(1)
thf(fact_129_mod__type_OM__times_I2_J,axiom,
! [M: int,X4: int,Y: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ ( times_times_int @ X4 @ ( vector_Matrix_mod_M @ M @ Y ) ) )
= ( vector_Matrix_mod_M @ M @ ( times_times_int @ X4 @ Y ) ) ) ) ).
% mod_type.M_times(2)
thf(fact_130_Matrix__Vector__Extras_Oinj__zero__hom_Ovec__hom__zero__iff,axiom,
! [Hom: int > int,X4: vec_int,N: nat] :
( ( ring_i7091729277519141597nt_int @ Hom )
=> ( ( ( map_vec_int_int @ Hom @ X4 )
= ( zero_vec_int @ N ) )
= ( X4
= ( zero_vec_int @ N ) ) ) ) ).
% Matrix_Vector_Extras.inj_zero_hom.vec_hom_zero_iff
thf(fact_131_Matrix__Vector__Extras_Oinj__zero__hom_Ovec__hom__zero__iff,axiom,
! [Hom: finite_mod_ring_a > int,X4: vec_Fi4296597256376197294ring_a,N: nat] :
( ( ring_i3475985505635201454_a_int @ Hom )
=> ( ( ( map_ve50164598268879916_a_int @ Hom @ X4 )
= ( zero_vec_int @ N ) )
= ( X4
= ( zero_v1122566495420860901ring_a @ N ) ) ) ) ).
% Matrix_Vector_Extras.inj_zero_hom.vec_hom_zero_iff
thf(fact_132_Matrix__Vector__Extras_Oinj__zero__hom_Ovec__hom__zero__iff,axiom,
! [Hom: int > finite_mod_ring_a,X4: vec_int,N: nat] :
( ( ring_i5052177498423121492ring_a @ Hom )
=> ( ( ( map_ve1626356591056799954ring_a @ Hom @ X4 )
= ( zero_v1122566495420860901ring_a @ N ) )
= ( X4
= ( zero_vec_int @ N ) ) ) ) ).
% Matrix_Vector_Extras.inj_zero_hom.vec_hom_zero_iff
thf(fact_133_Matrix__Vector__Extras_Oinj__zero__hom_Ovec__hom__zero__iff,axiom,
! [Hom: finite_mod_ring_a > finite_mod_ring_a,X4: vec_Fi4296597256376197294ring_a,N: nat] :
( ( ring_i7582611058579704515ring_a @ Hom )
=> ( ( ( map_ve1378714242357356997ring_a @ Hom @ X4 )
= ( zero_v1122566495420860901ring_a @ N ) )
= ( X4
= ( zero_v1122566495420860901ring_a @ N ) ) ) ) ).
% Matrix_Vector_Extras.inj_zero_hom.vec_hom_zero_iff
thf(fact_134_M__plus_I2_J,axiom,
! [X4: int,Y: int] :
( ( vector_Matrix_mod_M @ m @ ( plus_plus_int @ X4 @ ( vector_Matrix_mod_M @ m @ Y ) ) )
= ( vector_Matrix_mod_M @ m @ ( plus_plus_int @ X4 @ Y ) ) ) ).
% M_plus(2)
thf(fact_135_M__plus_I1_J,axiom,
! [X4: int,Y: int] :
( ( vector_Matrix_mod_M @ m @ ( plus_plus_int @ ( vector_Matrix_mod_M @ m @ X4 ) @ Y ) )
= ( vector_Matrix_mod_M @ m @ ( plus_plus_int @ X4 @ Y ) ) ) ).
% M_plus(1)
thf(fact_136_add__right__cancel,axiom,
! [B: int,A3: int,C: int] :
( ( ( plus_plus_int @ B @ A3 )
= ( plus_plus_int @ C @ A3 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_137_add__right__cancel,axiom,
! [B: finite_mod_ring_a,A3: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A3 )
= ( plus_p6165643967897163644ring_a @ C @ A3 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_138_add__right__cancel,axiom,
! [B: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_139_add__right__cancel,axiom,
! [B: real,A3: real,C: real] :
( ( ( plus_plus_real @ B @ A3 )
= ( plus_plus_real @ C @ A3 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_140_add__left__cancel,axiom,
! [A3: int,B: int,C: int] :
( ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ A3 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_141_add__left__cancel,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A3 @ B )
= ( plus_p6165643967897163644ring_a @ A3 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_142_add__left__cancel,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ A3 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_143_add__left__cancel,axiom,
! [A3: real,B: real,C: real] :
( ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ A3 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_144_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_145_mult__zero__left,axiom,
! [A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a @ A3 )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_left
thf(fact_146_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_147_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_148_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_149_mult__zero__right,axiom,
! [A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_right
thf(fact_150_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_151_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_152_mult__eq__0__iff,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_153_mult__eq__0__iff,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_154_mult__eq__0__iff,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_155_mult__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_156_mult__cancel__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_157_mult__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_158_mult__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_159_mult__cancel__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_160_mult__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_161_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_162_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_163_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_164_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_165_add_Oright__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.right_neutral
thf(fact_166_add_Oright__neutral,axiom,
! [A3: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A3 @ zero_z7902377541816115708ring_a )
= A3 ) ).
% add.right_neutral
thf(fact_167_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_168_add_Oright__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.right_neutral
thf(fact_169_double__zero__sym,axiom,
! [A3: int] :
( ( zero_zero_int
= ( plus_plus_int @ A3 @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_170_double__zero__sym,axiom,
! [A3: real] :
( ( zero_zero_real
= ( plus_plus_real @ A3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_171_add__cancel__left__left,axiom,
! [B: int,A3: int] :
( ( ( plus_plus_int @ B @ A3 )
= A3 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_172_add__cancel__left__left,axiom,
! [B: finite_mod_ring_a,A3: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A3 )
= A3 )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_left_left
thf(fact_173_add__cancel__left__left,axiom,
! [B: nat,A3: nat] :
( ( ( plus_plus_nat @ B @ A3 )
= A3 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_174_add__cancel__left__left,axiom,
! [B: real,A3: real] :
( ( ( plus_plus_real @ B @ A3 )
= A3 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_175_add__cancel__left__right,axiom,
! [A3: int,B: int] :
( ( ( plus_plus_int @ A3 @ B )
= A3 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_176_add__cancel__left__right,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A3 @ B )
= A3 )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_left_right
thf(fact_177_add__cancel__left__right,axiom,
! [A3: nat,B: nat] :
( ( ( plus_plus_nat @ A3 @ B )
= A3 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_178_add__cancel__left__right,axiom,
! [A3: real,B: real] :
( ( ( plus_plus_real @ A3 @ B )
= A3 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_179_add__cancel__right__left,axiom,
! [A3: int,B: int] :
( ( A3
= ( plus_plus_int @ B @ A3 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_180_add__cancel__right__left,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A3
= ( plus_p6165643967897163644ring_a @ B @ A3 ) )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_right_left
thf(fact_181_add__cancel__right__left,axiom,
! [A3: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ B @ A3 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_182_add__cancel__right__left,axiom,
! [A3: real,B: real] :
( ( A3
= ( plus_plus_real @ B @ A3 ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_183_add__cancel__right__right,axiom,
! [A3: int,B: int] :
( ( A3
= ( plus_plus_int @ A3 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_184_add__cancel__right__right,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A3
= ( plus_p6165643967897163644ring_a @ A3 @ B ) )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_right_right
thf(fact_185_add__cancel__right__right,axiom,
! [A3: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_186_add__cancel__right__right,axiom,
! [A3: real,B: real] :
( ( A3
= ( plus_plus_real @ A3 @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_187_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_188_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y ) )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_189_add__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add_0
thf(fact_190_add__0,axiom,
! [A3: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A3 )
= A3 ) ).
% add_0
thf(fact_191_add__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add_0
thf(fact_192_add__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add_0
thf(fact_193_double__eq__0__iff,axiom,
! [A3: int] :
( ( ( plus_plus_int @ A3 @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_194_double__eq__0__iff,axiom,
! [A3: real] :
( ( ( plus_plus_real @ A3 @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_195_mult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% mult_1
thf(fact_196_mult__1,axiom,
! [A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A3 )
= A3 ) ).
% mult_1
thf(fact_197_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_198_mult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% mult_1
thf(fact_199_mult_Oright__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.right_neutral
thf(fact_200_mult_Oright__neutral,axiom,
! [A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ one_on2109788427901206336ring_a )
= A3 ) ).
% mult.right_neutral
thf(fact_201_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_202_mult_Oright__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.right_neutral
thf(fact_203_mod__add__self1,axiom,
! [B: int,A3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A3 ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% mod_add_self1
thf(fact_204_mod__add__self1,axiom,
! [B: nat,A3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A3 ) @ B )
= ( modulo_modulo_nat @ A3 @ B ) ) ).
% mod_add_self1
thf(fact_205_mod__add__self2,axiom,
! [A3: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% mod_add_self2
thf(fact_206_mod__add__self2,axiom,
! [A3: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B ) @ B )
= ( modulo_modulo_nat @ A3 @ B ) ) ).
% mod_add_self2
thf(fact_207_mult__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ( times_times_int @ A3 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_208_mult__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ( times_times_real @ A3 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_209_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_210_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_211_mult__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ( times_times_int @ C @ A3 )
= C )
= ( ( C = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_212_mult__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ( times_times_real @ C @ A3 )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_213_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_214_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_215_mod__mult__self1__is__0,axiom,
! [B: int,A3: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A3 ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_216_mod__mult__self1__is__0,axiom,
! [B: nat,A3: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A3 ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_217_mod__mult__self2__is__0,axiom,
! [A3: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A3 @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_218_mod__mult__self2__is__0,axiom,
! [A3: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_219_mod__by__1,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_220_mod__by__1,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_221_bits__mod__by__1,axiom,
! [A3: int] :
( ( modulo_modulo_int @ A3 @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_222_bits__mod__by__1,axiom,
! [A3: nat] :
( ( modulo_modulo_nat @ A3 @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_223_mod__mult__self4,axiom,
! [B: int,C: int,A3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A3 ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% mod_mult_self4
thf(fact_224_mod__mult__self4,axiom,
! [B: nat,C: nat,A3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A3 ) @ B )
= ( modulo_modulo_nat @ A3 @ B ) ) ).
% mod_mult_self4
thf(fact_225_mod__mult__self3,axiom,
! [C: int,B: int,A3: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A3 ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% mod_mult_self3
thf(fact_226_mod__mult__self3,axiom,
! [C: nat,B: nat,A3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A3 ) @ B )
= ( modulo_modulo_nat @ A3 @ B ) ) ).
% mod_mult_self3
thf(fact_227_mod__mult__self2,axiom,
! [A3: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( times_times_int @ B @ C ) ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% mod_mult_self2
thf(fact_228_mod__mult__self2,axiom,
! [A3: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A3 @ B ) ) ).
% mod_mult_self2
thf(fact_229_mod__mult__self1,axiom,
! [A3: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( times_times_int @ C @ B ) ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% mod_mult_self1
thf(fact_230_mod__mult__self1,axiom,
! [A3: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A3 @ B ) ) ).
% mod_mult_self1
thf(fact_231_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_232_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_233_of__int__hom_Ohom__1__iff,axiom,
! [X4: int] :
( ( ( ring_1_of_int_int @ X4 )
= one_one_int )
= ( X4 = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_234_of__int__hom_Ohom__1__iff,axiom,
! [X4: int] :
( ( ( ring_1_of_int_real @ X4 )
= one_one_real )
= ( X4 = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_235_of__int__hom_Ohom__one,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_hom.hom_one
thf(fact_236_of__int__hom_Ohom__one,axiom,
( ( ring_18169885480643366966ring_a @ one_one_int )
= one_on2109788427901206336ring_a ) ).
% of_int_hom.hom_one
thf(fact_237_of__int__hom_Ohom__one,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_hom.hom_one
thf(fact_238_of__int__hom_Ohom__mult,axiom,
! [X4: int,Y: int] :
( ( ring_1_of_int_int @ ( times_times_int @ X4 @ Y ) )
= ( times_times_int @ ( ring_1_of_int_int @ X4 ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_239_of__int__hom_Ohom__mult,axiom,
! [X4: int,Y: int] :
( ( ring_18169885480643366966ring_a @ ( times_times_int @ X4 @ Y ) )
= ( times_5121417576591743744ring_a @ ( ring_18169885480643366966ring_a @ X4 ) @ ( ring_18169885480643366966ring_a @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_240_of__int__hom_Ohom__mult,axiom,
! [X4: int,Y: int] :
( ( ring_1_of_int_real @ ( times_times_int @ X4 @ Y ) )
= ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_241_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_242_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_18169885480643366966ring_a @ ( times_times_int @ W @ Z ) )
= ( times_5121417576591743744ring_a @ ( ring_18169885480643366966ring_a @ W ) @ ( ring_18169885480643366966ring_a @ Z ) ) ) ).
% of_int_mult
thf(fact_243_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_244_of__int__hom_Ohom__add,axiom,
! [X4: int,Y: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ X4 @ Y ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ X4 ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_add
thf(fact_245_of__int__hom_Ohom__add,axiom,
! [X4: int,Y: int] :
( ( ring_18169885480643366966ring_a @ ( plus_plus_int @ X4 @ Y ) )
= ( plus_p6165643967897163644ring_a @ ( ring_18169885480643366966ring_a @ X4 ) @ ( ring_18169885480643366966ring_a @ Y ) ) ) ).
% of_int_hom.hom_add
thf(fact_246_of__int__hom_Ohom__add,axiom,
! [X4: int,Y: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ Y ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_add
thf(fact_247_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_248_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_18169885480643366966ring_a @ ( plus_plus_int @ W @ Z ) )
= ( plus_p6165643967897163644ring_a @ ( ring_18169885480643366966ring_a @ W ) @ ( ring_18169885480643366966ring_a @ Z ) ) ) ).
% of_int_add
thf(fact_249_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_add
thf(fact_250_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_251_mult__hom_Ohom__add__eq__zero,axiom,
! [X4: int,Y: int,C: int] :
( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( times_times_int @ C @ X4 ) @ ( times_times_int @ C @ Y ) )
= zero_zero_int ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_252_mult__hom_Ohom__add__eq__zero,axiom,
! [X4: finite_mod_ring_a,Y: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ X4 @ Y )
= zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ X4 ) @ ( times_5121417576591743744ring_a @ C @ Y ) )
= zero_z7902377541816115708ring_a ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_253_mult__hom_Ohom__add__eq__zero,axiom,
! [X4: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X4 ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_254_mult__hom_Ohom__add__eq__zero,axiom,
! [X4: real,Y: real,C: real] :
( ( ( plus_plus_real @ X4 @ Y )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C @ X4 ) @ ( times_times_real @ C @ Y ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_255_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_256_one__reorient,axiom,
! [X4: finite_mod_ring_a] :
( ( one_on2109788427901206336ring_a = X4 )
= ( X4 = one_on2109788427901206336ring_a ) ) ).
% one_reorient
thf(fact_257_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_258_one__reorient,axiom,
! [X4: real] :
( ( one_one_real = X4 )
= ( X4 = one_one_real ) ) ).
% one_reorient
thf(fact_259_add__right__imp__eq,axiom,
! [B: int,A3: int,C: int] :
( ( ( plus_plus_int @ B @ A3 )
= ( plus_plus_int @ C @ A3 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_260_add__right__imp__eq,axiom,
! [B: finite_mod_ring_a,A3: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A3 )
= ( plus_p6165643967897163644ring_a @ C @ A3 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_261_add__right__imp__eq,axiom,
! [B: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_262_add__right__imp__eq,axiom,
! [B: real,A3: real,C: real] :
( ( ( plus_plus_real @ B @ A3 )
= ( plus_plus_real @ C @ A3 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_263_add__left__imp__eq,axiom,
! [A3: int,B: int,C: int] :
( ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ A3 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_264_add__left__imp__eq,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A3 @ B )
= ( plus_p6165643967897163644ring_a @ A3 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_265_add__left__imp__eq,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_266_add__left__imp__eq,axiom,
! [A3: real,B: real,C: real] :
( ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ A3 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_267_mult_Oleft__commute,axiom,
! [B: int,A3: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A3 @ C ) )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_268_mult_Oleft__commute,axiom,
! [B: finite_mod_ring_a,A3: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ B @ ( times_5121417576591743744ring_a @ A3 @ C ) )
= ( times_5121417576591743744ring_a @ A3 @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_269_mult_Oleft__commute,axiom,
! [B: nat,A3: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A3 @ C ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_270_mult_Oleft__commute,axiom,
! [B: real,A3: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A3 @ C ) )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_271_mult_Ocomm__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.comm_neutral
thf(fact_272_mult_Ocomm__neutral,axiom,
! [A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ one_on2109788427901206336ring_a )
= A3 ) ).
% mult.comm_neutral
thf(fact_273_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_274_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_275_add_Oleft__commute,axiom,
! [B: int,A3: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A3 @ C ) )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_276_add_Oleft__commute,axiom,
! [B: finite_mod_ring_a,A3: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ B @ ( plus_p6165643967897163644ring_a @ A3 @ C ) )
= ( plus_p6165643967897163644ring_a @ A3 @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_277_add_Oleft__commute,axiom,
! [B: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_278_add_Oleft__commute,axiom,
! [B: real,A3: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A3 @ C ) )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_279_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B3: int] : ( times_times_int @ B3 @ A5 ) ) ) ).
% mult.commute
thf(fact_280_mult_Ocommute,axiom,
( times_5121417576591743744ring_a
= ( ^ [A5: finite_mod_ring_a,B3: finite_mod_ring_a] : ( times_5121417576591743744ring_a @ B3 @ A5 ) ) ) ).
% mult.commute
thf(fact_281_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B3: nat] : ( times_times_nat @ B3 @ A5 ) ) ) ).
% mult.commute
thf(fact_282_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B3: real] : ( times_times_real @ B3 @ A5 ) ) ) ).
% mult.commute
thf(fact_283_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B3: int] : ( plus_plus_int @ B3 @ A5 ) ) ) ).
% add.commute
thf(fact_284_add_Ocommute,axiom,
( plus_p6165643967897163644ring_a
= ( ^ [A5: finite_mod_ring_a,B3: finite_mod_ring_a] : ( plus_p6165643967897163644ring_a @ B3 @ A5 ) ) ) ).
% add.commute
thf(fact_285_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B3: nat] : ( plus_plus_nat @ B3 @ A5 ) ) ) ).
% add.commute
thf(fact_286_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A5: real,B3: real] : ( plus_plus_real @ B3 @ A5 ) ) ) ).
% add.commute
thf(fact_287_add_Oright__cancel,axiom,
! [B: int,A3: int,C: int] :
( ( ( plus_plus_int @ B @ A3 )
= ( plus_plus_int @ C @ A3 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_288_add_Oright__cancel,axiom,
! [B: finite_mod_ring_a,A3: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A3 )
= ( plus_p6165643967897163644ring_a @ C @ A3 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_289_add_Oright__cancel,axiom,
! [B: real,A3: real,C: real] :
( ( ( plus_plus_real @ B @ A3 )
= ( plus_plus_real @ C @ A3 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_290_mult_Oassoc,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_291_mult_Oassoc,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A3 @ B ) @ C )
= ( times_5121417576591743744ring_a @ A3 @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_292_mult_Oassoc,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_293_mult_Oassoc,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B ) @ C )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_294_add_Oleft__cancel,axiom,
! [A3: int,B: int,C: int] :
( ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ A3 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_295_add_Oleft__cancel,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A3 @ B )
= ( plus_p6165643967897163644ring_a @ A3 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_296_add_Oleft__cancel,axiom,
! [A3: real,B: real,C: real] :
( ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ A3 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_297_add_Oassoc,axiom,
! [A3: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_298_add_Oassoc,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A3 @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_299_add_Oassoc,axiom,
! [A3: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_300_add_Oassoc,axiom,
! [A3: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_301_comm__monoid__mult__class_Omult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_302_comm__monoid__mult__class_Omult__1,axiom,
! [A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_303_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_304_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_305_mult__hom_Ohom__add,axiom,
! [C: int,X4: int,Y: int] :
( ( times_times_int @ C @ ( plus_plus_int @ X4 @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C @ X4 ) @ ( times_times_int @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_306_mult__hom_Ohom__add,axiom,
! [C: finite_mod_ring_a,X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ C @ ( plus_p6165643967897163644ring_a @ X4 @ Y ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ X4 ) @ ( times_5121417576591743744ring_a @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_307_mult__hom_Ohom__add,axiom,
! [C: nat,X4: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X4 @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X4 ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_308_mult__hom_Ohom__add,axiom,
! [C: real,X4: real,Y: real] :
( ( times_times_real @ C @ ( plus_plus_real @ X4 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C @ X4 ) @ ( times_times_real @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_309_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A3: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A3 @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_310_group__cancel_Oadd2,axiom,
! [B4: finite_mod_ring_a,K: finite_mod_ring_a,B: finite_mod_ring_a,A3: finite_mod_ring_a] :
( ( B4
= ( plus_p6165643967897163644ring_a @ K @ B ) )
=> ( ( plus_p6165643967897163644ring_a @ A3 @ B4 )
= ( plus_p6165643967897163644ring_a @ K @ ( plus_p6165643967897163644ring_a @ A3 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_311_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A3: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A3 @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_312_group__cancel_Oadd2,axiom,
! [B4: real,K: real,B: real,A3: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A3 @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_313_group__cancel_Oadd1,axiom,
! [A: int,K: int,A3: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A3 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_314_group__cancel_Oadd1,axiom,
! [A: finite_mod_ring_a,K: finite_mod_ring_a,A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A
= ( plus_p6165643967897163644ring_a @ K @ A3 ) )
=> ( ( plus_p6165643967897163644ring_a @ A @ B )
= ( plus_p6165643967897163644ring_a @ K @ ( plus_p6165643967897163644ring_a @ A3 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_315_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_316_group__cancel_Oadd1,axiom,
! [A: real,K: real,A3: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A3 ) )
=> ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_317_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_318_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_319_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_320_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_321_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A3 @ B ) @ C )
= ( times_5121417576591743744ring_a @ A3 @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_322_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_323_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B ) @ C )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_324_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_325_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A3 @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_326_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_327_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_328_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_329_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_330_mod__eqE,axiom,
! [A3: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ B @ C ) )
=> ~ ! [D: int] :
( B
!= ( plus_plus_int @ A3 @ ( times_times_int @ C @ D ) ) ) ) ).
% mod_eqE
thf(fact_331_combine__common__factor,axiom,
! [A3: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A3 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_332_combine__common__factor,axiom,
! [A3: finite_mod_ring_a,E: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ E ) @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ B @ E ) @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_333_combine__common__factor,axiom,
! [A3: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A3 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_334_combine__common__factor,axiom,
! [A3: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A3 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_335_distrib__right,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_336_distrib__right,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% distrib_right
thf(fact_337_distrib__right,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_338_distrib__right,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_339_distrib__left,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B ) @ ( times_times_int @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_340_distrib__left,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ B ) @ ( times_5121417576591743744ring_a @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_341_distrib__left,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ A3 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ B ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_342_distrib__left,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B ) @ ( times_times_real @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_343_comm__semiring__class_Odistrib,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_344_comm__semiring__class_Odistrib,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_345_comm__semiring__class_Odistrib,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_346_comm__semiring__class_Odistrib,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_347_ring__class_Oring__distribs_I1_J,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B ) @ ( times_times_int @ A3 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_348_ring__class_Oring__distribs_I1_J,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ B ) @ ( times_5121417576591743744ring_a @ A3 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_349_ring__class_Oring__distribs_I1_J,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B ) @ ( times_times_real @ A3 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_350_ring__class_Oring__distribs_I2_J,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_351_ring__class_Oring__distribs_I2_J,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_352_ring__class_Oring__distribs_I2_J,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_353_of__int__hom_Ohom__1,axiom,
! [X4: int] :
( ( ( ring_1_of_int_int @ X4 )
= one_one_int )
=> ( X4 = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_354_of__int__hom_Ohom__1,axiom,
! [X4: int] :
( ( ( ring_1_of_int_real @ X4 )
= one_one_real )
=> ( X4 = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_355_Matrix__Vector__Extras_Oof__int__hom_Ovec__hom__inj,axiom,
! [V: vec_int,W: vec_int] :
( ( ( map_vec_int_real @ ring_1_of_int_real @ V )
= ( map_vec_int_real @ ring_1_of_int_real @ W ) )
=> ( V = W ) ) ).
% Matrix_Vector_Extras.of_int_hom.vec_hom_inj
thf(fact_356_comm__monoid__add__class_Oadd__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_357_comm__monoid__add__class_Oadd__0,axiom,
! [A3: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_358_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_359_comm__monoid__add__class_Oadd__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_360_add_Ocomm__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.comm_neutral
thf(fact_361_add_Ocomm__neutral,axiom,
! [A3: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A3 @ zero_z7902377541816115708ring_a )
= A3 ) ).
% add.comm_neutral
thf(fact_362_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_363_add_Ocomm__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.comm_neutral
thf(fact_364_add_Ogroup__left__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_365_add_Ogroup__left__neutral,axiom,
! [A3: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_366_add_Ogroup__left__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_367_mult__not__zero,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_368_mult__not__zero,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A3 @ B )
!= zero_z7902377541816115708ring_a )
=> ( ( A3 != zero_z7902377541816115708ring_a )
& ( B != zero_z7902377541816115708ring_a ) ) ) ).
% mult_not_zero
thf(fact_369_mult__not__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_370_mult__not__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_371_divisors__zero,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_372_divisors__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_373_divisors__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_374_no__zero__divisors,axiom,
! [A3: int,B: int] :
( ( A3 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A3 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_375_no__zero__divisors,axiom,
! [A3: nat,B: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_376_no__zero__divisors,axiom,
! [A3: real,B: real] :
( ( A3 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A3 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_377_mult__left__cancel,axiom,
! [C: int,A3: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_378_mult__left__cancel,axiom,
! [C: nat,A3: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_379_mult__left__cancel,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_380_mult__right__cancel,axiom,
! [C: int,A3: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_381_mult__right__cancel,axiom,
! [C: nat,A3: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_382_mult__right__cancel,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_383_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_384_zero__neq__one,axiom,
zero_z7902377541816115708ring_a != one_on2109788427901206336ring_a ).
% zero_neq_one
thf(fact_385_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_386_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_387_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_388_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_389_mod__add__eq,axiom,
! [A3: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_390_mod__add__eq,axiom,
! [A3: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_391_mod__add__cong,axiom,
! [A3: int,C: int,A6: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ A6 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_392_mod__add__cong,axiom,
! [A3: nat,C: nat,A6: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A3 @ C )
= ( modulo_modulo_nat @ A6 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_393_mod__add__left__eq,axiom,
! [A3: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_394_mod__add__left__eq,axiom,
! [A3: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_395_mod__add__right__eq,axiom,
! [A3: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_396_mod__add__right__eq,axiom,
! [A3: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_397_mult__of__int__commute,axiom,
! [X4: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X4 ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X4 ) ) ) ).
% mult_of_int_commute
thf(fact_398_mult__of__int__commute,axiom,
! [X4: int,Y: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( ring_18169885480643366966ring_a @ X4 ) @ Y )
= ( times_5121417576591743744ring_a @ Y @ ( ring_18169885480643366966ring_a @ X4 ) ) ) ).
% mult_of_int_commute
thf(fact_399_mult__of__int__commute,axiom,
! [X4: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X4 ) ) ) ).
% mult_of_int_commute
thf(fact_400_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_401_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_402_mod__mult__eq,axiom,
! [A3: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_403_mod__mult__eq,axiom,
! [A3: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_404_mod__mult__cong,axiom,
! [A3: int,C: int,A6: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A3 @ C )
= ( modulo_modulo_int @ A6 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A3 @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A6 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_405_mod__mult__cong,axiom,
! [A3: nat,C: nat,A6: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A3 @ C )
= ( modulo_modulo_nat @ A6 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A6 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_406_mod__mult__mult2,axiom,
! [A3: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A3 @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_407_mod__mult__mult2,axiom,
! [A3: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A3 @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_408_mult__mod__right,axiom,
! [C: int,A3: int,B: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A3 @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_409_mult__mod__right,axiom,
! [C: nat,A3: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A3 @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_410_mod__mult__left__eq,axiom,
! [A3: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A3 @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_411_mod__mult__left__eq,axiom,
! [A3: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A3 @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_412_mod__mult__right__eq,axiom,
! [A3: int,B: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A3 @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A3 @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_413_mod__mult__right__eq,axiom,
! [A3: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_414_of__int__hom_Ohom__add__eq__zero,axiom,
! [X4: int,Y: int] :
( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( ring_1_of_int_int @ X4 ) @ ( ring_1_of_int_int @ Y ) )
= zero_zero_int ) ) ).
% of_int_hom.hom_add_eq_zero
thf(fact_415_of__int__hom_Ohom__add__eq__zero,axiom,
! [X4: int,Y: int] :
( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
=> ( ( plus_p6165643967897163644ring_a @ ( ring_18169885480643366966ring_a @ X4 ) @ ( ring_18169885480643366966ring_a @ Y ) )
= zero_z7902377541816115708ring_a ) ) ).
% of_int_hom.hom_add_eq_zero
thf(fact_416_of__int__hom_Ohom__add__eq__zero,axiom,
! [X4: int,Y: int] :
( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
=> ( ( plus_plus_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ Y ) )
= zero_zero_real ) ) ).
% of_int_hom.hom_add_eq_zero
thf(fact_417_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X4: int,Y: int] :
( ( ( times_times_int @ X4 @ Y )
= zero_zero_int )
=> ( ( times_times_int @ ( ring_1_of_int_int @ X4 ) @ ( ring_1_of_int_int @ Y ) )
= zero_zero_int ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_418_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X4: int,Y: int] :
( ( ( times_times_int @ X4 @ Y )
= zero_zero_int )
=> ( ( times_5121417576591743744ring_a @ ( ring_18169885480643366966ring_a @ X4 ) @ ( ring_18169885480643366966ring_a @ Y ) )
= zero_z7902377541816115708ring_a ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_419_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X4: int,Y: int] :
( ( ( times_times_int @ X4 @ Y )
= zero_zero_int )
=> ( ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ Y ) )
= zero_zero_real ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_420_of__int__hom_Oinj__zero__hom__axioms,axiom,
ring_i2058719125409607389t_real @ ring_1_of_int_real ).
% of_int_hom.inj_zero_hom_axioms
thf(fact_421_to__int__mod__ring__hom_Oinj__zero__hom__axioms,axiom,
ring_i3475985505635201454_a_int @ finite1095367895020317408ring_a ).
% to_int_mod_ring_hom.inj_zero_hom_axioms
thf(fact_422_Matrix__Vector__Extras_Oinj__zero__hom_Ovec__hom__inj,axiom,
! [Hom: finite_mod_ring_a > int,V: vec_Fi4296597256376197294ring_a,W: vec_Fi4296597256376197294ring_a] :
( ( ring_i3475985505635201454_a_int @ Hom )
=> ( ( ( map_ve50164598268879916_a_int @ Hom @ V )
= ( map_ve50164598268879916_a_int @ Hom @ W ) )
=> ( V = W ) ) ) ).
% Matrix_Vector_Extras.inj_zero_hom.vec_hom_inj
thf(fact_423_mod__type_OM__plus_I2_J,axiom,
! [M: int,X4: int,Y: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ ( plus_plus_int @ X4 @ ( vector_Matrix_mod_M @ M @ Y ) ) )
= ( vector_Matrix_mod_M @ M @ ( plus_plus_int @ X4 @ Y ) ) ) ) ).
% mod_type.M_plus(2)
thf(fact_424_mod__type_OM__plus_I1_J,axiom,
! [M: int,X4: int,Y: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( vector_Matrix_mod_M @ M @ ( plus_plus_int @ ( vector_Matrix_mod_M @ M @ X4 ) @ Y ) )
= ( vector_Matrix_mod_M @ M @ ( plus_plus_int @ X4 @ Y ) ) ) ) ).
% mod_type.M_plus(1)
thf(fact_425_sum__squares__eq__zero__iff,axiom,
! [X4: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_426_sum__squares__eq__zero__iff,axiom,
! [X4: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_427_ideal_Oscale__one,axiom,
! [X4: int] :
( ( times_times_int @ one_one_int @ X4 )
= X4 ) ).
% ideal.scale_one
thf(fact_428_ideal_Oscale__one,axiom,
! [X4: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ X4 )
= X4 ) ).
% ideal.scale_one
thf(fact_429_ideal_Oscale__one,axiom,
! [X4: real] :
( ( times_times_real @ one_one_real @ X4 )
= X4 ) ).
% ideal.scale_one
thf(fact_430_of__int__hom_Ovec__hom__zero,axiom,
! [N: nat] :
( ( map_vec_int_int @ ring_1_of_int_int @ ( zero_vec_int @ N ) )
= ( zero_vec_int @ N ) ) ).
% of_int_hom.vec_hom_zero
thf(fact_431_of__int__hom_Ovec__hom__zero,axiom,
! [N: nat] :
( ( map_ve1626356591056799954ring_a @ ring_18169885480643366966ring_a @ ( zero_vec_int @ N ) )
= ( zero_v1122566495420860901ring_a @ N ) ) ).
% of_int_hom.vec_hom_zero
thf(fact_432_of__int__hom_Ovec__hom__zero,axiom,
! [N: nat] :
( ( map_vec_int_real @ ring_1_of_int_real @ ( zero_vec_int @ N ) )
= ( zero_vec_real @ N ) ) ).
% of_int_hom.vec_hom_zero
thf(fact_433_ideal_Oscale__zero__left,axiom,
! [X4: int] :
( ( times_times_int @ zero_zero_int @ X4 )
= zero_zero_int ) ).
% ideal.scale_zero_left
thf(fact_434_ideal_Oscale__zero__left,axiom,
! [X4: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a @ X4 )
= zero_z7902377541816115708ring_a ) ).
% ideal.scale_zero_left
thf(fact_435_ideal_Oscale__zero__left,axiom,
! [X4: real] :
( ( times_times_real @ zero_zero_real @ X4 )
= zero_zero_real ) ).
% ideal.scale_zero_left
thf(fact_436_ideal_Oscale__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% ideal.scale_zero_right
thf(fact_437_ideal_Oscale__zero__right,axiom,
! [A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% ideal.scale_zero_right
thf(fact_438_ideal_Oscale__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% ideal.scale_zero_right
thf(fact_439_to__int__mod__ring__times,axiom,
! [X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( finite1095367895020317408ring_a @ ( times_5121417576591743744ring_a @ X4 @ Y ) )
= ( modulo_modulo_int @ ( times_times_int @ ( finite1095367895020317408ring_a @ X4 ) @ ( finite1095367895020317408ring_a @ Y ) ) @ m ) ) ).
% to_int_mod_ring_times
thf(fact_440_to__int__mod__ring__plus,axiom,
! [X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( finite1095367895020317408ring_a @ ( plus_p6165643967897163644ring_a @ X4 @ Y ) )
= ( modulo_modulo_int @ ( plus_plus_int @ ( finite1095367895020317408ring_a @ X4 ) @ ( finite1095367895020317408ring_a @ Y ) ) @ m ) ) ).
% to_int_mod_ring_plus
thf(fact_441_one__M__Rel,axiom,
vector3024231343486012919_Rel_a @ m @ one_one_int @ one_on2109788427901206336ring_a ).
% one_M_Rel
thf(fact_442_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo_int @ M @ D2 )
= zero_zero_int )
=> ? [Q: int] :
( M
= ( times_times_int @ D2 @ Q ) ) ) ).
% zmod_eq_0D
thf(fact_443_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_444_to__int__mod__ring__hom_Ohom__1__iff,axiom,
! [X4: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X4 )
= one_one_int )
= ( X4 = one_on2109788427901206336ring_a ) ) ).
% to_int_mod_ring_hom.hom_1_iff
thf(fact_445_to__int__mod__ring__hom_Ohom__1,axiom,
! [X4: finite_mod_ring_a] :
( ( ( finite1095367895020317408ring_a @ X4 )
= one_one_int )
=> ( X4 = one_on2109788427901206336ring_a ) ) ).
% to_int_mod_ring_hom.hom_1
thf(fact_446_ideal_Oscale__left__commute,axiom,
! [A3: int,B: int,X4: int] :
( ( times_times_int @ A3 @ ( times_times_int @ B @ X4 ) )
= ( times_times_int @ B @ ( times_times_int @ A3 @ X4 ) ) ) ).
% ideal.scale_left_commute
thf(fact_447_ideal_Oscale__left__commute,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,X4: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ ( times_5121417576591743744ring_a @ B @ X4 ) )
= ( times_5121417576591743744ring_a @ B @ ( times_5121417576591743744ring_a @ A3 @ X4 ) ) ) ).
% ideal.scale_left_commute
thf(fact_448_ideal_Oscale__left__commute,axiom,
! [A3: real,B: real,X4: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B @ X4 ) )
= ( times_times_real @ B @ ( times_times_real @ A3 @ X4 ) ) ) ).
% ideal.scale_left_commute
thf(fact_449_ideal_Oscale__scale,axiom,
! [A3: int,B: int,X4: int] :
( ( times_times_int @ A3 @ ( times_times_int @ B @ X4 ) )
= ( times_times_int @ ( times_times_int @ A3 @ B ) @ X4 ) ) ).
% ideal.scale_scale
thf(fact_450_ideal_Oscale__scale,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,X4: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ ( times_5121417576591743744ring_a @ B @ X4 ) )
= ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A3 @ B ) @ X4 ) ) ).
% ideal.scale_scale
thf(fact_451_ideal_Oscale__scale,axiom,
! [A3: real,B: real,X4: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B @ X4 ) )
= ( times_times_real @ ( times_times_real @ A3 @ B ) @ X4 ) ) ).
% ideal.scale_scale
thf(fact_452_mod__type_Oone__M__Rel,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( vector3024231343486012919_Rel_a @ M @ one_one_int @ one_on2109788427901206336ring_a ) ) ).
% mod_type.one_M_Rel
thf(fact_453_mat__mod__type_Oto__int__mod__ring__plus,axiom,
! [M: int,X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( ( finite1095367895020317408ring_a @ ( plus_p6165643967897163644ring_a @ X4 @ Y ) )
= ( modulo_modulo_int @ ( plus_plus_int @ ( finite1095367895020317408ring_a @ X4 ) @ ( finite1095367895020317408ring_a @ Y ) ) @ M ) ) ) ).
% mat_mod_type.to_int_mod_ring_plus
thf(fact_454_mod__type_Oto__int__mod__ring__plus,axiom,
! [M: int,X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( finite1095367895020317408ring_a @ ( plus_p6165643967897163644ring_a @ X4 @ Y ) )
= ( vector_Matrix_mod_M @ M @ ( plus_plus_int @ ( finite1095367895020317408ring_a @ X4 ) @ ( finite1095367895020317408ring_a @ Y ) ) ) ) ) ).
% mod_type.to_int_mod_ring_plus
thf(fact_455_mat__mod__type_Oto__int__mod__ring__times,axiom,
! [M: int,X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( vector2974885785920725647type_a @ type_a @ M )
=> ( ( finite1095367895020317408ring_a @ ( times_5121417576591743744ring_a @ X4 @ Y ) )
= ( modulo_modulo_int @ ( times_times_int @ ( finite1095367895020317408ring_a @ X4 ) @ ( finite1095367895020317408ring_a @ Y ) ) @ M ) ) ) ).
% mat_mod_type.to_int_mod_ring_times
thf(fact_456_mod__type_Oto__int__mod__ring__times,axiom,
! [M: int,X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ( finite1095367895020317408ring_a @ ( times_5121417576591743744ring_a @ X4 @ Y ) )
= ( vector_Matrix_mod_M @ M @ ( times_times_int @ ( finite1095367895020317408ring_a @ X4 ) @ ( finite1095367895020317408ring_a @ Y ) ) ) ) ) ).
% mod_type.to_int_mod_ring_times
thf(fact_457_ideal_Oscale__right__distrib,axiom,
! [A3: int,X4: int,Y: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ X4 @ Y ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ X4 ) @ ( times_times_int @ A3 @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_458_ideal_Oscale__right__distrib,axiom,
! [A3: finite_mod_ring_a,X4: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ ( plus_p6165643967897163644ring_a @ X4 @ Y ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ X4 ) @ ( times_5121417576591743744ring_a @ A3 @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_459_ideal_Oscale__right__distrib,axiom,
! [A3: real,X4: real,Y: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ X4 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ X4 ) @ ( times_times_real @ A3 @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_460_ideal_Oscale__left__distrib,axiom,
! [A3: int,B: int,X4: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B ) @ X4 )
= ( plus_plus_int @ ( times_times_int @ A3 @ X4 ) @ ( times_times_int @ B @ X4 ) ) ) ).
% ideal.scale_left_distrib
thf(fact_461_ideal_Oscale__left__distrib,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,X4: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ X4 )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A3 @ X4 ) @ ( times_5121417576591743744ring_a @ B @ X4 ) ) ) ).
% ideal.scale_left_distrib
thf(fact_462_ideal_Oscale__left__distrib,axiom,
! [A3: real,B: real,X4: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B ) @ X4 )
= ( plus_plus_real @ ( times_times_real @ A3 @ X4 ) @ ( times_times_real @ B @ X4 ) ) ) ).
% ideal.scale_left_distrib
thf(fact_463_add__scale__eq__noteq,axiom,
! [R3: int,A3: int,B: int,C: int,D2: int] :
( ( R3 != zero_zero_int )
=> ( ( ( A3 = B )
& ( C != D2 ) )
=> ( ( plus_plus_int @ A3 @ ( times_times_int @ R3 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_464_add__scale__eq__noteq,axiom,
! [R3: nat,A3: nat,B: nat,C: nat,D2: nat] :
( ( R3 != zero_zero_nat )
=> ( ( ( A3 = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A3 @ ( times_times_nat @ R3 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_465_add__scale__eq__noteq,axiom,
! [R3: real,A3: real,B: real,C: real,D2: real] :
( ( R3 != zero_zero_real )
=> ( ( ( A3 = B )
& ( C != D2 ) )
=> ( ( plus_plus_real @ A3 @ ( times_times_real @ R3 @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_466_non__triv__m,axiom,
ord_less_int @ one_one_int @ m ).
% non_triv_m
thf(fact_467_crossproduct__noteq,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ( A3 != B )
& ( C != D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D2 ) )
!= ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_468_crossproduct__noteq,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ( A3 != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A3 @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_469_crossproduct__noteq,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ( A3 != B )
& ( C != D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D2 ) )
!= ( plus_plus_real @ ( times_times_real @ A3 @ D2 ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_470_crossproduct__eq,axiom,
! [W: int,Y: int,X4: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X4 @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X4 @ Y ) ) )
= ( ( W = X4 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_471_crossproduct__eq,axiom,
! [W: nat,Y: nat,X4: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X4 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X4 @ Y ) ) )
= ( ( W = X4 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_472_crossproduct__eq,axiom,
! [W: real,Y: real,X4: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X4 @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X4 @ Y ) ) )
= ( ( W = X4 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_473_additive__implies__homogenous,axiom,
! [F: int > int] :
( ! [X: int,Y2: int] :
( ( F @ ( plus_plus_int @ X @ Y2 ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_int )
= zero_zero_int ) ) ).
% additive_implies_homogenous
thf(fact_474_additive__implies__homogenous,axiom,
! [F: int > nat] :
( ! [X: int,Y2: int] :
( ( F @ ( plus_plus_int @ X @ Y2 ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_int )
= zero_zero_nat ) ) ).
% additive_implies_homogenous
thf(fact_475_additive__implies__homogenous,axiom,
! [F: int > real] :
( ! [X: int,Y2: int] :
( ( F @ ( plus_plus_int @ X @ Y2 ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_int )
= zero_zero_real ) ) ).
% additive_implies_homogenous
thf(fact_476_additive__implies__homogenous,axiom,
! [F: nat > int] :
( ! [X: nat,Y2: nat] :
( ( F @ ( plus_plus_nat @ X @ Y2 ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_nat )
= zero_zero_int ) ) ).
% additive_implies_homogenous
thf(fact_477_additive__implies__homogenous,axiom,
! [F: nat > nat] :
( ! [X: nat,Y2: nat] :
( ( F @ ( plus_plus_nat @ X @ Y2 ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_nat )
= zero_zero_nat ) ) ).
% additive_implies_homogenous
thf(fact_478_additive__implies__homogenous,axiom,
! [F: nat > real] :
( ! [X: nat,Y2: nat] :
( ( F @ ( plus_plus_nat @ X @ Y2 ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_nat )
= zero_zero_real ) ) ).
% additive_implies_homogenous
thf(fact_479_additive__implies__homogenous,axiom,
! [F: real > int] :
( ! [X: real,Y2: real] :
( ( F @ ( plus_plus_real @ X @ Y2 ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_real )
= zero_zero_int ) ) ).
% additive_implies_homogenous
thf(fact_480_additive__implies__homogenous,axiom,
! [F: real > nat] :
( ! [X: real,Y2: real] :
( ( F @ ( plus_plus_real @ X @ Y2 ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_real )
= zero_zero_nat ) ) ).
% additive_implies_homogenous
thf(fact_481_additive__implies__homogenous,axiom,
! [F: real > real] :
( ! [X: real,Y2: real] :
( ( F @ ( plus_plus_real @ X @ Y2 ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_real )
= zero_zero_real ) ) ).
% additive_implies_homogenous
thf(fact_482_additive__implies__homogenous,axiom,
! [F: int > finite_mod_ring_a] :
( ! [X: int,Y2: int] :
( ( F @ ( plus_plus_int @ X @ Y2 ) )
= ( plus_p6165643967897163644ring_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ( F @ zero_zero_int )
= zero_z7902377541816115708ring_a ) ) ).
% additive_implies_homogenous
thf(fact_483_verit__sum__simplify,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% verit_sum_simplify
thf(fact_484_verit__sum__simplify,axiom,
! [A3: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A3 @ zero_z7902377541816115708ring_a )
= A3 ) ).
% verit_sum_simplify
thf(fact_485_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_486_verit__sum__simplify,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% verit_sum_simplify
thf(fact_487_add__0__iff,axiom,
! [B: int,A3: int] :
( ( B
= ( plus_plus_int @ B @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_488_add__0__iff,axiom,
! [B: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ B @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_489_add__0__iff,axiom,
! [B: real,A3: real] :
( ( B
= ( plus_plus_real @ B @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_490_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_491_add__less__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A3 @ B ) ) ).
% add_less_cancel_right
thf(fact_492_add__less__cancel__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A3 @ B ) ) ).
% add_less_cancel_right
thf(fact_493_add__less__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A3 @ B ) ) ).
% add_less_cancel_right
thf(fact_494_add__less__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A3 @ B ) ) ).
% add_less_cancel_left
thf(fact_495_add__less__cancel__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A3 @ B ) ) ).
% add_less_cancel_left
thf(fact_496_add__less__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A3 @ B ) ) ).
% add_less_cancel_left
thf(fact_497_add__less__same__cancel1,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A3 ) @ B )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_498_add__less__same__cancel1,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A3 ) @ B )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_499_add__less__same__cancel1,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A3 ) @ B )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_500_add__less__same__cancel2,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ B ) @ B )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_501_add__less__same__cancel2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B ) @ B )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_502_add__less__same__cancel2,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ B ) @ B )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_503_less__add__same__cancel1,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_504_less__add__same__cancel1,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_505_less__add__same__cancel1,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_506_less__add__same__cancel2,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ B @ A3 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_507_less__add__same__cancel2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_508_less__add__same__cancel2,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ B @ A3 ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_509_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_510_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_511_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_512_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_513_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_514_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_515_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_516_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_517_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_518_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_519_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_520_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_521_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_522_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_523_linorder__neqE__linordered__idom,axiom,
! [X4: int,Y: int] :
( ( X4 != Y )
=> ( ~ ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_524_linorder__neqE__linordered__idom,axiom,
! [X4: real,Y: real] :
( ( X4 != Y )
=> ( ~ ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ Y @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_525_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_526_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_527_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_528_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_529_add__less__imp__less__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A3 @ B ) ) ).
% add_less_imp_less_right
thf(fact_530_add__less__imp__less__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A3 @ B ) ) ).
% add_less_imp_less_right
thf(fact_531_add__less__imp__less__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A3 @ B ) ) ).
% add_less_imp_less_right
thf(fact_532_add__less__imp__less__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A3 @ B ) ) ).
% add_less_imp_less_left
thf(fact_533_add__less__imp__less__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A3 @ B ) ) ).
% add_less_imp_less_left
thf(fact_534_add__less__imp__less__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A3 @ B ) ) ).
% add_less_imp_less_left
thf(fact_535_add__strict__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_536_add__strict__right__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_537_add__strict__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_538_add__strict__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_539_add__strict__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_540_add__strict__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_541_add__strict__mono,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_542_add__strict__mono,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_543_add__strict__mono,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_544_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_545_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_546_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_547_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_548_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_549_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_550_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_551_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_552_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_553_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_554_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_555_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_556_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_557_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_558_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_559_mult__less__cancel__right__disj,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_560_mult__less__cancel__right__disj,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_561_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_562_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_563_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_564_mult__strict__right__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_565_mult__strict__right__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_566_mult__less__cancel__left__disj,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_567_mult__less__cancel__left__disj,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_568_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_569_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_570_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_571_mult__strict__left__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_572_mult__strict__left__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_573_mult__less__cancel__left__pos,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A3 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_574_mult__less__cancel__left__pos,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A3 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_575_mult__less__cancel__left__neg,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_576_mult__less__cancel__left__neg,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_577_zero__less__mult__pos2,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_578_zero__less__mult__pos2,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_579_zero__less__mult__pos2,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_580_zero__less__mult__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_581_zero__less__mult__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_582_zero__less__mult__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_583_zero__less__mult__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_584_zero__less__mult__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_585_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A3 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_586_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_587_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_588_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_589_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_590_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_591_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_592_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_593_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_594_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_595_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_596_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_597_mult__less__0__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_598_mult__less__0__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_599_not__square__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( times_times_int @ A3 @ A3 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_600_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_601_mult__neg__neg,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_602_mult__neg__neg,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_603_pos__add__strict,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_604_pos__add__strict,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_605_pos__add__strict,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_606_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A3 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_607_add__pos__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_608_add__pos__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_609_add__pos__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_610_add__neg__neg,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_611_add__neg__neg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_612_add__neg__neg,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_613_add__less__zeroD,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X4 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X4 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_614_add__less__zeroD,axiom,
! [X4: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X4 @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X4 @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_615_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_616_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_617_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_618_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_619_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_620_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_621_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_622_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_623_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_624_add__mono1,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_625_add__mono1,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_626_add__mono1,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_627_less__add__one,axiom,
! [A3: int] : ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ one_one_int ) ) ).
% less_add_one
thf(fact_628_less__add__one,axiom,
! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).
% less_add_one
thf(fact_629_less__add__one,axiom,
! [A3: real] : ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ one_one_real ) ) ).
% less_add_one
thf(fact_630_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_631_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_632_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_633_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_634_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_635_mat__mod_Ointro,axiom,
! [M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( vector2311527483241216340at_mod @ M ) ) ).
% mat_mod.intro
thf(fact_636_mat__mod_Onon__triv__m,axiom,
! [M: int] :
( ( vector2311527483241216340at_mod @ M )
=> ( ord_less_int @ one_one_int @ M ) ) ).
% mat_mod.non_triv_m
thf(fact_637_Vector__Matrix__Mod_Omat__mod__def,axiom,
( vector2311527483241216340at_mod
= ( ord_less_int @ one_one_int ) ) ).
% Vector_Matrix_Mod.mat_mod_def
thf(fact_638_not__sum__squares__lt__zero,axiom,
! [X4: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_639_not__sum__squares__lt__zero,axiom,
! [X4: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_640_sum__squares__gt__zero__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X4 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_641_sum__squares__gt__zero__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X4 != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_642_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_643_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_644_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_645_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_646_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_647_mod__type_Om1,axiom,
! [M: int] :
( ( vector4745807456731380595type_a @ type_a @ M )
=> ( ord_less_int @ one_one_int @ M ) ) ).
% mod_type.m1
thf(fact_648_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_649_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_650_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_651_mult__less__iff1,axiom,
! [Z: int,X4: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X4 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_652_mult__less__iff1,axiom,
! [Z: real,X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X4 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_653_unit__vec__eq,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( unit_vec_int @ N @ I )
= ( unit_vec_int @ N @ J ) )
= ( I = J ) ) ) ).
% unit_vec_eq
thf(fact_654_unit__vec__eq,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( unit_v6648808783362777473ring_a @ N @ I )
= ( unit_v6648808783362777473ring_a @ N @ J ) )
= ( I = J ) ) ) ).
% unit_vec_eq
thf(fact_655_unit__vec__nonzero,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( unit_vec_int @ N @ I )
!= ( zero_vec_int @ N ) ) ) ).
% unit_vec_nonzero
thf(fact_656_unit__vec__nonzero,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( unit_v6648808783362777473ring_a @ N @ I )
!= ( zero_v1122566495420860901ring_a @ N ) ) ) ).
% unit_vec_nonzero
thf(fact_657_is__num__normalize_I1_J,axiom,
! [A3: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_658_is__num__normalize_I1_J,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A3 @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_659_is__num__normalize_I1_J,axiom,
! [A3: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_660_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_661_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_662_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_663_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_664_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_665_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_666_ex__less__of__int,axiom,
! [X4: real] :
? [Z2: int] : ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).
% ex_less_of_int
thf(fact_667_ex__of__int__less,axiom,
! [X4: real] :
? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X4 ) ).
% ex_of_int_less
thf(fact_668_plus__eq__zero__2,axiom,
! [S: nat,T: nat] :
( ( ( plus_plus_nat @ S @ T )
= zero_zero_nat )
=> ( T = zero_zero_nat ) ) ).
% plus_eq_zero_2
thf(fact_669_plus__eq__zero,axiom,
! [S: nat,T: nat] :
( ( ( plus_plus_nat @ S @ T )
= zero_zero_nat )
=> ( S = zero_zero_nat ) ) ).
% plus_eq_zero
thf(fact_670_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_671_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_672_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5901776551076858996ring_a @ zero_z7902377541816115708ring_a )
= one_on2109788427901206336ring_a ) ).
% dbl_inc_simps(2)
thf(fact_673_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_674_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_675_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 )
=> ! [I3: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J2 ) ) )
=> ( Q2 @ J2 ) ) ) ) ) ).
% split_mod
thf(fact_676_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_677_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_678_dbl__inc__def,axiom,
( neg_nu5901776551076858996ring_a
= ( ^ [X3: finite_mod_ring_a] : ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ X3 @ X3 ) @ one_on2109788427901206336ring_a ) ) ) ).
% dbl_inc_def
thf(fact_679_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_680_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_681_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_682_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_683_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_684_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_685_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_686_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_687_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_688_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_689_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_690_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_691_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_692_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_693_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_694_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_695_nat__mod__eq__iff,axiom,
! [X4: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X4 @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X4 @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_696_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_697_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_698_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_699_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_700_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_701_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_702_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_703_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_704_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_705_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_706_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_707_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_708_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_709_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_710_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_711_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_712_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_713_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_714_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_715_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_716_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_717_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_718_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( P @ A4 @ B2 )
= ( P @ B2 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B2: nat] :
( ( P @ A4 @ B2 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B2 ) ) )
=> ( P @ A3 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_719_split__zmod,axiom,
! [Q2: int > $o,N: int,K: int] :
( ( Q2 @ ( modulo_modulo_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( Q2 @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( Q2 @ J2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( Q2 @ J2 ) ) ) ) ) ).
% split_zmod
thf(fact_720_int__mod__neg__eq,axiom,
! [A3: int,B: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq_int @ R3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R3 )
=> ( ( modulo_modulo_int @ A3 @ B )
= R3 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_721_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_722_add__le__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A3 @ B ) ) ).
% add_le_cancel_right
thf(fact_723_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A3 @ B ) ) ).
% add_le_cancel_right
thf(fact_724_add__le__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A3 @ B ) ) ).
% add_le_cancel_right
thf(fact_725_add__le__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A3 @ B ) ) ).
% add_le_cancel_left
thf(fact_726_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A3 @ B ) ) ).
% add_le_cancel_left
thf(fact_727_add__le__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A3 @ B ) ) ).
% add_le_cancel_left
thf(fact_728_add__le__same__cancel1,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A3 ) @ B )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_729_add__le__same__cancel1,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A3 ) @ B )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_730_add__le__same__cancel1,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A3 ) @ B )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_731_add__le__same__cancel2,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B ) @ B )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_732_add__le__same__cancel2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B ) @ B )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_733_add__le__same__cancel2,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B ) @ B )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_734_le__add__same__cancel1,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ A3 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_735_le__add__same__cancel1,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_736_le__add__same__cancel1,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ A3 @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_737_le__add__same__cancel2,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ B @ A3 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_738_le__add__same__cancel2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_739_le__add__same__cancel2,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ B @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_740_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_741_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_742_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_743_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_744_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_745_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_746_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_747_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_748_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_749_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_750_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_751_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_752_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_753_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_754_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_755_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_756_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_757_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_758_zero__min,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_min
thf(fact_759_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_760_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_761_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_762_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_763_leq__add__left,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( plus_plus_nat @ Y @ X4 ) ) ).
% leq_add_left
thf(fact_764_leq__add__right,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( plus_plus_nat @ X4 @ Y ) ) ).
% leq_add_right
thf(fact_765_le__imp__add,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ? [C2: int] :
( B
= ( plus_plus_int @ A3 @ C2 ) ) ) ).
% le_imp_add
thf(fact_766_le__imp__add,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ? [C2: nat] :
( B
= ( plus_plus_nat @ A3 @ C2 ) ) ) ).
% le_imp_add
thf(fact_767_le__imp__add,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ? [C2: real] :
( B
= ( plus_plus_real @ A3 @ C2 ) ) ) ).
% le_imp_add
thf(fact_768_add__le__imp__le__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A3 @ B ) ) ).
% add_le_imp_le_right
thf(fact_769_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A3 @ B ) ) ).
% add_le_imp_le_right
thf(fact_770_add__le__imp__le__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A3 @ B ) ) ).
% add_le_imp_le_right
thf(fact_771_add__le__imp__le__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A3 @ B ) ) ).
% add_le_imp_le_left
thf(fact_772_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A3 @ B ) ) ).
% add_le_imp_le_left
thf(fact_773_add__le__imp__le__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A3 @ B ) ) ).
% add_le_imp_le_left
thf(fact_774_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_775_add__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_776_add__right__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_777_add__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_778_less__eqE,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A3 @ C2 ) ) ) ).
% less_eqE
thf(fact_779_add__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_780_add__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_781_add__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_782_add__mono,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_783_add__mono,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_784_add__mono,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_785_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_786_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_787_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_788_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_789_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_790_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_791_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_792_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_793_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_794_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_795_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_796_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_797_ex__le__of__int,axiom,
! [X4: real] :
? [Z2: int] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).
% ex_le_of_int
thf(fact_798_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_799_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_800_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_801_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_802_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_803_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_804_zero__le__mult__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_805_zero__le__mult__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_806_mult__nonneg__nonpos2,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_807_mult__nonneg__nonpos2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_808_mult__nonneg__nonpos2,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_809_mult__nonpos__nonneg,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_810_mult__nonpos__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_811_mult__nonpos__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_812_mult__nonneg__nonpos,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_813_mult__nonneg__nonpos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_814_mult__nonneg__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_815_mult__nonneg__nonneg,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_816_mult__nonneg__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_817_mult__nonneg__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_818_split__mult__neg__le,axiom,
! [A3: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_819_split__mult__neg__le,axiom,
! [A3: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_820_split__mult__neg__le,axiom,
! [A3: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_821_mult__le__0__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_822_mult__le__0__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_823_mult__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_824_mult__right__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_825_mult__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_826_mult__right__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_827_mult__right__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_828_mult__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_829_mult__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_830_mult__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_831_mult__nonpos__nonpos,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_832_mult__nonpos__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_833_mult__left__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_834_mult__left__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_835_split__mult__pos__le,axiom,
! [A3: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_836_split__mult__pos__le,axiom,
! [A3: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_837_zero__le__square,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_838_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_839_mult__mono_H,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_840_mult__mono_H,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_841_mult__mono_H,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_842_mult__mono,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_843_mult__mono,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_844_mult__mono,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_845_add__nonpos__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_846_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_847_add__nonpos__eq__0__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X4 @ Y )
= zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_848_add__nonneg__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_849_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_850_add__nonneg__eq__0__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X4 @ Y )
= zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_851_add__nonpos__nonpos,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_852_add__nonpos__nonpos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_853_add__nonpos__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_854_add__nonneg__nonneg,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_855_add__nonneg__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_856_add__nonneg__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_857_add__increasing2,axiom,
! [C: int,B: int,A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A3 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_858_add__increasing2,axiom,
! [C: nat,B: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A3 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_859_add__increasing2,axiom,
! [C: real,B: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A3 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_860_add__decreasing2,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_861_add__decreasing2,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_862_add__decreasing2,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_863_add__increasing,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_864_add__increasing,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_865_add__increasing,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_866_add__decreasing,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_867_add__decreasing,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_868_add__decreasing,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_869_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_870_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_871_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_872_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_873_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_874_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_875_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_876_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_877_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_878_add__less__le__mono,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_879_add__less__le__mono,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_880_add__less__le__mono,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_881_add__le__less__mono,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_882_add__le__less__mono,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_883_add__le__less__mono,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_884_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_885_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_886_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_887_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_888_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_889_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_890_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_891_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_892_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_893_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_894_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_895_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_896_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_897_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_898_mult__right__le__imp__le,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_899_mult__right__le__imp__le,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_900_mult__right__le__imp__le,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_901_mult__left__le__imp__le,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_902_mult__left__le__imp__le,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_903_mult__left__le__imp__le,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_904_mult__le__cancel__left__pos,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A3 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_905_mult__le__cancel__left__pos,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A3 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_906_mult__le__cancel__left__neg,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_907_mult__le__cancel__left__neg,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_908_mult__less__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_909_mult__less__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_910_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_911_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_912_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_913_mult__right__less__imp__less,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_914_mult__right__less__imp__less,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_915_mult__right__less__imp__less,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_916_mult__less__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_917_mult__less__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_918_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_919_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_920_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_921_mult__left__less__imp__less,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_922_mult__left__less__imp__less,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_923_mult__left__less__imp__less,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_924_mult__le__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_925_mult__le__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_926_mult__le__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_927_mult__le__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_928_mult__le__cancel__iff1,axiom,
! [Z: int,X4: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X4 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_929_mult__le__cancel__iff1,axiom,
! [Z: real,X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X4 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_930_mult__le__cancel__iff2,axiom,
! [Z: int,X4: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X4 ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X4 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_931_mult__le__cancel__iff2,axiom,
! [Z: real,X4: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X4 ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X4 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_932_add__strict__increasing2,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_933_add__strict__increasing2,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_934_add__strict__increasing2,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_935_add__strict__increasing,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_936_add__strict__increasing,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_937_add__strict__increasing,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_938_add__pos__nonneg,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_939_add__pos__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_940_add__pos__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_941_add__nonpos__neg,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_942_add__nonpos__neg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_943_add__nonpos__neg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_944_add__nonneg__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_945_add__nonneg__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_946_add__nonneg__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_947_add__neg__nonpos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_948_add__neg__nonpos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_949_add__neg__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_950_sum__squares__ge__zero,axiom,
! [X4: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_951_sum__squares__ge__zero,axiom,
! [X4: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_952_sum__squares__le__zero__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_953_sum__squares__le__zero__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_954_mult__left__le,axiom,
! [C: int,A3: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_955_mult__left__le,axiom,
! [C: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_956_mult__left__le,axiom,
! [C: real,A3: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_957_mult__le__one,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ 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 @ A3 @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_958_mult__le__one,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ 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 @ A3 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_959_mult__le__one,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_960_mult__right__le__one__le,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X4 @ Y ) @ X4 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_961_mult__right__le__one__le,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X4 @ Y ) @ X4 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_962_mult__left__le__one__le,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X4 ) @ X4 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_963_mult__left__le__one__le,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X4 ) @ X4 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_964_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_965_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_966_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_967_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_968_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_969_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_970_mult__less__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_971_mult__less__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_972_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_973_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_974_mult__less__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_975_mult__less__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_976_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_977_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_978_mult__le__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_979_mult__le__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_980_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_981_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_982_mult__le__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_983_mult__le__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_984_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_985_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_986_convex__bound__le,axiom,
! [X4: int,A3: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X4 @ A3 )
=> ( ( ord_less_eq_int @ Y @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_987_convex__bound__le,axiom,
! [X4: real,A3: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X4 @ A3 )
=> ( ( ord_less_eq_real @ Y @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_988_floor__exists,axiom,
! [X4: real] :
? [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X4 )
& ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_989_floor__exists1,axiom,
! [X4: real] :
? [X: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ X4 )
& ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ X @ one_one_int ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X4 )
& ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
=> ( Y4 = X ) ) ) ).
% floor_exists1
thf(fact_990_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_991_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
= ( plus_plus_int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_992_convex__bound__lt,axiom,
! [X4: int,A3: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X4 @ A3 )
=> ( ( ord_less_int @ Y @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_993_convex__bound__lt,axiom,
! [X4: real,A3: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X4 @ A3 )
=> ( ( ord_less_real @ Y @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_994_int__mod__pos__eq,axiom,
! [A3: int,B: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ R3 @ B )
=> ( ( modulo_modulo_int @ A3 @ B )
= R3 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_995_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_996_field__le__mult__one__interval,axiom,
! [X4: real,Y: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X4 ) @ Y ) ) )
=> ( ord_less_eq_real @ X4 @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_997_inv__M__def,axiom,
! [X4: int] :
( ( ( ord_less_eq_int @ ( plus_plus_int @ X4 @ X4 ) @ m )
=> ( ( vector2349909385534816566_inv_M @ m @ X4 )
= X4 ) )
& ( ~ ( ord_less_eq_int @ ( plus_plus_int @ X4 @ X4 ) @ m )
=> ( ( vector2349909385534816566_inv_M @ m @ X4 )
= ( minus_minus_int @ X4 @ m ) ) ) ) ).
% inv_M_def
thf(fact_998_field__le__epsilon,axiom,
! [X4: real,Y: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X4 @ ( plus_plus_real @ Y @ E2 ) ) )
=> ( ord_less_eq_real @ X4 @ Y ) ) ).
% field_le_epsilon
thf(fact_999_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_1000_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A3 @ A3 )
= zero_z7902377541816115708ring_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1001_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1002_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1003_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1004_diff__zero,axiom,
! [A3: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A3 @ zero_z7902377541816115708ring_a )
= A3 ) ).
% diff_zero
thf(fact_1005_diff__zero,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_zero
thf(fact_1006_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_1007_diff__zero,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_zero
thf(fact_1008_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1009_diff__0__right,axiom,
! [A3: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A3 @ zero_z7902377541816115708ring_a )
= A3 ) ).
% diff_0_right
thf(fact_1010_diff__0__right,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_0_right
thf(fact_1011_diff__0__right,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_0_right
thf(fact_1012_diff__self,axiom,
! [A3: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A3 @ A3 )
= zero_z7902377541816115708ring_a ) ).
% diff_self
thf(fact_1013_diff__self,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% diff_self
thf(fact_1014_diff__self,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% diff_self
thf(fact_1015_add__diff__cancel,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ B )
= A3 ) ).
% add_diff_cancel
thf(fact_1016_add__diff__cancel,axiom,
! [A3: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B ) @ B )
= A3 ) ).
% add_diff_cancel
thf(fact_1017_add__diff__cancel,axiom,
! [A3: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B ) @ B )
= A3 ) ).
% add_diff_cancel
thf(fact_1018_diff__add__cancel,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( minus_3609261664126569004ring_a @ A3 @ B ) @ B )
= A3 ) ).
% diff_add_cancel
thf(fact_1019_diff__add__cancel,axiom,
! [A3: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A3 @ B ) @ B )
= A3 ) ).
% diff_add_cancel
thf(fact_1020_diff__add__cancel,axiom,
! [A3: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A3 @ B ) @ B )
= A3 ) ).
% diff_add_cancel
thf(fact_1021_add__diff__cancel__left,axiom,
! [C: finite_mod_ring_a,A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ C @ A3 ) @ ( plus_p6165643967897163644ring_a @ C @ B ) )
= ( minus_3609261664126569004ring_a @ A3 @ B ) ) ).
% add_diff_cancel_left
thf(fact_1022_add__diff__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A3 @ B ) ) ).
% add_diff_cancel_left
thf(fact_1023_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A3 @ B ) ) ).
% add_diff_cancel_left
thf(fact_1024_add__diff__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A3 @ B ) ) ).
% add_diff_cancel_left
thf(fact_1025_add__diff__cancel__left_H,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ A3 )
= B ) ).
% add_diff_cancel_left'
thf(fact_1026_add__diff__cancel__left_H,axiom,
! [A3: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B ) @ A3 )
= B ) ).
% add_diff_cancel_left'
thf(fact_1027_add__diff__cancel__left_H,axiom,
! [A3: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B ) @ A3 )
= B ) ).
% add_diff_cancel_left'
thf(fact_1028_add__diff__cancel__left_H,axiom,
! [A3: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B ) @ A3 )
= B ) ).
% add_diff_cancel_left'
thf(fact_1029_add__diff__cancel__right,axiom,
! [A3: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ C ) @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ A3 @ B ) ) ).
% add_diff_cancel_right
thf(fact_1030_add__diff__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A3 @ B ) ) ).
% add_diff_cancel_right
thf(fact_1031_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A3 @ B ) ) ).
% add_diff_cancel_right
thf(fact_1032_add__diff__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A3 @ B ) ) ).
% add_diff_cancel_right
thf(fact_1033_add__diff__cancel__right_H,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A3 @ B ) @ B )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1034_add__diff__cancel__right_H,axiom,
! [A3: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B ) @ B )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1035_add__diff__cancel__right_H,axiom,
! [A3: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B ) @ B )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1036_add__diff__cancel__right_H,axiom,
! [A3: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B ) @ B )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1037_minus__mod__self2,axiom,
! [A3: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B ) @ B )
= ( modulo_modulo_int @ A3 @ B ) ) ).
% minus_mod_self2
thf(fact_1038_diff__ge__0__iff__ge,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B ) )
= ( ord_less_eq_int @ B @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1039_diff__ge__0__iff__ge,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B ) )
= ( ord_less_eq_real @ B @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1040_diff__gt__0__iff__gt,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B ) )
= ( ord_less_int @ B @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1041_diff__gt__0__iff__gt,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B ) )
= ( ord_less_real @ B @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1042_diff__add__zero,axiom,
! [A3: nat,B: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1043_diff__numeral__special_I9_J,axiom,
( ( minus_3609261664126569004ring_a @ one_on2109788427901206336ring_a @ one_on2109788427901206336ring_a )
= zero_z7902377541816115708ring_a ) ).
% diff_numeral_special(9)
thf(fact_1044_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1045_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1046_le__add__diff__inverse,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A3 @ B ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1047_le__add__diff__inverse,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A3 @ B ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1048_le__add__diff__inverse,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A3 @ B ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1049_le__add__diff__inverse2,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A3 @ B ) @ B )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1050_le__add__diff__inverse2,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B ) @ B )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1051_le__add__diff__inverse2,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A3 @ B ) @ B )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1052_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_1053_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_1054_of__int__hom_Ohom__minus,axiom,
! [X4: int,Y: int] :
( ( ring_18169885480643366966ring_a @ ( minus_minus_int @ X4 @ Y ) )
= ( minus_3609261664126569004ring_a @ ( ring_18169885480643366966ring_a @ X4 ) @ ( ring_18169885480643366966ring_a @ Y ) ) ) ).
% of_int_hom.hom_minus
thf(fact_1055_of__int__hom_Ohom__minus,axiom,
! [X4: int,Y: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ X4 @ Y ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ X4 ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_minus
thf(fact_1056_of__int__hom_Ohom__minus,axiom,
! [X4: int,Y: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ X4 @ Y ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_minus
thf(fact_1057_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_18169885480643366966ring_a @ ( minus_minus_int @ W @ Z ) )
= ( minus_3609261664126569004ring_a @ ( ring_18169885480643366966ring_a @ W ) @ ( ring_18169885480643366966ring_a @ Z ) ) ) ).
% of_int_diff
thf(fact_1058_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_1059_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_diff
thf(fact_1060_M__minus_I2_J,axiom,
! [X4: int,Y: int] :
( ( vector_Matrix_mod_M @ m @ ( minus_minus_int @ X4 @ ( vector_Matrix_mod_M @ m @ Y ) ) )
= ( vector_Matrix_mod_M @ m @ ( minus_minus_int @ X4 @ Y ) ) ) ).
% M_minus(2)
thf(fact_1061_M__minus_I1_J,axiom,
! [X4: int,Y: int] :
( ( vector_Matrix_mod_M @ m @ ( minus_minus_int @ ( vector_Matrix_mod_M @ m @ X4 ) @ Y ) )
= ( vector_Matrix_mod_M @ m @ ( minus_minus_int @ X4 @ Y ) ) ) ).
% M_minus(1)
thf(fact_1062_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1063_diff__eq__diff__less__eq,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A3 @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1064_diff__eq__diff__less__eq,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A3 @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ A3 @ B )
= ( ord_less_eq_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1065_diff__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1066_diff__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1067_diff__left__mono,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1068_diff__left__mono,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1069_diff__mono,axiom,
! [A3: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_1070_diff__mono,axiom,
! [A3: real,B: real,D2: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ D2 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_1071_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: finite_mod_ring_a,Z4: finite_mod_ring_a] : ( Y5 = Z4 ) )
= ( ^ [A5: finite_mod_ring_a,B3: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A5 @ B3 )
= zero_z7902377541816115708ring_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1072_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A5: int,B3: int] :
( ( minus_minus_int @ A5 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1073_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
= ( ^ [A5: real,B3: real] :
( ( minus_minus_real @ A5 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1074_diff__strict__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1075_diff__strict__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1076_diff__strict__left__mono,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_int @ B @ A3 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1077_diff__strict__left__mono,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_real @ B @ A3 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1078_diff__eq__diff__less,axiom,
! [A3: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A3 @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_1079_diff__eq__diff__less,axiom,
! [A3: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A3 @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_real @ A3 @ B )
= ( ord_less_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_1080_diff__strict__mono,axiom,
! [A3: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_1081_diff__strict__mono,axiom,
! [A3: real,B: real,D2: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ D2 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_1082_right__diff__distrib_H,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ A3 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ B ) @ ( times_times_int @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1083_right__diff__distrib_H,axiom,
! [A3: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A3 @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A3 @ B ) @ ( times_5121417576591743744ring_a @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1084_right__diff__distrib_H,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1085_right__diff__distrib_H,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B ) @ ( times_times_real @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1086_left__diff__distrib_H,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A3: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ B @ C ) @ A3 )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ B @ A3 ) @ ( times_5121417576591743744ring_a @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_1087_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A3 ) @ ( times_times_nat @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_1088_left__diff__distrib_H,axiom,
! [B: real,C: real,A3: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A3 )
= ( minus_minus_real @ ( times_times_real @ B @ A3 ) @ ( times_times_real @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_1089_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1090_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1091_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1092_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_1093_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1094_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1095_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_1096_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_1097_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1098_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_1099_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_1100_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_1101_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_1102_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1103_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1104_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1105_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_1106_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_1107_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_1108_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_1109_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_1110_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X: int,K2: int] :
( ( P1 @ X )
= ( P1 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( P @ X )
= ( P1 @ X ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1111_plusinfinity,axiom,
! [D2: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X: int,K2: int] :
( ( P2 @ X )
= ( P2 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [X_12: int] : ( P2 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1112_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_1113_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_1114_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1115_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( minus_minus_int @ X @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1116_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_1117_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1118_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_1119_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1120_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1121_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_1122_conj__le__cong,axiom,
! [X4: int,X6: int,P: $o,P2: $o] :
( ( X4 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1123_imp__le__cong,axiom,
! [X4: int,X6: int,P: $o,P2: $o] :
( ( X4 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1124_not__real__square__gt__zero,axiom,
! [X4: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X4 @ X4 ) ) )
= ( X4 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1125_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1126_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_1127_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_1128_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_1129_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_1130_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1140_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1141_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_1142_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_1143_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_1155_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% abs_mod_less
thf(fact_1156_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B ) )
= ( ( ( ord_less_nat @ A3 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A3
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1157_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A3
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1158_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_1159_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_1160_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1161_incr__lemma,axiom,
! [D2: int,Z: int,X4: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1162_decr__lemma,axiom,
! [D2: int,X4: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1163_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1164_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1165_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1166_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1167_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1168_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1169_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1170_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1171_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1172_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1173_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1174_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1175_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1176_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1177_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1178_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1179_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1180_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1181_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1182_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1183_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1184_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1185_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1186_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1187_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_1188_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1189_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1190_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1191_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1192_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1193_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1194_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1195_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1196_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1197_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1198_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1199_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1200_mod__induct,axiom,
! [P: nat > $o,N: nat,P3: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P3 )
=> ( ( ord_less_nat @ M @ P3 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P3 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P3 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1201_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1202_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1203_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1204_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1205_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1206_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1207_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1208_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1209_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1210_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1211_sin__bound__lemma,axiom,
! [X4: real,Y: real,U: real,V: real] :
( ( X4 = Y )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X4 @ U ) @ Y ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_1212_real__divide__square__eq,axiom,
! [R3: real,A3: real] :
( ( divide_divide_real @ ( times_times_real @ R3 @ A3 ) @ ( times_times_real @ R3 @ R3 ) )
= ( divide_divide_real @ A3 @ R3 ) ) ).
% real_divide_square_eq
thf(fact_1213_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1214_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1215_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1216_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1217_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1218_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1219_real__of__int__div3,axiom,
! [N: int,X4: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_1220_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1221_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1222_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1223_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1224_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1225_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1226_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A3 @ B ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1227_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A3 @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1228_div__neg__pos__less0,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1229_real__of__int__div__aux,axiom,
! [X4: int,D2: int] :
( ( divide_divide_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ D2 ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X4 @ D2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X4 @ D2 ) ) @ ( ring_1_of_int_real @ D2 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_1230_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_1231_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1232_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1233_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1234_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1235_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1236_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1237_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1238_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1239_div__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( suc @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% div_Suc
thf(fact_1240_int__div__less__self,axiom,
! [X4: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X4 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).
% int_div_less_self
thf(fact_1241_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B ) )
= ( ( ord_less_eq_int @ B @ A3 )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1242_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1243_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1244_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1245_div__nonpos__pos__le0,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1246_div__nonneg__neg__le0,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1247_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1248_zdiv__mono2__neg,axiom,
! [A3: int,B5: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B5 ) @ ( divide_divide_int @ A3 @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1249_zdiv__mono1__neg,axiom,
! [A3: int,A6: int,B: int] :
( ( ord_less_eq_int @ A3 @ A6 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A3 @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1250_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1251_zdiv__mono2,axiom,
! [A3: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ ( divide_divide_int @ A3 @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1252_zdiv__mono1,axiom,
! [A3: int,A6: int,B: int] :
( ( ord_less_eq_int @ A3 @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1253_zdiv__zmult2__eq,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1254_div__mod__decomp,axiom,
! [A: nat,N: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N ) @ N ) @ ( modulo_modulo_nat @ A @ N ) ) ) ).
% div_mod_decomp
thf(fact_1255_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1256_zdiv__mono__strict,axiom,
! [A: int,B4: int,N: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B4 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1257_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N2: nat] : ( minus_minus_nat @ M2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% modulo_nat_def
thf(fact_1258_div__mod__decomp__int,axiom,
! [A: int,N: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ N ) @ N ) @ ( modulo_modulo_int @ A @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1259_eq__diff__eq_H,axiom,
! [X4: real,Y: real,Z: real] :
( ( X4
= ( minus_minus_real @ Y @ Z ) )
= ( Y
= ( plus_plus_real @ X4 @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_1260_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1261_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1262_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1263_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1264_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1265_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J2 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_1266_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1267_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1268_int__div__pos__eq,axiom,
! [A3: int,B: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ R3 @ B )
=> ( ( divide_divide_int @ A3 @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1269_int__div__neg__eq,axiom,
! [A3: int,B: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq_int @ R3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R3 )
=> ( ( divide_divide_int @ A3 @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1270_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1271_zmod__zmult2__eq,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A3 @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A3 @ B ) @ C ) ) @ ( modulo_modulo_int @ A3 @ B ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1272_verit__le__mono__div,axiom,
! [A: nat,B4: nat,N: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B4 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B4 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1273_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1274_verit__le__mono__div__int,axiom,
! [A: int,B4: int,N: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B4 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B4 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1275_div__mult__mono,axiom,
! [A3: nat,D2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ D2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( times_times_nat @ A3 @ B ) @ D2 ) @ B ) ) ) ).
% div_mult_mono
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
right_7341149013462710248ring_a @ ( vector5230790432342831627_Rel_a @ m ) ).
%------------------------------------------------------------------------------