TPTP Problem File: SLH0324^1.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 : Commuting_Hermitian/0002_Commuting_Hermitian/prob_03036_120743__19666020_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1169 ( 444 unt; 86 typ; 0 def)
% Number of atoms : 2916 (1149 equ; 0 cnn)
% Maximal formula atoms : 18 ( 2 avg)
% Number of connectives : 9255 ( 349 ~; 115 |; 155 &;7247 @)
% ( 0 <=>;1389 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 393 ( 393 >; 0 *; 0 +; 0 <<)
% Number of symbols : 78 ( 75 usr; 20 con; 0-4 aty)
% Number of variables : 2944 ( 101 ^;2706 !; 137 ?;2944 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:38:44.863
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__List__Olist_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
list_list_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
set_list_mat_a: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
list_mat_a: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_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 (75)
thf(sy_c_Determinant_Odelete__index,type,
delete_index: nat > nat > nat ).
thf(sy_c_Determinant_Omat__delete_001tf__a,type,
mat_delete_a: mat_a > nat > nat > mat_a ).
thf(sy_c_Determinant_Opermutation__delete,type,
permutation_delete: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Int__Oint,type,
permut3692553072317293667rt_int: int > nat > ( int > nat ) > int > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Nat__Onat,type,
permut3695043542826343943rt_nat: nat > nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Real__Oreal,type,
permut4060954620988167523t_real: real > nat > ( real > nat ) > real > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_Itf__a_J,type,
minus_minus_mat_a: mat_a > mat_a > mat_a ).
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__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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_Itf__a_J,type,
times_times_mat_a: mat_a > mat_a > mat_a ).
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__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_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_List_Ogen__length_001t__Matrix__Omat_Itf__a_J,type,
gen_length_mat_a: nat > list_mat_a > nat ).
thf(sy_c_List_Olast_001t__Matrix__Omat_Itf__a_J,type,
last_mat_a: list_mat_a > mat_a ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
cons_list_mat_a: list_mat_a > list_list_mat_a > list_list_mat_a ).
thf(sy_c_List_Olist_OCons_001t__Matrix__Omat_Itf__a_J,type,
cons_mat_a: mat_a > list_mat_a > list_mat_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
nil_list_mat_a: list_list_mat_a ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Omat_Itf__a_J,type,
nil_mat_a: list_mat_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_Ohd_001t__Matrix__Omat_Itf__a_J,type,
hd_mat_a: list_mat_a > mat_a ).
thf(sy_c_List_Olist_Otl_001t__Matrix__Omat_Itf__a_J,type,
tl_mat_a: list_mat_a > list_mat_a ).
thf(sy_c_List_Olist__update_001t__Matrix__Omat_Itf__a_J,type,
list_update_mat_a: list_mat_a > nat > mat_a > list_mat_a ).
thf(sy_c_List_On__lists_001t__Matrix__Omat_Itf__a_J,type,
n_lists_mat_a: nat > list_mat_a > list_list_mat_a ).
thf(sy_c_List_Onth_001t__Matrix__Omat_Itf__a_J,type,
nth_mat_a: list_mat_a > nat > mat_a ).
thf(sy_c_List_Oproduct__lists_001t__Matrix__Omat_Itf__a_J,type,
product_lists_mat_a: list_list_mat_a > list_list_mat_a ).
thf(sy_c_List_Osubseqs_001t__Matrix__Omat_Itf__a_J,type,
subseqs_mat_a: list_mat_a > list_list_mat_a ).
thf(sy_c_Matrix_Odiag__block__mat_001tf__a,type,
diag_block_mat_a: list_mat_a > mat_a ).
thf(sy_c_Matrix_Odim__col_001tf__a,type,
dim_col_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Omk__diagonal_001tf__a,type,
mk_diagonal_a: list_a > mat_a ).
thf(sy_c_Matrix_Oundef__vec_001t__Matrix__Omat_Itf__a_J,type,
undef_vec_mat_a: nat > mat_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
size_s6656407794899724303_mat_a: list_list_mat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
size_size_list_mat_a: list_mat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
bot_bo2759726786008686517_mat_a: set_list_mat_a ).
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_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
ord_le3279973697895081845_mat_a: set_list_mat_a > set_list_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
ord_le4771995077433322369_mat_a: set_list_mat_a > set_list_mat_a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
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_Spectral__Theory__Complements_Oreal__diag__decomp_001tf__a,type,
spectr3403749184330357196comp_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_Sublist_Osublists_001t__Matrix__Omat_Itf__a_J,type,
sublists_mat_a: list_mat_a > list_list_mat_a ).
thf(sy_c_member_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
member_list_mat_a: list_mat_a > set_list_mat_a > $o ).
thf(sy_v_Als,type,
als: set_list_mat_a ).
thf(sy_v_Alsa____,type,
alsa: set_list_mat_a ).
thf(sy_v_U0____,type,
u0: mat_a ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_na____,type,
na: nat ).
% Relevant facts (1075)
thf(fact_0_Suc_I2_J,axiom,
alsa != bot_bo2759726786008686517_mat_a ).
% Suc(2)
thf(fact_1_assms_I3_J,axiom,
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ als )
=> ( ( size_size_list_mat_a @ X )
= n ) ) ).
% assms(3)
thf(fact_2_eq__comps_Ocases,axiom,
! [X2: list_mat_a] :
( ( X2 != nil_mat_a )
=> ( ! [X3: mat_a] :
( X2
!= ( cons_mat_a @ X3 @ nil_mat_a ) )
=> ~ ! [X3: mat_a,Y: mat_a,L: list_mat_a] :
( X2
!= ( cons_mat_a @ X3 @ ( cons_mat_a @ Y @ L ) ) ) ) ) ).
% eq_comps.cases
thf(fact_3_eq__comps_Oinduct,axiom,
! [P: list_mat_a > $o,A0: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X3: mat_a] : ( P @ ( cons_mat_a @ X3 @ nil_mat_a ) )
=> ( ! [X3: mat_a,Y: mat_a,L: list_mat_a] :
( ( P @ ( cons_mat_a @ Y @ L ) )
=> ( P @ ( cons_mat_a @ X3 @ ( cons_mat_a @ Y @ L ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_4_diag__block__mat__singleton,axiom,
! [A: mat_a] :
( ( diag_block_mat_a @ ( cons_mat_a @ A @ nil_mat_a ) )
= A ) ).
% diag_block_mat_singleton
thf(fact_5_list__induct2,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,P: list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( P @ nil_mat_a @ nil_mat_a )
=> ( ! [X3: mat_a,Xs2: list_mat_a,Y: mat_a,Ys2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ ( cons_mat_a @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_6_list__induct3,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_a,P: list_mat_a > list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_a )
=> ( ! [X3: mat_a,Xs2: list_mat_a,Y: mat_a,Ys2: list_mat_a,Z: mat_a,Zs2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ ( cons_mat_a @ Y @ Ys2 ) @ ( cons_mat_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_7_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_a,Ws: list_mat_a,P: list_mat_a > list_mat_a > list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_size_list_mat_a @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_a @ nil_mat_a )
=> ( ! [X3: mat_a,Xs2: list_mat_a,Y: mat_a,Ys2: list_mat_a,Z: mat_a,Zs2: list_mat_a,W: mat_a,Ws2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ ( cons_mat_a @ Y @ Ys2 ) @ ( cons_mat_a @ Z @ Zs2 ) @ ( cons_mat_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_8_diag__block__mat_Ocases,axiom,
! [X2: list_mat_a] :
( ( X2 != nil_mat_a )
=> ~ ! [A2: mat_a,As: list_mat_a] :
( X2
!= ( cons_mat_a @ A2 @ As ) ) ) ).
% diag_block_mat.cases
thf(fact_9_diag__block__mat_Oinduct,axiom,
! [P: list_mat_a > $o,A0: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [A2: mat_a,As: list_mat_a] :
( ( P @ As )
=> ( P @ ( cons_mat_a @ A2 @ As ) ) )
=> ( P @ A0 ) ) ) ).
% diag_block_mat.induct
thf(fact_10_u0,axiom,
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ? [B: mat_a] : ( spectr3403749184330357196comp_a @ ( nth_mat_a @ X @ zero_zero_nat ) @ B @ u0 ) ) ).
% u0
thf(fact_11_list_Osimps_I3_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( ( cons_mat_a @ X21 @ X22 )
!= nil_mat_a ) ).
% list.simps(3)
thf(fact_12_list_OdiscI,axiom,
! [List: list_mat_a,X21: mat_a,X22: list_mat_a] :
( ( List
= ( cons_mat_a @ X21 @ X22 ) )
=> ( List != nil_mat_a ) ) ).
% list.discI
thf(fact_13_assms_I1_J,axiom,
als != bot_bo2759726786008686517_mat_a ).
% assms(1)
thf(fact_14__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062U0_O_A_092_060forall_062Al_092_060in_062Als_O_A_092_060exists_062B_O_Areal__diag__decomp_A_IAl_A_B_A0_J_AB_AU0_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [U0: mat_a] :
~ ! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ? [B: mat_a] : ( spectr3403749184330357196comp_a @ ( nth_mat_a @ X @ zero_zero_nat ) @ B @ U0 ) ) ).
% \<open>\<And>thesis. (\<And>U0. \<forall>Al\<in>Als. \<exists>B. real_diag_decomp (Al ! 0) B U0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_15__092_060open_062_092_060exists_062U_O_A_092_060forall_062Al_092_060in_062Als_O_A_092_060exists_062B_O_Areal__diag__decomp_A_IAl_A_B_A0_J_AB_AU_092_060close_062,axiom,
? [U: mat_a] :
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ? [B: mat_a] : ( spectr3403749184330357196comp_a @ ( nth_mat_a @ X @ zero_zero_nat ) @ B @ U ) ) ).
% \<open>\<exists>U. \<forall>Al\<in>Als. \<exists>B. real_diag_decomp (Al ! 0) B U\<close>
thf(fact_16__092_060open_062_092_060forall_062Al_092_060in_062Als_O_Adiag__block__mat_AAl_A_061_AAl_A_B_A0_092_060close_062,axiom,
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ( ( diag_block_mat_a @ X )
= ( nth_mat_a @ X @ zero_zero_nat ) ) ) ).
% \<open>\<forall>Al\<in>Als. diag_block_mat Al = Al ! 0\<close>
thf(fact_17_True,axiom,
na = zero_zero_nat ).
% True
thf(fact_18_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(2)
thf(fact_19_assms_I5_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ n )
=> ? [U: mat_a] :
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ als )
=> ? [B: mat_a] : ( spectr3403749184330357196comp_a @ ( nth_mat_a @ X @ I ) @ B @ U ) ) ) ).
% assms(5)
thf(fact_20_nth__Cons__0,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_21_length__0__conv,axiom,
! [Xs: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_mat_a ) ) ).
% length_0_conv
thf(fact_22_list_Osize_I3_J,axiom,
( ( size_size_list_mat_a @ nil_mat_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_23_diag__block__mat__length__1,axiom,
! [Al: list_mat_a] :
( ( ( size_size_list_mat_a @ Al )
= one_one_nat )
=> ( ( diag_block_mat_a @ Al )
= ( nth_mat_a @ Al @ zero_zero_nat ) ) ) ).
% diag_block_mat_length_1
thf(fact_24_not__Cons__self2,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( cons_mat_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_25_list_Oinject,axiom,
! [X21: mat_a,X22: list_mat_a,Y21: mat_a,Y22: list_mat_a] :
( ( ( cons_mat_a @ X21 @ X22 )
= ( cons_mat_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_26_neq__if__length__neq,axiom,
! [Xs: list_mat_a,Ys: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs )
!= ( size_size_list_mat_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_27_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_mat_a] :
( ( size_size_list_mat_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_28_list__nonempty__induct,axiom,
! [Xs: list_mat_a,P: list_mat_a > $o] :
( ( Xs != nil_mat_a )
=> ( ! [X3: mat_a] : ( P @ ( cons_mat_a @ X3 @ nil_mat_a ) )
=> ( ! [X3: mat_a,Xs2: list_mat_a] :
( ( Xs2 != nil_mat_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_29_induct__list012,axiom,
! [P: list_mat_a > $o,Xs: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X3: mat_a] : ( P @ ( cons_mat_a @ X3 @ nil_mat_a ) )
=> ( ! [X3: mat_a,Y: mat_a,Zs2: list_mat_a] :
( ( P @ Zs2 )
=> ( ( P @ ( cons_mat_a @ Y @ Zs2 ) )
=> ( P @ ( cons_mat_a @ X3 @ ( cons_mat_a @ Y @ Zs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_30_list__induct2_H,axiom,
! [P: list_mat_a > list_mat_a > $o,Xs: list_mat_a,Ys: list_mat_a] :
( ( P @ nil_mat_a @ nil_mat_a )
=> ( ! [X3: mat_a,Xs2: list_mat_a] : ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ nil_mat_a )
=> ( ! [Y: mat_a,Ys2: list_mat_a] : ( P @ nil_mat_a @ ( cons_mat_a @ Y @ Ys2 ) )
=> ( ! [X3: mat_a,Xs2: list_mat_a,Y: mat_a,Ys2: list_mat_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ ( cons_mat_a @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_31_neq__Nil__conv,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
= ( ? [Y2: mat_a,Ys3: list_mat_a] :
( Xs
= ( cons_mat_a @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_32_map__tailrec__rev_Oinduct,axiom,
! [P: ( mat_a > mat_a ) > list_mat_a > list_mat_a > $o,A0: mat_a > mat_a,A1: list_mat_a,A22: list_mat_a] :
( ! [F: mat_a > mat_a,X_1: list_mat_a] : ( P @ F @ nil_mat_a @ X_1 )
=> ( ! [F: mat_a > mat_a,A3: mat_a,As2: list_mat_a,Bs: list_mat_a] :
( ( P @ F @ As2 @ ( cons_mat_a @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons_mat_a @ A3 @ As2 ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_33_successively_Oinduct,axiom,
! [P: ( mat_a > mat_a > $o ) > list_mat_a > $o,A0: mat_a > mat_a > $o,A1: list_mat_a] :
( ! [P2: mat_a > mat_a > $o] : ( P @ P2 @ nil_mat_a )
=> ( ! [P2: mat_a > mat_a > $o,X3: mat_a] : ( P @ P2 @ ( cons_mat_a @ X3 @ nil_mat_a ) )
=> ( ! [P2: mat_a > mat_a > $o,X3: mat_a,Y: mat_a,Xs2: list_mat_a] :
( ( P @ P2 @ ( cons_mat_a @ Y @ Xs2 ) )
=> ( P @ P2 @ ( cons_mat_a @ X3 @ ( cons_mat_a @ Y @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_34_remdups__adj_Oinduct,axiom,
! [P: list_mat_a > $o,A0: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X3: mat_a] : ( P @ ( cons_mat_a @ X3 @ nil_mat_a ) )
=> ( ! [X3: mat_a,Y: mat_a,Xs2: list_mat_a] :
( ( ( X3 = Y )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y )
=> ( P @ ( cons_mat_a @ Y @ Xs2 ) ) )
=> ( P @ ( cons_mat_a @ X3 @ ( cons_mat_a @ Y @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_35_sorted__wrt_Oinduct,axiom,
! [P: ( mat_a > mat_a > $o ) > list_mat_a > $o,A0: mat_a > mat_a > $o,A1: list_mat_a] :
( ! [P2: mat_a > mat_a > $o] : ( P @ P2 @ nil_mat_a )
=> ( ! [P2: mat_a > mat_a > $o,X3: mat_a,Ys2: list_mat_a] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_mat_a @ X3 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_36_List_Otranspose_Ocases,axiom,
! [X2: list_list_mat_a] :
( ( X2 != nil_list_mat_a )
=> ( ! [Xss: list_list_mat_a] :
( X2
!= ( cons_list_mat_a @ nil_mat_a @ Xss ) )
=> ~ ! [X3: mat_a,Xs2: list_mat_a,Xss: list_list_mat_a] :
( X2
!= ( cons_list_mat_a @ ( cons_mat_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_37_shuffles_Oinduct,axiom,
! [P: list_mat_a > list_mat_a > $o,A0: list_mat_a,A1: list_mat_a] :
( ! [X_1: list_mat_a] : ( P @ nil_mat_a @ X_1 )
=> ( ! [Xs2: list_mat_a] : ( P @ Xs2 @ nil_mat_a )
=> ( ! [X3: mat_a,Xs2: list_mat_a,Y: mat_a,Ys2: list_mat_a] :
( ( P @ Xs2 @ ( cons_mat_a @ Y @ Ys2 ) )
=> ( ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ ( cons_mat_a @ Y @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_38_splice_Oinduct,axiom,
! [P: list_mat_a > list_mat_a > $o,A0: list_mat_a,A1: list_mat_a] :
( ! [X_1: list_mat_a] : ( P @ nil_mat_a @ X_1 )
=> ( ! [X3: mat_a,Xs2: list_mat_a,Ys2: list_mat_a] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_39_list_Oinducts,axiom,
! [P: list_mat_a > $o,List: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X1: mat_a,X23: list_mat_a] :
( ( P @ X23 )
=> ( P @ ( cons_mat_a @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_40_list_Oexhaust,axiom,
! [Y3: list_mat_a] :
( ( Y3 != nil_mat_a )
=> ~ ! [X212: mat_a,X222: list_mat_a] :
( Y3
!= ( cons_mat_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_41_undef__vec__def,axiom,
( undef_vec_mat_a
= ( nth_mat_a @ nil_mat_a ) ) ).
% undef_vec_def
thf(fact_42_u0__dim,axiom,
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ( ( ( dim_row_a @ ( nth_mat_a @ ( cons_mat_a @ u0 @ nil_mat_a ) @ zero_zero_nat ) )
= ( dim_row_a @ ( nth_mat_a @ X @ zero_zero_nat ) ) )
& ( ( dim_col_a @ ( nth_mat_a @ ( cons_mat_a @ u0 @ nil_mat_a ) @ zero_zero_nat ) )
= ( dim_col_a @ ( nth_mat_a @ X @ zero_zero_nat ) ) ) ) ) ).
% u0_dim
thf(fact_43_longest__common__prefix_Oinduct,axiom,
! [P: list_mat_a > list_mat_a > $o,A0: list_mat_a,A1: list_mat_a] :
( ! [X3: mat_a,Xs2: list_mat_a,Y: mat_a,Ys2: list_mat_a] :
( ( ( X3 = Y )
=> ( P @ Xs2 @ Ys2 ) )
=> ( P @ ( cons_mat_a @ X3 @ Xs2 ) @ ( cons_mat_a @ Y @ Ys2 ) ) )
=> ( ! [X_1: list_mat_a] : ( P @ nil_mat_a @ X_1 )
=> ( ! [Uu: list_mat_a] : ( P @ Uu @ nil_mat_a )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_44_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_45_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_46_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_47_class__field_Ozero__not__one,axiom,
zero_zero_real != one_one_real ).
% class_field.zero_not_one
thf(fact_48_assms_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ n )
=> ! [X: list_mat_a] :
( ( member_list_mat_a @ X @ als )
=> ( ( dim_row_a @ ( nth_mat_a @ X @ I ) )
= ( dim_col_a @ ( nth_mat_a @ X @ I ) ) ) ) ) ).
% assms(4)
thf(fact_49_Suc_Oprems_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ ( suc @ na ) ).
% Suc.prems(2)
thf(fact_50_Suc_I1_J,axiom,
! [Als: set_list_mat_a] :
( ( Als != bot_bo2759726786008686517_mat_a )
=> ( ( ord_less_nat @ zero_zero_nat @ na )
=> ( ! [X3: list_mat_a] :
( ( member_list_mat_a @ X3 @ Als )
=> ( ( size_size_list_mat_a @ X3 )
= na ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ na )
=> ! [X3: list_mat_a] :
( ( member_list_mat_a @ X3 @ Als )
=> ( ( dim_row_a @ ( nth_mat_a @ X3 @ I2 ) )
= ( dim_col_a @ ( nth_mat_a @ X3 @ I2 ) ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ na )
=> ? [U2: mat_a] :
! [X3: list_mat_a] :
( ( member_list_mat_a @ X3 @ Als )
=> ? [B2: mat_a] : ( spectr3403749184330357196comp_a @ ( nth_mat_a @ X3 @ I2 ) @ B2 @ U2 ) ) )
=> ? [Ul: list_mat_a] :
( ( ( size_size_list_mat_a @ Ul )
= na )
& ! [I: nat] :
( ( ord_less_nat @ I @ na )
=> ! [X: list_mat_a] :
( ( member_list_mat_a @ X @ Als )
=> ( ( ( dim_row_a @ ( nth_mat_a @ Ul @ I ) )
= ( dim_row_a @ ( nth_mat_a @ X @ I ) ) )
& ( ( dim_col_a @ ( nth_mat_a @ Ul @ I ) )
= ( dim_col_a @ ( nth_mat_a @ X @ I ) ) ) ) ) )
& ! [X: list_mat_a] :
( ( member_list_mat_a @ X @ Als )
=> ? [Bl: list_mat_a] :
( ( ( size_size_list_mat_a @ Bl )
= na )
& ( spectr3403749184330357196comp_a @ ( diag_block_mat_a @ X ) @ ( diag_block_mat_a @ Bl ) @ ( diag_block_mat_a @ Ul ) ) ) ) ) ) ) ) ) ) ).
% Suc(1)
thf(fact_51_Suc_I4_J,axiom,
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ( ( size_size_list_mat_a @ X )
= ( suc @ na ) ) ) ).
% Suc(4)
thf(fact_52_Suc_I6_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ na ) )
=> ? [U: mat_a] :
! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ? [B: mat_a] : ( spectr3403749184330357196comp_a @ ( nth_mat_a @ X @ I ) @ B @ U ) ) ) ).
% Suc(6)
thf(fact_53_Suc_Oprems_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ na ) )
=> ! [X: list_mat_a] :
( ( member_list_mat_a @ X @ alsa )
=> ( ( dim_row_a @ ( nth_mat_a @ X @ I ) )
= ( dim_col_a @ ( nth_mat_a @ X @ I ) ) ) ) ) ).
% Suc.prems(4)
thf(fact_54_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_55_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_56_semiring__norm_I137_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% semiring_norm(137)
thf(fact_57_semiring__norm_I137_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% semiring_norm(137)
thf(fact_58_semiring__norm_I137_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% semiring_norm(137)
thf(fact_59_semiring__norm_I138_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% semiring_norm(138)
thf(fact_60_semiring__norm_I138_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% semiring_norm(138)
thf(fact_61_semiring__norm_I138_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% semiring_norm(138)
thf(fact_62_diag__block__mat__cong__comp,axiom,
! [Al: list_mat_a,Bl2: list_mat_a,J: nat] :
( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_row_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Bl2 @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_col_a @ ( nth_mat_a @ Bl2 @ I2 ) ) ) )
=> ( ( ( diag_block_mat_a @ Al )
= ( diag_block_mat_a @ Bl2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_mat_a @ Al ) )
=> ( ( nth_mat_a @ Al @ J )
= ( nth_mat_a @ Bl2 @ J ) ) ) ) ) ) ) ).
% diag_block_mat_cong_comp
thf(fact_63_diag__block__mat__dim__row__col__eq,axiom,
! [Al: list_mat_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_row_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_col_a @ ( nth_mat_a @ Al @ I2 ) ) ) )
=> ( ( dim_row_a @ ( diag_block_mat_a @ Al ) )
= ( dim_col_a @ ( diag_block_mat_a @ Al ) ) ) ) ).
% diag_block_mat_dim_row_col_eq
thf(fact_64_length__induct,axiom,
! [P: list_mat_a > $o,Xs: list_mat_a] :
( ! [Xs2: list_mat_a] :
( ! [Ys4: list_mat_a] :
( ( ord_less_nat @ ( size_size_list_mat_a @ Ys4 ) @ ( size_size_list_mat_a @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_65_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_66_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_67_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_68_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_69_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_70_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_71_semiring__norm_I135_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% semiring_norm(135)
thf(fact_72_semiring__norm_I135_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% semiring_norm(135)
thf(fact_73_semiring__norm_I135_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% semiring_norm(135)
thf(fact_74_diag__block__mat__dim__col__cong,axiom,
! [Ul2: list_mat_a,Bl2: list_mat_a] :
( ( ( size_size_list_mat_a @ Ul2 )
= ( size_size_list_mat_a @ Bl2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Bl2 ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Bl2 @ I2 ) )
= ( dim_col_a @ ( nth_mat_a @ Ul2 @ I2 ) ) ) )
=> ( ( dim_col_a @ ( diag_block_mat_a @ Ul2 ) )
= ( dim_col_a @ ( diag_block_mat_a @ Bl2 ) ) ) ) ) ).
% diag_block_mat_dim_col_cong
thf(fact_75_diag__block__mat__dim__row__cong,axiom,
! [Ul2: list_mat_a,Bl2: list_mat_a] :
( ( ( size_size_list_mat_a @ Ul2 )
= ( size_size_list_mat_a @ Bl2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Bl2 ) )
=> ( ( dim_row_a @ ( nth_mat_a @ Bl2 @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Ul2 @ I2 ) ) ) )
=> ( ( dim_row_a @ ( diag_block_mat_a @ Ul2 ) )
= ( dim_row_a @ ( diag_block_mat_a @ Bl2 ) ) ) ) ) ).
% diag_block_mat_dim_row_cong
thf(fact_76_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_mat_a,Z2: list_mat_a] : ( Y4 = Z2 ) )
= ( ^ [Xs3: list_mat_a,Ys3: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs3 ) )
=> ( ( nth_mat_a @ Xs3 @ I3 )
= ( nth_mat_a @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_77_Skolem__list__nth,axiom,
! [K: nat,P: nat > mat_a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: mat_a] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_mat_a @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_78_nth__equalityI,axiom,
! [Xs: list_mat_a,Ys: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( nth_mat_a @ Xs @ I2 )
= ( nth_mat_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_79_length__greater__0__conv,axiom,
! [Xs: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Xs ) )
= ( Xs != nil_mat_a ) ) ).
% length_greater_0_conv
thf(fact_80_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_81_sorted__list__subset_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [A3: nat,As2: list_nat,B3: nat,Bs: list_nat] :
( ( ( A3 = B3 )
=> ( P @ As2 @ ( cons_nat @ B3 @ Bs ) ) )
=> ( ( ( A3 != B3 )
=> ( ( ord_less_nat @ B3 @ A3 )
=> ( P @ ( cons_nat @ A3 @ As2 ) @ Bs ) ) )
=> ( P @ ( cons_nat @ A3 @ As2 ) @ ( cons_nat @ B3 @ Bs ) ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [A3: nat,Uv: list_nat] : ( P @ ( cons_nat @ A3 @ Uv ) @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sorted_list_subset.induct
thf(fact_82_sorted__list__subset_Oinduct,axiom,
! [P: list_real > list_real > $o,A0: list_real,A1: list_real] :
( ! [A3: real,As2: list_real,B3: real,Bs: list_real] :
( ( ( A3 = B3 )
=> ( P @ As2 @ ( cons_real @ B3 @ Bs ) ) )
=> ( ( ( A3 != B3 )
=> ( ( ord_less_real @ B3 @ A3 )
=> ( P @ ( cons_real @ A3 @ As2 ) @ Bs ) ) )
=> ( P @ ( cons_real @ A3 @ As2 ) @ ( cons_real @ B3 @ Bs ) ) ) )
=> ( ! [X_1: list_real] : ( P @ nil_real @ X_1 )
=> ( ! [A3: real,Uv: list_real] : ( P @ ( cons_real @ A3 @ Uv ) @ nil_real )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sorted_list_subset.induct
thf(fact_83_sorted__list__subset_Oinduct,axiom,
! [P: list_int > list_int > $o,A0: list_int,A1: list_int] :
( ! [A3: int,As2: list_int,B3: int,Bs: list_int] :
( ( ( A3 = B3 )
=> ( P @ As2 @ ( cons_int @ B3 @ Bs ) ) )
=> ( ( ( A3 != B3 )
=> ( ( ord_less_int @ B3 @ A3 )
=> ( P @ ( cons_int @ A3 @ As2 ) @ Bs ) ) )
=> ( P @ ( cons_int @ A3 @ As2 ) @ ( cons_int @ B3 @ Bs ) ) ) )
=> ( ! [X_1: list_int] : ( P @ nil_int @ X_1 )
=> ( ! [A3: int,Uv: list_int] : ( P @ ( cons_int @ A3 @ Uv ) @ nil_int )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sorted_list_subset.induct
thf(fact_84_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_mat_a @ N @ nil_mat_a )
= ( cons_list_mat_a @ nil_mat_a @ nil_list_mat_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_mat_a @ N @ nil_mat_a )
= nil_list_mat_a ) ) ) ).
% n_lists_Nil
thf(fact_85_n__lists_Osimps_I1_J,axiom,
! [Xs: list_mat_a] :
( ( n_lists_mat_a @ zero_zero_nat @ Xs )
= ( cons_list_mat_a @ nil_mat_a @ nil_list_mat_a ) ) ).
% n_lists.simps(1)
thf(fact_86_nat_Osimps_I1_J,axiom,
! [X24: nat,Y23: nat] :
( ( ( suc @ X24 )
= ( suc @ Y23 ) )
= ( X24 = Y23 ) ) ).
% nat.simps(1)
thf(fact_87_old_Onat_Osimps_I1_J,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.simps(1)
thf(fact_88_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_89_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
% Suc_n_not_n
thf(fact_90_nat_Osimps_I3_J,axiom,
! [X24: nat] :
( ( suc @ X24 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_91_old_Onat_Osimps_I3_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.simps(3)
thf(fact_92_old_Onat_Osimps_I2_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.simps(2)
thf(fact_93_nat_OdiscI,axiom,
! [Nat: nat,X24: nat] :
( ( Nat
= ( suc @ X24 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_94_nat_Oinduct,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% nat.induct
thf(fact_95_nat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [X23: nat] :
( Y3
!= ( suc @ X23 ) ) ) ).
% nat.exhaust
thf(fact_96_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_97_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X3: nat,Y: nat] :
( ( P @ X3 @ Y )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_98_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_99_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_100_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_not_Zero
thf(fact_101_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_102_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_103_not__less__simps_I1_J,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(1)
thf(fact_104_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_105_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_106_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_107_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_108_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_109_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_110_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_111_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_112_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_113_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_114_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_115_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_116_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_117_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_118_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_119_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_120_less__trans__Suc,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_121_less__Suc__induct,axiom,
! [I4: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I4 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_122_strict__inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_123_unit__vecs__last_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N2: nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat,I2: nat] :
( ( P @ N2 @ I2 )
=> ( P @ N2 @ ( suc @ I2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_last.induct
thf(fact_124_lift__Suc__mono__less,axiom,
! [F2: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_125_lift__Suc__mono__less,axiom,
! [F2: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_126_lift__Suc__mono__less,axiom,
! [F2: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_127_lift__Suc__mono__less__iff,axiom,
! [F2: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_128_lift__Suc__mono__less__iff,axiom,
! [F2: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_129_lift__Suc__mono__less__iff,axiom,
! [F2: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_130_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_131_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_132_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_133_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_134_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_135_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_136_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_137_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_138_numeral__nat_I7_J,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% numeral_nat(7)
thf(fact_139_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_140_length__Suc__conv,axiom,
! [Xs: list_mat_a,N: nat] :
( ( ( size_size_list_mat_a @ Xs )
= ( suc @ N ) )
= ( ? [Y2: mat_a,Ys3: list_mat_a] :
( ( Xs
= ( cons_mat_a @ Y2 @ Ys3 ) )
& ( ( size_size_list_mat_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_141_Suc__length__conv,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ( suc @ N )
= ( size_size_list_mat_a @ Xs ) )
= ( ? [Y2: mat_a,Ys3: list_mat_a] :
( ( Xs
= ( cons_mat_a @ Y2 @ Ys3 ) )
& ( ( size_size_list_mat_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_142_nth__Cons__Suc,axiom,
! [X2: mat_a,Xs: list_mat_a,N: nat] :
( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_mat_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_143_verit__comp__simplify_I1_J,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% verit_comp_simplify(1)
thf(fact_144_verit__comp__simplify_I1_J,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% verit_comp_simplify(1)
thf(fact_145_verit__comp__simplify_I1_J,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% verit_comp_simplify(1)
thf(fact_146_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_147_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_148_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_149_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_150_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_151_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_152_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_153_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_154_size__neq__size__imp__neq,axiom,
! [X2: list_mat_a,Y3: list_mat_a] :
( ( ( size_size_list_mat_a @ X2 )
!= ( size_size_list_mat_a @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_155_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_156_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_157_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_158_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_159_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_160_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_161_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_162_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_163_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_164_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_165_length__Cons,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( size_size_list_mat_a @ ( cons_mat_a @ X2 @ Xs ) )
= ( suc @ ( size_size_list_mat_a @ Xs ) ) ) ).
% length_Cons
thf(fact_166_diag__block__mat__commute,axiom,
! [Al: list_mat_a,Bl2: list_mat_a] :
( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( times_times_mat_a @ ( nth_mat_a @ Al @ I2 ) @ ( nth_mat_a @ Bl2 @ I2 ) )
= ( times_times_mat_a @ ( nth_mat_a @ Bl2 @ I2 ) @ ( nth_mat_a @ Al @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Bl2 @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Bl2 @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Al @ I2 ) ) ) )
=> ( ( times_times_mat_a @ ( diag_block_mat_a @ Al ) @ ( diag_block_mat_a @ Bl2 ) )
= ( times_times_mat_a @ ( diag_block_mat_a @ Bl2 ) @ ( diag_block_mat_a @ Al ) ) ) ) ) ) ) ).
% diag_block_mat_commute
thf(fact_167_diag__block__mat__commute__comp,axiom,
! [Al: list_mat_a,Bl2: list_mat_a,I4: nat] :
( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_row_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_col_a @ ( nth_mat_a @ Al @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_row_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Bl2 @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_col_a @ ( nth_mat_a @ Bl2 @ I2 ) ) ) )
=> ( ( ( times_times_mat_a @ ( diag_block_mat_a @ Al ) @ ( diag_block_mat_a @ Bl2 ) )
= ( times_times_mat_a @ ( diag_block_mat_a @ Bl2 ) @ ( diag_block_mat_a @ Al ) ) )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_mat_a @ Al ) )
=> ( ( times_times_mat_a @ ( nth_mat_a @ Al @ I4 ) @ ( nth_mat_a @ Bl2 @ I4 ) )
= ( times_times_mat_a @ ( nth_mat_a @ Bl2 @ I4 ) @ ( nth_mat_a @ Al @ I4 ) ) ) ) ) ) ) ) ) ).
% diag_block_mat_commute_comp
thf(fact_168_inf__concat_Oinduct,axiom,
! [P: ( nat > nat ) > nat > $o,A0: nat > nat,A1: nat] :
( ! [N2: nat > nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat > nat,K2: nat] :
( ( P @ N2 @ K2 )
=> ( P @ N2 @ ( suc @ K2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inf_concat.induct
thf(fact_169_nth__item_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% nth_item.cases
thf(fact_170_more__arith__simps_I11_J,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A4 @ B4 ) @ C )
= ( times_times_nat @ A4 @ ( times_times_nat @ B4 @ C ) ) ) ).
% more_arith_simps(11)
thf(fact_171_more__arith__simps_I11_J,axiom,
! [A4: real,B4: real,C: real] :
( ( times_times_real @ ( times_times_real @ A4 @ B4 ) @ C )
= ( times_times_real @ A4 @ ( times_times_real @ B4 @ C ) ) ) ).
% more_arith_simps(11)
thf(fact_172_more__arith__simps_I11_J,axiom,
! [A4: int,B4: int,C: int] :
( ( times_times_int @ ( times_times_int @ A4 @ B4 ) @ C )
= ( times_times_int @ A4 @ ( times_times_int @ B4 @ C ) ) ) ).
% more_arith_simps(11)
thf(fact_173_class__cring_Ofactors__equal,axiom,
! [A4: real,B4: real,C: real,D: real] :
( ( A4 = B4 )
=> ( ( C = D )
=> ( ( times_times_real @ A4 @ C )
= ( times_times_real @ B4 @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_174_class__cring_Ofactors__equal,axiom,
! [A4: int,B4: int,C: int,D: int] :
( ( A4 = B4 )
=> ( ( C = D )
=> ( ( times_times_int @ A4 @ C )
= ( times_times_int @ B4 @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_175_mult__right__cancel,axiom,
! [C: nat,A4: nat,B4: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A4 @ C )
= ( times_times_nat @ B4 @ C ) )
= ( A4 = B4 ) ) ) ).
% mult_right_cancel
thf(fact_176_mult__right__cancel,axiom,
! [C: real,A4: real,B4: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A4 @ C )
= ( times_times_real @ B4 @ C ) )
= ( A4 = B4 ) ) ) ).
% mult_right_cancel
thf(fact_177_mult__right__cancel,axiom,
! [C: int,A4: int,B4: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A4 @ C )
= ( times_times_int @ B4 @ C ) )
= ( A4 = B4 ) ) ) ).
% mult_right_cancel
thf(fact_178_mult__cancel__right,axiom,
! [A4: nat,C: nat,B4: nat] :
( ( ( times_times_nat @ A4 @ C )
= ( times_times_nat @ B4 @ C ) )
= ( ( C = zero_zero_nat )
| ( A4 = B4 ) ) ) ).
% mult_cancel_right
thf(fact_179_mult__cancel__right,axiom,
! [A4: real,C: real,B4: real] :
( ( ( times_times_real @ A4 @ C )
= ( times_times_real @ B4 @ C ) )
= ( ( C = zero_zero_real )
| ( A4 = B4 ) ) ) ).
% mult_cancel_right
thf(fact_180_mult__cancel__right,axiom,
! [A4: int,C: int,B4: int] :
( ( ( times_times_int @ A4 @ C )
= ( times_times_int @ B4 @ C ) )
= ( ( C = zero_zero_int )
| ( A4 = B4 ) ) ) ).
% mult_cancel_right
thf(fact_181_mult__left__cancel,axiom,
! [C: nat,A4: nat,B4: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A4 )
= ( times_times_nat @ C @ B4 ) )
= ( A4 = B4 ) ) ) ).
% mult_left_cancel
thf(fact_182_mult__left__cancel,axiom,
! [C: real,A4: real,B4: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A4 )
= ( times_times_real @ C @ B4 ) )
= ( A4 = B4 ) ) ) ).
% mult_left_cancel
thf(fact_183_mult__left__cancel,axiom,
! [C: int,A4: int,B4: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A4 )
= ( times_times_int @ C @ B4 ) )
= ( A4 = B4 ) ) ) ).
% mult_left_cancel
thf(fact_184_mult__cancel__left,axiom,
! [C: nat,A4: nat,B4: nat] :
( ( ( times_times_nat @ C @ A4 )
= ( times_times_nat @ C @ B4 ) )
= ( ( C = zero_zero_nat )
| ( A4 = B4 ) ) ) ).
% mult_cancel_left
thf(fact_185_mult__cancel__left,axiom,
! [C: real,A4: real,B4: real] :
( ( ( times_times_real @ C @ A4 )
= ( times_times_real @ C @ B4 ) )
= ( ( C = zero_zero_real )
| ( A4 = B4 ) ) ) ).
% mult_cancel_left
thf(fact_186_mult__cancel__left,axiom,
! [C: int,A4: int,B4: int] :
( ( ( times_times_int @ C @ A4 )
= ( times_times_int @ C @ B4 ) )
= ( ( C = zero_zero_int )
| ( A4 = B4 ) ) ) ).
% mult_cancel_left
thf(fact_187_no__zero__divisors,axiom,
! [A4: nat,B4: nat] :
( ( A4 != zero_zero_nat )
=> ( ( B4 != zero_zero_nat )
=> ( ( times_times_nat @ A4 @ B4 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_188_no__zero__divisors,axiom,
! [A4: real,B4: real] :
( ( A4 != zero_zero_real )
=> ( ( B4 != zero_zero_real )
=> ( ( times_times_real @ A4 @ B4 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_189_no__zero__divisors,axiom,
! [A4: int,B4: int] :
( ( A4 != zero_zero_int )
=> ( ( B4 != zero_zero_int )
=> ( ( times_times_int @ A4 @ B4 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_190_mult__eq__0__iff,axiom,
! [A4: nat,B4: nat] :
( ( ( times_times_nat @ A4 @ B4 )
= zero_zero_nat )
= ( ( A4 = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_191_mult__eq__0__iff,axiom,
! [A4: real,B4: real] :
( ( ( times_times_real @ A4 @ B4 )
= zero_zero_real )
= ( ( A4 = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_192_mult__eq__0__iff,axiom,
! [A4: int,B4: int] :
( ( ( times_times_int @ A4 @ B4 )
= zero_zero_int )
= ( ( A4 = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_193_divisors__zero,axiom,
! [A4: nat,B4: nat] :
( ( ( times_times_nat @ A4 @ B4 )
= zero_zero_nat )
=> ( ( A4 = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_194_divisors__zero,axiom,
! [A4: real,B4: real] :
( ( ( times_times_real @ A4 @ B4 )
= zero_zero_real )
=> ( ( A4 = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_195_divisors__zero,axiom,
! [A4: int,B4: int] :
( ( ( times_times_int @ A4 @ B4 )
= zero_zero_int )
=> ( ( A4 = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_196_mult__zero__right,axiom,
! [A4: nat] :
( ( times_times_nat @ A4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_197_mult__zero__right,axiom,
! [A4: real] :
( ( times_times_real @ A4 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_198_mult__zero__right,axiom,
! [A4: int] :
( ( times_times_int @ A4 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_199_mult__zero__left,axiom,
! [A4: nat] :
( ( times_times_nat @ zero_zero_nat @ A4 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_200_mult__zero__left,axiom,
! [A4: real] :
( ( times_times_real @ zero_zero_real @ A4 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_201_mult__zero__left,axiom,
! [A4: int] :
( ( times_times_int @ zero_zero_int @ A4 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_202_mult__not__zero,axiom,
! [A4: nat,B4: nat] :
( ( ( times_times_nat @ A4 @ B4 )
!= zero_zero_nat )
=> ( ( A4 != zero_zero_nat )
& ( B4 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_203_mult__not__zero,axiom,
! [A4: real,B4: real] :
( ( ( times_times_real @ A4 @ B4 )
!= zero_zero_real )
=> ( ( A4 != zero_zero_real )
& ( B4 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_204_mult__not__zero,axiom,
! [A4: int,B4: int] :
( ( ( times_times_int @ A4 @ B4 )
!= zero_zero_int )
=> ( ( A4 != zero_zero_int )
& ( B4 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_205_more__arith__simps_I6_J,axiom,
! [A4: nat] :
( ( times_times_nat @ A4 @ one_one_nat )
= A4 ) ).
% more_arith_simps(6)
thf(fact_206_more__arith__simps_I6_J,axiom,
! [A4: real] :
( ( times_times_real @ A4 @ one_one_real )
= A4 ) ).
% more_arith_simps(6)
thf(fact_207_more__arith__simps_I6_J,axiom,
! [A4: int] :
( ( times_times_int @ A4 @ one_one_int )
= A4 ) ).
% more_arith_simps(6)
thf(fact_208_more__arith__simps_I5_J,axiom,
! [A4: nat] :
( ( times_times_nat @ one_one_nat @ A4 )
= A4 ) ).
% more_arith_simps(5)
thf(fact_209_more__arith__simps_I5_J,axiom,
! [A4: real] :
( ( times_times_real @ one_one_real @ A4 )
= A4 ) ).
% more_arith_simps(5)
thf(fact_210_more__arith__simps_I5_J,axiom,
! [A4: int] :
( ( times_times_int @ one_one_int @ A4 )
= A4 ) ).
% more_arith_simps(5)
thf(fact_211_index__mult__mat_I2_J,axiom,
! [A: mat_a,B5: mat_a] :
( ( dim_row_a @ ( times_times_mat_a @ A @ B5 ) )
= ( dim_row_a @ A ) ) ).
% index_mult_mat(2)
thf(fact_212_index__mult__mat_I3_J,axiom,
! [A: mat_a,B5: mat_a] :
( ( dim_col_a @ ( times_times_mat_a @ A @ B5 ) )
= ( dim_col_a @ B5 ) ) ).
% index_mult_mat(3)
thf(fact_213_mult__sign__intros_I8_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B4 ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_214_mult__sign__intros_I8_J,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A4 @ B4 ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_215_mult__sign__intros_I7_J,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_216_mult__sign__intros_I7_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ord_less_real @ ( times_times_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(7)
thf(fact_217_mult__sign__intros_I7_J,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ ( times_times_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% mult_sign_intros(7)
thf(fact_218_mult__sign__intros_I6_J,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_219_mult__sign__intros_I6_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(6)
thf(fact_220_mult__sign__intros_I6_J,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% mult_sign_intros(6)
thf(fact_221_mult__sign__intros_I5_J,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A4 @ B4 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_222_mult__sign__intros_I5_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B4 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_223_mult__sign__intros_I5_J,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A4 @ B4 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_224_not__square__less__zero,axiom,
! [A4: real] :
~ ( ord_less_real @ ( times_times_real @ A4 @ A4 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_225_not__square__less__zero,axiom,
! [A4: int] :
~ ( ord_less_int @ ( times_times_int @ A4 @ A4 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_226_mult__less__0__iff,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A4 @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B4 ) ) ) ) ).
% mult_less_0_iff
thf(fact_227_mult__less__0__iff,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A4 @ B4 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A4 )
& ( ord_less_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_int @ A4 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B4 ) ) ) ) ).
% mult_less_0_iff
thf(fact_228_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B4 @ A4 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_229_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B4 @ A4 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_230_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B4 @ A4 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_231_zero__less__mult__iff,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_232_zero__less__mult__iff,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A4 @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A4 )
& ( ord_less_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_int @ A4 @ zero_zero_int )
& ( ord_less_int @ B4 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_233_zero__less__mult__pos,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A4 @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_234_zero__less__mult__pos,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B4 ) )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ zero_zero_real @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_235_zero__less__mult__pos,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A4 @ B4 ) )
=> ( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ord_less_int @ zero_zero_int @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_236_zero__less__mult__pos2,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B4 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_237_zero__less__mult__pos2,axiom,
! [B4: real,A4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B4 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ zero_zero_real @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_238_zero__less__mult__pos2,axiom,
! [B4: int,A4: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B4 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ord_less_int @ zero_zero_int @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_239_mult__less__cancel__left__neg,axiom,
! [C: real,A4: real,B4: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A4 ) @ ( times_times_real @ C @ B4 ) )
= ( ord_less_real @ B4 @ A4 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_240_mult__less__cancel__left__neg,axiom,
! [C: int,A4: int,B4: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A4 ) @ ( times_times_int @ C @ B4 ) )
= ( ord_less_int @ B4 @ A4 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_241_mult__less__cancel__left__pos,axiom,
! [C: real,A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A4 ) @ ( times_times_real @ C @ B4 ) )
= ( ord_less_real @ A4 @ B4 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_242_mult__less__cancel__left__pos,axiom,
! [C: int,A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A4 ) @ ( times_times_int @ C @ B4 ) )
= ( ord_less_int @ A4 @ B4 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_243_mult__strict__left__mono__neg,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A4 ) @ ( times_times_real @ C @ B4 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_244_mult__strict__left__mono__neg,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A4 ) @ ( times_times_int @ C @ B4 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_245_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A4 ) @ ( times_times_nat @ C @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_246_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A4 ) @ ( times_times_real @ C @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_247_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A4 ) @ ( times_times_int @ C @ B4 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_248_mult__less__cancel__left__disj,axiom,
! [C: real,A4: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ C @ A4 ) @ ( times_times_real @ C @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A4 @ B4 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B4 @ A4 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_249_mult__less__cancel__left__disj,axiom,
! [C: int,A4: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ C @ A4 ) @ ( times_times_int @ C @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A4 @ B4 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B4 @ A4 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_250_mult__strict__right__mono__neg,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A4 @ C ) @ ( times_times_real @ B4 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_251_mult__strict__right__mono__neg,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A4 @ C ) @ ( times_times_int @ B4 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_252_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ C ) @ ( times_times_nat @ B4 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_253_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A4 @ C ) @ ( times_times_real @ B4 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_254_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A4 @ C ) @ ( times_times_int @ B4 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_255_mult__less__cancel__right__disj,axiom,
! [A4: real,C: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A4 @ C ) @ ( times_times_real @ B4 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A4 @ B4 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B4 @ A4 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_256_mult__less__cancel__right__disj,axiom,
! [A4: int,C: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A4 @ C ) @ ( times_times_int @ B4 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A4 @ B4 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B4 @ A4 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_257_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A4 ) @ ( times_times_nat @ C @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_258_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A4 ) @ ( times_times_real @ C @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_259_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A4 ) @ ( times_times_int @ C @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_260_mult__cancel__right2,axiom,
! [A4: real,C: real] :
( ( ( times_times_real @ A4 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A4 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_261_mult__cancel__right2,axiom,
! [A4: int,C: int] :
( ( ( times_times_int @ A4 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A4 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_262_mult__cancel__right1,axiom,
! [C: real,B4: real] :
( ( C
= ( times_times_real @ B4 @ C ) )
= ( ( C = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_263_mult__cancel__right1,axiom,
! [C: int,B4: int] :
( ( C
= ( times_times_int @ B4 @ C ) )
= ( ( C = zero_zero_int )
| ( B4 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_264_mult__cancel__left2,axiom,
! [C: real,A4: real] :
( ( ( times_times_real @ C @ A4 )
= C )
= ( ( C = zero_zero_real )
| ( A4 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_265_mult__cancel__left2,axiom,
! [C: int,A4: int] :
( ( ( times_times_int @ C @ A4 )
= C )
= ( ( C = zero_zero_int )
| ( A4 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_266_mult__cancel__left1,axiom,
! [C: real,B4: real] :
( ( C
= ( times_times_real @ C @ B4 ) )
= ( ( C = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_267_mult__cancel__left1,axiom,
! [C: int,B4: int] :
( ( C
= ( times_times_int @ C @ B4 ) )
= ( ( C = zero_zero_int )
| ( B4 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_268_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_269_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_270_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_271_mult__if__delta,axiom,
! [P: $o,Q: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_272_mult__if__delta,axiom,
! [P: $o,Q: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_273_mult__if__delta,axiom,
! [P: $o,Q: int] :
( ( P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_274_mult__less__iff1,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ ( times_times_real @ X2 @ Z3 ) @ ( times_times_real @ Y3 @ Z3 ) )
= ( ord_less_real @ X2 @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_275_mult__less__iff1,axiom,
! [Z3: int,X2: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_int @ ( times_times_int @ X2 @ Z3 ) @ ( times_times_int @ Y3 @ Z3 ) )
= ( ord_less_int @ X2 @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_276_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_277_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_278_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_279_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_280_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_281_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_282_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_283_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_284_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_285_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_286_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_287_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_288_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_289_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_290_mult__less__cancel1,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 ) ) ) ).
% mult_less_cancel1
thf(fact_291_mult__less__mono2,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_292_mult__less__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_bot__less,axiom,
! [A4: set_list_mat_a] :
( ( A4 != bot_bo2759726786008686517_mat_a )
= ( ord_le3279973697895081845_mat_a @ bot_bo2759726786008686517_mat_a @ A4 ) ) ).
% bot_less
thf(fact_303_bot__less,axiom,
! [A4: nat] :
( ( A4 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A4 ) ) ).
% bot_less
thf(fact_304_not__less__bot,axiom,
! [A4: set_list_mat_a] :
~ ( ord_le3279973697895081845_mat_a @ A4 @ bot_bo2759726786008686517_mat_a ) ).
% not_less_bot
thf(fact_305_not__less__bot,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ bot_bot_nat ) ).
% not_less_bot
thf(fact_306_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_307_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_308_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_309_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_310_mult_Oleft__commute,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( times_times_nat @ B4 @ ( times_times_nat @ A4 @ C ) )
= ( times_times_nat @ A4 @ ( times_times_nat @ B4 @ C ) ) ) ).
% mult.left_commute
thf(fact_311_mult_Oleft__commute,axiom,
! [B4: real,A4: real,C: real] :
( ( times_times_real @ B4 @ ( times_times_real @ A4 @ C ) )
= ( times_times_real @ A4 @ ( times_times_real @ B4 @ C ) ) ) ).
% mult.left_commute
thf(fact_312_mult_Oleft__commute,axiom,
! [B4: int,A4: int,C: int] :
( ( times_times_int @ B4 @ ( times_times_int @ A4 @ C ) )
= ( times_times_int @ A4 @ ( times_times_int @ B4 @ C ) ) ) ).
% mult.left_commute
thf(fact_313_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B6: nat] : ( times_times_nat @ B6 @ A5 ) ) ) ).
% mult.commute
thf(fact_314_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B6: real] : ( times_times_real @ B6 @ A5 ) ) ) ).
% mult.commute
thf(fact_315_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B6: int] : ( times_times_int @ B6 @ A5 ) ) ) ).
% mult.commute
thf(fact_316_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A4 @ B4 ) @ C )
= ( times_times_nat @ A4 @ ( times_times_nat @ B4 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_317_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: real,B4: real,C: real] :
( ( times_times_real @ ( times_times_real @ A4 @ B4 ) @ C )
= ( times_times_real @ A4 @ ( times_times_real @ B4 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_318_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: int,B4: int,C: int] :
( ( times_times_int @ ( times_times_int @ A4 @ B4 ) @ C )
= ( times_times_int @ A4 @ ( times_times_int @ B4 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_319_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_320_order__less__imp__not__less,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_321_order__less__imp__not__less,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_322_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_323_order__less__imp__not__eq2,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_324_order__less__imp__not__eq2,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_325_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_326_order__less__imp__not__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_327_order__less__imp__not__eq,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_328_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_329_linorder__less__linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_real @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_330_linorder__less__linear,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_int @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_331_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_332_order__less__imp__triv,axiom,
! [X2: real,Y3: real,P: $o] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_real @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_333_order__less__imp__triv,axiom,
! [X2: int,Y3: int,P: $o] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_int @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_334_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_335_order__less__not__sym,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_336_order__less__not__sym,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_337_order__less__subst2,axiom,
! [A4: nat,B4: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_338_order__less__subst2,axiom,
! [A4: nat,B4: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_339_order__less__subst2,axiom,
! [A4: nat,B4: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_340_order__less__subst2,axiom,
! [A4: real,B4: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_341_order__less__subst2,axiom,
! [A4: real,B4: real,F2: real > real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_342_order__less__subst2,axiom,
! [A4: real,B4: real,F2: real > int,C: int] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_343_order__less__subst2,axiom,
! [A4: int,B4: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_344_order__less__subst2,axiom,
! [A4: int,B4: int,F2: int > real,C: real] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_345_order__less__subst2,axiom,
! [A4: int,B4: int,F2: int > int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_346_order__less__subst1,axiom,
! [A4: nat,F2: nat > nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_347_order__less__subst1,axiom,
! [A4: nat,F2: real > nat,B4: real,C: real] :
( ( ord_less_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_348_order__less__subst1,axiom,
! [A4: nat,F2: int > nat,B4: int,C: int] :
( ( ord_less_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_349_order__less__subst1,axiom,
! [A4: real,F2: nat > real,B4: nat,C: nat] :
( ( ord_less_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_350_order__less__subst1,axiom,
! [A4: real,F2: real > real,B4: real,C: real] :
( ( ord_less_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_351_order__less__subst1,axiom,
! [A4: real,F2: int > real,B4: int,C: int] :
( ( ord_less_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_352_order__less__subst1,axiom,
! [A4: int,F2: nat > int,B4: nat,C: nat] :
( ( ord_less_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_353_order__less__subst1,axiom,
! [A4: int,F2: real > int,B4: real,C: real] :
( ( ord_less_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_354_order__less__subst1,axiom,
! [A4: int,F2: int > int,B4: int,C: int] :
( ( ord_less_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_355_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_356_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_357_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_358_ord__less__eq__subst,axiom,
! [A4: nat,B4: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_359_ord__less__eq__subst,axiom,
! [A4: nat,B4: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_360_ord__less__eq__subst,axiom,
! [A4: nat,B4: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_361_ord__less__eq__subst,axiom,
! [A4: real,B4: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_362_ord__less__eq__subst,axiom,
! [A4: real,B4: real,F2: real > real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_363_ord__less__eq__subst,axiom,
! [A4: real,B4: real,F2: real > int,C: int] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_364_ord__less__eq__subst,axiom,
! [A4: int,B4: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_365_ord__less__eq__subst,axiom,
! [A4: int,B4: int,F2: int > real,C: real] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_366_ord__less__eq__subst,axiom,
! [A4: int,B4: int,F2: int > int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_367_ord__eq__less__subst,axiom,
! [A4: nat,F2: nat > nat,B4: nat,C: nat] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_368_ord__eq__less__subst,axiom,
! [A4: real,F2: nat > real,B4: nat,C: nat] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_369_ord__eq__less__subst,axiom,
! [A4: int,F2: nat > int,B4: nat,C: nat] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_370_ord__eq__less__subst,axiom,
! [A4: nat,F2: real > nat,B4: real,C: real] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_371_ord__eq__less__subst,axiom,
! [A4: real,F2: real > real,B4: real,C: real] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_372_ord__eq__less__subst,axiom,
! [A4: int,F2: real > int,B4: real,C: real] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_373_ord__eq__less__subst,axiom,
! [A4: nat,F2: int > nat,B4: int,C: int] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_374_ord__eq__less__subst,axiom,
! [A4: real,F2: int > real,B4: int,C: int] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_375_ord__eq__less__subst,axiom,
! [A4: int,F2: int > int,B4: int,C: int] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_376_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_377_order__less__trans,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_378_order__less__trans,axiom,
! [X2: int,Y3: int,Z3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_379_order__less__asym_H,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ~ ( ord_less_nat @ B4 @ A4 ) ) ).
% order_less_asym'
thf(fact_380_order__less__asym_H,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ~ ( ord_less_real @ B4 @ A4 ) ) ).
% order_less_asym'
thf(fact_381_order__less__asym_H,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ~ ( ord_less_int @ B4 @ A4 ) ) ).
% order_less_asym'
thf(fact_382_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_383_linorder__neq__iff,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
= ( ( ord_less_real @ X2 @ Y3 )
| ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_384_linorder__neq__iff,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
= ( ( ord_less_int @ X2 @ Y3 )
| ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_385_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_386_order__less__asym,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_387_order__less__asym,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_388_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_389_linorder__neqE,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_390_linorder__neqE,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_391_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( A4 != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_392_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( A4 != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_393_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( A4 != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_394_order_Ostrict__implies__not__eq,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( A4 != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_395_order_Ostrict__implies__not__eq,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( A4 != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_396_order_Ostrict__implies__not__eq,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( A4 != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_397_dual__order_Ostrict__trans,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( ( ord_less_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_398_dual__order_Ostrict__trans,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ( ord_less_real @ C @ B4 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_399_dual__order_Ostrict__trans,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ( ord_less_int @ C @ B4 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_400_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_401_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y3: real] :
( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( ( ord_less_real @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_402_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y3: int] :
( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( ( ord_less_int @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_403_order_Ostrict__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_404_order_Ostrict__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_405_order_Ostrict__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_406_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B4: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_407_linorder__less__wlog,axiom,
! [P: real > real > $o,A4: real,B4: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_408_linorder__less__wlog,axiom,
! [P: int > int > $o,A4: int,B4: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_409_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P4 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_410_dual__order_Oirrefl,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_411_dual__order_Oirrefl,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_412_dual__order_Oirrefl,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_413_dual__order_Oasym,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ~ ( ord_less_nat @ A4 @ B4 ) ) ).
% dual_order.asym
thf(fact_414_dual__order_Oasym,axiom,
! [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
=> ~ ( ord_less_real @ A4 @ B4 ) ) ).
% dual_order.asym
thf(fact_415_dual__order_Oasym,axiom,
! [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
=> ~ ( ord_less_int @ A4 @ B4 ) ) ).
% dual_order.asym
thf(fact_416_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_417_linorder__cases,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_418_linorder__cases,axiom,
! [X2: int,Y3: int] :
( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_419_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_420_antisym__conv3,axiom,
! [Y3: real,X2: real] :
( ~ ( ord_less_real @ Y3 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_421_antisym__conv3,axiom,
! [Y3: int,X2: int] :
( ~ ( ord_less_int @ Y3 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_422_less__induct,axiom,
! [P: nat > $o,A4: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ).
% less_induct
thf(fact_423_ord__less__eq__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_424_ord__less__eq__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_425_ord__less__eq__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_426_ord__eq__less__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( A4 = B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_427_ord__eq__less__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( A4 = B4 )
=> ( ( ord_less_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_428_ord__eq__less__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( A4 = B4 )
=> ( ( ord_less_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_429_order_Oasym,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ~ ( ord_less_nat @ B4 @ A4 ) ) ).
% order.asym
thf(fact_430_order_Oasym,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ~ ( ord_less_real @ B4 @ A4 ) ) ).
% order.asym
thf(fact_431_order_Oasym,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ~ ( ord_less_int @ B4 @ A4 ) ) ).
% order.asym
thf(fact_432_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_433_less__imp__neq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_434_less__imp__neq,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_435_dense,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ? [Z: real] :
( ( ord_less_real @ X2 @ Z )
& ( ord_less_real @ Z @ Y3 ) ) ) ).
% dense
thf(fact_436_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_437_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_438_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_439_lt__ex,axiom,
! [X2: real] :
? [Y: real] : ( ord_less_real @ Y @ X2 ) ).
% lt_ex
thf(fact_440_lt__ex,axiom,
! [X2: int] :
? [Y: int] : ( ord_less_int @ Y @ X2 ) ).
% lt_ex
thf(fact_441_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_442_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_443_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_444_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_445_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_446_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_447_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_448_comm__monoid__mult__class_Omult__1,axiom,
! [A4: nat] :
( ( times_times_nat @ one_one_nat @ A4 )
= A4 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_449_comm__monoid__mult__class_Omult__1,axiom,
! [A4: real] :
( ( times_times_real @ one_one_real @ A4 )
= A4 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_450_comm__monoid__mult__class_Omult__1,axiom,
! [A4: int] :
( ( times_times_int @ one_one_int @ A4 )
= A4 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_451_mult_Ocomm__neutral,axiom,
! [A4: nat] :
( ( times_times_nat @ A4 @ one_one_nat )
= A4 ) ).
% mult.comm_neutral
thf(fact_452_mult_Ocomm__neutral,axiom,
! [A4: real] :
( ( times_times_real @ A4 @ one_one_real )
= A4 ) ).
% mult.comm_neutral
thf(fact_453_mult_Ocomm__neutral,axiom,
! [A4: int] :
( ( times_times_int @ A4 @ one_one_int )
= A4 ) ).
% mult.comm_neutral
thf(fact_454_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_455_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_456_mult__delta__right,axiom,
! [B4: $o,X2: nat,Y3: nat] :
( ( B4
=> ( ( times_times_nat @ X2 @ ( if_nat @ B4 @ Y3 @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y3 ) ) )
& ( ~ B4
=> ( ( times_times_nat @ X2 @ ( if_nat @ B4 @ Y3 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_457_mult__delta__right,axiom,
! [B4: $o,X2: real,Y3: real] :
( ( B4
=> ( ( times_times_real @ X2 @ ( if_real @ B4 @ Y3 @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y3 ) ) )
& ( ~ B4
=> ( ( times_times_real @ X2 @ ( if_real @ B4 @ Y3 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_458_mult__delta__right,axiom,
! [B4: $o,X2: int,Y3: int] :
( ( B4
=> ( ( times_times_int @ X2 @ ( if_int @ B4 @ Y3 @ zero_zero_int ) )
= ( times_times_int @ X2 @ Y3 ) ) )
& ( ~ B4
=> ( ( times_times_int @ X2 @ ( if_int @ B4 @ Y3 @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_459_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A4: real,B4: real,X2: real] :
( ( times_times_real @ A4 @ ( times_times_real @ B4 @ X2 ) )
= ( times_times_real @ ( times_times_real @ A4 @ B4 ) @ X2 ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_460_vector__space__over__itself_Oscale__left__commute,axiom,
! [A4: real,B4: real,X2: real] :
( ( times_times_real @ A4 @ ( times_times_real @ B4 @ X2 ) )
= ( times_times_real @ B4 @ ( times_times_real @ A4 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_461_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A4: real,X2: real] :
( ( ( times_times_real @ A4 @ X2 )
= zero_zero_real )
= ( ( A4 = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_462_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_463_vector__space__over__itself_Oscale__zero__right,axiom,
! [A4: real] :
( ( times_times_real @ A4 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_464_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A4: real,X2: real,Y3: real] :
( ( ( times_times_real @ A4 @ X2 )
= ( times_times_real @ A4 @ Y3 ) )
= ( ( X2 = Y3 )
| ( A4 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_465_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A4: real,X2: real,Y3: real] :
( ( A4 != zero_zero_real )
=> ( ( ( times_times_real @ A4 @ X2 )
= ( times_times_real @ A4 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_466_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A4: real,X2: real,B4: real] :
( ( ( times_times_real @ A4 @ X2 )
= ( times_times_real @ B4 @ X2 ) )
= ( ( A4 = B4 )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_467_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A4: real,B4: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A4 @ X2 )
= ( times_times_real @ B4 @ X2 ) )
=> ( A4 = B4 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_468_mult__delta__left,axiom,
! [B4: $o,X2: nat,Y3: nat] :
( ( B4
=> ( ( times_times_nat @ ( if_nat @ B4 @ X2 @ zero_zero_nat ) @ Y3 )
= ( times_times_nat @ X2 @ Y3 ) ) )
& ( ~ B4
=> ( ( times_times_nat @ ( if_nat @ B4 @ X2 @ zero_zero_nat ) @ Y3 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_469_mult__delta__left,axiom,
! [B4: $o,X2: real,Y3: real] :
( ( B4
=> ( ( times_times_real @ ( if_real @ B4 @ X2 @ zero_zero_real ) @ Y3 )
= ( times_times_real @ X2 @ Y3 ) ) )
& ( ~ B4
=> ( ( times_times_real @ ( if_real @ B4 @ X2 @ zero_zero_real ) @ Y3 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_470_mult__delta__left,axiom,
! [B4: $o,X2: int,Y3: int] :
( ( B4
=> ( ( times_times_int @ ( if_int @ B4 @ X2 @ zero_zero_int ) @ Y3 )
= ( times_times_int @ X2 @ Y3 ) ) )
& ( ~ B4
=> ( ( times_times_int @ ( if_int @ B4 @ X2 @ zero_zero_int ) @ Y3 )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_471_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_472_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_473_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_474_sublists_Osimps_I1_J,axiom,
( ( sublists_mat_a @ nil_mat_a )
= ( cons_list_mat_a @ nil_mat_a @ nil_list_mat_a ) ) ).
% sublists.simps(1)
thf(fact_475_product__lists_Osimps_I1_J,axiom,
( ( product_lists_mat_a @ nil_list_mat_a )
= ( cons_list_mat_a @ nil_mat_a @ nil_list_mat_a ) ) ).
% product_lists.simps(1)
thf(fact_476_mk__diagonal__dim_I2_J,axiom,
! [As3: list_a] :
( ( dim_col_a @ ( mk_diagonal_a @ As3 ) )
= ( size_size_list_a @ As3 ) ) ).
% mk_diagonal_dim(2)
thf(fact_477_mk__diagonal__dim_I1_J,axiom,
! [As3: list_a] :
( ( dim_row_a @ ( mk_diagonal_a @ As3 ) )
= ( size_size_list_a @ As3 ) ) ).
% mk_diagonal_dim(1)
thf(fact_478_diag__block__mat__cong__hd,axiom,
! [Al: list_mat_a,Bl2: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Al ) )
=> ( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl2 ) )
=> ( ( ( dim_row_a @ ( hd_mat_a @ Al ) )
= ( dim_row_a @ ( hd_mat_a @ Bl2 ) ) )
=> ( ( ( dim_col_a @ ( hd_mat_a @ Al ) )
= ( dim_col_a @ ( hd_mat_a @ Bl2 ) ) )
=> ( ( ( diag_block_mat_a @ Al )
= ( diag_block_mat_a @ Bl2 ) )
=> ( ( hd_mat_a @ Al )
= ( hd_mat_a @ Bl2 ) ) ) ) ) ) ) ).
% diag_block_mat_cong_hd
thf(fact_479_nth__non__equal__first__eq,axiom,
! [X2: mat_a,Y3: mat_a,Xs: list_mat_a,N: nat] :
( ( X2 != Y3 )
=> ( ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= Y3 )
= ( ( ( nth_mat_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_480_nth__Cons__pos,axiom,
! [N: nat,X2: mat_a,Xs: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= ( nth_mat_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_481_length__code,axiom,
( size_size_list_mat_a
= ( gen_length_mat_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_482_nat__distrib_I4_J,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 ) ) ) ).
% nat_distrib(4)
thf(fact_483_nat__distrib_I3_J,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 ) ) ) ).
% nat_distrib(3)
thf(fact_484_diff__commute,axiom,
! [I4: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J ) ) ).
% diff_commute
thf(fact_485_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I4: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).
% zero_induct_lemma
thf(fact_486_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_487_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_488_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_489_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_490_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_491_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_492_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_493_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_494_diff__strict__mono,axiom,
! [A4: real,B4: real,D: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A4 @ C ) @ ( minus_minus_real @ B4 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_495_diff__strict__mono,axiom,
! [A4: int,B4: int,D: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A4 @ C ) @ ( minus_minus_int @ B4 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_496_diff__eq__diff__less,axiom,
! [A4: real,B4: real,C: real,D: real] :
( ( ( minus_minus_real @ A4 @ B4 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A4 @ B4 )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_497_diff__eq__diff__less,axiom,
! [A4: int,B4: int,C: int,D: int] :
( ( ( minus_minus_int @ A4 @ B4 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A4 @ B4 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_498_diff__strict__left__mono,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A4 ) @ ( minus_minus_real @ C @ B4 ) ) ) ).
% diff_strict_left_mono
thf(fact_499_diff__strict__left__mono,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A4 ) @ ( minus_minus_int @ C @ B4 ) ) ) ).
% diff_strict_left_mono
thf(fact_500_diff__strict__right__mono,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ord_less_real @ ( minus_minus_real @ A4 @ C ) @ ( minus_minus_real @ B4 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_501_diff__strict__right__mono,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ord_less_int @ ( minus_minus_int @ A4 @ C ) @ ( minus_minus_int @ B4 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_502_Rings_Oring__distribs_I4_J,axiom,
! [A4: real,B4: real,C: real] :
( ( times_times_real @ A4 @ ( minus_minus_real @ B4 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A4 @ B4 ) @ ( times_times_real @ A4 @ C ) ) ) ).
% Rings.ring_distribs(4)
thf(fact_503_Rings_Oring__distribs_I4_J,axiom,
! [A4: int,B4: int,C: int] :
( ( times_times_int @ A4 @ ( minus_minus_int @ B4 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A4 @ B4 ) @ ( times_times_int @ A4 @ C ) ) ) ).
% Rings.ring_distribs(4)
thf(fact_504_Rings_Oring__distribs_I3_J,axiom,
! [A4: real,B4: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A4 @ B4 ) @ C )
= ( minus_minus_real @ ( times_times_real @ A4 @ C ) @ ( times_times_real @ B4 @ C ) ) ) ).
% Rings.ring_distribs(3)
thf(fact_505_Rings_Oring__distribs_I3_J,axiom,
! [A4: int,B4: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A4 @ B4 ) @ C )
= ( minus_minus_int @ ( times_times_int @ A4 @ C ) @ ( times_times_int @ B4 @ C ) ) ) ).
% Rings.ring_distribs(3)
thf(fact_506_left__diff__distrib_H,axiom,
! [B4: nat,C: nat,A4: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B4 @ C ) @ A4 )
= ( minus_minus_nat @ ( times_times_nat @ B4 @ A4 ) @ ( times_times_nat @ C @ A4 ) ) ) ).
% left_diff_distrib'
thf(fact_507_left__diff__distrib_H,axiom,
! [B4: real,C: real,A4: real] :
( ( times_times_real @ ( minus_minus_real @ B4 @ C ) @ A4 )
= ( minus_minus_real @ ( times_times_real @ B4 @ A4 ) @ ( times_times_real @ C @ A4 ) ) ) ).
% left_diff_distrib'
thf(fact_508_left__diff__distrib_H,axiom,
! [B4: int,C: int,A4: int] :
( ( times_times_int @ ( minus_minus_int @ B4 @ C ) @ A4 )
= ( minus_minus_int @ ( times_times_int @ B4 @ A4 ) @ ( times_times_int @ C @ A4 ) ) ) ).
% left_diff_distrib'
thf(fact_509_right__diff__distrib_H,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( times_times_nat @ A4 @ ( minus_minus_nat @ B4 @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A4 @ B4 ) @ ( times_times_nat @ A4 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_510_right__diff__distrib_H,axiom,
! [A4: real,B4: real,C: real] :
( ( times_times_real @ A4 @ ( minus_minus_real @ B4 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A4 @ B4 ) @ ( times_times_real @ A4 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_511_right__diff__distrib_H,axiom,
! [A4: int,B4: int,C: int] :
( ( times_times_int @ A4 @ ( minus_minus_int @ B4 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A4 @ B4 ) @ ( times_times_int @ A4 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_512_diff__self,axiom,
! [A4: real] :
( ( minus_minus_real @ A4 @ A4 )
= zero_zero_real ) ).
% diff_self
thf(fact_513_diff__self,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ A4 )
= zero_zero_int ) ).
% diff_self
thf(fact_514_diff__0__right,axiom,
! [A4: real] :
( ( minus_minus_real @ A4 @ zero_zero_real )
= A4 ) ).
% diff_0_right
thf(fact_515_diff__0__right,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ zero_zero_int )
= A4 ) ).
% diff_0_right
thf(fact_516_right__minus__eq,axiom,
! [A4: real,B4: real] :
( ( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real )
= ( A4 = B4 ) ) ).
% right_minus_eq
thf(fact_517_right__minus__eq,axiom,
! [A4: int,B4: int] :
( ( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int )
= ( A4 = B4 ) ) ).
% right_minus_eq
thf(fact_518_zero__diff,axiom,
! [A4: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A4 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_519_diff__zero,axiom,
! [A4: real] :
( ( minus_minus_real @ A4 @ zero_zero_real )
= A4 ) ).
% diff_zero
thf(fact_520_diff__zero,axiom,
! [A4: nat] :
( ( minus_minus_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% diff_zero
thf(fact_521_diff__zero,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ zero_zero_int )
= A4 ) ).
% diff_zero
thf(fact_522_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A4: real] :
( ( minus_minus_real @ A4 @ A4 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_523_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A4: nat] :
( ( minus_minus_nat @ A4 @ A4 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_524_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ A4 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_525_list_Osel_I1_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( ( hd_mat_a @ ( cons_mat_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_526_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A4: real,B4: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A4 @ B4 ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A4 @ X2 ) @ ( times_times_real @ B4 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_527_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A4: real,X2: real,Y3: real] :
( ( times_times_real @ A4 @ ( minus_minus_real @ X2 @ Y3 ) )
= ( minus_minus_real @ ( times_times_real @ A4 @ X2 ) @ ( times_times_real @ A4 @ Y3 ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_528_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B6: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B6 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_529_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B6: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B6 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_530_diff__gt__0__iff__gt,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A4 @ B4 ) )
= ( ord_less_real @ B4 @ A4 ) ) ).
% diff_gt_0_iff_gt
thf(fact_531_diff__gt__0__iff__gt,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A4 @ B4 ) )
= ( ord_less_int @ B4 @ A4 ) ) ).
% diff_gt_0_iff_gt
thf(fact_532_arith__special_I21_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% arith_special(21)
thf(fact_533_arith__special_I21_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% arith_special(21)
thf(fact_534_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_535_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_536_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_537_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_538_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_539_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_540_hd__conv__nth,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( hd_mat_a @ Xs )
= ( nth_mat_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_541_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_mat_a @ N @ nil_mat_a )
= N ) ).
% gen_length_code(1)
thf(fact_542_diff__Suc__less,axiom,
! [N: nat,I4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_543_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_544_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_545_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_546_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_547_nth__Cons_H,axiom,
! [N: nat,X2: mat_a,Xs: list_mat_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= ( nth_mat_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_548_gen__length__code_I2_J,axiom,
! [N: nat,X2: mat_a,Xs: list_mat_a] :
( ( gen_length_mat_a @ N @ ( cons_mat_a @ X2 @ Xs ) )
= ( gen_length_mat_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_549_poly__cancel__eq__conv,axiom,
! [X2: real,A4: real,Y3: real,B4: real] :
( ( X2 = zero_zero_real )
=> ( ( A4 != zero_zero_real )
=> ( ( Y3 = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A4 @ Y3 ) @ ( times_times_real @ B4 @ X2 ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_550_permutation__insert__expand,axiom,
( permut3695043542826343943rt_nat
= ( ^ [I3: nat,J3: nat,P5: nat > nat,I5: nat] : ( if_nat @ ( ord_less_nat @ I5 @ I3 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I5 ) @ J3 ) @ ( P5 @ I5 ) @ ( suc @ ( P5 @ I5 ) ) ) @ ( if_nat @ ( I5 = I3 ) @ J3 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) @ J3 ) @ ( P5 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) @ ( suc @ ( P5 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_551_permutation__insert__expand,axiom,
( permut4060954620988167523t_real
= ( ^ [I3: real,J3: nat,P5: real > nat,I5: real] : ( if_nat @ ( ord_less_real @ I5 @ I3 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I5 ) @ J3 ) @ ( P5 @ I5 ) @ ( suc @ ( P5 @ I5 ) ) ) @ ( if_nat @ ( I5 = I3 ) @ J3 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_real @ I5 @ one_one_real ) ) @ J3 ) @ ( P5 @ ( minus_minus_real @ I5 @ one_one_real ) ) @ ( suc @ ( P5 @ ( minus_minus_real @ I5 @ one_one_real ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_552_permutation__insert__expand,axiom,
( permut3692553072317293667rt_int
= ( ^ [I3: int,J3: nat,P5: int > nat,I5: int] : ( if_nat @ ( ord_less_int @ I5 @ I3 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I5 ) @ J3 ) @ ( P5 @ I5 ) @ ( suc @ ( P5 @ I5 ) ) ) @ ( if_nat @ ( I5 = I3 ) @ J3 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_int @ I5 @ one_one_int ) ) @ J3 ) @ ( P5 @ ( minus_minus_int @ I5 @ one_one_int ) ) @ ( suc @ ( P5 @ ( minus_minus_int @ I5 @ one_one_int ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_553_diag__block__mat__cong__tl,axiom,
! [Al: list_mat_a,Bl2: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Al ) )
=> ( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl2 ) )
=> ( ( ( dim_row_a @ ( hd_mat_a @ Al ) )
= ( dim_row_a @ ( hd_mat_a @ Bl2 ) ) )
=> ( ( ( dim_col_a @ ( hd_mat_a @ Al ) )
= ( dim_col_a @ ( hd_mat_a @ Bl2 ) ) )
=> ( ( ( diag_block_mat_a @ Al )
= ( diag_block_mat_a @ Bl2 ) )
=> ( ( diag_block_mat_a @ ( tl_mat_a @ Al ) )
= ( diag_block_mat_a @ ( tl_mat_a @ Bl2 ) ) ) ) ) ) ) ) ).
% diag_block_mat_cong_tl
thf(fact_554_inf__period_I1_J,axiom,
! [P: real > $o,D3: real,Q2: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ! [X: real,K3: real] :
( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P @ ( minus_minus_real @ X @ ( times_times_real @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_minus_real @ X @ ( times_times_real @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_555_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q2: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X: int,K3: int] :
( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P @ ( minus_minus_int @ X @ ( times_times_int @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_minus_int @ X @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_556_inf__period_I2_J,axiom,
! [P: real > $o,D3: real,Q2: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ! [X: real,K3: real] :
( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P @ ( minus_minus_real @ X @ ( times_times_real @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_minus_real @ X @ ( times_times_real @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_557_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q2: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X: int,K3: int] :
( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P @ ( minus_minus_int @ X @ ( times_times_int @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_minus_int @ X @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_558_index__minus__mat_I3_J,axiom,
! [A: mat_a,B5: mat_a] :
( ( dim_col_a @ ( minus_minus_mat_a @ A @ B5 ) )
= ( dim_col_a @ B5 ) ) ).
% index_minus_mat(3)
thf(fact_559_index__minus__mat_I2_J,axiom,
! [A: mat_a,B5: mat_a] :
( ( dim_row_a @ ( minus_minus_mat_a @ A @ B5 ) )
= ( dim_row_a @ B5 ) ) ).
% index_minus_mat(2)
thf(fact_560_list_Osel_I3_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( ( tl_mat_a @ ( cons_mat_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_561_list_Osel_I2_J,axiom,
( ( tl_mat_a @ nil_mat_a )
= nil_mat_a ) ).
% list.sel(2)
thf(fact_562_Nil__tl,axiom,
! [Xs: list_mat_a] :
( ( nil_mat_a
= ( tl_mat_a @ Xs ) )
= ( ( Xs = nil_mat_a )
| ? [X6: mat_a] :
( Xs
= ( cons_mat_a @ X6 @ nil_mat_a ) ) ) ) ).
% Nil_tl
thf(fact_563_tl__Nil,axiom,
! [Xs: list_mat_a] :
( ( ( tl_mat_a @ Xs )
= nil_mat_a )
= ( ( Xs = nil_mat_a )
| ? [X6: mat_a] :
( Xs
= ( cons_mat_a @ X6 @ nil_mat_a ) ) ) ) ).
% tl_Nil
thf(fact_564_list_Oexpand,axiom,
! [List: list_mat_a,List2: list_mat_a] :
( ( ( List = nil_mat_a )
= ( List2 = nil_mat_a ) )
=> ( ( ( List != nil_mat_a )
=> ( ( List2 != nil_mat_a )
=> ( ( ( hd_mat_a @ List )
= ( hd_mat_a @ List2 ) )
& ( ( tl_mat_a @ List )
= ( tl_mat_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_565_length__tl,axiom,
! [Xs: list_mat_a] :
( ( size_size_list_mat_a @ ( tl_mat_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_mat_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_566_hd__Cons__tl,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( cons_mat_a @ ( hd_mat_a @ Xs ) @ ( tl_mat_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_567_list_Oexhaust__sel,axiom,
! [List: list_mat_a] :
( ( List != nil_mat_a )
=> ( List
= ( cons_mat_a @ ( hd_mat_a @ List ) @ ( tl_mat_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_568_list_Ocollapse,axiom,
! [List: list_mat_a] :
( ( List != nil_mat_a )
=> ( ( cons_mat_a @ ( hd_mat_a @ List ) @ ( tl_mat_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_569_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_570_pinf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_571_pinf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_572_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_573_pinf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_574_pinf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_575_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_576_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_577_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_578_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_579_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_580_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_581_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_nat @ X @ T ) ) ).
% pinf(5)
thf(fact_582_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ~ ( ord_less_real @ X @ T ) ) ).
% pinf(5)
thf(fact_583_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ~ ( ord_less_int @ X @ T ) ) ).
% pinf(5)
thf(fact_584_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_nat @ T @ X ) ) ).
% pinf(7)
thf(fact_585_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_real @ T @ X ) ) ).
% pinf(7)
thf(fact_586_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ord_less_int @ T @ X ) ) ).
% pinf(7)
thf(fact_587_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_588_minf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_589_minf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_590_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_591_minf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_592_minf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_593_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_594_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_595_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_596_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_597_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_598_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_599_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ T ) ) ).
% minf(5)
thf(fact_600_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ord_less_real @ X @ T ) ) ).
% minf(5)
thf(fact_601_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ord_less_int @ X @ T ) ) ).
% minf(5)
thf(fact_602_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_nat @ T @ X ) ) ).
% minf(7)
thf(fact_603_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ~ ( ord_less_real @ T @ X ) ) ).
% minf(7)
thf(fact_604_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ~ ( ord_less_int @ T @ X ) ) ).
% minf(7)
thf(fact_605_nth__tl,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ ( tl_mat_a @ Xs ) ) )
=> ( ( nth_mat_a @ ( tl_mat_a @ Xs ) @ N )
= ( nth_mat_a @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_606_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_mat_a
= ( ^ [Xs3: list_mat_a] : ( if_nat @ ( Xs3 = nil_mat_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_mat_a @ ( tl_mat_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_607_delete__index__def,axiom,
( delete_index
= ( ^ [I3: nat,I5: nat] : ( if_nat @ ( ord_less_nat @ I5 @ I3 ) @ I5 @ ( minus_minus_nat @ I5 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% delete_index_def
thf(fact_608_permutation__delete__expand,axiom,
( permutation_delete
= ( ^ [P5: nat > nat,I3: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ ( P5 @ ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) @ ( P5 @ I3 ) ) @ ( P5 @ ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) @ ( minus_minus_nat @ ( P5 @ ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% permutation_delete_expand
thf(fact_609_mat__delete__dim_I2_J,axiom,
! [A: mat_a,I4: nat,J: nat] :
( ( dim_col_a @ ( mat_delete_a @ A @ I4 @ J ) )
= ( minus_minus_nat @ ( dim_col_a @ A ) @ one_one_nat ) ) ).
% mat_delete_dim(2)
thf(fact_610_mat__delete__dim_I1_J,axiom,
! [A: mat_a,I4: nat,J: nat] :
( ( dim_row_a @ ( mat_delete_a @ A @ I4 @ J ) )
= ( minus_minus_nat @ ( dim_row_a @ A ) @ one_one_nat ) ) ).
% mat_delete_dim(1)
thf(fact_611_last__conv__nth,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( last_mat_a @ Xs )
= ( nth_mat_a @ Xs @ ( minus_minus_nat @ ( size_size_list_mat_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_612_power__minus__mult,axiom,
! [N: nat,A4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A4 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A4 )
= ( power_power_nat @ A4 @ N ) ) ) ).
% power_minus_mult
thf(fact_613_power__minus__mult,axiom,
! [N: nat,A4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A4 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A4 )
= ( power_power_real @ A4 @ N ) ) ) ).
% power_minus_mult
thf(fact_614_power__minus__mult,axiom,
! [N: nat,A4: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A4 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A4 )
= ( power_power_int @ A4 @ N ) ) ) ).
% power_minus_mult
thf(fact_615_power__eq__if,axiom,
( power_power_nat
= ( ^ [P5: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_616_power__eq__if,axiom,
( power_power_real
= ( ^ [P5: real,M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_617_power__eq__if,axiom,
( power_power_int
= ( ^ [P5: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_618_power__not__zero,axiom,
! [A4: nat,N: nat] :
( ( A4 != zero_zero_nat )
=> ( ( power_power_nat @ A4 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_619_power__not__zero,axiom,
! [A4: real,N: nat] :
( ( A4 != zero_zero_real )
=> ( ( power_power_real @ A4 @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_620_power__not__zero,axiom,
! [A4: int,N: nat] :
( ( A4 != zero_zero_int )
=> ( ( power_power_int @ A4 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_621_power__commuting__commutes,axiom,
! [X2: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y3 )
= ( times_times_nat @ Y3 @ X2 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y3 )
= ( times_times_nat @ Y3 @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_622_power__commuting__commutes,axiom,
! [X2: real,Y3: real,N: nat] :
( ( ( times_times_real @ X2 @ Y3 )
= ( times_times_real @ Y3 @ X2 ) )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ Y3 )
= ( times_times_real @ Y3 @ ( power_power_real @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_623_power__commuting__commutes,axiom,
! [X2: int,Y3: int,N: nat] :
( ( ( times_times_int @ X2 @ Y3 )
= ( times_times_int @ Y3 @ X2 ) )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y3 )
= ( times_times_int @ Y3 @ ( power_power_int @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_624_power__mult__distrib,axiom,
! [A4: nat,B4: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A4 @ B4 ) @ N )
= ( times_times_nat @ ( power_power_nat @ A4 @ N ) @ ( power_power_nat @ B4 @ N ) ) ) ).
% power_mult_distrib
thf(fact_625_power__mult__distrib,axiom,
! [A4: real,B4: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A4 @ B4 ) @ N )
= ( times_times_real @ ( power_power_real @ A4 @ N ) @ ( power_power_real @ B4 @ N ) ) ) ).
% power_mult_distrib
thf(fact_626_power__mult__distrib,axiom,
! [A4: int,B4: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A4 @ B4 ) @ N )
= ( times_times_int @ ( power_power_int @ A4 @ N ) @ ( power_power_int @ B4 @ N ) ) ) ).
% power_mult_distrib
thf(fact_627_power__commutes,axiom,
! [A4: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A4 @ N ) @ A4 )
= ( times_times_nat @ A4 @ ( power_power_nat @ A4 @ N ) ) ) ).
% power_commutes
thf(fact_628_power__commutes,axiom,
! [A4: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A4 @ N ) @ A4 )
= ( times_times_real @ A4 @ ( power_power_real @ A4 @ N ) ) ) ).
% power_commutes
thf(fact_629_power__commutes,axiom,
! [A4: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A4 @ N ) @ A4 )
= ( times_times_int @ A4 @ ( power_power_int @ A4 @ N ) ) ) ).
% power_commutes
thf(fact_630_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_631_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_632_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_633_class__semiring_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% class_semiring.nat_pow_one
thf(fact_634_class__semiring_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% class_semiring.nat_pow_one
thf(fact_635_class__semiring_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% class_semiring.nat_pow_one
thf(fact_636_power__one__right,axiom,
! [A4: nat] :
( ( power_power_nat @ A4 @ one_one_nat )
= A4 ) ).
% power_one_right
thf(fact_637_power__one__right,axiom,
! [A4: real] :
( ( power_power_real @ A4 @ one_one_nat )
= A4 ) ).
% power_one_right
thf(fact_638_power__one__right,axiom,
! [A4: int] :
( ( power_power_int @ A4 @ one_one_nat )
= A4 ) ).
% power_one_right
thf(fact_639_power__mult,axiom,
! [A4: nat,M: nat,N: nat] :
( ( power_power_nat @ A4 @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A4 @ M ) @ N ) ) ).
% power_mult
thf(fact_640_power__mult,axiom,
! [A4: real,M: nat,N: nat] :
( ( power_power_real @ A4 @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A4 @ M ) @ N ) ) ).
% power_mult
thf(fact_641_power__mult,axiom,
! [A4: int,M: nat,N: nat] :
( ( power_power_int @ A4 @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A4 @ M ) @ N ) ) ).
% power_mult
thf(fact_642_zero__less__power,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A4 @ N ) ) ) ).
% zero_less_power
thf(fact_643_zero__less__power,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A4 @ N ) ) ) ).
% zero_less_power
thf(fact_644_zero__less__power,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A4 @ N ) ) ) ).
% zero_less_power
thf(fact_645_left__right__inverse__power,axiom,
! [X2: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y3 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y3 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_646_left__right__inverse__power,axiom,
! [X2: real,Y3: real,N: nat] :
( ( ( times_times_real @ X2 @ Y3 )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_647_left__right__inverse__power,axiom,
! [X2: int,Y3: int,N: nat] :
( ( ( times_times_int @ X2 @ Y3 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_648_power__inject__exp,axiom,
! [A4: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ( ( power_power_nat @ A4 @ M )
= ( power_power_nat @ A4 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_649_power__inject__exp,axiom,
! [A4: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ( power_power_real @ A4 @ M )
= ( power_power_real @ A4 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_650_power__inject__exp,axiom,
! [A4: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A4 )
=> ( ( ( power_power_int @ A4 @ M )
= ( power_power_int @ A4 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_651_class__semiring_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% class_semiring.nat_pow_zero
thf(fact_652_class__semiring_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% class_semiring.nat_pow_zero
thf(fact_653_class__semiring_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% class_semiring.nat_pow_zero
thf(fact_654_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_655_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_656_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_657_class__semiring_Onat__pow__0,axiom,
! [X2: nat] :
( ( power_power_nat @ X2 @ zero_zero_nat )
= one_one_nat ) ).
% class_semiring.nat_pow_0
thf(fact_658_class__semiring_Onat__pow__0,axiom,
! [X2: real] :
( ( power_power_real @ X2 @ zero_zero_nat )
= one_one_real ) ).
% class_semiring.nat_pow_0
thf(fact_659_class__semiring_Onat__pow__0,axiom,
! [X2: int] :
( ( power_power_int @ X2 @ zero_zero_nat )
= one_one_int ) ).
% class_semiring.nat_pow_0
thf(fact_660_power__0,axiom,
! [A4: nat] :
( ( power_power_nat @ A4 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_661_power__0,axiom,
! [A4: real] :
( ( power_power_real @ A4 @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_662_power__0,axiom,
! [A4: int] :
( ( power_power_int @ A4 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_663_class__semiring_Onat__pow__Suc,axiom,
! [X2: nat,N: nat] :
( ( power_power_nat @ X2 @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ X2 ) ) ).
% class_semiring.nat_pow_Suc
thf(fact_664_class__semiring_Onat__pow__Suc,axiom,
! [X2: real,N: nat] :
( ( power_power_real @ X2 @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ X2 @ N ) @ X2 ) ) ).
% class_semiring.nat_pow_Suc
thf(fact_665_class__semiring_Onat__pow__Suc,axiom,
! [X2: int,N: nat] :
( ( power_power_int @ X2 @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ X2 @ N ) @ X2 ) ) ).
% class_semiring.nat_pow_Suc
thf(fact_666_power_Osimps_I2_J,axiom,
! [A4: nat,N: nat] :
( ( power_power_nat @ A4 @ ( suc @ N ) )
= ( times_times_nat @ A4 @ ( power_power_nat @ A4 @ N ) ) ) ).
% power.simps(2)
thf(fact_667_power_Osimps_I2_J,axiom,
! [A4: real,N: nat] :
( ( power_power_real @ A4 @ ( suc @ N ) )
= ( times_times_real @ A4 @ ( power_power_real @ A4 @ N ) ) ) ).
% power.simps(2)
thf(fact_668_power_Osimps_I2_J,axiom,
! [A4: int,N: nat] :
( ( power_power_int @ A4 @ ( suc @ N ) )
= ( times_times_int @ A4 @ ( power_power_int @ A4 @ N ) ) ) ).
% power.simps(2)
thf(fact_669_power__Suc2,axiom,
! [A4: nat,N: nat] :
( ( power_power_nat @ A4 @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A4 @ N ) @ A4 ) ) ).
% power_Suc2
thf(fact_670_power__Suc2,axiom,
! [A4: real,N: nat] :
( ( power_power_real @ A4 @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A4 @ N ) @ A4 ) ) ).
% power_Suc2
thf(fact_671_power__Suc2,axiom,
! [A4: int,N: nat] :
( ( power_power_int @ A4 @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A4 @ N ) @ A4 ) ) ).
% power_Suc2
thf(fact_672_power__Suc0__right,axiom,
! [A4: nat] :
( ( power_power_nat @ A4 @ ( suc @ zero_zero_nat ) )
= A4 ) ).
% power_Suc0_right
thf(fact_673_power__Suc0__right,axiom,
! [A4: real] :
( ( power_power_real @ A4 @ ( suc @ zero_zero_nat ) )
= A4 ) ).
% power_Suc0_right
thf(fact_674_power__Suc0__right,axiom,
! [A4: int] :
( ( power_power_int @ A4 @ ( suc @ zero_zero_nat ) )
= A4 ) ).
% power_Suc0_right
thf(fact_675_nat__power__less__imp__less,axiom,
! [I4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I4 )
=> ( ( ord_less_nat @ ( power_power_nat @ I4 @ M ) @ ( power_power_nat @ I4 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_676_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_677_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power_nat @ X2 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_678_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_679_last_Osimps,axiom,
! [Xs: list_mat_a,X2: mat_a] :
( ( ( Xs = nil_mat_a )
=> ( ( last_mat_a @ ( cons_mat_a @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_mat_a )
=> ( ( last_mat_a @ ( cons_mat_a @ X2 @ Xs ) )
= ( last_mat_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_680_last__ConsL,axiom,
! [Xs: list_mat_a,X2: mat_a] :
( ( Xs = nil_mat_a )
=> ( ( last_mat_a @ ( cons_mat_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_681_last__ConsR,axiom,
! [Xs: list_mat_a,X2: mat_a] :
( ( Xs != nil_mat_a )
=> ( ( last_mat_a @ ( cons_mat_a @ X2 @ Xs ) )
= ( last_mat_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_682_hd__Nil__eq__last,axiom,
( ( hd_mat_a @ nil_mat_a )
= ( last_mat_a @ nil_mat_a ) ) ).
% hd_Nil_eq_last
thf(fact_683_last__tl,axiom,
! [Xs: list_mat_a] :
( ( ( Xs = nil_mat_a )
| ( ( tl_mat_a @ Xs )
!= nil_mat_a ) )
=> ( ( last_mat_a @ ( tl_mat_a @ Xs ) )
= ( last_mat_a @ Xs ) ) ) ).
% last_tl
thf(fact_684_power__less__power__Suc,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ord_less_nat @ ( power_power_nat @ A4 @ N ) @ ( times_times_nat @ A4 @ ( power_power_nat @ A4 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_685_power__less__power__Suc,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ord_less_real @ ( power_power_real @ A4 @ N ) @ ( times_times_real @ A4 @ ( power_power_real @ A4 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_686_power__less__power__Suc,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ one_one_int @ A4 )
=> ( ord_less_int @ ( power_power_int @ A4 @ N ) @ ( times_times_int @ A4 @ ( power_power_int @ A4 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_687_power__gt1__lemma,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A4 @ ( power_power_nat @ A4 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_688_power__gt1__lemma,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A4 @ ( power_power_real @ A4 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_689_power__gt1__lemma,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ one_one_int @ A4 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A4 @ ( power_power_int @ A4 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_690_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_691_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_692_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_693_power__strict__increasing__iff,axiom,
! [B4: nat,X2: nat,Y3: nat] :
( ( ord_less_nat @ one_one_nat @ B4 )
=> ( ( ord_less_nat @ ( power_power_nat @ B4 @ X2 ) @ ( power_power_nat @ B4 @ Y3 ) )
= ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% power_strict_increasing_iff
thf(fact_694_power__strict__increasing__iff,axiom,
! [B4: real,X2: nat,Y3: nat] :
( ( ord_less_real @ one_one_real @ B4 )
=> ( ( ord_less_real @ ( power_power_real @ B4 @ X2 ) @ ( power_power_real @ B4 @ Y3 ) )
= ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% power_strict_increasing_iff
thf(fact_695_power__strict__increasing__iff,axiom,
! [B4: int,X2: nat,Y3: nat] :
( ( ord_less_int @ one_one_int @ B4 )
=> ( ( ord_less_int @ ( power_power_int @ B4 @ X2 ) @ ( power_power_int @ B4 @ Y3 ) )
= ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% power_strict_increasing_iff
thf(fact_696_power__strict__increasing,axiom,
! [N: nat,N5: nat,A4: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ord_less_nat @ ( power_power_nat @ A4 @ N ) @ ( power_power_nat @ A4 @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_697_power__strict__increasing,axiom,
! [N: nat,N5: nat,A4: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ A4 )
=> ( ord_less_real @ ( power_power_real @ A4 @ N ) @ ( power_power_real @ A4 @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_698_power__strict__increasing,axiom,
! [N: nat,N5: nat,A4: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A4 )
=> ( ord_less_int @ ( power_power_int @ A4 @ N ) @ ( power_power_int @ A4 @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_699_power__less__imp__less__exp,axiom,
! [A4: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ( ord_less_nat @ ( power_power_nat @ A4 @ M ) @ ( power_power_nat @ A4 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_700_power__less__imp__less__exp,axiom,
! [A4: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ ( power_power_real @ A4 @ M ) @ ( power_power_real @ A4 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_701_power__less__imp__less__exp,axiom,
! [A4: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A4 )
=> ( ( ord_less_int @ ( power_power_int @ A4 @ M ) @ ( power_power_int @ A4 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_702_power__gt1,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A4 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_703_power__gt1,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A4 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_704_power__gt1,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ one_one_int @ A4 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A4 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_705_power__eq__0__iff,axiom,
! [A4: nat,N: nat] :
( ( ( power_power_nat @ A4 @ N )
= zero_zero_nat )
= ( ( A4 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_706_power__eq__0__iff,axiom,
! [A4: real,N: nat] :
( ( ( power_power_real @ A4 @ N )
= zero_zero_real )
= ( ( A4 = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_707_power__eq__0__iff,axiom,
! [A4: int,N: nat] :
( ( ( power_power_int @ A4 @ N )
= zero_zero_int )
= ( ( A4 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_708_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_709_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_710_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_711_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_712_power__Suc__less,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ A4 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ ( power_power_nat @ A4 @ N ) ) @ ( power_power_nat @ A4 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_713_power__Suc__less,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ A4 @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A4 @ ( power_power_real @ A4 @ N ) ) @ ( power_power_real @ A4 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_714_power__Suc__less,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ A4 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A4 @ ( power_power_int @ A4 @ N ) ) @ ( power_power_int @ A4 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_715_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A4: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ A4 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A4 @ N5 ) @ ( power_power_nat @ A4 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_716_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A4: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ A4 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A4 @ N5 ) @ ( power_power_real @ A4 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_717_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A4: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ A4 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A4 @ N5 ) @ ( power_power_int @ A4 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_718_power__strict__decreasing__iff,axiom,
! [B4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_nat @ B4 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B4 @ M ) @ ( power_power_nat @ B4 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_719_power__strict__decreasing__iff,axiom,
! [B4: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ( ord_less_real @ B4 @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B4 @ M ) @ ( power_power_real @ B4 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_720_power__strict__decreasing__iff,axiom,
! [B4: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_int @ B4 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B4 @ M ) @ ( power_power_int @ B4 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_721_power__Suc__less__one,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ A4 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A4 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_722_power__Suc__less__one,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ A4 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A4 @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_723_power__Suc__less__one,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ A4 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A4 @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_724_one__less__power,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A4 @ N ) ) ) ) ).
% one_less_power
thf(fact_725_one__less__power,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A4 @ N ) ) ) ) ).
% one_less_power
thf(fact_726_one__less__power,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ one_one_int @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A4 @ N ) ) ) ) ).
% one_less_power
thf(fact_727_length__n__lists,axiom,
! [N: nat,Xs: list_mat_a] :
( ( size_s6656407794899724303_mat_a @ ( n_lists_mat_a @ N @ Xs ) )
= ( power_power_nat @ ( size_size_list_mat_a @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_728_realpow__pos__nth__unique,axiom,
! [N: nat,A4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A4 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A4 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_729_realpow__pos__nth,axiom,
! [N: nat,A4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A4 ) ) ) ) ).
% realpow_pos_nth
thf(fact_730_last__list__update,axiom,
! [Xs: list_mat_a,K: nat,X2: mat_a] :
( ( Xs != nil_mat_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_mat_a @ Xs ) @ one_one_nat ) )
=> ( ( last_mat_a @ ( list_update_mat_a @ Xs @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_mat_a @ Xs ) @ one_one_nat ) )
=> ( ( last_mat_a @ ( list_update_mat_a @ Xs @ K @ X2 ) )
= ( last_mat_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_731_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_732_nth__list__update__neq,axiom,
! [I4: nat,J: nat,Xs: list_mat_a,X2: mat_a] :
( ( I4 != J )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I4 @ X2 ) @ J )
= ( nth_mat_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_733_list__update__id,axiom,
! [Xs: list_mat_a,I4: nat] :
( ( list_update_mat_a @ Xs @ I4 @ ( nth_mat_a @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_734_length__list__update,axiom,
! [Xs: list_mat_a,I4: nat,X2: mat_a] :
( ( size_size_list_mat_a @ ( list_update_mat_a @ Xs @ I4 @ X2 ) )
= ( size_size_list_mat_a @ Xs ) ) ).
% length_list_update
thf(fact_735_list__update__code_I1_J,axiom,
! [I4: nat,Y3: mat_a] :
( ( list_update_mat_a @ nil_mat_a @ I4 @ Y3 )
= nil_mat_a ) ).
% list_update_code(1)
thf(fact_736_list__update_Osimps_I1_J,axiom,
! [I4: nat,V: mat_a] :
( ( list_update_mat_a @ nil_mat_a @ I4 @ V )
= nil_mat_a ) ).
% list_update.simps(1)
thf(fact_737_list__update__nonempty,axiom,
! [Xs: list_mat_a,K: nat,X2: mat_a] :
( ( ( list_update_mat_a @ Xs @ K @ X2 )
= nil_mat_a )
= ( Xs = nil_mat_a ) ) ).
% list_update_nonempty
thf(fact_738_list__update__code_I2_J,axiom,
! [X2: mat_a,Xs: list_mat_a,Y3: mat_a] :
( ( list_update_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
= ( cons_mat_a @ Y3 @ Xs ) ) ).
% list_update_code(2)
thf(fact_739_list__update__code_I3_J,axiom,
! [X2: mat_a,Xs: list_mat_a,I4: nat,Y3: mat_a] :
( ( list_update_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ ( suc @ I4 ) @ Y3 )
= ( cons_mat_a @ X2 @ ( list_update_mat_a @ Xs @ I4 @ Y3 ) ) ) ).
% list_update_code(3)
thf(fact_740_nth__list__update,axiom,
! [I4: nat,Xs: list_mat_a,J: nat,X2: mat_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( ( I4 = J )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I4 @ X2 ) @ J )
= X2 ) )
& ( ( I4 != J )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I4 @ X2 ) @ J )
= ( nth_mat_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_741_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_mat_a,X2: mat_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I4 @ X2 ) @ I4 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_742_list__update__same__conv,axiom,
! [I4: nat,Xs: list_mat_a,X2: mat_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( ( list_update_mat_a @ Xs @ I4 @ X2 )
= Xs )
= ( ( nth_mat_a @ Xs @ I4 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_743_realpow__pos__nth2,axiom,
! [A4: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A4 ) ) ) ).
% realpow_pos_nth2
thf(fact_744_reals__power__lt__ex,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ one_one_real @ Y3 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y3 ) @ K2 ) @ X2 ) ) ) ) ).
% reals_power_lt_ex
thf(fact_745_power__strict__mono,axiom,
! [A4: nat,B4: nat,N: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A4 @ N ) @ ( power_power_nat @ B4 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_746_power__strict__mono,axiom,
! [A4: real,B4: real,N: nat] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A4 @ N ) @ ( power_power_real @ B4 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_747_power__strict__mono,axiom,
! [A4: int,B4: int,N: nat] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A4 @ N ) @ ( power_power_int @ B4 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_748_subseqs_Osimps_I1_J,axiom,
( ( subseqs_mat_a @ nil_mat_a )
= ( cons_list_mat_a @ nil_mat_a @ nil_list_mat_a ) ) ).
% subseqs.simps(1)
thf(fact_749_power__diff,axiom,
! [A4: real,N: nat,M: nat] :
( ( A4 != zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_real @ A4 @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_real @ ( power_power_real @ A4 @ M ) @ ( power_power_real @ A4 @ N ) ) ) ) ) ).
% power_diff
thf(fact_750_power__diff,axiom,
! [A4: nat,N: nat,M: nat] :
( ( A4 != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_nat @ A4 @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A4 @ M ) @ ( power_power_nat @ A4 @ N ) ) ) ) ) ).
% power_diff
thf(fact_751_power__diff,axiom,
! [A4: int,N: nat,M: nat] :
( ( A4 != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_int @ A4 @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_int @ ( power_power_int @ A4 @ M ) @ ( power_power_int @ A4 @ N ) ) ) ) ) ).
% power_diff
thf(fact_752_power__increasing,axiom,
! [N: nat,N5: nat,A4: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A4 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A4 @ N ) @ ( power_power_nat @ A4 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_753_power__increasing,axiom,
! [N: nat,N5: nat,A4: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ A4 )
=> ( ord_less_eq_real @ ( power_power_real @ A4 @ N ) @ ( power_power_real @ A4 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_754_power__increasing,axiom,
! [N: nat,N5: nat,A4: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A4 )
=> ( ord_less_eq_int @ ( power_power_int @ A4 @ N ) @ ( power_power_int @ A4 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_755_power__divide,axiom,
! [A4: real,B4: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A4 @ B4 ) @ N )
= ( divide_divide_real @ ( power_power_real @ A4 @ N ) @ ( power_power_real @ B4 @ N ) ) ) ).
% power_divide
thf(fact_756_power__le__imp__le__exp,axiom,
! [A4: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A4 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A4 @ M ) @ ( power_power_nat @ A4 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_757_power__le__imp__le__exp,axiom,
! [A4: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A4 @ M ) @ ( power_power_real @ A4 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_758_power__le__imp__le__exp,axiom,
! [A4: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A4 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A4 @ M ) @ ( power_power_int @ A4 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_759_power__increasing__iff,axiom,
! [B4: nat,X2: nat,Y3: nat] :
( ( ord_less_nat @ one_one_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B4 @ X2 ) @ ( power_power_nat @ B4 @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% power_increasing_iff
thf(fact_760_power__increasing__iff,axiom,
! [B4: real,X2: nat,Y3: nat] :
( ( ord_less_real @ one_one_real @ B4 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B4 @ X2 ) @ ( power_power_real @ B4 @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% power_increasing_iff
thf(fact_761_power__increasing__iff,axiom,
! [B4: int,X2: nat,Y3: nat] :
( ( ord_less_int @ one_one_int @ B4 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B4 @ X2 ) @ ( power_power_int @ B4 @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% power_increasing_iff
thf(fact_762_power__decreasing,axiom,
! [N: nat,N5: nat,A4: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A4 @ N5 ) @ ( power_power_nat @ A4 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_763_power__decreasing,axiom,
! [N: nat,N5: nat,A4: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ A4 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A4 @ N5 ) @ ( power_power_real @ A4 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_764_power__decreasing,axiom,
! [N: nat,N5: nat,A4: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_eq_int @ A4 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A4 @ N5 ) @ ( power_power_int @ A4 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_765_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_766_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ~ ( ord_less_eq_real @ T @ X ) ) ).
% minf(8)
thf(fact_767_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ~ ( ord_less_eq_int @ T @ X ) ) ).
% minf(8)
thf(fact_768_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_769_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ord_less_eq_real @ X @ T ) ) ).
% minf(6)
thf(fact_770_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ord_less_eq_int @ X @ T ) ) ).
% minf(6)
thf(fact_771_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_eq_nat @ T @ X ) ) ).
% pinf(8)
thf(fact_772_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_eq_real @ T @ X ) ) ).
% pinf(8)
thf(fact_773_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ord_less_eq_int @ T @ X ) ) ).
% pinf(8)
thf(fact_774_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ T ) ) ).
% pinf(6)
thf(fact_775_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ T ) ) ).
% pinf(6)
thf(fact_776_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ T ) ) ).
% pinf(6)
thf(fact_777_real__divide__square__eq,axiom,
! [R2: real,A4: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A4 ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A4 @ R2 ) ) ).
% real_divide_square_eq
thf(fact_778_zero__compare__simps_I9_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A4 @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ) ) ).
% zero_compare_simps(9)
thf(fact_779_zero__compare__simps_I5_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A4 @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(5)
thf(fact_780_inf__pigeonhole__principle,axiom,
! [N: nat,F2: nat > nat > $o] :
( ! [K2: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( F2 @ K2 @ I ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F2 @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_781_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I4: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F2 @ I4 ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_782_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_783_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_784_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_785_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_786_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_787_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_788_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_789_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_790_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_791_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_792_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_793_order__trans__rules_I22_J,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_trans_rules(22)
thf(fact_794_order__trans__rules_I22_J,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_trans_rules(22)
thf(fact_795_order__trans__rules_I22_J,axiom,
! [X2: int,Y3: int,Z3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_trans_rules(22)
thf(fact_796_order__trans__rules_I21_J,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_trans_rules(21)
thf(fact_797_order__trans__rules_I21_J,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_trans_rules(21)
thf(fact_798_order__trans__rules_I21_J,axiom,
! [X2: int,Y3: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_trans_rules(21)
thf(fact_799_order__trans__rules_I18_J,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_nat @ A4 @ B4 ) ) ) ).
% order_trans_rules(18)
thf(fact_800_order__trans__rules_I18_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_real @ A4 @ B4 ) ) ) ).
% order_trans_rules(18)
thf(fact_801_order__trans__rules_I18_J,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_int @ A4 @ B4 ) ) ) ).
% order_trans_rules(18)
thf(fact_802_order__trans__rules_I17_J,axiom,
! [A4: nat,B4: nat] :
( ( A4 != B4 )
=> ( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ord_less_nat @ A4 @ B4 ) ) ) ).
% order_trans_rules(17)
thf(fact_803_order__trans__rules_I17_J,axiom,
! [A4: real,B4: real] :
( ( A4 != B4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ord_less_real @ A4 @ B4 ) ) ) ).
% order_trans_rules(17)
thf(fact_804_order__trans__rules_I17_J,axiom,
! [A4: int,B4: int] :
( ( A4 != B4 )
=> ( ( ord_less_eq_int @ A4 @ B4 )
=> ( ord_less_int @ A4 @ B4 ) ) ) ).
% order_trans_rules(17)
thf(fact_805_order__trans__rules_I6_J,axiom,
! [A4: nat,F2: nat > nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_806_order__trans__rules_I6_J,axiom,
! [A4: real,F2: nat > real,B4: nat,C: nat] :
( ( ord_less_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_807_order__trans__rules_I6_J,axiom,
! [A4: int,F2: nat > int,B4: nat,C: nat] :
( ( ord_less_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_808_order__trans__rules_I6_J,axiom,
! [A4: nat,F2: real > nat,B4: real,C: real] :
( ( ord_less_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_809_order__trans__rules_I6_J,axiom,
! [A4: real,F2: real > real,B4: real,C: real] :
( ( ord_less_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_810_order__trans__rules_I6_J,axiom,
! [A4: int,F2: real > int,B4: real,C: real] :
( ( ord_less_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_811_order__trans__rules_I6_J,axiom,
! [A4: nat,F2: int > nat,B4: int,C: int] :
( ( ord_less_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_812_order__trans__rules_I6_J,axiom,
! [A4: real,F2: int > real,B4: int,C: int] :
( ( ord_less_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_813_order__trans__rules_I6_J,axiom,
! [A4: int,F2: int > int,B4: int,C: int] :
( ( ord_less_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_814_order__trans__rules_I5_J,axiom,
! [A4: nat,B4: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_815_order__trans__rules_I5_J,axiom,
! [A4: real,B4: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_816_order__trans__rules_I5_J,axiom,
! [A4: int,B4: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_817_order__trans__rules_I5_J,axiom,
! [A4: nat,B4: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_818_order__trans__rules_I5_J,axiom,
! [A4: real,B4: real,F2: real > real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_819_order__trans__rules_I5_J,axiom,
! [A4: int,B4: int,F2: int > real,C: real] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_820_order__trans__rules_I5_J,axiom,
! [A4: nat,B4: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_821_order__trans__rules_I5_J,axiom,
! [A4: real,B4: real,F2: real > int,C: int] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_822_order__trans__rules_I5_J,axiom,
! [A4: int,B4: int,F2: int > int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(5)
thf(fact_823_order__trans__rules_I4_J,axiom,
! [A4: nat,F2: nat > nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_824_order__trans__rules_I4_J,axiom,
! [A4: nat,F2: real > nat,B4: real,C: real] :
( ( ord_less_eq_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_825_order__trans__rules_I4_J,axiom,
! [A4: nat,F2: int > nat,B4: int,C: int] :
( ( ord_less_eq_nat @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_826_order__trans__rules_I4_J,axiom,
! [A4: real,F2: nat > real,B4: nat,C: nat] :
( ( ord_less_eq_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_827_order__trans__rules_I4_J,axiom,
! [A4: real,F2: real > real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_828_order__trans__rules_I4_J,axiom,
! [A4: real,F2: int > real,B4: int,C: int] :
( ( ord_less_eq_real @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_829_order__trans__rules_I4_J,axiom,
! [A4: int,F2: nat > int,B4: nat,C: nat] :
( ( ord_less_eq_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_830_order__trans__rules_I4_J,axiom,
! [A4: int,F2: real > int,B4: real,C: real] :
( ( ord_less_eq_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_831_order__trans__rules_I4_J,axiom,
! [A4: int,F2: int > int,B4: int,C: int] :
( ( ord_less_eq_int @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_832_order__trans__rules_I3_J,axiom,
! [A4: nat,B4: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_833_order__trans__rules_I3_J,axiom,
! [A4: nat,B4: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_834_order__trans__rules_I3_J,axiom,
! [A4: nat,B4: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_835_order__trans__rules_I3_J,axiom,
! [A4: real,B4: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_836_order__trans__rules_I3_J,axiom,
! [A4: real,B4: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_837_order__trans__rules_I3_J,axiom,
! [A4: real,B4: real,F2: real > int,C: int] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_838_order__trans__rules_I3_J,axiom,
! [A4: int,B4: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_839_order__trans__rules_I3_J,axiom,
! [A4: int,B4: int,F2: int > real,C: real] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_real @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_real @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_840_order__trans__rules_I3_J,axiom,
! [A4: int,B4: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_int @ ( F2 @ B4 ) @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_trans_rules(3)
thf(fact_841_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_842_leD,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq_real @ Y3 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y3 ) ) ).
% leD
thf(fact_843_leD,axiom,
! [Y3: int,X2: int] :
( ( ord_less_eq_int @ Y3 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y3 ) ) ).
% leD
thf(fact_844_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_845_leI,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% leI
thf(fact_846_leI,axiom,
! [X2: int,Y3: int] :
( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% leI
thf(fact_847_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X6: nat,Y2: nat] :
( ( ord_less_nat @ X6 @ Y2 )
| ( X6 = Y2 ) ) ) ) ).
% le_less
thf(fact_848_le__less,axiom,
( ord_less_eq_real
= ( ^ [X6: real,Y2: real] :
( ( ord_less_real @ X6 @ Y2 )
| ( X6 = Y2 ) ) ) ) ).
% le_less
thf(fact_849_le__less,axiom,
( ord_less_eq_int
= ( ^ [X6: int,Y2: int] :
( ( ord_less_int @ X6 @ Y2 )
| ( X6 = Y2 ) ) ) ) ).
% le_less
thf(fact_850_less__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y2: nat] :
( ( ord_less_eq_nat @ X6 @ Y2 )
& ( X6 != Y2 ) ) ) ) ).
% less_le
thf(fact_851_less__le,axiom,
( ord_less_real
= ( ^ [X6: real,Y2: real] :
( ( ord_less_eq_real @ X6 @ Y2 )
& ( X6 != Y2 ) ) ) ) ).
% less_le
thf(fact_852_less__le,axiom,
( ord_less_int
= ( ^ [X6: int,Y2: int] :
( ( ord_less_eq_int @ X6 @ Y2 )
& ( X6 != Y2 ) ) ) ) ).
% less_le
thf(fact_853_nless__le,axiom,
! [A4: nat,B4: nat] :
( ( ~ ( ord_less_nat @ A4 @ B4 ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% nless_le
thf(fact_854_nless__le,axiom,
! [A4: real,B4: real] :
( ( ~ ( ord_less_real @ A4 @ B4 ) )
= ( ~ ( ord_less_eq_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% nless_le
thf(fact_855_nless__le,axiom,
! [A4: int,B4: int] :
( ( ~ ( ord_less_int @ A4 @ B4 ) )
= ( ~ ( ord_less_eq_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% nless_le
thf(fact_856_not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% not_le
thf(fact_857_not__le,axiom,
! [X2: real,Y3: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y3 ) )
= ( ord_less_real @ Y3 @ X2 ) ) ).
% not_le
thf(fact_858_not__le,axiom,
! [X2: int,Y3: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y3 ) )
= ( ord_less_int @ Y3 @ X2 ) ) ).
% not_le
thf(fact_859_not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% not_less
thf(fact_860_not__less,axiom,
! [X2: real,Y3: real] :
( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% not_less
thf(fact_861_not__less,axiom,
! [X2: int,Y3: int] :
( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% not_less
thf(fact_862_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_863_antisym__conv1,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_864_antisym__conv1,axiom,
! [X2: int,Y3: int] :
( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_eq_int @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_865_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_866_antisym__conv2,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_867_antisym__conv2,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_868_less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_869_less__imp__le,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_870_less__imp__le,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_871_dense__ge,axiom,
! [Z3: real,Y3: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y3 @ X3 ) )
=> ( ord_less_eq_real @ Y3 @ Z3 ) ) ).
% dense_ge
thf(fact_872_dense__le,axiom,
! [Y3: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y3 @ Z3 ) ) ).
% dense_le
thf(fact_873_le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% le_less_linear
thf(fact_874_le__less__linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
| ( ord_less_real @ Y3 @ X2 ) ) ).
% le_less_linear
thf(fact_875_le__less__linear,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
| ( ord_less_int @ Y3 @ X2 ) ) ).
% le_less_linear
thf(fact_876_le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% le_imp_less_or_eq
thf(fact_877_le__imp__less__or__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% le_imp_less_or_eq
thf(fact_878_le__imp__less__or__eq,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% le_imp_less_or_eq
thf(fact_879_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y2: nat] :
( ( ord_less_eq_nat @ X6 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_880_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X6: real,Y2: real] :
( ( ord_less_eq_real @ X6 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_881_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X6: int,Y2: int] :
( ( ord_less_eq_int @ X6 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_882_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_883_not__le__imp__less,axiom,
! [Y3: real,X2: real] :
( ~ ( ord_less_eq_real @ Y3 @ X2 )
=> ( ord_less_real @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_884_not__le__imp__less,axiom,
! [Y3: int,X2: int] :
( ~ ( ord_less_eq_int @ Y3 @ X2 )
=> ( ord_less_int @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_885_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_886_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B6: real] :
( ( ord_less_real @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_887_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B6: int] :
( ( ord_less_int @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_888_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_889_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_890_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_891_order_Ostrict__trans1,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_892_order_Ostrict__trans1,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_893_order_Ostrict__trans1,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_894_order_Ostrict__trans2,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_895_order_Ostrict__trans2,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_896_order_Ostrict__trans2,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_897_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_898_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
& ~ ( ord_less_eq_real @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_899_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
& ~ ( ord_less_eq_int @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_900_dense__ge__bounded,axiom,
! [Z3: real,X2: real,Y3: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y3 @ W ) ) )
=> ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_901_dense__le__bounded,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y3 )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_902_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_nat @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_903_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B6: real,A5: real] :
( ( ord_less_real @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_904_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B6: int,A5: int] :
( ( ord_less_int @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_905_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_906_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B6: real,A5: real] :
( ( ord_less_eq_real @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_907_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B6: int,A5: int] :
( ( ord_less_eq_int @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_908_dual__order_Ostrict__trans1,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
=> ( ( ord_less_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_909_dual__order_Ostrict__trans1,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_real @ C @ B4 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_910_dual__order_Ostrict__trans1,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_eq_int @ B4 @ A4 )
=> ( ( ord_less_int @ C @ B4 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_911_dual__order_Ostrict__trans2,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_912_dual__order_Ostrict__trans2,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ C @ B4 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_913_dual__order_Ostrict__trans2,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ( ord_less_eq_int @ C @ B4 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_914_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_915_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B6: real,A5: real] :
( ( ord_less_eq_real @ B6 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_916_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B6: int,A5: int] :
( ( ord_less_eq_int @ B6 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_917_order_Ostrict__implies__order,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ord_less_eq_nat @ A4 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_918_order_Ostrict__implies__order,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ord_less_eq_real @ A4 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_919_order_Ostrict__implies__order,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ord_less_eq_int @ A4 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_920_dual__order_Ostrict__implies__order,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( ord_less_eq_nat @ B4 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_921_dual__order_Ostrict__implies__order,axiom,
! [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ord_less_eq_real @ B4 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_922_dual__order_Ostrict__implies__order,axiom,
! [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ord_less_eq_int @ B4 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_923_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_924_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
= ( ord_less_real @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_925_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_926_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_927_zero__order_I1_J,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_order(1)
thf(fact_928_semiring__norm_I113_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% semiring_norm(113)
thf(fact_929_semiring__norm_I113_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% semiring_norm(113)
thf(fact_930_semiring__norm_I113_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% semiring_norm(113)
thf(fact_931_div__by__0,axiom,
! [A4: real] :
( ( divide_divide_real @ A4 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_932_div__by__0,axiom,
! [A4: nat] :
( ( divide_divide_nat @ A4 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_933_div__by__0,axiom,
! [A4: int] :
( ( divide_divide_int @ A4 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_934_div__0,axiom,
! [A4: real] :
( ( divide_divide_real @ zero_zero_real @ A4 )
= zero_zero_real ) ).
% div_0
thf(fact_935_div__0,axiom,
! [A4: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A4 )
= zero_zero_nat ) ).
% div_0
thf(fact_936_div__0,axiom,
! [A4: int] :
( ( divide_divide_int @ zero_zero_int @ A4 )
= zero_zero_int ) ).
% div_0
thf(fact_937_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_938_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_939_le__diff__iff_H,axiom,
! [A4: nat,C: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A4 ) @ ( minus_minus_nat @ C @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_940_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_941_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_942_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_943_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_944_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_945_div__by__1,axiom,
! [A4: real] :
( ( divide_divide_real @ A4 @ one_one_real )
= A4 ) ).
% div_by_1
thf(fact_946_div__by__1,axiom,
! [A4: nat] :
( ( divide_divide_nat @ A4 @ one_one_nat )
= A4 ) ).
% div_by_1
thf(fact_947_div__by__1,axiom,
! [A4: int] :
( ( divide_divide_int @ A4 @ one_one_int )
= A4 ) ).
% div_by_1
thf(fact_948_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_949_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_950_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_951_bot_Oextremum__uniqueI,axiom,
! [A4: set_list_mat_a] :
( ( ord_le4771995077433322369_mat_a @ A4 @ bot_bo2759726786008686517_mat_a )
=> ( A4 = bot_bo2759726786008686517_mat_a ) ) ).
% bot.extremum_uniqueI
thf(fact_952_bot_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
=> ( A4 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_953_bot_Oextremum__unique,axiom,
! [A4: set_list_mat_a] :
( ( ord_le4771995077433322369_mat_a @ A4 @ bot_bo2759726786008686517_mat_a )
= ( A4 = bot_bo2759726786008686517_mat_a ) ) ).
% bot.extremum_unique
thf(fact_954_bot_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
= ( A4 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_955_bot_Oextremum,axiom,
! [A4: set_list_mat_a] : ( ord_le4771995077433322369_mat_a @ bot_bo2759726786008686517_mat_a @ A4 ) ).
% bot.extremum
thf(fact_956_bot_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A4 ) ).
% bot.extremum
thf(fact_957_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_958_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_959_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_960_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_961_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_962_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_963_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_964_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_965_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_966_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_967_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z: nat] :
( ( R3 @ X3 @ Y )
=> ( ( R3 @ Y @ Z )
=> ( R3 @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_968_verit__eq__simplify_I6_J,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% verit_eq_simplify(6)
thf(fact_969_verit__eq__simplify_I6_J,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% verit_eq_simplify(6)
thf(fact_970_verit__eq__simplify_I6_J,axiom,
! [X2: int,Y3: int] :
( ( X2 = Y3 )
=> ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% verit_eq_simplify(6)
thf(fact_971_verit__comp__simplify_I2_J,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% verit_comp_simplify(2)
thf(fact_972_verit__comp__simplify_I2_J,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% verit_comp_simplify(2)
thf(fact_973_verit__comp__simplify_I2_J,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% verit_comp_simplify(2)
thf(fact_974_lift__Suc__mono__le,axiom,
! [F2: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_975_lift__Suc__mono__le,axiom,
! [F2: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_976_lift__Suc__mono__le,axiom,
! [F2: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_977_lift__Suc__antimono__le,axiom,
! [F2: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_978_lift__Suc__antimono__le,axiom,
! [F2: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_979_lift__Suc__antimono__le,axiom,
! [F2: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_980_verit__la__disequality,axiom,
! [A4: nat,B4: nat] :
( ( A4 = B4 )
| ~ ( ord_less_eq_nat @ A4 @ B4 )
| ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ).
% verit_la_disequality
thf(fact_981_verit__la__disequality,axiom,
! [A4: real,B4: real] :
( ( A4 = B4 )
| ~ ( ord_less_eq_real @ A4 @ B4 )
| ~ ( ord_less_eq_real @ B4 @ A4 ) ) ).
% verit_la_disequality
thf(fact_982_verit__la__disequality,axiom,
! [A4: int,B4: int] :
( ( A4 = B4 )
| ~ ( ord_less_eq_int @ A4 @ B4 )
| ~ ( ord_less_eq_int @ B4 @ A4 ) ) ).
% verit_la_disequality
thf(fact_983_mult__le__mono2,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_984_mult__le__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_985_mult__le__mono,axiom,
! [I4: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_986_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_987_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_988_power__decreasing__iff,axiom,
! [B4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_nat @ B4 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B4 @ M ) @ ( power_power_nat @ B4 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_989_power__decreasing__iff,axiom,
! [B4: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ( ord_less_real @ B4 @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B4 @ M ) @ ( power_power_real @ B4 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_990_power__decreasing__iff,axiom,
! [B4: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_int @ B4 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B4 @ M ) @ ( power_power_int @ B4 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_991_nonzero__mult__div__cancel__right,axiom,
! [B4: real,A4: real] :
( ( B4 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A4 @ B4 ) @ B4 )
= A4 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_992_nonzero__mult__div__cancel__right,axiom,
! [B4: nat,A4: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A4 @ B4 ) @ B4 )
= A4 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_993_nonzero__mult__div__cancel__right,axiom,
! [B4: int,A4: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A4 @ B4 ) @ B4 )
= A4 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_994_nonzero__mult__div__cancel__left,axiom,
! [A4: real,B4: real] :
( ( A4 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A4 @ B4 ) @ A4 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_995_nonzero__mult__div__cancel__left,axiom,
! [A4: nat,B4: nat] :
( ( A4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A4 @ B4 ) @ A4 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_996_nonzero__mult__div__cancel__left,axiom,
! [A4: int,B4: int] :
( ( A4 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A4 @ B4 ) @ A4 )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_997_zero__compare__simps_I11_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ ( divide_divide_real @ A4 @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B4 ) ) ) ) ).
% zero_compare_simps(11)
thf(fact_998_zero__compare__simps_I7_J,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A4 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(7)
thf(fact_999_div__self,axiom,
! [A4: real] :
( ( A4 != zero_zero_real )
=> ( ( divide_divide_real @ A4 @ A4 )
= one_one_real ) ) ).
% div_self
thf(fact_1000_div__self,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
=> ( ( divide_divide_nat @ A4 @ A4 )
= one_one_nat ) ) ).
% div_self
thf(fact_1001_div__self,axiom,
! [A4: int] :
( ( A4 != zero_zero_int )
=> ( ( divide_divide_int @ A4 @ A4 )
= one_one_int ) ) ).
% div_self
thf(fact_1002_power__one__over,axiom,
! [A4: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A4 ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A4 @ N ) ) ) ).
% power_one_over
thf(fact_1003_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A4 ) @ ( times_times_nat @ C @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1004_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A4 ) @ ( times_times_real @ C @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1005_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A4 ) @ ( times_times_int @ C @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1006_zero__le__mult__iff,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A4 @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1007_zero__le__mult__iff,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A4 @ B4 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A4 )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_eq_int @ A4 @ zero_zero_int )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1008_mult__nonneg__nonpos2,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B4 @ A4 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1009_mult__nonneg__nonpos2,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B4 @ A4 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1010_mult__nonneg__nonpos2,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B4 @ A4 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1011_mult__nonpos__nonneg,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1012_mult__nonpos__nonneg,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1013_mult__nonpos__nonneg,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( times_times_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1014_mult__nonneg__nonpos,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1015_mult__nonneg__nonpos,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1016_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1017_le__simps_I3_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% le_simps(3)
thf(fact_1018_le__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(2)
thf(fact_1019_not__less__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(2)
thf(fact_1020_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1021_dec__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ I4 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1022_inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% inc_induct
thf(fact_1023_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1024_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1025_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1026_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1027_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1028_diff__less__mono,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_nat @ ( minus_minus_nat @ A4 @ C ) @ ( minus_minus_nat @ B4 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1029_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1030_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1031_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K2: nat,M2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ K2 )
=> ( P @ ( suc @ K2 ) @ ( minus_minus_nat @ M2 @ ( suc @ K2 ) ) ) )
=> ( P @ K2 @ M2 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_1032_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_1033_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1034_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_1035_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_1036_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_1037_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_1038_nat__one__le__power,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I4 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I4 @ N ) ) ) ).
% nat_one_le_power
thf(fact_1039_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1040_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_1041_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1042_less__mult__imp__div__less,axiom,
! [M: nat,I4: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I4 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I4 ) ) ).
% less_mult_imp_div_less
thf(fact_1043_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_1044_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_1045_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1046_le__trans,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_1047_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1048_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1049_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1050_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1051_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_1052_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1060_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1061_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1062_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_1063_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1064_div__if,axiom,
( divide_divide_nat
= ( ^ [M4: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M4 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M4 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_1065_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_1066_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_1067_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_1068_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F2: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F2 @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
& ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
=> ? [L: ( nat > real ) > nat > nat] :
( ! [X: nat > real,I: nat] : ( ord_less_eq_nat @ ( L @ X @ I ) @ one_one_nat )
& ! [X: nat > real,I: nat] :
( ( ( P @ X )
& ( Q2 @ I )
& ( ( X @ I )
= zero_zero_real ) )
=> ( ( L @ X @ I )
= zero_zero_nat ) )
& ! [X: nat > real,I: nat] :
( ( ( P @ X )
& ( Q2 @ I )
& ( ( X @ I )
= one_one_real ) )
=> ( ( L @ X @ I )
= one_one_nat ) )
& ! [X: nat > real,I: nat] :
( ( ( P @ X )
& ( Q2 @ I )
& ( ( L @ X @ I )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X @ I ) @ ( F2 @ X @ I ) ) )
& ! [X: nat > real,I: nat] :
( ( ( P @ X )
& ( Q2 @ I )
& ( ( L @ X @ I )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F2 @ X @ I ) @ ( X @ I ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1069_power__le__one__iff,axiom,
! [A4: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A4 @ N ) @ one_one_real )
= ( ( N = zero_zero_nat )
| ( ord_less_eq_real @ A4 @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1070_zdiv__zmult2__eq,axiom,
! [C: int,A4: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A4 @ ( times_times_int @ B4 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A4 @ B4 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1071_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X: int] :
( ( P @ X )
=> ( P @ ( minus_minus_int @ X @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1072_plusinfinity,axiom,
! [D: int,P6: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [X_12: int] : ( P6 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1073_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1074_verit__la__generic,axiom,
! [A4: int,X2: int] :
( ( ord_less_eq_int @ A4 @ X2 )
| ( A4 = X2 )
| ( ord_less_eq_int @ X2 @ A4 ) ) ).
% verit_la_generic
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: list_mat_a] :
( ( member_list_mat_a @ X3 @ alsa )
=> ? [Bl3: list_mat_a] :
( ( ( size_size_list_mat_a @ Bl3 )
= one_one_nat )
& ( spectr3403749184330357196comp_a @ ( diag_block_mat_a @ X3 ) @ ( diag_block_mat_a @ Bl3 ) @ ( diag_block_mat_a @ ( cons_mat_a @ u0 @ nil_mat_a ) ) ) ) ) ).
%------------------------------------------------------------------------------