TPTP Problem File: SLH0231^1.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_02229_087485__19602404_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1231 ( 397 unt; 183 typ; 0 def)
% Number of atoms : 3037 (1129 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10046 ( 347 ~; 62 |; 121 &;7817 @)
% ( 0 <=>;1699 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 708 ( 708 >; 0 *; 0 +; 0 <<)
% Number of symbols : 151 ( 148 usr; 19 con; 0-3 aty)
% Number of variables : 3345 ( 186 ^;3074 !; 85 ?;3345 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:38:07.107
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
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% Explicit typings (148)
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thf(sy_c_Matrix_Oupper__triangular_001t__Nat__Onat,type,
upper_triangular_nat: mat_nat > $o ).
thf(sy_c_Missing__Polynomial_Oexpand__powers_001t__Nat__Onat,type,
missin6482572040563731271rs_nat: list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
size_s5969786470865220249omplex: list_mat_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
size_s3423402466807558097omplex: list_P6605091754902497125omplex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
size_s2777191050033188080omplex: list_P1999334753057444956omplex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $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__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le5598786136212072115omplex: set_mat_complex > set_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le4845169857901429244at_nat: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $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__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le2819838839419867280at_nat: produc8199716216217303280at_nat > produc8199716216217303280at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le3632134057777142183omplex: set_mat_complex > set_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le3678578370064672496at_nat: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).
thf(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
produc101793102246108661omplex: complex > complex > produc4411394909380815293omplex ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Complex__Ocomplex_J_001t__List__Olist_It__Complex__Ocomplex_J,type,
produc2490022270806822549omplex: list_complex > list_complex > produc1631295542841207645omplex ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc3658446505030690647omplex: mat_complex > mat_complex > produc352478934956084711omplex ).
thf(sy_c_Product__Type_OPair_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
produc3916067632315525152ex_nat: mat_complex > nat > produc4941145339993070502ex_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Complex__Ocomplex,type,
produc6973218034000581911omplex: nat > complex > produc4863162743050822367omplex ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
produc8282810413953273033st_nat: nat > list_nat > produc4575160907756185873st_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc4998868960714853886omplex: nat > mat_complex > produc3259542890344722124omplex ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
collect_mat_complex: ( mat_complex > $o ) > set_mat_complex ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collec9201399625632817755at_nat: ( produc8199716216217303280at_nat > $o ) > set_Pr9093778441882193744at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Complex__Ocomplex_J_Mt__List__Olist_It__Complex__Ocomplex_J_J,type,
collec2134092743982711368omplex: ( produc1631295542841207645omplex > $o ) > set_Pr135664057150635453omplex ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
collec1570431334306492044st_nat: ( produc1828647624359046049st_nat > $o ) > set_Pr3451248702717554689st_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_member_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
member_mat_complex: mat_complex > set_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
member5793383173714906214omplex: produc4411394909380815293omplex > set_Pr5085853215250843933omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Complex__Ocomplex_J_Mt__List__Olist_It__Complex__Ocomplex_J_J,type,
member6068360845309590790omplex: produc1631295542841207645omplex > set_Pr135664057150635453omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_v_A,type,
a: mat_complex ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_k____,type,
k: complex ).
thf(sy_v_l____,type,
l: list_complex ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_p____,type,
p: nat ).
% Relevant facts (1044)
thf(fact_0_assms_I1_J,axiom,
diagonal_mat_complex @ a ).
% assms(1)
thf(fact_1_assms_I3_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(3)
thf(fact_2_assms_I2_J,axiom,
member_mat_complex @ a @ ( carrier_mat_complex @ n @ n ) ).
% assms(2)
thf(fact_3__092_060open_062sum__list_A_Ieq__comps_A_Idiag__mat_AA_J_J_A_061_Alength_A_Idiag__mat_AA_J_092_060close_062,axiom,
( ( groups4561878855575611511st_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ a ) ) )
= ( size_s3451745648224563538omplex @ ( diag_mat_complex @ a ) ) ) ).
% \<open>sum_list (eq_comps (diag_mat A)) = length (diag_mat A)\<close>
thf(fact_4_assms_I4_J,axiom,
ord_less_nat @ i @ ( size_size_list_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ a ) ) ) ).
% assms(4)
thf(fact_5_l__def,axiom,
( l
= ( diag_mat_complex @ a ) ) ).
% l_def
thf(fact_6_ne,axiom,
( ( commut93809757773076895omplex @ ( diag_mat_complex @ a ) )
!= nil_nat ) ).
% ne
thf(fact_7__092_060open_062p_A_060_Aeq__comps_A_Idiag__mat_AA_J_A_B_Ai_092_060close_062,axiom,
ord_less_nat @ p @ ( nth_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ a ) ) @ i ) ).
% \<open>p < eq_comps (diag_mat A) ! i\<close>
thf(fact_8_eq__comps__sum__list,axiom,
! [L: list_P6011104703257516679at_nat] :
( ( groups4561878855575611511st_nat @ ( commut2990393512377091280at_nat @ L ) )
= ( size_s5460976970255530739at_nat @ L ) ) ).
% eq_comps_sum_list
thf(fact_9_eq__comps__sum__list,axiom,
! [L: list_P6605091754902497125omplex] :
( ( groups4561878855575611511st_nat @ ( commut937718278427031726omplex @ L ) )
= ( size_s3423402466807558097omplex @ L ) ) ).
% eq_comps_sum_list
thf(fact_10_eq__comps__sum__list,axiom,
! [L: list_mat_complex] :
( ( groups4561878855575611511st_nat @ ( commut5736191610077499254omplex @ L ) )
= ( size_s5969786470865220249omplex @ L ) ) ).
% eq_comps_sum_list
thf(fact_11_eq__comps__sum__list,axiom,
! [L: list_nat] :
( ( groups4561878855575611511st_nat @ ( commut2436974278740741825ps_nat @ L ) )
= ( size_size_list_nat @ L ) ) ).
% eq_comps_sum_list
thf(fact_12_eq__comps__sum__list,axiom,
! [L: list_complex] :
( ( groups4561878855575611511st_nat @ ( commut93809757773076895omplex @ L ) )
= ( size_s3451745648224563538omplex @ L ) ) ).
% eq_comps_sum_list
thf(fact_13_eq__comps__length,axiom,
! [L: list_P6011104703257516679at_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( commut2990393512377091280at_nat @ L ) ) @ ( size_s5460976970255530739at_nat @ L ) ) ).
% eq_comps_length
thf(fact_14_eq__comps__length,axiom,
! [L: list_P6605091754902497125omplex] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( commut937718278427031726omplex @ L ) ) @ ( size_s3423402466807558097omplex @ L ) ) ).
% eq_comps_length
thf(fact_15_eq__comps__length,axiom,
! [L: list_mat_complex] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( commut5736191610077499254omplex @ L ) ) @ ( size_s5969786470865220249omplex @ L ) ) ).
% eq_comps_length
thf(fact_16_eq__comps__length,axiom,
! [L: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( commut2436974278740741825ps_nat @ L ) ) @ ( size_size_list_nat @ L ) ) ).
% eq_comps_length
thf(fact_17_eq__comps__length,axiom,
! [L: list_complex] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( commut93809757773076895omplex @ L ) ) @ ( size_s3451745648224563538omplex @ L ) ) ).
% eq_comps_length
thf(fact_18_eq__comp__sum__diag__mat,axiom,
! [A: mat_mat_complex] :
( ( groups4561878855575611511st_nat @ ( commut5736191610077499254omplex @ ( diag_mat_mat_complex @ A ) ) )
= ( dim_row_mat_complex @ A ) ) ).
% eq_comp_sum_diag_mat
thf(fact_19_eq__comp__sum__diag__mat,axiom,
! [A: mat_complex] :
( ( groups4561878855575611511st_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A ) ) )
= ( dim_row_complex @ A ) ) ).
% eq_comp_sum_diag_mat
thf(fact_20_eq__comp__sum__diag__mat,axiom,
! [A: mat_nat] :
( ( groups4561878855575611511st_nat @ ( commut2436974278740741825ps_nat @ ( diag_mat_nat @ A ) ) )
= ( dim_row_nat @ A ) ) ).
% eq_comp_sum_diag_mat
thf(fact_21_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_22_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_23_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_24_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_25_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_26_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_27_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_28_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_29_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_30_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_31_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_32_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_33_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_34_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_35_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_36_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_37_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_38_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_39_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_40_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_41_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_42_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_43_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_44_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_45_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_46_size__neq__size__imp__neq,axiom,
! [X2: list_mat_complex,Y3: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ X2 )
!= ( size_s5969786470865220249omplex @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_47_size__neq__size__imp__neq,axiom,
! [X2: list_P6011104703257516679at_nat,Y3: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ X2 )
!= ( size_s5460976970255530739at_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_48_size__neq__size__imp__neq,axiom,
! [X2: list_P6605091754902497125omplex,Y3: list_P6605091754902497125omplex] :
( ( ( size_s3423402466807558097omplex @ X2 )
!= ( size_s3423402466807558097omplex @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_49_size__neq__size__imp__neq,axiom,
! [X2: list_complex,Y3: list_complex] :
( ( ( size_s3451745648224563538omplex @ X2 )
!= ( size_s3451745648224563538omplex @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_50_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_51_eq__comps__not__empty,axiom,
! [L: list_P6011104703257516679at_nat] :
( ( L != nil_Pr5478986624290739719at_nat )
=> ( ( commut2990393512377091280at_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_52_eq__comps__not__empty,axiom,
! [L: list_list_nat] :
( ( L != nil_list_nat )
=> ( ( commut9114419477716286801st_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_53_eq__comps__not__empty,axiom,
! [L: list_mat_complex] :
( ( L != nil_mat_complex )
=> ( ( commut5736191610077499254omplex @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_54_eq__comps__not__empty,axiom,
! [L: list_complex] :
( ( L != nil_complex )
=> ( ( commut93809757773076895omplex @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_55_eq__comps__not__empty,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ( commut2436974278740741825ps_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_56_eq__comps__empty__if,axiom,
! [L: list_P6011104703257516679at_nat] :
( ( ( commut2990393512377091280at_nat @ L )
= nil_nat )
=> ( L = nil_Pr5478986624290739719at_nat ) ) ).
% eq_comps_empty_if
thf(fact_57_eq__comps__empty__if,axiom,
! [L: list_list_nat] :
( ( ( commut9114419477716286801st_nat @ L )
= nil_nat )
=> ( L = nil_list_nat ) ) ).
% eq_comps_empty_if
thf(fact_58_eq__comps__empty__if,axiom,
! [L: list_mat_complex] :
( ( ( commut5736191610077499254omplex @ L )
= nil_nat )
=> ( L = nil_mat_complex ) ) ).
% eq_comps_empty_if
thf(fact_59_eq__comps__empty__if,axiom,
! [L: list_complex] :
( ( ( commut93809757773076895omplex @ L )
= nil_nat )
=> ( L = nil_complex ) ) ).
% eq_comps_empty_if
thf(fact_60_eq__comps__empty__if,axiom,
! [L: list_nat] :
( ( ( commut2436974278740741825ps_nat @ L )
= nil_nat )
=> ( L = nil_nat ) ) ).
% eq_comps_empty_if
thf(fact_61_eq__comps_Osimps_I1_J,axiom,
( ( commut2990393512377091280at_nat @ nil_Pr5478986624290739719at_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_62_eq__comps_Osimps_I1_J,axiom,
( ( commut9114419477716286801st_nat @ nil_list_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_63_eq__comps_Osimps_I1_J,axiom,
( ( commut5736191610077499254omplex @ nil_mat_complex )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_64_eq__comps_Osimps_I1_J,axiom,
( ( commut93809757773076895omplex @ nil_complex )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_65_eq__comps_Osimps_I1_J,axiom,
( ( commut2436974278740741825ps_nat @ nil_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_66_mem__Collect__eq,axiom,
! [A2: produc1631295542841207645omplex,P: produc1631295542841207645omplex > $o] :
( ( member6068360845309590790omplex @ A2 @ ( collec2134092743982711368omplex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A2: produc1828647624359046049st_nat,P: produc1828647624359046049st_nat > $o] :
( ( member7340969449405702474st_nat @ A2 @ ( collec1570431334306492044st_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A2: produc8199716216217303280at_nat,P: produc8199716216217303280at_nat > $o] :
( ( member7226740684066999833at_nat @ A2 @ ( collec9201399625632817755at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A2: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A2 @ ( collect_mat_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_Pr135664057150635453omplex] :
( ( collec2134092743982711368omplex
@ ^ [X4: produc1631295542841207645omplex] : ( member6068360845309590790omplex @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_Pr3451248702717554689st_nat] :
( ( collec1570431334306492044st_nat
@ ^ [X4: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ( collec9201399625632817755at_nat
@ ^ [X4: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X4: mat_complex] : ( member_mat_complex @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_78_Collect__cong,axiom,
! [P: mat_complex > $o,Q: mat_complex > $o] :
( ! [X: mat_complex] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_mat_complex @ P )
= ( collect_mat_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_79_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_80_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_81_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_82_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_83_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_84_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_85_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_86_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_87_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_88_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_89_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_90_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_91_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_92_sum__list__geq__0,axiom,
! [L: list_complex] :
( ( L != nil_complex )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( nth_complex @ L @ J2 ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( groups486868518411355989omplex @ L ) ) ) ) ).
% sum_list_geq_0
thf(fact_93_sum__list__geq__0,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% sum_list_geq_0
thf(fact_94_sum__list__mono2,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ord_less_eq_complex @ ( nth_complex @ Xs @ I3 ) @ ( nth_complex @ Ys @ I3 ) ) )
=> ( ord_less_eq_complex @ ( groups486868518411355989omplex @ Xs ) @ ( groups486868518411355989omplex @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_95_sum__list__mono2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_96_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_97_sum__list__cong,axiom,
! [L: list_P6011104703257516679at_nat,M: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ L )
= ( size_s5460976970255530739at_nat @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ L ) )
=> ( ( nth_Pr7617993195940197384at_nat @ L @ I3 )
= ( nth_Pr7617993195940197384at_nat @ M @ I3 ) ) )
=> ( ( groups4206474380581351322at_nat @ L )
= ( groups4206474380581351322at_nat @ M ) ) ) ) ).
% sum_list_cong
thf(fact_98_sum__list__cong,axiom,
! [L: list_P6605091754902497125omplex,M: list_P6605091754902497125omplex] :
( ( ( size_s3423402466807558097omplex @ L )
= ( size_s3423402466807558097omplex @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3423402466807558097omplex @ L ) )
=> ( ( nth_Pr9135773202038487014omplex @ L @ I3 )
= ( nth_Pr9135773202038487014omplex @ M @ I3 ) ) )
=> ( ( groups6234704736776048504omplex @ L )
= ( groups6234704736776048504omplex @ M ) ) ) ) ).
% sum_list_cong
thf(fact_99_sum__list__cong,axiom,
! [L: list_complex,M: list_complex] :
( ( ( size_s3451745648224563538omplex @ L )
= ( size_s3451745648224563538omplex @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ( nth_complex @ L @ I3 )
= ( nth_complex @ M @ I3 ) ) )
=> ( ( groups486868518411355989omplex @ L )
= ( groups486868518411355989omplex @ M ) ) ) ) ).
% sum_list_cong
thf(fact_100_sum__list__cong,axiom,
! [L: list_nat,M: list_nat] :
( ( ( size_size_list_nat @ L )
= ( size_size_list_nat @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
=> ( ( nth_nat @ L @ I3 )
= ( nth_nat @ M @ I3 ) ) )
=> ( ( groups4561878855575611511st_nat @ L )
= ( groups4561878855575611511st_nat @ M ) ) ) ) ).
% sum_list_cong
thf(fact_101_length__greater__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) )
= ( Xs != nil_list_nat ) ) ).
% length_greater_0_conv
thf(fact_102_length__greater__0__conv,axiom,
! [Xs: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Xs ) )
= ( Xs != nil_mat_complex ) ) ).
% length_greater_0_conv
thf(fact_103_length__greater__0__conv,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) )
= ( Xs != nil_Pr5478986624290739719at_nat ) ) ).
% length_greater_0_conv
thf(fact_104_length__greater__0__conv,axiom,
! [Xs: list_P6605091754902497125omplex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3423402466807558097omplex @ Xs ) )
= ( Xs != nil_Pr910363667030563813omplex ) ) ).
% length_greater_0_conv
thf(fact_105_length__greater__0__conv,axiom,
! [Xs: list_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) )
= ( Xs != nil_complex ) ) ).
% length_greater_0_conv
thf(fact_106_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_107_diag__mat__length,axiom,
! [A: mat_mat_complex] :
( ( size_s5969786470865220249omplex @ ( diag_mat_mat_complex @ A ) )
= ( dim_row_mat_complex @ A ) ) ).
% diag_mat_length
thf(fact_108_diag__mat__length,axiom,
! [A: mat_Pr3994417008679617630at_nat] :
( ( size_s5460976970255530739at_nat @ ( diag_m245791634876247544at_nat @ A ) )
= ( dim_ro1249899285275649537at_nat @ A ) ) ).
% diag_mat_length
thf(fact_109_diag__mat__length,axiom,
! [A: mat_Pr6529108596446733116omplex] :
( ( size_s3423402466807558097omplex @ ( diag_m2992240987531481558omplex @ A ) )
= ( dim_ro7373274333410768863omplex @ A ) ) ).
% diag_mat_length
thf(fact_110_diag__mat__length,axiom,
! [A: mat_complex] :
( ( size_s3451745648224563538omplex @ ( diag_mat_complex @ A ) )
= ( dim_row_complex @ A ) ) ).
% diag_mat_length
thf(fact_111_diag__mat__length,axiom,
! [A: mat_nat] :
( ( size_size_list_nat @ ( diag_mat_nat @ A ) )
= ( dim_row_nat @ A ) ) ).
% diag_mat_length
thf(fact_112_nth__equalityI,axiom,
! [Xs: list_P1999334753057444956omplex,Ys: list_P1999334753057444956omplex] :
( ( ( size_s2777191050033188080omplex @ Xs )
= ( size_s2777191050033188080omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s2777191050033188080omplex @ Xs ) )
=> ( ( nth_Pr7800990439711681285omplex @ Xs @ I3 )
= ( nth_Pr7800990439711681285omplex @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_113_nth__equalityI,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
= ( nth_Pr7617993195940197384at_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_114_nth__equalityI,axiom,
! [Xs: list_P6605091754902497125omplex,Ys: list_P6605091754902497125omplex] :
( ( ( size_s3423402466807558097omplex @ Xs )
= ( size_s3423402466807558097omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3423402466807558097omplex @ Xs ) )
=> ( ( nth_Pr9135773202038487014omplex @ Xs @ I3 )
= ( nth_Pr9135773202038487014omplex @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_115_nth__equalityI,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ Xs @ I3 )
= ( nth_mat_complex @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_116_nth__equalityI,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( nth_complex @ Xs @ I3 )
= ( nth_complex @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_117_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_118_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc3259542890344722124omplex > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: produc3259542890344722124omplex] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs2: list_P1999334753057444956omplex] :
( ( ( size_s2777191050033188080omplex @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_Pr7800990439711681285omplex @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_119_Skolem__list__nth,axiom,
! [K: nat,P: nat > product_prod_nat_nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: product_prod_nat_nat] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_120_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc4863162743050822367omplex > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: produc4863162743050822367omplex] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs2: list_P6605091754902497125omplex] :
( ( ( size_s3423402466807558097omplex @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_Pr9135773202038487014omplex @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_121_Skolem__list__nth,axiom,
! [K: nat,P: nat > mat_complex > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: mat_complex] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_mat_complex @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_122_Skolem__list__nth,axiom,
! [K: nat,P: nat > complex > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: complex] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_complex @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_123_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: nat] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_124_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P1999334753057444956omplex,Z: list_P1999334753057444956omplex] : ( Y4 = Z ) )
= ( ^ [Xs2: list_P1999334753057444956omplex,Ys2: list_P1999334753057444956omplex] :
( ( ( size_s2777191050033188080omplex @ Xs2 )
= ( size_s2777191050033188080omplex @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s2777191050033188080omplex @ Xs2 ) )
=> ( ( nth_Pr7800990439711681285omplex @ Xs2 @ I4 )
= ( nth_Pr7800990439711681285omplex @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_125_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P6011104703257516679at_nat,Z: list_P6011104703257516679at_nat] : ( Y4 = Z ) )
= ( ^ [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 )
= ( nth_Pr7617993195940197384at_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_126_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P6605091754902497125omplex,Z: list_P6605091754902497125omplex] : ( Y4 = Z ) )
= ( ^ [Xs2: list_P6605091754902497125omplex,Ys2: list_P6605091754902497125omplex] :
( ( ( size_s3423402466807558097omplex @ Xs2 )
= ( size_s3423402466807558097omplex @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3423402466807558097omplex @ Xs2 ) )
=> ( ( nth_Pr9135773202038487014omplex @ Xs2 @ I4 )
= ( nth_Pr9135773202038487014omplex @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_127_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_mat_complex,Z: list_mat_complex] : ( Y4 = Z ) )
= ( ^ [Xs2: list_mat_complex,Ys2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs2 )
= ( size_s5969786470865220249omplex @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( ( nth_mat_complex @ Xs2 @ I4 )
= ( nth_mat_complex @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_128_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_complex,Z: list_complex] : ( Y4 = Z ) )
= ( ^ [Xs2: list_complex,Ys2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( ( nth_complex @ Xs2 @ I4 )
= ( nth_complex @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_129_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I4 )
= ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_130_sum__list_ONil,axiom,
( ( groups4206474380581351322at_nat @ nil_Pr5478986624290739719at_nat )
= zero_z3979849011205770936at_nat ) ).
% sum_list.Nil
thf(fact_131_sum__list_ONil,axiom,
( ( groups486868518411355989omplex @ nil_complex )
= zero_zero_complex ) ).
% sum_list.Nil
thf(fact_132_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_133_neq__if__length__neq,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
!= ( size_s5969786470865220249omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_134_neq__if__length__neq,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
!= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_135_neq__if__length__neq,axiom,
! [Xs: list_P6605091754902497125omplex,Ys: list_P6605091754902497125omplex] :
( ( ( size_s3423402466807558097omplex @ Xs )
!= ( size_s3423402466807558097omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_136_neq__if__length__neq,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
!= ( size_s3451745648224563538omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_137_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_138_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_mat_complex] :
( ( size_s5969786470865220249omplex @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_139_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P6011104703257516679at_nat] :
( ( size_s5460976970255530739at_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_140_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P6605091754902497125omplex] :
( ( size_s3423402466807558097omplex @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_141_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_complex] :
( ( size_s3451745648224563538omplex @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_142_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_143_length__induct,axiom,
! [P: list_mat_complex > $o,Xs: list_mat_complex] :
( ! [Xs3: list_mat_complex] :
( ! [Ys3: list_mat_complex] :
( ( ord_less_nat @ ( size_s5969786470865220249omplex @ Ys3 ) @ ( size_s5969786470865220249omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_144_length__induct,axiom,
! [P: list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ! [Xs3: list_P6011104703257516679at_nat] :
( ! [Ys3: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ys3 ) @ ( size_s5460976970255530739at_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_145_length__induct,axiom,
! [P: list_P6605091754902497125omplex > $o,Xs: list_P6605091754902497125omplex] :
( ! [Xs3: list_P6605091754902497125omplex] :
( ! [Ys3: list_P6605091754902497125omplex] :
( ( ord_less_nat @ ( size_s3423402466807558097omplex @ Ys3 ) @ ( size_s3423402466807558097omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_146_length__induct,axiom,
! [P: list_complex > $o,Xs: list_complex] :
( ! [Xs3: list_complex] :
( ! [Ys3: list_complex] :
( ( ord_less_nat @ ( size_s3451745648224563538omplex @ Ys3 ) @ ( size_s3451745648224563538omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_147_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_148_length__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_nat ) ) ).
% length_0_conv
thf(fact_149_length__0__conv,axiom,
! [Xs: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_mat_complex ) ) ).
% length_0_conv
thf(fact_150_length__0__conv,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% length_0_conv
thf(fact_151_length__0__conv,axiom,
! [Xs: list_P6605091754902497125omplex] :
( ( ( size_s3423402466807558097omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr910363667030563813omplex ) ) ).
% length_0_conv
thf(fact_152_length__0__conv,axiom,
! [Xs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_complex ) ) ).
% length_0_conv
thf(fact_153_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_154_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_155_list_Osize_I3_J,axiom,
( ( size_s5969786470865220249omplex @ nil_mat_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_156_list_Osize_I3_J,axiom,
( ( size_s5460976970255530739at_nat @ nil_Pr5478986624290739719at_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_157_list_Osize_I3_J,axiom,
( ( size_s3423402466807558097omplex @ nil_Pr910363667030563813omplex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_158_list_Osize_I3_J,axiom,
( ( size_s3451745648224563538omplex @ nil_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_159_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_160_k__def,axiom,
( k
= ( nth_complex @ l @ ( commut2019222099004354946um_nat @ i @ ( commut93809757773076895omplex @ l ) ) ) ) ).
% k_def
thf(fact_161_extract__subdiags__diagonal,axiom,
! [B2: mat_complex,N: nat,L: list_nat,I: nat] :
( ( diagonal_mat_complex @ B2 )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( L != nil_nat )
=> ( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ L ) @ N )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
=> ( diagonal_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ B2 @ L ) @ I ) ) ) ) ) ) ) ).
% extract_subdiags_diagonal
thf(fact_162_n__sum__sum__list,axiom,
! [I: nat,L: list_complex] :
( ( ord_less_eq_nat @ I @ ( size_s3451745648224563538omplex @ L ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( nth_complex @ L @ J2 ) ) )
=> ( ord_less_eq_complex @ ( commut6323218633641605728omplex @ I @ L ) @ ( groups486868518411355989omplex @ L ) ) ) ) ).
% n_sum_sum_list
thf(fact_163_n__sum__sum__list,axiom,
! [I: nat,L: list_nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ L ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J2 ) ) )
=> ( ord_less_eq_nat @ ( commut2019222099004354946um_nat @ I @ L ) @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% n_sum_sum_list
thf(fact_164_carrier__matD_I1_J,axiom,
! [A: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( dim_row_nat @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_165_carrier__matD_I1_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( dim_row_complex @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_166_extract__subdiags__carrier,axiom,
! [I: nat,L: list_nat,B2: mat_complex] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
=> ( member_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ B2 @ L ) @ I ) @ ( carrier_mat_complex @ ( nth_nat @ L @ I ) @ ( nth_nat @ L @ I ) ) ) ) ).
% extract_subdiags_carrier
thf(fact_167_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_168_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( F @ K2 @ I2 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_169_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_170_extract__subdiags_Osimps_I1_J,axiom,
! [B2: mat_complex] :
( ( commut6900707758132580272omplex @ B2 @ nil_nat )
= nil_mat_complex ) ).
% extract_subdiags.simps(1)
thf(fact_171_extract__subdiags__length,axiom,
! [B2: mat_complex,L: list_nat] :
( ( size_s5969786470865220249omplex @ ( commut6900707758132580272omplex @ B2 @ L ) )
= ( size_size_list_nat @ L ) ) ).
% extract_subdiags_length
thf(fact_172_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_173_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_174_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_175_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_176_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_177_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_178_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_179_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_180_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_181_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_182_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_183_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_184_n__sum_Osimps_I1_J,axiom,
! [L: list_P6011104703257516679at_nat] :
( ( commut2293368841785035983at_nat @ zero_zero_nat @ L )
= zero_z3979849011205770936at_nat ) ).
% n_sum.simps(1)
thf(fact_185_n__sum_Osimps_I1_J,axiom,
! [L: list_complex] :
( ( commut6323218633641605728omplex @ zero_zero_nat @ L )
= zero_zero_complex ) ).
% n_sum.simps(1)
thf(fact_186_n__sum_Osimps_I1_J,axiom,
! [L: list_nat] :
( ( commut2019222099004354946um_nat @ zero_zero_nat @ L )
= zero_zero_nat ) ).
% n_sum.simps(1)
thf(fact_187_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_188_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_189_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_190_undef__vec__def,axiom,
( undef_7074962713475498729omplex
= ( nth_Pr9135773202038487014omplex @ nil_Pr910363667030563813omplex ) ) ).
% undef_vec_def
thf(fact_191_undef__vec__def,axiom,
( undef_6336265055795355202omplex
= ( nth_Pr7800990439711681285omplex @ nil_Pr6446730170661106246omplex ) ) ).
% undef_vec_def
thf(fact_192_undef__vec__def,axiom,
( undef_7626143578040714507at_nat
= ( nth_Pr7617993195940197384at_nat @ nil_Pr5478986624290739719at_nat ) ) ).
% undef_vec_def
thf(fact_193_undef__vec__def,axiom,
( undef_vec_list_nat
= ( nth_list_nat @ nil_list_nat ) ) ).
% undef_vec_def
thf(fact_194_undef__vec__def,axiom,
( undef_vec_complex
= ( nth_complex @ nil_complex ) ) ).
% undef_vec_def
thf(fact_195_undef__vec__def,axiom,
( undef_2495355514574404529omplex
= ( nth_mat_complex @ nil_mat_complex ) ) ).
% undef_vec_def
thf(fact_196_undef__vec__def,axiom,
( undef_vec_nat
= ( nth_nat @ nil_nat ) ) ).
% undef_vec_def
thf(fact_197_eq__comps__elems__eq,axiom,
! [L: list_P6605091754902497125omplex,I: nat,J: nat] :
( ( L != nil_Pr910363667030563813omplex )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut937718278427031726omplex @ L ) ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ ( commut937718278427031726omplex @ L ) @ I ) )
=> ( ( nth_Pr9135773202038487014omplex @ L @ ( commut2019222099004354946um_nat @ I @ ( commut937718278427031726omplex @ L ) ) )
= ( nth_Pr9135773202038487014omplex @ L @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ ( commut937718278427031726omplex @ L ) ) @ J ) ) ) ) ) ) ).
% eq_comps_elems_eq
thf(fact_198_eq__comps__elems__eq,axiom,
! [L: list_P1999334753057444956omplex,I: nat,J: nat] :
( ( L != nil_Pr6446730170661106246omplex )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut2964314653314687037omplex @ L ) ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ ( commut2964314653314687037omplex @ L ) @ I ) )
=> ( ( nth_Pr7800990439711681285omplex @ L @ ( commut2019222099004354946um_nat @ I @ ( commut2964314653314687037omplex @ L ) ) )
= ( nth_Pr7800990439711681285omplex @ L @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ ( commut2964314653314687037omplex @ L ) ) @ J ) ) ) ) ) ) ).
% eq_comps_elems_eq
thf(fact_199_eq__comps__elems__eq,axiom,
! [L: list_P6011104703257516679at_nat,I: nat,J: nat] :
( ( L != nil_Pr5478986624290739719at_nat )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut2990393512377091280at_nat @ L ) ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ ( commut2990393512377091280at_nat @ L ) @ I ) )
=> ( ( nth_Pr7617993195940197384at_nat @ L @ ( commut2019222099004354946um_nat @ I @ ( commut2990393512377091280at_nat @ L ) ) )
= ( nth_Pr7617993195940197384at_nat @ L @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ ( commut2990393512377091280at_nat @ L ) ) @ J ) ) ) ) ) ) ).
% eq_comps_elems_eq
thf(fact_200_eq__comps__elems__eq,axiom,
! [L: list_list_nat,I: nat,J: nat] :
( ( L != nil_list_nat )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut9114419477716286801st_nat @ L ) ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ ( commut9114419477716286801st_nat @ L ) @ I ) )
=> ( ( nth_list_nat @ L @ ( commut2019222099004354946um_nat @ I @ ( commut9114419477716286801st_nat @ L ) ) )
= ( nth_list_nat @ L @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ ( commut9114419477716286801st_nat @ L ) ) @ J ) ) ) ) ) ) ).
% eq_comps_elems_eq
thf(fact_201_eq__comps__elems__eq,axiom,
! [L: list_mat_complex,I: nat,J: nat] :
( ( L != nil_mat_complex )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut5736191610077499254omplex @ L ) ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ ( commut5736191610077499254omplex @ L ) @ I ) )
=> ( ( nth_mat_complex @ L @ ( commut2019222099004354946um_nat @ I @ ( commut5736191610077499254omplex @ L ) ) )
= ( nth_mat_complex @ L @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ ( commut5736191610077499254omplex @ L ) ) @ J ) ) ) ) ) ) ).
% eq_comps_elems_eq
thf(fact_202_eq__comps__elems__eq,axiom,
! [L: list_complex,I: nat,J: nat] :
( ( L != nil_complex )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut93809757773076895omplex @ L ) ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ ( commut93809757773076895omplex @ L ) @ I ) )
=> ( ( nth_complex @ L @ ( commut2019222099004354946um_nat @ I @ ( commut93809757773076895omplex @ L ) ) )
= ( nth_complex @ L @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ ( commut93809757773076895omplex @ L ) ) @ J ) ) ) ) ) ) ).
% eq_comps_elems_eq
thf(fact_203_eq__comps__elems__eq,axiom,
! [L: list_nat,I: nat,J: nat] :
( ( L != nil_nat )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut2436974278740741825ps_nat @ L ) ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ ( commut2436974278740741825ps_nat @ L ) @ I ) )
=> ( ( nth_nat @ L @ ( commut2019222099004354946um_nat @ I @ ( commut2436974278740741825ps_nat @ L ) ) )
= ( nth_nat @ L @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ ( commut2436974278740741825ps_nat @ L ) ) @ J ) ) ) ) ) ) ).
% eq_comps_elems_eq
thf(fact_204_order__le__imp__less__or__eq,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_eq_complex @ X2 @ Y3 )
=> ( ( ord_less_complex @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_205_order__le__imp__less__or__eq,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
=> ( ( ord_le7866589430770878221at_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_206_order__le__imp__less__or__eq,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X2 @ Y3 )
=> ( ( ord_le4845169857901429244at_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_207_order__le__imp__less__or__eq,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y3 )
=> ( ( ord_le5598786136212072115omplex @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_208_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_209_order__le__imp__less__or__eq,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y3 )
=> ( ( ord_less_nat_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_210_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_211_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_212_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_213_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_complex @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_214_order__less__le__subst2,axiom,
! [A2: complex,B: complex,F: complex > complex,C: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_eq_complex @ ( F @ B ) @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_215_order__less__le__subst2,axiom,
! [A2: complex,B: complex,F: complex > nat,C: nat] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_216_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_217_order__less__le__subst2,axiom,
! [A2: complex,B: complex,F: complex > set_nat,C: set_nat] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_218_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_mat_complex,C: set_mat_complex] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le3632134057777142183omplex @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5598786136212072115omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5598786136212072115omplex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_219_order__less__le__subst2,axiom,
! [A2: complex,B: complex,F: complex > set_mat_complex,C: set_mat_complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_le3632134057777142183omplex @ ( F @ B ) @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_le5598786136212072115omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5598786136212072115omplex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_220_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_221_order__less__le__subst2,axiom,
! [A2: complex,B: complex,F: complex > nat > nat,C: nat > nat] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_eq_nat_nat @ ( F @ B ) @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_222_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_223_order__less__le__subst1,axiom,
! [A2: complex,F: nat > complex,B: nat,C: nat] :
( ( ord_less_complex @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_224_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_225_order__less__le__subst1,axiom,
! [A2: complex,F: set_nat > complex,B: set_nat,C: set_nat] :
( ( ord_less_complex @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_226_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_227_order__less__le__subst1,axiom,
! [A2: set_mat_complex,F: nat > set_mat_complex,B: nat,C: nat] :
( ( ord_le5598786136212072115omplex @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3632134057777142183omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5598786136212072115omplex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_228_order__less__le__subst1,axiom,
! [A2: nat > nat,F: nat > nat > nat,B: nat,C: nat] :
( ( ord_less_nat_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_229_order__less__le__subst1,axiom,
! [A2: complex,F: set_mat_complex > complex,B: set_mat_complex,C: set_mat_complex] :
( ( ord_less_complex @ A2 @ ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_230_order__less__le__subst1,axiom,
! [A2: nat,F: set_mat_complex > nat,B: set_mat_complex,C: set_mat_complex] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_231_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_232_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_233_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_complex @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_234_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_235_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > complex,C: complex] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_complex @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_236_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_237_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_mat_complex,C: set_mat_complex] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le5598786136212072115omplex @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3632134057777142183omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5598786136212072115omplex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_238_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_239_order__le__less__subst2,axiom,
! [A2: set_mat_complex,B: set_mat_complex,F: set_mat_complex > complex,C: complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( ord_less_complex @ ( F @ B ) @ C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_240_order__le__less__subst2,axiom,
! [A2: set_mat_complex,B: set_mat_complex,F: set_mat_complex > nat,C: nat] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_241_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_242_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_243_order__le__less__subst1,axiom,
! [A2: complex,F: nat > complex,B: nat,C: nat] :
( ( ord_less_eq_complex @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_244_order__le__less__subst1,axiom,
! [A2: complex,F: complex > complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A2 @ ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_245_order__le__less__subst1,axiom,
! [A2: nat,F: complex > nat,B: complex,C: complex] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_246_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_247_order__le__less__subst1,axiom,
! [A2: set_nat,F: complex > set_nat,B: complex,C: complex] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_248_order__le__less__subst1,axiom,
! [A2: set_mat_complex,F: nat > set_mat_complex,B: nat,C: nat] :
( ( ord_le3632134057777142183omplex @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5598786136212072115omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5598786136212072115omplex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_249_order__le__less__subst1,axiom,
! [A2: set_mat_complex,F: complex > set_mat_complex,B: complex,C: complex] :
( ( ord_le3632134057777142183omplex @ A2 @ ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_le5598786136212072115omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5598786136212072115omplex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_250_order__le__less__subst1,axiom,
! [A2: nat > nat,F: nat > nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_251_order__le__less__subst1,axiom,
! [A2: nat > nat,F: complex > nat > nat,B: complex,C: complex] :
( ( ord_less_eq_nat_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_252_order__less__le__trans,axiom,
! [X2: complex,Y3: complex,Z4: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( ( ord_less_eq_complex @ Y3 @ Z4 )
=> ( ord_less_complex @ X2 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_253_order__less__le__trans,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ X2 @ Y3 )
=> ( ( ord_le3146513528884898305at_nat @ Y3 @ Z4 )
=> ( ord_le7866589430770878221at_nat @ X2 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_254_order__less__le__trans,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z4: set_Pr9093778441882193744at_nat] :
( ( ord_le4845169857901429244at_nat @ X2 @ Y3 )
=> ( ( ord_le3678578370064672496at_nat @ Y3 @ Z4 )
=> ( ord_le4845169857901429244at_nat @ X2 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_255_order__less__le__trans,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex,Z4: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ X2 @ Y3 )
=> ( ( ord_le3632134057777142183omplex @ Y3 @ Z4 )
=> ( ord_le5598786136212072115omplex @ X2 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_256_order__less__le__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z4 )
=> ( ord_less_set_nat @ X2 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_257_order__less__le__trans,axiom,
! [X2: nat > nat,Y3: nat > nat,Z4: nat > nat] :
( ( ord_less_nat_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat_nat @ Y3 @ Z4 )
=> ( ord_less_nat_nat @ X2 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_258_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z4: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z4 )
=> ( ord_less_nat @ X2 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_259_nat__arith_Oadd1,axiom,
! [A: product_prod_nat_nat,K: product_prod_nat_nat,A2: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( A
= ( plus_p9057090461656269880at_nat @ K @ A2 ) )
=> ( ( plus_p9057090461656269880at_nat @ A @ B )
= ( plus_p9057090461656269880at_nat @ K @ ( plus_p9057090461656269880at_nat @ A2 @ B ) ) ) ) ).
% nat_arith.add1
thf(fact_260_nat__arith_Oadd1,axiom,
! [A: complex,K: complex,A2: complex,B: complex] :
( ( A
= ( plus_plus_complex @ K @ A2 ) )
=> ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A2 @ B ) ) ) ) ).
% nat_arith.add1
thf(fact_261_nat__arith_Oadd1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( plus_plus_set_nat @ K @ A2 ) )
=> ( ( plus_plus_set_nat @ A @ B )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A2 @ B ) ) ) ) ).
% nat_arith.add1
thf(fact_262_nat__arith_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% nat_arith.add1
thf(fact_263_nat__arith_Oadd2,axiom,
! [B2: product_prod_nat_nat,K: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat] :
( ( B2
= ( plus_p9057090461656269880at_nat @ K @ B ) )
=> ( ( plus_p9057090461656269880at_nat @ A2 @ B2 )
= ( plus_p9057090461656269880at_nat @ K @ ( plus_p9057090461656269880at_nat @ A2 @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_264_nat__arith_Oadd2,axiom,
! [B2: complex,K: complex,B: complex,A2: complex] :
( ( B2
= ( plus_plus_complex @ K @ B ) )
=> ( ( plus_plus_complex @ A2 @ B2 )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A2 @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_265_nat__arith_Oadd2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B2
= ( plus_plus_set_nat @ K @ B ) )
=> ( ( plus_plus_set_nat @ A2 @ B2 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A2 @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_266_nat__arith_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_267_nat__arith_Orule0,axiom,
! [A2: set_nat] :
( A2
= ( plus_plus_set_nat @ A2 @ zero_zero_set_nat ) ) ).
% nat_arith.rule0
thf(fact_268_nat__arith_Orule0,axiom,
! [A2: complex] :
( A2
= ( plus_plus_complex @ A2 @ zero_zero_complex ) ) ).
% nat_arith.rule0
thf(fact_269_nat__arith_Orule0,axiom,
! [A2: product_prod_nat_nat] :
( A2
= ( plus_p9057090461656269880at_nat @ A2 @ zero_z3979849011205770936at_nat ) ) ).
% nat_arith.rule0
thf(fact_270_nat__arith_Orule0,axiom,
! [A2: nat] :
( A2
= ( plus_plus_nat @ A2 @ zero_zero_nat ) ) ).
% nat_arith.rule0
thf(fact_271_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_272_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_273_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_274_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_275_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_276_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_277_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_278_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_279_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_280_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_281_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_282_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_283_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_284_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_285_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_286_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_287_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_288_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_289_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_290_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_291_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_292_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_293_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_294_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M5 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_295_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_296_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_297_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_298_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_299_order__trans__rules_I26_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( A2 = B )
=> ( ( ord_le3146513528884898305at_nat @ B @ C )
=> ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_300_order__trans__rules_I26_J,axiom,
! [A2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( A2 = B )
=> ( ( ord_le3678578370064672496at_nat @ B @ C )
=> ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_301_order__trans__rules_I26_J,axiom,
! [A2: set_mat_complex,B: set_mat_complex,C: set_mat_complex] :
( ( A2 = B )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ord_le3632134057777142183omplex @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_302_order__trans__rules_I26_J,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_303_order__trans__rules_I26_J,axiom,
! [A2: nat > nat,B: nat > nat,C: nat > nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat_nat @ B @ C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_304_order__trans__rules_I26_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_305_order__trans__rules_I25_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_306_order__trans__rules_I25_J,axiom,
! [A2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_307_order__trans__rules_I25_J,axiom,
! [A2: set_mat_complex,B: set_mat_complex,C: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3632134057777142183omplex @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_308_order__trans__rules_I25_J,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_309_order__trans__rules_I25_J,axiom,
! [A2: nat > nat,B: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_310_order__trans__rules_I25_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_311_order__trans__rules_I24_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ( ord_le3146513528884898305at_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_trans_rules(24)
thf(fact_312_order__trans__rules_I24_J,axiom,
! [A2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B )
=> ( ( ord_le3678578370064672496at_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_trans_rules(24)
thf(fact_313_order__trans__rules_I24_J,axiom,
! [A2: set_mat_complex,B: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( ord_le3632134057777142183omplex @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_trans_rules(24)
thf(fact_314_order__trans__rules_I24_J,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_trans_rules(24)
thf(fact_315_order__trans__rules_I24_J,axiom,
! [A2: nat > nat,B: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( ( ord_less_eq_nat_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_trans_rules(24)
thf(fact_316_order__trans__rules_I24_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_trans_rules(24)
thf(fact_317_order__trans__rules_I23_J,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
=> ( ( ord_le3146513528884898305at_nat @ Y3 @ Z4 )
=> ( ord_le3146513528884898305at_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(23)
thf(fact_318_order__trans__rules_I23_J,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z4: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X2 @ Y3 )
=> ( ( ord_le3678578370064672496at_nat @ Y3 @ Z4 )
=> ( ord_le3678578370064672496at_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(23)
thf(fact_319_order__trans__rules_I23_J,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex,Z4: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y3 )
=> ( ( ord_le3632134057777142183omplex @ Y3 @ Z4 )
=> ( ord_le3632134057777142183omplex @ X2 @ Z4 ) ) ) ).
% order_trans_rules(23)
thf(fact_320_order__trans__rules_I23_J,axiom,
! [X2: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z4 )
=> ( ord_less_eq_set_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(23)
thf(fact_321_order__trans__rules_I23_J,axiom,
! [X2: nat > nat,Y3: nat > nat,Z4: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat_nat @ Y3 @ Z4 )
=> ( ord_less_eq_nat_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(23)
thf(fact_322_order__trans__rules_I23_J,axiom,
! [X2: nat,Y3: nat,Z4: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z4 )
=> ( ord_less_eq_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(23)
thf(fact_323_order__trans__rules_I10_J,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_324_order__trans__rules_I10_J,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_325_order__trans__rules_I10_J,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_326_order__trans__rules_I10_J,axiom,
! [A2: set_mat_complex,F: nat > set_mat_complex,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3632134057777142183omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3632134057777142183omplex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_327_order__trans__rules_I10_J,axiom,
! [A2: nat > nat,F: nat > nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_328_order__trans__rules_I10_J,axiom,
! [A2: nat,F: set_mat_complex > nat,B: set_mat_complex,C: set_mat_complex] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_329_order__trans__rules_I10_J,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_330_order__trans__rules_I10_J,axiom,
! [A2: nat,F: ( nat > nat ) > nat,B: nat > nat,C: nat > nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat_nat @ B @ C )
=> ( ! [X: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_331_order__trans__rules_I10_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,F: nat > set_Pr1261947904930325089at_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3146513528884898305at_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3146513528884898305at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_332_order__trans__rules_I10_J,axiom,
! [A2: nat,F: set_Pr1261947904930325089at_nat > nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3146513528884898305at_nat @ B @ C )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_333_order__trans__rules_I9_J,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_334_order__trans__rules_I9_J,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_335_order__trans__rules_I9_J,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_336_order__trans__rules_I9_J,axiom,
! [A2: nat,B: nat,F: nat > set_mat_complex,C: set_mat_complex] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3632134057777142183omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3632134057777142183omplex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_337_order__trans__rules_I9_J,axiom,
! [A2: nat,B: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_338_order__trans__rules_I9_J,axiom,
! [A2: set_mat_complex,B: set_mat_complex,F: set_mat_complex > nat,C: nat] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_339_order__trans__rules_I9_J,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_340_order__trans__rules_I9_J,axiom,
! [A2: nat > nat,B: nat > nat,F: ( nat > nat ) > nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_341_order__trans__rules_I9_J,axiom,
! [A2: nat,B: nat,F: nat > set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3146513528884898305at_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3146513528884898305at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_342_order__trans__rules_I9_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat,C: nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_343_order__trans__rules_I8_J,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_344_order__trans__rules_I8_J,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_345_order__trans__rules_I8_J,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_346_order__trans__rules_I8_J,axiom,
! [A2: nat,F: set_mat_complex > nat,B: set_mat_complex,C: set_mat_complex] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_347_order__trans__rules_I8_J,axiom,
! [A2: nat,F: ( nat > nat ) > nat,B: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat_nat @ B @ C )
=> ( ! [X: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_348_order__trans__rules_I8_J,axiom,
! [A2: set_mat_complex,F: nat > set_mat_complex,B: nat,C: nat] :
( ( ord_le3632134057777142183omplex @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3632134057777142183omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3632134057777142183omplex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_349_order__trans__rules_I8_J,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_350_order__trans__rules_I8_J,axiom,
! [A2: nat > nat,F: nat > nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_351_order__trans__rules_I8_J,axiom,
! [A2: nat,F: set_Pr1261947904930325089at_nat > nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le3146513528884898305at_nat @ B @ C )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_352_order__trans__rules_I8_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,F: nat > set_Pr1261947904930325089at_nat,B: nat,C: nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3146513528884898305at_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3146513528884898305at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_353_order__trans__rules_I7_J,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_354_order__trans__rules_I7_J,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_355_order__trans__rules_I7_J,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_356_order__trans__rules_I7_J,axiom,
! [A2: nat,B: nat,F: nat > set_mat_complex,C: set_mat_complex] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le3632134057777142183omplex @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3632134057777142183omplex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3632134057777142183omplex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_357_order__trans__rules_I7_J,axiom,
! [A2: nat,B: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_358_order__trans__rules_I7_J,axiom,
! [A2: set_mat_complex,B: set_mat_complex,F: set_mat_complex > nat,C: nat] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_mat_complex,Y: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_359_order__trans__rules_I7_J,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_360_order__trans__rules_I7_J,axiom,
! [A2: nat > nat,B: nat > nat,F: ( nat > nat ) > nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_361_order__trans__rules_I7_J,axiom,
! [A2: nat,B: nat,F: nat > set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le3146513528884898305at_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3146513528884898305at_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le3146513528884898305at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_362_order__trans__rules_I7_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat,C: nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_363_linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linear
thf(fact_364_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_365_eq__refl,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( X2 = Y3 )
=> ( ord_le3146513528884898305at_nat @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_366_eq__refl,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( X2 = Y3 )
=> ( ord_le3678578370064672496at_nat @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_367_eq__refl,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex] :
( ( X2 = Y3 )
=> ( ord_le3632134057777142183omplex @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_368_eq__refl,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_369_eq__refl,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat_nat @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_370_eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_371_le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% le_cases
thf(fact_372_le__cases3,axiom,
! [X2: nat,Y3: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_373_antisym__conv,axiom,
! [Y3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ Y3 @ X2 )
=> ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_374_antisym__conv,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y3 @ X2 )
=> ( ( ord_le3678578370064672496at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_375_antisym__conv,axiom,
! [Y3: set_mat_complex,X2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ Y3 @ X2 )
=> ( ( ord_le3632134057777142183omplex @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_376_antisym__conv,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_377_antisym__conv,axiom,
! [Y3: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_378_antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_379_order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
& ( ord_le3146513528884898305at_nat @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_380_order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr9093778441882193744at_nat,Z: set_Pr9093778441882193744at_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_Pr9093778441882193744at_nat,B3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A3 @ B3 )
& ( ord_le3678578370064672496at_nat @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_381_order_Oeq__iff,axiom,
( ( ^ [Y4: set_mat_complex,Z: set_mat_complex] : ( Y4 = Z ) )
= ( ^ [A3: set_mat_complex,B3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A3 @ B3 )
& ( ord_le3632134057777142183omplex @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_382_order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_383_order_Oeq__iff,axiom,
( ( ^ [Y4: nat > nat,Z: nat > nat] : ( Y4 = Z ) )
= ( ^ [A3: nat > nat,B3: nat > nat] :
( ( ord_less_eq_nat_nat @ A3 @ B3 )
& ( ord_less_eq_nat_nat @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_384_order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.eq_iff
thf(fact_385_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y4 = Z ) )
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X4 @ Y5 )
& ( ord_le3146513528884898305at_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_386_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr9093778441882193744at_nat,Z: set_Pr9093778441882193744at_nat] : ( Y4 = Z ) )
= ( ^ [X4: set_Pr9093778441882193744at_nat,Y5: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X4 @ Y5 )
& ( ord_le3678578370064672496at_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_387_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_mat_complex,Z: set_mat_complex] : ( Y4 = Z ) )
= ( ^ [X4: set_mat_complex,Y5: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y5 )
& ( ord_le3632134057777142183omplex @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_388_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_389_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > nat,Z: nat > nat] : ( Y4 = Z ) )
= ( ^ [X4: nat > nat,Y5: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y5 )
& ( ord_less_eq_nat_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_390_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_391_order__antisym,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
=> ( ( ord_le3146513528884898305at_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_392_order__antisym,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X2 @ Y3 )
=> ( ( ord_le3678578370064672496at_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_393_order__antisym,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y3 )
=> ( ( ord_le3632134057777142183omplex @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_394_order__antisym,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_395_order__antisym,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_396_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_397_order_Orefl,axiom,
! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A2 @ A2 ) ).
% order.refl
thf(fact_398_order_Orefl,axiom,
! [A2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A2 @ A2 ) ).
% order.refl
thf(fact_399_order_Orefl,axiom,
! [A2: set_mat_complex] : ( ord_le3632134057777142183omplex @ A2 @ A2 ) ).
% order.refl
thf(fact_400_order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% order.refl
thf(fact_401_order_Orefl,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% order.refl
thf(fact_402_order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% order.refl
thf(fact_403_order__refl,axiom,
! [X2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_404_order__refl,axiom,
! [X2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_405_order__refl,axiom,
! [X2: set_mat_complex] : ( ord_le3632134057777142183omplex @ X2 @ X2 ) ).
% order_refl
thf(fact_406_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_407_order__refl,axiom,
! [X2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_408_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_409_order_Otrans,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ( ord_le3146513528884898305at_nat @ B @ C )
=> ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_410_order_Otrans,axiom,
! [A2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B )
=> ( ( ord_le3678578370064672496at_nat @ B @ C )
=> ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_411_order_Otrans,axiom,
! [A2: set_mat_complex,B: set_mat_complex,C: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ord_le3632134057777142183omplex @ A2 @ C ) ) ) ).
% order.trans
thf(fact_412_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_413_order_Otrans,axiom,
! [A2: nat > nat,B: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( ( ord_less_eq_nat_nat @ B @ C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_414_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_415_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_416_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
& ( ord_le3146513528884898305at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_417_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr9093778441882193744at_nat,Z: set_Pr9093778441882193744at_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_Pr9093778441882193744at_nat,B3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B3 @ A3 )
& ( ord_le3678578370064672496at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_418_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_mat_complex,Z: set_mat_complex] : ( Y4 = Z ) )
= ( ^ [A3: set_mat_complex,B3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ B3 @ A3 )
& ( ord_le3632134057777142183omplex @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_419_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_420_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat > nat,Z: nat > nat] : ( Y4 = Z ) )
= ( ^ [A3: nat > nat,B3: nat > nat] :
( ( ord_less_eq_nat_nat @ B3 @ A3 )
& ( ord_less_eq_nat_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_421_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_422_dual__order_Oantisym,axiom,
! [B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B @ A2 )
=> ( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_423_dual__order_Oantisym,axiom,
! [B: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B @ A2 )
=> ( ( ord_le3678578370064672496at_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_424_dual__order_Oantisym,axiom,
! [B: set_mat_complex,A2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ B @ A2 )
=> ( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_425_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_426_dual__order_Oantisym,axiom,
! [B: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B @ A2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_427_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_428_dual__order_Otrans,axiom,
! [B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B @ A2 )
=> ( ( ord_le3146513528884898305at_nat @ C @ B )
=> ( ord_le3146513528884898305at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_429_dual__order_Otrans,axiom,
! [B: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B @ A2 )
=> ( ( ord_le3678578370064672496at_nat @ C @ B )
=> ( ord_le3678578370064672496at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_430_dual__order_Otrans,axiom,
! [B: set_mat_complex,A2: set_mat_complex,C: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ B @ A2 )
=> ( ( ord_le3632134057777142183omplex @ C @ B )
=> ( ord_le3632134057777142183omplex @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_431_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_432_dual__order_Otrans,axiom,
! [B: nat > nat,A2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ B @ A2 )
=> ( ( ord_less_eq_nat_nat @ C @ B )
=> ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_433_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_434_le__funD,axiom,
! [F: nat > nat,G: nat > nat,X2: nat] :
( ( ord_less_eq_nat_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funD
thf(fact_435_le__funE,axiom,
! [F: nat > nat,G: nat > nat,X2: nat] :
( ( ord_less_eq_nat_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funE
thf(fact_436_le__funI,axiom,
! [F: nat > nat,G: nat > nat] :
( ! [X: nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq_nat_nat @ F @ G ) ) ).
% le_funI
thf(fact_437_le__fun__def,axiom,
( ord_less_eq_nat_nat
= ( ^ [F2: nat > nat,G2: nat > nat] :
! [X4: nat] : ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_438_order__less__imp__not__less,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ~ ( ord_less_complex @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_439_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_440_order__less__imp__not__eq2,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_441_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_442_order__less__imp__not__eq,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_443_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_444_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_445_order__less__imp__triv,axiom,
! [X2: complex,Y3: complex,P: $o] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( ( ord_less_complex @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_446_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_447_order__less__not__sym,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ~ ( ord_less_complex @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_448_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_449_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_complex @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_450_order__less__subst2,axiom,
! [A2: complex,B: complex,F: complex > nat,C: nat] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_451_order__less__subst2,axiom,
! [A2: complex,B: complex,F: complex > complex,C: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_complex @ ( F @ B ) @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_452_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_453_order__less__subst1,axiom,
! [A2: nat,F: complex > nat,B: complex,C: complex] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_454_order__less__subst1,axiom,
! [A2: complex,F: nat > complex,B: nat,C: nat] :
( ( ord_less_complex @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_455_order__less__subst1,axiom,
! [A2: complex,F: complex > complex,B: complex,C: complex] :
( ( ord_less_complex @ A2 @ ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_456_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_457_order__less__irrefl,axiom,
! [X2: complex] :
~ ( ord_less_complex @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_458_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_459_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_460_ord__less__eq__subst,axiom,
! [A2: complex,B: complex,F: complex > nat,C: nat] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_461_ord__less__eq__subst,axiom,
! [A2: complex,B: complex,F: complex > complex,C: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_462_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_463_ord__eq__less__subst,axiom,
! [A2: complex,F: nat > complex,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_464_ord__eq__less__subst,axiom,
! [A2: nat,F: complex > nat,B: complex,C: complex] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_465_ord__eq__less__subst,axiom,
! [A2: complex,F: complex > complex,B: complex,C: complex] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_complex @ B @ C )
=> ( ! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_466_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_467_order__less__trans,axiom,
! [X2: complex,Y3: complex,Z4: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( ( ord_less_complex @ Y3 @ Z4 )
=> ( ord_less_complex @ X2 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_468_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z4: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z4 )
=> ( ord_less_nat @ X2 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_469_order__less__asym_H,axiom,
! [A2: complex,B: complex] :
( ( ord_less_complex @ A2 @ B )
=> ~ ( ord_less_complex @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_470_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_471_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_472_order__less__asym,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ~ ( ord_less_complex @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_473_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_474_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_475_dual__order_Ostrict__implies__not__eq,axiom,
! [B: complex,A2: complex] :
( ( ord_less_complex @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_476_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_477_order_Ostrict__implies__not__eq,axiom,
! [A2: complex,B: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_478_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_479_dual__order_Ostrict__trans,axiom,
! [B: complex,A2: complex,C: complex] :
( ( ord_less_complex @ B @ A2 )
=> ( ( ord_less_complex @ C @ B )
=> ( ord_less_complex @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_480_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_481_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_482_order_Ostrict__trans,axiom,
! [A2: complex,B: complex,C: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ( ord_less_complex @ B @ C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_483_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_484_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_485_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P4 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_486_dual__order_Oirrefl,axiom,
! [A2: complex] :
~ ( ord_less_complex @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_487_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_488_dual__order_Oasym,axiom,
! [B: complex,A2: complex] :
( ( ord_less_complex @ B @ A2 )
=> ~ ( ord_less_complex @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_489_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_490_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_491_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_492_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X )
=> ( P @ Y2 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_493_ord__less__eq__trans,axiom,
! [A2: complex,B: complex,C: complex] :
( ( ord_less_complex @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_494_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_495_ord__eq__less__trans,axiom,
! [A2: complex,B: complex,C: complex] :
( ( A2 = B )
=> ( ( ord_less_complex @ B @ C )
=> ( ord_less_complex @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_496_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_497_order_Oasym,axiom,
! [A2: complex,B: complex] :
( ( ord_less_complex @ A2 @ B )
=> ~ ( ord_less_complex @ B @ A2 ) ) ).
% order.asym
thf(fact_498_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_499_less__imp__neq,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_500_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_501_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_502_order__trans__rules_I21_J,axiom,
! [X2: complex,Y3: complex,Z4: complex] :
( ( ord_less_eq_complex @ X2 @ Y3 )
=> ( ( ord_less_complex @ Y3 @ Z4 )
=> ( ord_less_complex @ X2 @ Z4 ) ) ) ).
% order_trans_rules(21)
thf(fact_503_order__trans__rules_I21_J,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
=> ( ( ord_le7866589430770878221at_nat @ Y3 @ Z4 )
=> ( ord_le7866589430770878221at_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(21)
thf(fact_504_order__trans__rules_I21_J,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z4: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X2 @ Y3 )
=> ( ( ord_le4845169857901429244at_nat @ Y3 @ Z4 )
=> ( ord_le4845169857901429244at_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(21)
thf(fact_505_order__trans__rules_I21_J,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex,Z4: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y3 )
=> ( ( ord_le5598786136212072115omplex @ Y3 @ Z4 )
=> ( ord_le5598786136212072115omplex @ X2 @ Z4 ) ) ) ).
% order_trans_rules(21)
thf(fact_506_order__trans__rules_I21_J,axiom,
! [X2: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z4 )
=> ( ord_less_set_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(21)
thf(fact_507_order__trans__rules_I21_J,axiom,
! [X2: nat > nat,Y3: nat > nat,Z4: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y3 )
=> ( ( ord_less_nat_nat @ Y3 @ Z4 )
=> ( ord_less_nat_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(21)
thf(fact_508_order__trans__rules_I21_J,axiom,
! [X2: nat,Y3: nat,Z4: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z4 )
=> ( ord_less_nat @ X2 @ Z4 ) ) ) ).
% order_trans_rules(21)
thf(fact_509_order__trans__rules_I18_J,axiom,
! [A2: complex,B: complex] :
( ( ord_less_eq_complex @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_complex @ A2 @ B ) ) ) ).
% order_trans_rules(18)
thf(fact_510_order__trans__rules_I18_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le7866589430770878221at_nat @ A2 @ B ) ) ) ).
% order_trans_rules(18)
thf(fact_511_order__trans__rules_I18_J,axiom,
! [A2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le4845169857901429244at_nat @ A2 @ B ) ) ) ).
% order_trans_rules(18)
thf(fact_512_order__trans__rules_I18_J,axiom,
! [A2: set_mat_complex,B: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le5598786136212072115omplex @ A2 @ B ) ) ) ).
% order_trans_rules(18)
thf(fact_513_order__trans__rules_I18_J,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_trans_rules(18)
thf(fact_514_order__trans__rules_I18_J,axiom,
! [A2: nat > nat,B: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat_nat @ A2 @ B ) ) ) ).
% order_trans_rules(18)
thf(fact_515_order__trans__rules_I18_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_trans_rules(18)
thf(fact_516_order__trans__rules_I17_J,axiom,
! [A2: complex,B: complex] :
( ( A2 != B )
=> ( ( ord_less_eq_complex @ A2 @ B )
=> ( ord_less_complex @ A2 @ B ) ) ) ).
% order_trans_rules(17)
thf(fact_517_order__trans__rules_I17_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( A2 != B )
=> ( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ord_le7866589430770878221at_nat @ A2 @ B ) ) ) ).
% order_trans_rules(17)
thf(fact_518_order__trans__rules_I17_J,axiom,
! [A2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( A2 != B )
=> ( ( ord_le3678578370064672496at_nat @ A2 @ B )
=> ( ord_le4845169857901429244at_nat @ A2 @ B ) ) ) ).
% order_trans_rules(17)
thf(fact_519_order__trans__rules_I17_J,axiom,
! [A2: set_mat_complex,B: set_mat_complex] :
( ( A2 != B )
=> ( ( ord_le3632134057777142183omplex @ A2 @ B )
=> ( ord_le5598786136212072115omplex @ A2 @ B ) ) ) ).
% order_trans_rules(17)
thf(fact_520_order__trans__rules_I17_J,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_trans_rules(17)
thf(fact_521_order__trans__rules_I17_J,axiom,
! [A2: nat > nat,B: nat > nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat_nat @ A2 @ B )
=> ( ord_less_nat_nat @ A2 @ B ) ) ) ).
% order_trans_rules(17)
thf(fact_522_order__trans__rules_I17_J,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_trans_rules(17)
thf(fact_523_leD,axiom,
! [Y3: complex,X2: complex] :
( ( ord_less_eq_complex @ Y3 @ X2 )
=> ~ ( ord_less_complex @ X2 @ Y3 ) ) ).
% leD
thf(fact_524_leD,axiom,
! [Y3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ Y3 @ X2 )
=> ~ ( ord_le7866589430770878221at_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_525_leD,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y3 @ X2 )
=> ~ ( ord_le4845169857901429244at_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_526_leD,axiom,
! [Y3: set_mat_complex,X2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ Y3 @ X2 )
=> ~ ( ord_le5598786136212072115omplex @ X2 @ Y3 ) ) ).
% leD
thf(fact_527_leD,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_528_leD,axiom,
! [Y3: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_529_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_530_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_531_le__less,axiom,
( ord_less_eq_complex
= ( ^ [X4: complex,Y5: complex] :
( ( ord_less_complex @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_532_le__less,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_533_le__less,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [X4: set_Pr9093778441882193744at_nat,Y5: set_Pr9093778441882193744at_nat] :
( ( ord_le4845169857901429244at_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_534_le__less,axiom,
( ord_le3632134057777142183omplex
= ( ^ [X4: set_mat_complex,Y5: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_535_le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_536_le__less,axiom,
( ord_less_eq_nat_nat
= ( ^ [X4: nat > nat,Y5: nat > nat] :
( ( ord_less_nat_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_537_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_538_less__le,axiom,
( ord_less_complex
= ( ^ [X4: complex,Y5: complex] :
( ( ord_less_eq_complex @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_539_less__le,axiom,
( ord_le7866589430770878221at_nat
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_540_less__le,axiom,
( ord_le4845169857901429244at_nat
= ( ^ [X4: set_Pr9093778441882193744at_nat,Y5: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_541_less__le,axiom,
( ord_le5598786136212072115omplex
= ( ^ [X4: set_mat_complex,Y5: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_542_less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_543_less__le,axiom,
( ord_less_nat_nat
= ( ^ [X4: nat > nat,Y5: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_544_less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_545_nless__le,axiom,
! [A2: complex,B: complex] :
( ( ~ ( ord_less_complex @ A2 @ B ) )
= ( ~ ( ord_less_eq_complex @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_546_nless__le,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ~ ( ord_le7866589430770878221at_nat @ A2 @ B ) )
= ( ~ ( ord_le3146513528884898305at_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_547_nless__le,axiom,
! [A2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ~ ( ord_le4845169857901429244at_nat @ A2 @ B ) )
= ( ~ ( ord_le3678578370064672496at_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_548_nless__le,axiom,
! [A2: set_mat_complex,B: set_mat_complex] :
( ( ~ ( ord_le5598786136212072115omplex @ A2 @ B ) )
= ( ~ ( ord_le3632134057777142183omplex @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_549_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_550_nless__le,axiom,
! [A2: nat > nat,B: nat > nat] :
( ( ~ ( ord_less_nat_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_551_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_552_not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% not_le
thf(fact_553_not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% not_less
thf(fact_554_antisym__conv1,axiom,
! [X2: complex,Y3: complex] :
( ~ ( ord_less_complex @ X2 @ Y3 )
=> ( ( ord_less_eq_complex @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_555_antisym__conv1,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ~ ( ord_le7866589430770878221at_nat @ X2 @ Y3 )
=> ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_556_antisym__conv1,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ~ ( ord_le4845169857901429244at_nat @ X2 @ Y3 )
=> ( ( ord_le3678578370064672496at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_557_antisym__conv1,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex] :
( ~ ( ord_le5598786136212072115omplex @ X2 @ Y3 )
=> ( ( ord_le3632134057777142183omplex @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_558_antisym__conv1,axiom,
! [X2: set_nat,Y3: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_559_antisym__conv1,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ~ ( ord_less_nat_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_560_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_561_antisym__conv2,axiom,
! [X2: complex,Y3: complex] :
( ( ord_less_eq_complex @ X2 @ Y3 )
=> ( ( ~ ( ord_less_complex @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_562_antisym__conv2,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_le7866589430770878221at_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_563_antisym__conv2,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_le4845169857901429244at_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_564_antisym__conv2,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y3 )
=> ( ( ~ ( ord_le5598786136212072115omplex @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_565_antisym__conv2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_566_antisym__conv2,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_567_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_568_less__imp__le,axiom,
! [X2: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ord_le4845169857901429244at_nat @ X2 @ Y3 )
=> ( ord_le3678578370064672496at_nat @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_569_less__imp__le,axiom,
! [X2: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ X2 @ Y3 )
=> ( ord_le3632134057777142183omplex @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_570_less__imp__le,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_571_less__imp__le,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ( ord_less_nat_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat_nat @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_572_less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% less_imp_le
thf(fact_573_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_574_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_575_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_576_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_577_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_578_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_579_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_580_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_581_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_582_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_583_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_584_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_585_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_586_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_587_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_588_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_589_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_590_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_591_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_592_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_593_comm__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ A @ B2 )
= ( plus_p8323303612493835998omplex @ B2 @ A ) ) ) ) ).
% comm_add_mat
thf(fact_594_assoc__add__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ A @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_595_add__carrier__mat,axiom,
! [B2: mat_complex,Nr: nat,Nc: nat,A: mat_complex] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_596_index__add__mat_I2_J,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( dim_row_complex @ ( plus_p8323303612493835998omplex @ A @ B2 ) )
= ( dim_row_complex @ B2 ) ) ).
% index_add_mat(2)
thf(fact_597_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_598_Groups_Oadd__ac_I3_J,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% Groups.add_ac(3)
thf(fact_599_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% Groups.add_ac(2)
thf(fact_600_Groups_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% Groups.add_ac(1)
thf(fact_601_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_602_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_603_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_604_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_605_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_606_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_607_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_608_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_609_zero__order_I5_J,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_610_zero__order_I4_J,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_order(4)
thf(fact_611_zero__order_I3_J,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% zero_order(3)
thf(fact_612_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_613_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_614_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_615_add__0__left,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0_left
thf(fact_616_add__0__right,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add_0_right
thf(fact_617_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_618_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_619_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_620_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_621_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_622_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_623_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_624_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_625_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_626_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_627_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_628_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_629_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_630_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_631_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_632_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_633_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_634_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_635_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_636_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_637_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_638_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_639_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_640_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_641_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_642_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_643_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_644_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_645_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_646_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_647_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_648_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_649_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_650_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_651_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_652_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_653_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_654_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_655_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_656_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_657_add__sign__intros_I6_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(6)
thf(fact_658_add__sign__intros_I2_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_sign_intros(2)
thf(fact_659_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_660_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_661_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_662_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_663_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_664_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_665_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_666_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_667_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_668_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_669_extract__subdiags__diag__elem,axiom,
! [B2: mat_complex,N: nat,L: list_nat,I: nat,J: nat] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( L != nil_nat )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ L @ I ) )
=> ( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ L ) @ N )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
=> ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ L @ J2 ) ) )
=> ( ( index_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ B2 @ L ) @ I ) @ ( product_Pair_nat_nat @ J @ J ) )
= ( nth_complex @ ( diag_mat_complex @ B2 ) @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ L ) @ J ) ) ) ) ) ) ) ) ) ) ).
% extract_subdiags_diag_elem
thf(fact_670_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_671_add__carrier__mat_H,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% add_carrier_mat'
thf(fact_672_swap__plus__mat,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C2 ) @ B2 ) ) ) ) ) ).
% swap_plus_mat
thf(fact_673_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_mat_complex,N: nat] :
( ( ord_less_nat @ M @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_Pr7800990439711681285omplex @ ( enumer6113296918280681228omplex @ N @ Xs ) @ M )
= ( produc4998868960714853886omplex @ ( plus_plus_nat @ N @ M ) @ ( nth_mat_complex @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_674_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_complex,N: nat] :
( ( ord_less_nat @ M @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( nth_Pr9135773202038487014omplex @ ( enumerate_complex @ N @ Xs ) @ M )
= ( produc6973218034000581911omplex @ ( plus_plus_nat @ N @ M ) @ ( nth_complex @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_675_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_676_add__Pair,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_677_add__Pair,axiom,
! [A2: nat,B: mat_complex,C: nat,D: mat_complex] :
( ( plus_p8221215230258962133omplex @ ( produc4998868960714853886omplex @ A2 @ B ) @ ( produc4998868960714853886omplex @ C @ D ) )
= ( produc4998868960714853886omplex @ ( plus_plus_nat @ A2 @ C ) @ ( plus_p8323303612493835998omplex @ B @ D ) ) ) ).
% add_Pair
thf(fact_678_add__Pair,axiom,
! [A2: mat_complex,B: nat,C: mat_complex,D: nat] :
( ( plus_p679445643052534703ex_nat @ ( produc3916067632315525152ex_nat @ A2 @ B ) @ ( produc3916067632315525152ex_nat @ C @ D ) )
= ( produc3916067632315525152ex_nat @ ( plus_p8323303612493835998omplex @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_679_add__Pair,axiom,
! [A2: mat_complex,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( plus_p6104634242915576478omplex @ ( produc3658446505030690647omplex @ A2 @ B ) @ ( produc3658446505030690647omplex @ C @ D ) )
= ( produc3658446505030690647omplex @ ( plus_p8323303612493835998omplex @ A2 @ C ) @ ( plus_p8323303612493835998omplex @ B @ D ) ) ) ).
% add_Pair
thf(fact_680_Pair__le,axiom,
! [A2: nat > nat,B: nat,C: nat > nat,D: nat] :
( ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ A2 @ B ) @ ( produc72220940542539688at_nat @ C @ D ) )
= ( ( ord_less_eq_nat_nat @ A2 @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_681_Pair__le,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_682_fold__atLeastAtMost__nat_Ocases,axiom,
! [X2: produc4471711990508489141at_nat] :
~ ! [F3: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
( X2
!= ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_683_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_nat @ N @ nil_nat )
= nil_Pr5478986624290739719at_nat ) ).
% enumerate_simps(1)
thf(fact_684_length__enumerate,axiom,
! [N: nat,Xs: list_complex] :
( ( size_s3423402466807558097omplex @ ( enumerate_complex @ N @ Xs ) )
= ( size_s3451745648224563538omplex @ Xs ) ) ).
% length_enumerate
thf(fact_685_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_686_mat__assoc__test_I15_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( plus_p8323303612493835998omplex @ C2 @ D2 ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C2 ) @ ( plus_p8323303612493835998omplex @ B2 @ D2 ) ) ) ) ) ) ) ).
% mat_assoc_test(15)
thf(fact_687_mat__assoc__test_I14_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ C2 @ B2 ) @ A ) ) ) ) ) ) ).
% mat_assoc_test(14)
thf(fact_688_mat__assoc__test_I13_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( plus_p8323303612493835998omplex @ A @ B2 )
= ( plus_p8323303612493835998omplex @ B2 @ A ) ) ) ) ) ) ).
% mat_assoc_test(13)
thf(fact_689_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_690_Pair__mono,axiom,
! [X2: nat > nat,X7: nat > nat,Y3: nat,Y6: nat] :
( ( ord_less_eq_nat_nat @ X2 @ X7 )
=> ( ( ord_less_eq_nat @ Y3 @ Y6 )
=> ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) @ ( produc72220940542539688at_nat @ X7 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_691_Pair__mono,axiom,
! [X2: nat,X7: nat,Y3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ X7 )
=> ( ( ord_less_eq_nat @ Y3 @ Y6 )
=> ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( product_Pair_nat_nat @ X7 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_692_rel__simps_I46_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% rel_simps(46)
thf(fact_693_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_694_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_695_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_696_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_697_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_698_verit__comp__simplify_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_699_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_700_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ C4 ) )
=> ( P @ X3 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_701_commute__diag__mat__zero__comp,axiom,
! [D2: mat_complex,N: nat,B2: mat_complex,I: nat,J: nat] :
( ( diagonal_mat_complex @ D2 )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( times_8009071140041733218omplex @ B2 @ D2 )
= ( times_8009071140041733218omplex @ D2 @ B2 ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( index_mat_complex @ D2 @ ( product_Pair_nat_nat @ I @ I ) )
!= ( index_mat_complex @ D2 @ ( product_Pair_nat_nat @ J @ J ) ) )
=> ( ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_complex ) ) ) ) ) ) ) ) ).
% commute_diag_mat_zero_comp
thf(fact_702_crossproduct__noteq,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( A2 != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_703_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X2: nat,Z4: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X2 @ Z4 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z4 ) @ ( times_times_nat @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z4 ) ) ) ).
% crossproduct_eq
thf(fact_704_mat__assoc__test_I1_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ C2 @ D2 ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C2 ) @ D2 ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_705_Groups_Omult__ac_I3_J,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_706_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_707_Groups_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_708_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_709_index__mult__mat_I2_J,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( dim_row_complex @ ( times_8009071140041733218omplex @ A @ B2 ) )
= ( dim_row_complex @ A ) ) ).
% index_mult_mat(2)
thf(fact_710_assoc__mult__mat,axiom,
! [A: mat_complex,N_1: nat,N_2: nat,B2: mat_complex,N_3: nat,C2: mat_complex,N_4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C2 )
= ( times_8009071140041733218omplex @ A @ ( times_8009071140041733218omplex @ B2 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_711_mult__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_712_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_713_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_714_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_715_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_716_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_717_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_718_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_719_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_720_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_721_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_722_nat__distrib_I2_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% nat_distrib(2)
thf(fact_723_Rings_Oring__distribs_I2_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_724_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_725_combine__common__factor,axiom,
! [A2: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_726_mult__sign__intros_I3_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_727_mult__sign__intros_I2_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_728_mult__sign__intros_I1_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_729_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_730_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_731_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_732_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_733_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_734_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_735_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_736_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_737_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_738_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_739_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_740_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_741_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_742_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_743_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_744_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_745_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_746_add__mult__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,C2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C2 )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ C2 ) @ ( times_8009071140041733218omplex @ B2 @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_747_mult__add__distrib__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_748_mat__assoc__test_I7_J,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( plus_p8323303612493835998omplex @ B2 @ C2 ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ B2 @ B2 ) ) @ ( times_8009071140041733218omplex @ A @ C2 ) ) @ ( times_8009071140041733218omplex @ B2 @ C2 ) ) ) ) ) ) ) ).
% mat_assoc_test(7)
thf(fact_749_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_750_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_751_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_752_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_753_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_754_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_755_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_756_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_757_diagonal__mat__sq__index,axiom,
! [B2: mat_complex,N: nat,I: nat,J: nat] :
( ( diagonal_mat_complex @ B2 )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ B2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_complex @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I @ I ) ) @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ J @ I ) ) ) ) ) ) ) ) ).
% diagonal_mat_sq_index
thf(fact_758_diagonal__mat__sq__index_H,axiom,
! [B2: mat_complex,N: nat,I: nat,J: nat] :
( ( diagonal_mat_complex @ B2 )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ B2 @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_complex @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ).
% diagonal_mat_sq_index'
thf(fact_759_diagonal__mat__mult__index,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,I: nat,J: nat] :
( ( diagonal_mat_complex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_complex @ ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I @ I ) ) @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index
thf(fact_760_diagonal__mat__mult__index_H,axiom,
! [A: mat_complex,N: nat,B2: mat_complex,J: nat,I: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ B2 )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ N )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_complex @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ J @ J ) ) @ ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index'
thf(fact_761_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_762_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_763_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_764_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_765_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_766_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_767_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_768_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_769_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_770_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_771_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_772_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_773_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_774_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_775_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_776_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_777_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_778_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_779_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_780_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_781_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_782_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_783_mult__le__cancel1,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 ) ) ) ).
% mult_le_cancel1
thf(fact_784_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_785_diagonal__mat__commute,axiom,
! [A: mat_complex,N: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B2 )
=> ( ( times_8009071140041733218omplex @ A @ B2 )
= ( times_8009071140041733218omplex @ B2 @ A ) ) ) ) ) ) ).
% diagonal_mat_commute
thf(fact_786_diagonal__mat__sq__diag,axiom,
! [B2: mat_complex,N: nat] :
( ( diagonal_mat_complex @ B2 )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ B2 @ B2 ) ) ) ) ).
% diagonal_mat_sq_diag
thf(fact_787_diagonal__mat__times__diag,axiom,
! [A: mat_complex,N: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B2 )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ A @ B2 ) ) ) ) ) ) ).
% diagonal_mat_times_diag
thf(fact_788_mult__hom_Ohom__add__eq__zero,axiom,
! [X2: nat,Y3: nat,C: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X2 ) @ ( times_times_nat @ C @ Y3 ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_789_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_790_mult__hom_Ohom__add,axiom,
! [C: nat,X2: nat,Y3: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X2 @ Y3 ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X2 ) @ ( times_times_nat @ C @ Y3 ) ) ) ).
% mult_hom.hom_add
thf(fact_791_mult__delta__right,axiom,
! [B: $o,X2: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_792_mult__delta__left,axiom,
! [B: $o,X2: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y3 )
= ( times_times_nat @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y3 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_793_n__sum__last__lt,axiom,
! [J: complex,L: list_complex,I: nat] :
( ( ord_less_complex @ J @ ( nth_complex @ L @ I ) )
=> ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ ( commut6323218633641605728omplex @ I @ L ) @ J ) @ ( commut6323218633641605728omplex @ ( suc @ I ) @ L ) ) ) ) ).
% n_sum_last_lt
thf(fact_794_n__sum__last__lt,axiom,
! [J: nat,L: list_nat,I: nat] :
( ( ord_less_nat @ J @ ( nth_nat @ L @ I ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ L ) @ J ) @ ( commut2019222099004354946um_nat @ ( suc @ I ) @ L ) ) ) ) ).
% n_sum_last_lt
thf(fact_795_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_796_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_797_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_798_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_799_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_800_semiring__norm_I163_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% semiring_norm(163)
thf(fact_801_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_802_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_803_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_804_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_805_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_806_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_807_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_808_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_809_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_810_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_811_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_812_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_813_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_814_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_815_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_816_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_817_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_818_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_819_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_820_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_821_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_822_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_823_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_824_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_825_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_826_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_827_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_828_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_829_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_830_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_831_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R2 @ X @ X )
=> ( ! [X: nat,Y: nat,Z2: nat] :
( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X @ Z2 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_832_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_833_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_834_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_835_unit__vecs__last_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N2: nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat,I3: nat] :
( ( P @ N2 @ I3 )
=> ( P @ N2 @ ( suc @ I3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_last.induct
thf(fact_836_nat_Osimps_I3_J,axiom,
! [X22: nat] :
( ( suc @ X22 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_837_old_Onat_Osimps_I3_J,axiom,
! [Nat: nat] :
( ( suc @ Nat )
!= zero_zero_nat ) ).
% old.nat.simps(3)
thf(fact_838_old_Onat_Osimps_I2_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
% old.nat.simps(2)
thf(fact_839_nat_OdiscI,axiom,
! [Nat2: nat,X22: nat] :
( ( Nat2
= ( suc @ X22 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_840_nat_Oinduct,axiom,
! [P: nat > $o,Nat2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat2 ) ) ) ).
% nat.induct
thf(fact_841_nat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [X23: nat] :
( Y3
!= ( suc @ X23 ) ) ) ).
% nat.exhaust
thf(fact_842_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_843_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X: nat,Y: nat] :
( ( P @ X @ Y )
=> ( P @ ( suc @ X ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_844_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_845_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_846_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_not_Zero
thf(fact_847_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_848_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_849_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_850_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_851_old_Onat_Oinject,axiom,
! [Nat2: nat,Nat: nat] :
( ( ( suc @ Nat2 )
= ( suc @ Nat ) )
= ( Nat2 = Nat ) ) ).
% old.nat.inject
thf(fact_852_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_853_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_854_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_855_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_856_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_857_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_858_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_859_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_860_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_861_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_862_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_863_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_864_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_865_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_866_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_867_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_868_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_869_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_870_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_871_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_872_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_873_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
? [K5: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M5 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_874_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_875_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_876_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_877_unit__vecs__first_Ocases,axiom,
! [X2: product_prod_nat_nat] :
( ! [N2: nat] :
( X2
!= ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) )
=> ~ ! [N2: nat,I3: nat] :
( X2
!= ( product_Pair_nat_nat @ N2 @ ( suc @ I3 ) ) ) ) ).
% unit_vecs_first.cases
thf(fact_878_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_879_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_880_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_881_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_882_times__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% times_nat.simps(2)
thf(fact_883_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_884_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_885_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_886_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_887_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_888_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_889_n__sum__last,axiom,
! [I: nat,L: list_complex] :
( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L ) )
=> ( ( commut6323218633641605728omplex @ ( suc @ I ) @ L )
= ( plus_plus_complex @ ( commut6323218633641605728omplex @ I @ L ) @ ( nth_complex @ L @ I ) ) ) ) ).
% n_sum_last
thf(fact_890_n__sum__last,axiom,
! [I: nat,L: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
=> ( ( commut2019222099004354946um_nat @ ( suc @ I ) @ L )
= ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I @ L ) @ ( nth_nat @ L @ I ) ) ) ) ).
% n_sum_last
thf(fact_891_pivot__positions__main__gen_Oinduct,axiom,
! [Nr: nat,Nc: nat,A: mat_complex,Zero: complex,P: nat > nat > $o,A0: nat,A1: nat] :
( ! [I3: nat,J2: nat] :
( ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J2 @ Nc )
=> ( ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= Zero )
=> ( P @ I3 @ ( suc @ J2 ) ) ) ) )
=> ( ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J2 @ Nc )
=> ( ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I3 @ J2 ) )
!= Zero )
=> ( P @ ( suc @ I3 ) @ ( suc @ J2 ) ) ) ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ).
% pivot_positions_main_gen.induct
thf(fact_892_lookup__other__ev_Oinduct,axiom,
! [P: complex > nat > mat_complex > $o,A0: complex,A1: nat,A22: mat_complex] :
( ! [Ev: complex,X_1: mat_complex] : ( P @ Ev @ zero_zero_nat @ X_1 )
=> ( ! [Ev: complex,I3: nat,A6: mat_complex] :
( ( ( ( index_mat_complex @ A6 @ ( product_Pair_nat_nat @ I3 @ I3 ) )
= Ev )
=> ( P @ Ev @ I3 @ A6 ) )
=> ( P @ Ev @ ( suc @ I3 ) @ A6 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_other_ev.induct
thf(fact_893_lookup__ev_Oinduct,axiom,
! [P: complex > nat > mat_complex > $o,A0: complex,A1: nat,A22: mat_complex] :
( ! [Ev: complex,X_1: mat_complex] : ( P @ Ev @ zero_zero_nat @ X_1 )
=> ( ! [Ev: complex,I3: nat,A6: mat_complex] :
( ( ( ( index_mat_complex @ A6 @ ( product_Pair_nat_nat @ I3 @ I3 ) )
!= Ev )
=> ( P @ Ev @ I3 @ A6 ) )
=> ( P @ Ev @ ( suc @ I3 ) @ A6 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_ev.induct
thf(fact_894_ev__blocks__part__def,axiom,
( jordan4637981584770492064omplex
= ( ^ [M5: nat,A7: mat_complex] :
! [I4: nat,J3: nat,K5: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K5 )
=> ( ( ord_less_nat @ K5 @ M5 )
=> ( ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ K5 @ K5 ) )
= ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) ) )
=> ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ J3 @ J3 ) )
= ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) ) ) ) ) ) ) ) ) ).
% ev_blocks_part_def
thf(fact_895_same__diag__def,axiom,
( jordan2620430285385836103omplex
= ( ^ [N3: nat,A7: mat_complex,B6: mat_complex] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) )
= ( index_mat_complex @ B6 @ ( product_Pair_nat_nat @ I4 @ I4 ) ) ) ) ) ) ).
% same_diag_def
thf(fact_896_inf__concat_Ocases,axiom,
! [X2: produc8199716216217303280at_nat] :
( ! [N2: nat > nat] :
( X2
!= ( produc72220940542539688at_nat @ N2 @ zero_zero_nat ) )
=> ~ ! [N2: nat > nat,K2: nat] :
( X2
!= ( produc72220940542539688at_nat @ N2 @ ( suc @ K2 ) ) ) ) ).
% inf_concat.cases
thf(fact_897_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_898_diff__ev__def,axiom,
( jordan8650160714669549932omplex
= ( ^ [A7: mat_complex,I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) )
!= ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ J3 @ J3 ) ) )
=> ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_complex ) ) ) ) ) ).
% diff_ev_def
thf(fact_899_uppert__def,axiom,
( jordan3528196489273997576omplex
= ( ^ [A7: mat_complex,I4: nat,J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_complex ) ) ) ) ).
% uppert_def
thf(fact_900_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_901_nth__item_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% nth_item.cases
thf(fact_902_Abstract__Rewriting_Ochain__mono,axiom,
! [R3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ R2 )
=> ( ! [I3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R3 )
=> ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R2 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_903_upper__triangular__def,axiom,
( upper_4850907204721561915omplex
= ( ^ [A7: mat_complex] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_complex @ A7 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_complex ) ) ) ) ) ).
% upper_triangular_def
thf(fact_904_upper__triangular__def,axiom,
( upper_triangular_nat
= ( ^ [A7: mat_nat] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_nat @ A7 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( index_mat_nat @ A7 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_nat ) ) ) ) ) ).
% upper_triangular_def
thf(fact_905_Set_Obasic__monos_I7_J,axiom,
! [A: set_mat_complex,B2: set_mat_complex,X2: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B2 )
=> ( ( member_mat_complex @ X2 @ A )
=> ( member_mat_complex @ X2 @ B2 ) ) ) ).
% Set.basic_monos(7)
thf(fact_906_basic__trans__rules_I31_J,axiom,
! [A: set_mat_complex,B2: set_mat_complex,C: mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B2 )
=> ( ( member_mat_complex @ C @ A )
=> ( member_mat_complex @ C @ B2 ) ) ) ).
% basic_trans_rules(31)
thf(fact_907_subsetI,axiom,
! [A: set_mat_complex,B2: set_mat_complex] :
( ! [X: mat_complex] :
( ( member_mat_complex @ X @ A )
=> ( member_mat_complex @ X @ B2 ) )
=> ( ord_le3632134057777142183omplex @ A @ B2 ) ) ).
% subsetI
thf(fact_908_subset__eq,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A7: set_mat_complex,B6: set_mat_complex] :
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A7 )
=> ( member_mat_complex @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_909_subset__iff,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A7: set_mat_complex,B6: set_mat_complex] :
! [T2: mat_complex] :
( ( member_mat_complex @ T2 @ A7 )
=> ( member_mat_complex @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_910_diagonal__imp__upper__triangular,axiom,
! [A: mat_complex,N: nat] :
( ( diagonal_mat_complex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( upper_4850907204721561915omplex @ A ) ) ) ).
% diagonal_imp_upper_triangular
thf(fact_911_upper__triangularD,axiom,
! [A: mat_complex,J: nat,I: nat] :
( ( upper_4850907204721561915omplex @ A )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_complex @ A ) )
=> ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_complex ) ) ) ) ).
% upper_triangularD
thf(fact_912_upper__triangularD,axiom,
! [A: mat_nat,J: nat,I: nat] :
( ( upper_triangular_nat @ A )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ A ) )
=> ( ( index_mat_nat @ A @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ).
% upper_triangularD
thf(fact_913_upper__triangularI,axiom,
! [A: mat_complex] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I3 )
=> ( ( ord_less_nat @ I3 @ ( dim_row_complex @ A ) )
=> ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_complex ) ) )
=> ( upper_4850907204721561915omplex @ A ) ) ).
% upper_triangularI
thf(fact_914_upper__triangularI,axiom,
! [A: mat_nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I3 )
=> ( ( ord_less_nat @ I3 @ ( dim_row_nat @ A ) )
=> ( ( index_mat_nat @ A @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_nat ) ) )
=> ( upper_triangular_nat @ A ) ) ).
% upper_triangularI
thf(fact_915_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_916_subrelI,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ! [X: nat > nat,Y: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ S ) )
=> ( ord_le3678578370064672496at_nat @ R @ S ) ) ).
% subrelI
thf(fact_917_set__zero__plus2,axiom,
! [A: set_nat,B2: set_nat] :
( ( member_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_set_nat @ B2 @ ( plus_plus_set_nat @ A @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_918_set__plus__elim,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ X2 @ ( plus_plus_set_nat @ A @ B2 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A )
=> ~ ( member_nat @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_919_set__plus__elim,axiom,
! [X2: mat_complex,A: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ X2 @ ( plus_p4229080058245121342omplex @ A @ B2 ) )
=> ~ ! [A4: mat_complex,B4: mat_complex] :
( ( X2
= ( plus_p8323303612493835998omplex @ A4 @ B4 ) )
=> ( ( member_mat_complex @ A4 @ A )
=> ~ ( member_mat_complex @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_920_set__plus__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_921_set__plus__intro,axiom,
! [A2: mat_complex,C2: set_mat_complex,B: mat_complex,D2: set_mat_complex] :
( ( member_mat_complex @ A2 @ C2 )
=> ( ( member_mat_complex @ B @ D2 )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A2 @ B ) @ ( plus_p4229080058245121342omplex @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_922_eq__comps__singleton__elems,axiom,
! [L: list_mat_complex,A2: nat] :
( ( ( commut5736191610077499254omplex @ L )
= ( cons_nat @ A2 @ nil_nat ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ L ) )
=> ( ( nth_mat_complex @ L @ I2 )
= ( nth_mat_complex @ L @ zero_zero_nat ) ) ) ) ).
% eq_comps_singleton_elems
thf(fact_923_eq__comps__singleton__elems,axiom,
! [L: list_complex,A2: nat] :
( ( ( commut93809757773076895omplex @ L )
= ( cons_nat @ A2 @ nil_nat ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ( nth_complex @ L @ I2 )
= ( nth_complex @ L @ zero_zero_nat ) ) ) ) ).
% eq_comps_singleton_elems
thf(fact_924_eq__comps__singleton__elems,axiom,
! [L: list_nat,A2: nat] :
( ( ( commut2436974278740741825ps_nat @ L )
= ( cons_nat @ A2 @ nil_nat ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
=> ( ( nth_nat @ L @ I2 )
= ( nth_nat @ L @ zero_zero_nat ) ) ) ) ).
% eq_comps_singleton_elems
thf(fact_925_diag__compat__diagonal,axiom,
! [B2: mat_complex,L: list_nat] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ ( dim_row_complex @ B2 ) @ ( dim_row_complex @ B2 ) ) )
=> ( ( diagonal_mat_complex @ B2 )
=> ( ( ( dim_row_complex @ B2 )
= ( groups4561878855575611511st_nat @ L ) )
=> ( commut5261563022830629508omplex @ B2 @ L ) ) ) ) ).
% diag_compat_diagonal
thf(fact_926_impossible__Cons,axiom,
! [Xs: list_complex,Ys: list_complex,X2: complex] :
( ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( Xs
!= ( cons_complex @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_927_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_928_sum__list_OCons,axiom,
! [X2: nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat @ ( cons_nat @ X2 @ Xs ) )
= ( plus_plus_nat @ X2 @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% sum_list.Cons
thf(fact_929_sorted__list__subset_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [A4: nat,As: list_nat,B4: nat,Bs: list_nat] :
( ( ( A4 = B4 )
=> ( P @ As @ ( cons_nat @ B4 @ Bs ) ) )
=> ( ( ( A4 != B4 )
=> ( ( ord_less_nat @ B4 @ A4 )
=> ( P @ ( cons_nat @ A4 @ As ) @ Bs ) ) )
=> ( P @ ( cons_nat @ A4 @ As ) @ ( cons_nat @ B4 @ Bs ) ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [A4: nat,Uv: list_nat] : ( P @ ( cons_nat @ A4 @ Uv ) @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sorted_list_subset.induct
thf(fact_930_list__encode_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_931_list__encode_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X @ Xs3 ) ) )
=> ( P @ A0 ) ) ) ).
% list_encode.induct
thf(fact_932_extract__subdiags__neq__Nil,axiom,
! [B2: mat_complex,A2: nat,L: list_nat] :
( ( commut6900707758132580272omplex @ B2 @ ( cons_nat @ A2 @ L ) )
!= nil_mat_complex ) ).
% extract_subdiags_neq_Nil
thf(fact_933_Suc__length__conv,axiom,
! [N: nat,Xs: list_complex] :
( ( ( suc @ N )
= ( size_s3451745648224563538omplex @ Xs ) )
= ( ? [Y5: complex,Ys2: list_complex] :
( ( Xs
= ( cons_complex @ Y5 @ Ys2 ) )
& ( ( size_s3451745648224563538omplex @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_934_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y5: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_935_length__Suc__conv,axiom,
! [Xs: list_complex,N: nat] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( suc @ N ) )
= ( ? [Y5: complex,Ys2: list_complex] :
( ( Xs
= ( cons_complex @ Y5 @ Ys2 ) )
& ( ( size_s3451745648224563538omplex @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_936_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_937_nth__Cons__Suc,axiom,
! [X2: complex,Xs: list_complex,N: nat] :
( ( nth_complex @ ( cons_complex @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_complex @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_938_nth__Cons__Suc,axiom,
! [X2: mat_complex,Xs: list_mat_complex,N: nat] :
( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_mat_complex @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_939_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_940_list__induct2,axiom,
! [Xs: list_complex,Ys: list_complex,P: list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( P @ nil_complex @ nil_complex )
=> ( ! [X: complex,Xs3: list_complex,Y: complex,Ys4: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_941_list__induct2,axiom,
! [Xs: list_complex,Ys: list_nat,P: list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_complex @ nil_nat )
=> ( ! [X: complex,Xs3: list_complex,Y: nat,Ys4: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_942_list__induct2,axiom,
! [Xs: list_nat,Ys: list_complex,P: list_nat > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( P @ nil_nat @ nil_complex )
=> ( ! [X: nat,Xs3: list_nat,Y: complex,Ys4: list_complex] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_943_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_944_list__induct3,axiom,
! [Xs: list_complex,Ys: list_complex,Zs: list_complex,P: list_complex > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X: complex,Xs3: list_complex,Y: complex,Ys4: list_complex,Z2: complex,Zs2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_945_list__induct3,axiom,
! [Xs: list_complex,Ys: list_complex,Zs: list_nat,P: list_complex > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X: complex,Xs3: list_complex,Y: complex,Ys4: list_complex,Z2: nat,Zs2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_946_list__induct3,axiom,
! [Xs: list_complex,Ys: list_nat,Zs: list_complex,P: list_complex > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex )
=> ( ! [X: complex,Xs3: list_complex,Y: nat,Ys4: list_nat,Z2: complex,Zs2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_947_list__induct3,axiom,
! [Xs: list_complex,Ys: list_nat,Zs: list_nat,P: list_complex > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_nat )
=> ( ! [X: complex,Xs3: list_complex,Y: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_948_list__induct3,axiom,
! [Xs: list_nat,Ys: list_complex,Zs: list_complex,P: list_nat > list_complex > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_complex )
=> ( ! [X: nat,Xs3: list_nat,Y: complex,Ys4: list_complex,Z2: complex,Zs2: list_complex] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_949_list__induct3,axiom,
! [Xs: list_nat,Ys: list_complex,Zs: list_nat,P: list_nat > list_complex > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y: complex,Ys4: list_complex,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_950_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_complex,P: list_nat > list_nat > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_complex )
=> ( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat,Z2: complex,Zs2: list_complex] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_951_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_952_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs: list_complex,Ws: list_complex,P: list_complex > list_complex > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X: complex,Xs3: list_complex,Y: complex,Ys4: list_complex,Z2: complex,Zs2: list_complex,W2: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs2 )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) @ ( cons_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_953_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs: list_complex,Ws: list_nat,P: list_complex > list_complex > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X: complex,Xs3: list_complex,Y: complex,Ys4: list_complex,Z2: complex,Zs2: list_complex,W2: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_954_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs: list_nat,Ws: list_complex,P: list_complex > list_complex > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat @ nil_complex )
=> ( ! [X: complex,Xs3: list_complex,Y: complex,Ys4: list_complex,Z2: nat,Zs2: list_nat,W2: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_955_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs: list_nat,Ws: list_nat,P: list_complex > list_complex > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat @ nil_nat )
=> ( ! [X: complex,Xs3: list_complex,Y: complex,Ys4: list_complex,Z2: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_956_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs: list_complex,Ws: list_complex,P: list_complex > list_nat > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex @ nil_complex )
=> ( ! [X: complex,Xs3: list_complex,Y: nat,Ys4: list_nat,Z2: complex,Zs2: list_complex,W2: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs2 )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) @ ( cons_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_957_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs: list_complex,Ws: list_nat,P: list_complex > list_nat > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex @ nil_nat )
=> ( ! [X: complex,Xs3: list_complex,Y: nat,Ys4: list_nat,Z2: complex,Zs2: list_complex,W2: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_958_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs: list_nat,Ws: list_complex,P: list_complex > list_nat > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_nat @ nil_complex )
=> ( ! [X: complex,Xs3: list_complex,Y: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W2: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_959_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_complex > list_nat > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: complex,Xs3: list_complex,Y: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_complex @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_960_list__induct4,axiom,
! [Xs: list_nat,Ys: list_complex,Zs: list_complex,Ws: list_complex,P: list_nat > list_complex > list_complex > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X: nat,Xs3: list_nat,Y: complex,Ys4: list_complex,Z2: complex,Zs2: list_complex,W2: complex,Ws2: list_complex] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs2 )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) @ ( cons_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_961_list__induct4,axiom,
! [Xs: list_nat,Ys: list_complex,Zs: list_complex,Ws: list_nat,P: list_nat > list_complex > list_complex > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y: complex,Ys4: list_complex,Z2: complex,Zs2: list_complex,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys4 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys4 )
= ( size_s3451745648224563538omplex @ Zs2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_complex @ Y @ Ys4 ) @ ( cons_complex @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_962_nth__Cons__0,axiom,
! [X2: complex,Xs: list_complex] :
( ( nth_complex @ ( cons_complex @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_963_nth__Cons__0,axiom,
! [X2: mat_complex,Xs: list_mat_complex] :
( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_964_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_965_list_Osimps_I1_J,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.simps(1)
thf(fact_966_not__Cons__self,axiom,
! [Xs: list_nat,X2: nat] :
( Xs
!= ( cons_nat @ X2 @ Xs ) ) ).
% not_Cons_self
thf(fact_967_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_968_induct__list012,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Y: nat,Zs2: list_nat] :
( ( P @ Zs2 )
=> ( ( P @ ( cons_nat @ Y @ Zs2 ) )
=> ( P @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_969_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X @ Xs3 ) @ nil_nat )
=> ( ! [Y: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y @ Ys4 ) )
=> ( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_970_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y5: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y5 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_971_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > nat ) > list_nat > list_nat > $o,A0: nat > nat,A1: list_nat,A22: list_nat] :
( ! [F3: nat > nat,X_1: list_nat] : ( P @ F3 @ nil_nat @ X_1 )
=> ( ! [F3: nat > nat,A4: nat,As: list_nat,Bs: list_nat] :
( ( P @ F3 @ As @ ( cons_nat @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_nat @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_972_successively_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P5: nat > nat > $o] : ( P @ P5 @ nil_nat )
=> ( ! [P5: nat > nat > $o,X: nat] : ( P @ P5 @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [P5: nat > nat > $o,X: nat,Y: nat,Xs3: list_nat] :
( ( P @ P5 @ ( cons_nat @ Y @ Xs3 ) )
=> ( P @ P5 @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_973_remdups__adj_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Y: nat,Xs3: list_nat] :
( ( ( X = Y )
=> ( P @ ( cons_nat @ X @ Xs3 ) ) )
=> ( ( ( X != Y )
=> ( P @ ( cons_nat @ Y @ Xs3 ) ) )
=> ( P @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_974_sorted__wrt_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P5: nat > nat > $o] : ( P @ P5 @ nil_nat )
=> ( ! [P5: nat > nat > $o,X: nat,Ys4: list_nat] :
( ( P @ P5 @ Ys4 )
=> ( P @ P5 @ ( cons_nat @ X @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_975_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ Y @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_976_shuffles_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( ( P @ Xs3 @ ( cons_nat @ Y @ Ys4 ) )
=> ( ( P @ ( cons_nat @ X @ Xs3 ) @ Ys4 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_977_min__list_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X: nat,Xs3: list_nat] :
( ! [X212: nat,X223: list_nat] :
( ( Xs3
= ( cons_nat @ X212 @ X223 ) )
=> ( P @ Xs3 ) )
=> ( P @ ( cons_nat @ X @ Xs3 ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_978_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_979_splice_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [X: nat,Xs3: list_nat,Ys4: list_nat] :
( ( P @ Ys4 @ Xs3 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ Ys4 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_980_list_Oinducts,axiom,
! [P: list_nat > $o,List: list_nat] :
( ( P @ nil_nat )
=> ( ! [X1: nat,X23: list_nat] :
( ( P @ X23 )
=> ( P @ ( cons_nat @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_981_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X213: nat,X224: list_nat] :
( Y3
!= ( cons_nat @ X213 @ X224 ) ) ) ).
% list.exhaust
thf(fact_982_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_983_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_984_eq__comps_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Y: nat,L2: list_nat] :
( ( P @ ( cons_nat @ Y @ L2 ) )
=> ( P @ ( cons_nat @ X @ ( cons_nat @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_985_List_Otranspose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X @ Xs3 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_986_eq__comps__neq__0,axiom,
! [A2: nat,M: list_nat,L: list_complex] :
( ( ( cons_nat @ A2 @ M )
= ( commut93809757773076895omplex @ L ) )
=> ( A2 != zero_zero_nat ) ) ).
% eq_comps_neq_0
thf(fact_987_eq__comps__neq__0,axiom,
! [A2: nat,M: list_nat,L: list_nat] :
( ( ( cons_nat @ A2 @ M )
= ( commut2436974278740741825ps_nat @ L ) )
=> ( A2 != zero_zero_nat ) ) ).
% eq_comps_neq_0
thf(fact_988_plus__coeffs_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
=> ( ! [V: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V @ Va ) )
=> ( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_989_enumerate__simps_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X2 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_990_sorted__list__subset_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [A4: nat,As: list_nat,B4: nat,Bs: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ ( cons_nat @ B4 @ Bs ) ) )
=> ( ! [Uu: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Uu ) )
=> ~ ! [A4: nat,Uv: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ Uv ) @ nil_nat ) ) ) ) ).
% sorted_list_subset.cases
thf(fact_991_subset__eq__mset__impl_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ~ ! [X: nat,Xs3: list_nat,Ys4: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ Ys4 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_992_shuffles_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ( ! [Xs3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
=> ~ ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_993_pderiv__coeffs__code_Ocases,axiom,
! [X2: produc4575160907756185873st_nat] :
( ! [F3: nat,X: nat,Xs3: list_nat] :
( X2
!= ( produc8282810413953273033st_nat @ F3 @ ( cons_nat @ X @ Xs3 ) ) )
=> ~ ! [F3: nat] :
( X2
!= ( produc8282810413953273033st_nat @ F3 @ nil_nat ) ) ) ).
% pderiv_coeffs_code.cases
thf(fact_994_plus__coeffs_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Xs3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
=> ( ! [V: nat,Va: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V @ Va ) ) )
=> ~ ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_995_successively_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P5: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
=> ( ! [P5: nat > nat > $o,X: nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ nil_nat ) ) )
=> ~ ! [P5: nat > nat > $o,X: nat,Y: nat,Xs3: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_996_sorted__wrt_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P5: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
=> ~ ! [P5: nat > nat > $o,X: nat,Ys4: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_997_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_complex] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s3451745648224563538omplex @ Xs ) )
= ( ? [X4: complex,Ys2: list_complex] :
( ( Xs
= ( cons_complex @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s3451745648224563538omplex @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_998_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_999_eq__comps__elem__le__length,axiom,
! [A2: nat,M: list_nat,L: list_complex] :
( ( ( cons_nat @ A2 @ M )
= ( commut93809757773076895omplex @ L ) )
=> ( ord_less_eq_nat @ A2 @ ( size_s3451745648224563538omplex @ L ) ) ) ).
% eq_comps_elem_le_length
thf(fact_1000_eq__comps__elem__le__length,axiom,
! [A2: nat,M: list_nat,L: list_nat] :
( ( ( cons_nat @ A2 @ M )
= ( commut2436974278740741825ps_nat @ L ) )
=> ( ord_less_eq_nat @ A2 @ ( size_size_list_nat @ L ) ) ) ).
% eq_comps_elem_le_length
thf(fact_1001_eq__comps__singleton,axiom,
! [A2: nat,L: list_complex] :
( ( ( cons_nat @ A2 @ nil_nat )
= ( commut93809757773076895omplex @ L ) )
=> ( A2
= ( size_s3451745648224563538omplex @ L ) ) ) ).
% eq_comps_singleton
thf(fact_1002_eq__comps__singleton,axiom,
! [A2: nat,L: list_nat] :
( ( ( cons_nat @ A2 @ nil_nat )
= ( commut2436974278740741825ps_nat @ L ) )
=> ( A2
= ( size_size_list_nat @ L ) ) ) ).
% eq_comps_singleton
thf(fact_1003_list_Osize_I4_J,axiom,
! [X21: complex,X222: list_complex] :
( ( size_s3451745648224563538omplex @ ( cons_complex @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_s3451745648224563538omplex @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1004_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1005_length__Cons,axiom,
! [X2: complex,Xs: list_complex] :
( ( size_s3451745648224563538omplex @ ( cons_complex @ X2 @ Xs ) )
= ( suc @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_Cons
thf(fact_1006_length__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_1007_diag__diff__hd__diff,axiom,
! [D2: mat_complex,A2: nat,Xs: list_nat,N: nat,I: nat,J: nat] :
( ( commut4502369927624756007omplex @ D2 @ ( cons_nat @ A2 @ Xs ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ J )
=> ( ( ord_less_nat @ J @ N )
=> ( ( index_mat_complex @ D2 @ ( product_Pair_nat_nat @ I @ I ) )
!= ( index_mat_complex @ D2 @ ( product_Pair_nat_nat @ J @ J ) ) ) ) ) ) ) ) ).
% diag_diff_hd_diff
thf(fact_1008_commute__diag__compat,axiom,
! [D2: mat_complex,N: nat,B2: mat_complex,L: list_nat] :
( ( diagonal_mat_complex @ D2 )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( times_8009071140041733218omplex @ B2 @ D2 )
= ( times_8009071140041733218omplex @ D2 @ B2 ) )
=> ( ( commut4502369927624756007omplex @ D2 @ L )
=> ( commut5261563022830629508omplex @ B2 @ L ) ) ) ) ) ) ).
% commute_diag_compat
thf(fact_1009_expand__powers_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [N2: nat,A4: nat,Ps: list_P6011104703257516679at_nat] :
( ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N2 @ A4 ) @ Ps ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N2 ) @ A4 ) @ Ps ) ) )
=> ( ! [A4: nat,Ps: list_P6011104703257516679at_nat] :
( ( P @ Ps )
=> ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A4 ) @ Ps ) ) )
=> ( P @ A0 ) ) ) ) ).
% expand_powers.induct
thf(fact_1010_expand__powers_Ocases,axiom,
! [X2: list_P6011104703257516679at_nat] :
( ( X2 != nil_Pr5478986624290739719at_nat )
=> ( ! [N2: nat,A4: nat,Ps: list_P6011104703257516679at_nat] :
( X2
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N2 ) @ A4 ) @ Ps ) )
=> ~ ! [A4: nat,Ps: list_P6011104703257516679at_nat] :
( X2
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A4 ) @ Ps ) ) ) ) ).
% expand_powers.cases
thf(fact_1011_expand__powers_Oelims,axiom,
! [X2: list_P6011104703257516679at_nat,Y3: list_nat] :
( ( ( missin6482572040563731271rs_nat @ X2 )
= Y3 )
=> ( ( ( X2 = nil_Pr5478986624290739719at_nat )
=> ( Y3 != nil_nat ) )
=> ( ! [N2: nat,A4: nat,Ps: list_P6011104703257516679at_nat] :
( ( X2
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N2 ) @ A4 ) @ Ps ) )
=> ( Y3
!= ( cons_nat @ A4 @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N2 @ A4 ) @ Ps ) ) ) ) )
=> ~ ! [A4: nat,Ps: list_P6011104703257516679at_nat] :
( ( X2
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A4 ) @ Ps ) )
=> ( Y3
!= ( missin6482572040563731271rs_nat @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_1012_expand__powers_Osimps_I1_J,axiom,
( ( missin6482572040563731271rs_nat @ nil_Pr5478986624290739719at_nat )
= nil_nat ) ).
% expand_powers.simps(1)
thf(fact_1013_expand__powers_Osimps_I3_J,axiom,
! [A2: nat,Ps2: list_P6011104703257516679at_nat] :
( ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A2 ) @ Ps2 ) )
= ( missin6482572040563731271rs_nat @ Ps2 ) ) ).
% expand_powers.simps(3)
thf(fact_1014_longest__common__prefix_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) )
=> ( ! [Uv: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
=> ~ ! [Uu: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Uu @ nil_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_1015_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1016_longest__common__prefix_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X: nat,Xs3: list_nat,Y: nat,Ys4: list_nat] :
( ( ( X = Y )
=> ( P @ Xs3 @ Ys4 ) )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Uu: list_nat] : ( P @ Uu @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_1017_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_1018_eq__comps__compare,axiom,
! [L: list_nat,A2: nat,M: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
=> ( ( ( cons_nat @ A2 @ M )
= ( commut2436974278740741825ps_nat @ L ) )
=> ( ( ord_less_nat @ I @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
=> ( ord_less_nat @ ( nth_nat @ L @ I ) @ ( nth_nat @ L @ J ) ) ) ) ) ) ) ).
% eq_comps_compare
thf(fact_1019_Cons__lenlex__iff,axiom,
! [M: complex,Ms: list_complex,N: complex,Ns: list_complex,R: set_Pr5085853215250843933omplex] :
( ( member6068360845309590790omplex @ ( produc2490022270806822549omplex @ ( cons_complex @ M @ Ms ) @ ( cons_complex @ N @ Ns ) ) @ ( lenlex_complex @ R ) )
= ( ( ord_less_nat @ ( size_s3451745648224563538omplex @ Ms ) @ ( size_s3451745648224563538omplex @ Ns ) )
| ( ( ( size_s3451745648224563538omplex @ Ms )
= ( size_s3451745648224563538omplex @ Ns ) )
& ( member5793383173714906214omplex @ ( produc101793102246108661omplex @ M @ N ) @ R ) )
| ( ( M = N )
& ( member6068360845309590790omplex @ ( produc2490022270806822549omplex @ Ms @ Ns ) @ ( lenlex_complex @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1020_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1021_sorted2,axiom,
! [X2: nat,Y3: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y3 @ Zs ) ) )
= ( ( ord_less_eq_nat @ X2 @ Y3 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y3 @ Zs ) ) ) ) ).
% sorted2
thf(fact_1022_sorted2__simps_I1_J,axiom,
! [X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted2_simps(1)
thf(fact_1023_sorted__wrt1,axiom,
! [P: nat > nat > $o,X2: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1024_sorted__wrt__nth__less,axiom,
! [P: mat_complex > mat_complex > $o,Xs: list_mat_complex,I: nat,J: nat] :
( ( sorted7304172569062733844omplex @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( P @ ( nth_mat_complex @ Xs @ I ) @ ( nth_mat_complex @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1025_sorted__wrt__nth__less,axiom,
! [P: complex > complex > $o,Xs: list_complex,I: nat,J: nat] :
( ( sorted_wrt_complex @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P @ ( nth_complex @ Xs @ I ) @ ( nth_complex @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1026_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1027_sorted__wrt__iff__nth__less,axiom,
( sorted7304172569062733844omplex
= ( ^ [P4: mat_complex > mat_complex > $o,Xs2: list_mat_complex] :
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( P4 @ ( nth_mat_complex @ Xs2 @ I4 ) @ ( nth_mat_complex @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1028_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_complex
= ( ^ [P4: complex > complex > $o,Xs2: list_complex] :
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P4 @ ( nth_complex @ Xs2 @ I4 ) @ ( nth_complex @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1029_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P4: nat > nat > $o,Xs2: list_nat] :
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P4 @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1030_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_1031_sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted_simps(1)
thf(fact_1032_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1033_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1034_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_1035_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_1036_lenlex__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_1037_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1038_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1039_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1040_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1041_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1042_lenlex__length,axiom,
! [Ms: list_complex,Ns: list_complex,R: set_Pr5085853215250843933omplex] :
( ( member6068360845309590790omplex @ ( produc2490022270806822549omplex @ Ms @ Ns ) @ ( lenlex_complex @ R ) )
=> ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Ms ) @ ( size_s3451745648224563538omplex @ Ns ) ) ) ).
% lenlex_length
thf(fact_1043_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ a ) ) ) @ n ).
%------------------------------------------------------------------------------