TPTP Problem File: SLH0774^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Commuting_Hermitian/0002_Commuting_Hermitian/prob_01377_056656__19475414_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1324 ( 519 unt; 248 typ; 0 def)
% Number of atoms : 2674 (1383 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10596 ( 412 ~; 56 |; 154 &;8513 @)
% ( 0 <=>;1461 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 57 ( 56 usr)
% Number of type conns : 699 ( 699 >; 0 *; 0 +; 0 <<)
% Number of symbols : 195 ( 192 usr; 43 con; 0-6 aty)
% Number of variables : 3337 ( 57 ^;3215 !; 65 ?;3337 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:37:32.895
%------------------------------------------------------------------------------
% Could-be-implicit typings (56)
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thf(ty_n_t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
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% Explicit typings (192)
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thf(sy_c_Commuting__Hermitian__Misc_On__sum_001t__Complex__Ocomplex,type,
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thf(sy_c_Gauss__Jordan__Elimination_Opivot__positions__main__gen_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_OPair_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__List__Olist_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_J,type,
produc5287803926896192989omplex: mat_complex > list_P7556462501187456557omplex > produc4940938296835214317omplex ).
thf(sy_c_Product__Type_OPair_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__List__Olist_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_J_J,type,
produc240115804040334493omplex: mat_complex > list_P6834526116611618029omplex > produc1686404485030744877omplex ).
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__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc2861545499953221015omplex: mat_complex > produc352478934956084711omplex > produc5677646155008957607omplex ).
thf(sy_c_Product__Type_OPair_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
produc1901862033385395287omplex: mat_complex > produc5677646155008957607omplex > produc1634985270395358183omplex ).
thf(sy_c_Product__Type_OPair_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc8200022702634260657at_nat: mat_complex > product_prod_nat_nat > produc6744924073142918721at_nat ).
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__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc6109913384486294878at_nat: nat > list_P6011104703257516679at_nat > produc8472197452120411308at_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc1709345877921393766at_nat: nat > produc2687737633280426365at_nat > produc6121082497140218670at_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_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
produc2291548248119593221at_nat: nat > produc6121082497140218670at_nat > produc5405368317271509971at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
produc7837566107596912789t_real: nat > real > produc7716430852924023517t_real ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1593612501639298397at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > produc7489448085829838189at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc8183940560428367229omplex: product_prod_nat_nat > mat_complex > produc8449972529899364365omplex ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc6350711070570205562at_nat: product_prod_nat_nat > nat > produc8373899037510109440at_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc6156676138143019412at_nat: produc8199716216217303280at_nat > nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc9163778666669654339omplex: produc352478934956084711omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc2697000228617323907omplex: produc5677646155008957607omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
produc8911724726559533635omplex: produc1634985270395358183omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > 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__Matrix__Omat_It__Real__Oreal_J,type,
collect_mat_real: ( mat_real > $o ) > set_mat_real ).
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__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_v_B,type,
b: mat_complex ).
thf(sy_v_B1____,type,
b1: mat_complex ).
thf(sy_v_B2____,type,
b2: mat_complex ).
thf(sy_v_B3____,type,
b3: mat_complex ).
thf(sy_v_B4____,type,
b4: mat_complex ).
thf(sy_v_Ba____,type,
ba: mat_complex ).
thf(sy_v_a____,type,
a: nat ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_l,type,
l: list_nat ).
thf(sy_v_la____,type,
la: list_nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_na____,type,
na: nat ).
% Relevant facts (1068)
thf(fact_0_Suc_I4_J,axiom,
la != nil_nat ).
% Suc(4)
thf(fact_1__092_060open_062extract__subdiags_AB4_A_Itl_Al_J_A_B_Ai_A_E_E_A_Ij_M_Aj_J_A_061_Adiag__mat_AB4_A_B_A_In__sum_Ai_A_Itl_Al_J_A_L_Aj_J_092_060close_062,axiom,
( ( index_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ b4 @ ( tl_nat @ la ) ) @ ia ) @ ( product_Pair_nat_nat @ j @ j ) )
= ( nth_complex @ ( diag_mat_complex @ b4 ) @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) @ j ) ) ) ).
% \<open>extract_subdiags B4 (tl l) ! i $$ (j, j) = diag_mat B4 ! (n_sum i (tl l) + j)\<close>
thf(fact_2_add__Pair,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_3_add__Pair,axiom,
! [A: mat_complex,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( plus_p6104634242915576478omplex @ ( produc3658446505030690647omplex @ A @ B ) @ ( produc3658446505030690647omplex @ C @ D ) )
= ( produc3658446505030690647omplex @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p8323303612493835998omplex @ B @ D ) ) ) ).
% add_Pair
thf(fact_4_add__Pair,axiom,
! [A: mat_complex,B: produc352478934956084711omplex,C: mat_complex,D: produc352478934956084711omplex] :
( ( plus_p4451486566768871134omplex @ ( produc2861545499953221015omplex @ A @ B ) @ ( produc2861545499953221015omplex @ C @ D ) )
= ( produc2861545499953221015omplex @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p6104634242915576478omplex @ B @ D ) ) ) ).
% add_Pair
thf(fact_5_add__Pair,axiom,
! [A: mat_complex,B: produc5677646155008957607omplex,C: mat_complex,D: produc5677646155008957607omplex] :
( ( plus_p1405069618600264606omplex @ ( produc1901862033385395287omplex @ A @ B ) @ ( produc1901862033385395287omplex @ C @ D ) )
= ( produc1901862033385395287omplex @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p4451486566768871134omplex @ B @ D ) ) ) ).
% add_Pair
thf(fact_6_add__Pair,axiom,
! [A: nat,B: mat_complex,C: nat,D: mat_complex] :
( ( plus_p8221215230258962133omplex @ ( produc4998868960714853886omplex @ A @ B ) @ ( produc4998868960714853886omplex @ C @ D ) )
= ( produc4998868960714853886omplex @ ( plus_plus_nat @ A @ C ) @ ( plus_p8323303612493835998omplex @ B @ D ) ) ) ).
% add_Pair
thf(fact_7_add__Pair,axiom,
! [A: mat_complex,B: nat,C: mat_complex,D: nat] :
( ( plus_p679445643052534703ex_nat @ ( produc3916067632315525152ex_nat @ A @ B ) @ ( produc3916067632315525152ex_nat @ C @ D ) )
= ( produc3916067632315525152ex_nat @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_8_add__Pair,axiom,
! [A: nat,B: product_prod_nat_nat,C: nat,D: product_prod_nat_nat] :
( ( plus_p6845766728538896175at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ C @ D ) )
= ( produc487386426758144856at_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_p9057090461656269880at_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_9_add__Pair,axiom,
! [A: product_prod_nat_nat,B: nat,C: product_prod_nat_nat,D: nat] :
( ( plus_p7971253712506197257at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ C @ D ) )
= ( produc6350711070570205562at_nat @ ( plus_p9057090461656269880at_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_10_add__Pair,axiom,
! [A: product_prod_nat_nat,B: mat_complex,C: product_prod_nat_nat,D: mat_complex] :
( ( plus_p4712150957712810180omplex @ ( produc8183940560428367229omplex @ A @ B ) @ ( produc8183940560428367229omplex @ C @ D ) )
= ( produc8183940560428367229omplex @ ( plus_p9057090461656269880at_nat @ A @ C ) @ ( plus_p8323303612493835998omplex @ B @ D ) ) ) ).
% add_Pair
thf(fact_11_add__Pair,axiom,
! [A: mat_complex,B: product_prod_nat_nat,C: mat_complex,D: product_prod_nat_nat] :
( ( plus_p3007102500956364536at_nat @ ( produc8200022702634260657at_nat @ A @ B ) @ ( produc8200022702634260657at_nat @ C @ D ) )
= ( produc8200022702634260657at_nat @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p9057090461656269880at_nat @ B @ D ) ) ) ).
% add_Pair
thf(fact_12__092_060open_062n__sum_A_ISuc_Ai_J_Al_A_L_Aj_A_060_An__sum_A_ISuc_A_ISuc_Ai_J_J_Al_092_060close_062,axiom,
ord_less_nat @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ( suc @ ia ) @ la ) @ j ) @ ( commut2019222099004354946um_nat @ ( suc @ ( suc @ ia ) ) @ la ) ).
% \<open>n_sum (Suc i) l + j < n_sum (Suc (Suc i)) l\<close>
thf(fact_13_calculation,axiom,
( ( index_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ ba @ la ) @ ( suc @ ia ) ) @ ( product_Pair_nat_nat @ j @ j ) )
= ( nth_complex @ ( diag_mat_complex @ b4 ) @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) @ j ) ) ) ).
% calculation
thf(fact_14__092_060open_062a_A_L_An__sum_Ai_A_Itl_Al_J_A_L_Aj_A_060_An_092_060close_062,axiom,
ord_less_nat @ ( plus_plus_nat @ ( plus_plus_nat @ a @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) ) @ j ) @ na ).
% \<open>a + n_sum i (tl l) + j < n\<close>
thf(fact_15__092_060open_062a_A_L_An__sum_Ai_A_Itl_Al_J_A_L_Aj_A_060_An__sum_A_ISuc_A_ISuc_Ai_J_J_Al_092_060close_062,axiom,
ord_less_nat @ ( plus_plus_nat @ ( plus_plus_nat @ a @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) ) @ j ) @ ( commut2019222099004354946um_nat @ ( suc @ ( suc @ ia ) ) @ la ) ).
% \<open>a + n_sum i (tl l) + j < n_sum (Suc (Suc i)) l\<close>
thf(fact_16__092_060open_062extract__subdiags_AB_Al_A_B_ASuc_Ai_A_E_E_A_Ij_M_Aj_J_A_061_Aextract__subdiags_AB4_A_Itl_Al_J_A_B_Ai_A_E_E_A_Ij_M_Aj_J_092_060close_062,axiom,
( ( index_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ ba @ la ) @ ( suc @ ia ) ) @ ( product_Pair_nat_nat @ j @ j ) )
= ( index_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ b4 @ ( tl_nat @ la ) ) @ ia ) @ ( product_Pair_nat_nat @ j @ j ) ) ) ).
% \<open>extract_subdiags B l ! Suc i $$ (j, j) = extract_subdiags B4 (tl l) ! i $$ (j, j)\<close>
thf(fact_17_bezw_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [X2: nat,Y: nat] :
( X
!= ( product_Pair_nat_nat @ X2 @ Y ) ) ).
% bezw.cases
thf(fact_18_assms_I3_J,axiom,
l != nil_nat ).
% assms(3)
thf(fact_19_Suc_I3_J,axiom,
ord_less_nat @ zero_zero_nat @ na ).
% Suc(3)
thf(fact_20__092_060open_062extract__subdiags_AB_Al_A_B_ASuc_Ai_A_061_Aextract__subdiags_AB4_A_Itl_Al_J_A_B_Ai_092_060close_062,axiom,
( ( nth_mat_complex @ ( commut6900707758132580272omplex @ ba @ la ) @ ( suc @ ia ) )
= ( nth_mat_complex @ ( commut6900707758132580272omplex @ b4 @ ( tl_nat @ la ) ) @ ia ) ) ).
% \<open>extract_subdiags B l ! Suc i = extract_subdiags B4 (tl l) ! i\<close>
thf(fact_21_Suc_I2_J,axiom,
member_mat_complex @ ba @ ( carrier_mat_complex @ na @ na ) ).
% Suc(2)
thf(fact_22_Suc_Oprems_I4_J,axiom,
ord_less_nat @ ( suc @ ia ) @ ( size_size_list_nat @ la ) ).
% Suc.prems(4)
thf(fact_23_Suc_Oprems_I5_J,axiom,
ord_less_nat @ j @ ( nth_nat @ la @ ( suc @ ia ) ) ).
% Suc.prems(5)
thf(fact_24_pivot__positions__main__gen_Oinduct,axiom,
! [Nr: nat,Nc: nat,A2: mat_real,Zero: real,P: nat > nat > $o,A0: nat,A1: nat] :
( ! [I: nat,J: nat] :
( ( ( ord_less_nat @ I @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= Zero )
=> ( P @ I @ ( suc @ J ) ) ) ) )
=> ( ( ( ord_less_nat @ I @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
!= Zero )
=> ( P @ ( suc @ I ) @ ( suc @ J ) ) ) ) )
=> ( P @ I @ J ) ) )
=> ( P @ A0 @ A1 ) ) ).
% pivot_positions_main_gen.induct
thf(fact_25_pivot__positions__main__gen_Oinduct,axiom,
! [Nr: nat,Nc: nat,A2: mat_nat,Zero: nat,P: nat > nat > $o,A0: nat,A1: nat] :
( ! [I: nat,J: nat] :
( ( ( ord_less_nat @ I @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= Zero )
=> ( P @ I @ ( suc @ J ) ) ) ) )
=> ( ( ( ord_less_nat @ I @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
!= Zero )
=> ( P @ ( suc @ I ) @ ( suc @ J ) ) ) ) )
=> ( P @ I @ J ) ) )
=> ( P @ A0 @ A1 ) ) ).
% pivot_positions_main_gen.induct
thf(fact_26_pivot__positions__main__gen_Oinduct,axiom,
! [Nr: nat,Nc: nat,A2: mat_complex,Zero: complex,P: nat > nat > $o,A0: nat,A1: nat] :
( ! [I: nat,J: nat] :
( ( ( ord_less_nat @ I @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= Zero )
=> ( P @ I @ ( suc @ J ) ) ) ) )
=> ( ( ( ord_less_nat @ I @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
!= Zero )
=> ( P @ ( suc @ I ) @ ( suc @ J ) ) ) ) )
=> ( P @ I @ J ) ) )
=> ( P @ A0 @ A1 ) ) ).
% pivot_positions_main_gen.induct
thf(fact_27__092_060open_062n__sum_Ai_A_Itl_Al_J_A_L_Aj_A_060_An_A_N_Aa_092_060close_062,axiom,
ord_less_nat @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) @ j ) @ ( minus_minus_nat @ na @ a ) ).
% \<open>n_sum i (tl l) + j < n - a\<close>
thf(fact_28__092_060open_062l_A_061_Aa_A_D_Atl_Al_092_060close_062,axiom,
( la
= ( cons_nat @ a @ ( tl_nat @ la ) ) ) ).
% \<open>l = a # tl l\<close>
thf(fact_29__092_060open_062a_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ a @ na ).
% \<open>a \<le> n\<close>
thf(fact_30_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_31_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_32_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_33_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_34_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_35_lift__Suc__mono__less,axiom,
! [F: nat > complex,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_complex @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_complex @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_36_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_37_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_38_lift__Suc__mono__less__iff,axiom,
! [F: nat > complex,N: nat,M: nat] :
( ! [N4: nat] : ( ord_less_complex @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_complex @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_39_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_40_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_41_a__def,axiom,
( a
= ( hd_nat @ la ) ) ).
% a_def
thf(fact_42_list_Osel_I2_J,axiom,
( ( tl_complex @ nil_complex )
= nil_complex ) ).
% list.sel(2)
thf(fact_43_list_Osel_I2_J,axiom,
( ( tl_mat_complex @ nil_mat_complex )
= nil_mat_complex ) ).
% list.sel(2)
thf(fact_44_list_Osel_I2_J,axiom,
( ( tl_real @ nil_real )
= nil_real ) ).
% list.sel(2)
thf(fact_45_list_Osel_I2_J,axiom,
( ( tl_lis4587848864434500726at_nat @ nil_li8973309667444810893at_nat )
= nil_li8973309667444810893at_nat ) ).
% list.sel(2)
thf(fact_46_list_Osel_I2_J,axiom,
( ( tl_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% list.sel(2)
thf(fact_47_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_48_list_Osel_I2_J,axiom,
( ( tl_Pro4228036916689694320at_nat @ nil_Pr5478986624290739719at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% list.sel(2)
thf(fact_49_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(2)
thf(fact_50_assms_I1_J,axiom,
member_mat_complex @ b @ ( carrier_mat_complex @ n @ n ) ).
% assms(1)
thf(fact_51_Suc_I8_J,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ la ) )
=> ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ la @ J2 ) ) ) ).
% Suc(8)
thf(fact_52_assms_I7_J,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ l ) )
=> ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ l @ J2 ) ) ) ).
% assms(7)
thf(fact_53_assms_I4_J,axiom,
ord_less_nat @ i @ ( size_size_list_nat @ l ) ).
% assms(4)
thf(fact_54_assms_I5_J,axiom,
ord_less_nat @ j @ ( nth_nat @ l @ i ) ).
% assms(5)
thf(fact_55_mem__Collect__eq,axiom,
! [A: mat_real,P: mat_real > $o] :
( ( member_mat_real @ A @ ( collect_mat_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A @ ( collect_mat_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_mat_real] :
( ( collect_mat_real
@ ^ [X3: mat_real] : ( member_mat_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X3: mat_complex] : ( member_mat_complex @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__cong,axiom,
! [P: mat_complex > $o,Q2: mat_complex > $o] :
( ! [X2: mat_complex] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_mat_complex @ P )
= ( collect_mat_complex @ Q2 ) ) ) ).
% Collect_cong
thf(fact_60_False,axiom,
( ( size_size_list_nat @ la )
!= one_one_nat ) ).
% False
thf(fact_61__092_060open_062B4_A_092_060in_062_Acarrier__mat_A_In_A_N_Aa_J_A_In_A_N_Aa_J_092_060close_062,axiom,
member_mat_complex @ b4 @ ( carrier_mat_complex @ ( minus_minus_nat @ na @ a ) @ ( minus_minus_nat @ na @ a ) ) ).
% \<open>B4 \<in> carrier_mat (n - a) (n - a)\<close>
thf(fact_62__092_060open_062B1_A_092_060in_062_Acarrier__mat_Aa_Aa_092_060close_062,axiom,
member_mat_complex @ b1 @ ( carrier_mat_complex @ a @ a ) ).
% \<open>B1 \<in> carrier_mat a a\<close>
thf(fact_63_Suc_I7_J,axiom,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ la ) @ na ).
% Suc(7)
thf(fact_64_list_Osize_I3_J,axiom,
( ( size_s4563920969210122745at_nat @ nil_li8973309667444810893at_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_65_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_66_list_Osize_I3_J,axiom,
( ( size_s5969786470865220249omplex @ nil_mat_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_67_list_Osize_I3_J,axiom,
( ( size_s3451745648224563538omplex @ nil_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_68_list_Osize_I3_J,axiom,
( ( size_size_list_real @ nil_real )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_69_list_Osize_I3_J,axiom,
( ( size_s5460976970255530739at_nat @ nil_Pr5478986624290739719at_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_70_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_71_List_Otranspose_Ocases,axiom,
! [X: list_list_complex] :
( ( X != nil_list_complex )
=> ( ! [Xss: list_list_complex] :
( X
!= ( cons_list_complex @ nil_complex @ Xss ) )
=> ~ ! [X2: complex,Xs: list_complex,Xss: list_list_complex] :
( X
!= ( cons_list_complex @ ( cons_complex @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_72_List_Otranspose_Ocases,axiom,
! [X: list_l5436439031154120755omplex] :
( ( X != nil_list_mat_complex )
=> ( ! [Xss: list_l5436439031154120755omplex] :
( X
!= ( cons_l4198107141827137507omplex @ nil_mat_complex @ Xss ) )
=> ~ ! [X2: mat_complex,Xs: list_mat_complex,Xss: list_l5436439031154120755omplex] :
( X
!= ( cons_l4198107141827137507omplex @ ( cons_mat_complex @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_73_List_Otranspose_Ocases,axiom,
! [X: list_list_real] :
( ( X != nil_list_real )
=> ( ! [Xss: list_list_real] :
( X
!= ( cons_list_real @ nil_real @ Xss ) )
=> ~ ! [X2: real,Xs: list_real,Xss: list_list_real] :
( X
!= ( cons_list_real @ ( cons_real @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_74_List_Otranspose_Ocases,axiom,
! [X: list_l4521591454032021523at_nat] :
( ( X != nil_li626268335206573715at_nat )
=> ( ! [Xss: list_l4521591454032021523at_nat] :
( X
!= ( cons_l6443770723755874883at_nat @ nil_li8973309667444810893at_nat @ Xss ) )
=> ~ ! [X2: list_P6011104703257516679at_nat,Xs: list_l3264859301627795341at_nat,Xss: list_l4521591454032021523at_nat] :
( X
!= ( cons_l6443770723755874883at_nat @ ( cons_l7612840610449961021at_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_75_List_Otranspose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X2: list_nat,Xs: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_76_List_Otranspose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_77_List_Otranspose_Ocases,axiom,
! [X: list_l3264859301627795341at_nat] :
( ( X != nil_li8973309667444810893at_nat )
=> ( ! [Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ Xss ) )
=> ~ ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_78_lift__Suc__mono__le,axiom,
! [F: nat > complex,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_complex @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_complex @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_79_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_80_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_81_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_82_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_83_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_84_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_85_nth__Cons__0,axiom,
! [X: complex,Xs2: list_complex] :
( ( nth_complex @ ( cons_complex @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_86_nth__Cons__0,axiom,
! [X: mat_complex,Xs2: list_mat_complex] :
( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_87_nth__Cons__0,axiom,
! [X: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_88_nth__Cons__0,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_89_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_90_le__diff__conv,axiom,
! [J3: nat,K3: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K3 ) @ I2 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I2 @ K3 ) ) ) ).
% le_diff_conv
thf(fact_91_list__induct2,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_92_list__induct2,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_nat,P: list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: nat,Ys2: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_93_list__induct2,axiom,
! [Xs2: list_nat,Ys: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_94_list__induct2,axiom,
! [Xs2: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_95_list__induct3,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_96_list__induct3,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_97_list__induct3,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: nat,Ys2: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_98_list__induct3,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: nat,Ys2: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_99_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_100_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat,P: list_nat > list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_101_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat,P: list_nat > list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_102_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_103_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( 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_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_104_list__induct4,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( 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_Pr5478986624290739719at_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_105_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_P6011104703257516679at_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_106_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat,Ws: list_nat,P: list_nat > list_nat > list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( ( size_s5460976970255530739at_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_107_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_P6011104703257516679at_nat,P: list_nat > list_nat > list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s5460976970255530739at_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: product_prod_nat_nat,Ws2: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s5460976970255530739at_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_P6512896166579812791at_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_108_list__induct4,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat,Ws: list_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_109_list__induct4,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat,Ws: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( ( size_s5460976970255530739at_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: nat,Ys2: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat,W: nat,Ws2: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_110_list__induct4,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_nat,Ws: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s5460976970255530739at_nat @ Ws ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: product_prod_nat_nat,Ws2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s5460976970255530739at_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_P6512896166579812791at_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_111_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ws: list_nat,P: list_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( ( size_s5460976970255530739at_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_112_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat,Ws: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s5460976970255530739at_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat,W: product_prod_nat_nat,Ws2: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s5460976970255530739at_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_P6512896166579812791at_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_113_Nat_Ole__diff__conv2,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J3 @ K3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_114_less__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_115_length__0__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_Pr5478986624290739719at_nat ) ) ).
% length_0_conv
thf(fact_116_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_117_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J3 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J3 ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_118_Nat_Odiff__add__assoc,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J3 ) @ K3 )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J3 @ K3 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_119_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_120_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_121_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K3 ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I2 ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_122_Nat_Odiff__add__assoc2,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I2 ) @ K3 )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K3 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_123_Nat_Odiff__diff__right,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J3 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K3 ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_124_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_125_Suc__length__conv,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat] :
( ( ( suc @ N )
= ( size_s5460976970255530739at_nat @ Xs2 ) )
= ( ? [Y2: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ Y2 @ Ys3 ) )
& ( ( size_s5460976970255530739at_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_126_Suc__length__conv,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs2 ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_127_length__Suc__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat,N: nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y2: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ Y2 @ Ys3 ) )
& ( ( size_s5460976970255530739at_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_128_length__Suc__conv,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_129_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s5460976970255530739at_nat @ Xs2 ) )
= ( ? [X3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_130_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
= ( ? [X3: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_131_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I2 )
= K3 )
= ( J3
= ( plus_plus_nat @ K3 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_132_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y3 ) )
=> ( X != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_133_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_134_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_135_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_136_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_137_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_138_impossible__Cons,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ ( size_s5460976970255530739at_nat @ Ys ) )
=> ( Xs2
!= ( cons_P6512896166579812791at_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_139_impossible__Cons,axiom,
! [Xs2: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs2
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_140_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_141_not__Cons__self2,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( cons_P6512896166579812791at_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_142_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_143_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_144_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_145_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_146_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_147_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_148_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_149_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_150_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_151_diff__commute,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J3 ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K3 ) @ J3 ) ) ).
% diff_commute
thf(fact_152_le__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_153_eq__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ( minus_minus_nat @ M @ K3 )
= ( minus_minus_nat @ N @ K3 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_154_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_155_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_156_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_157_le__trans,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K3 )
=> ( ord_less_eq_nat @ I2 @ K3 ) ) ) ).
% le_trans
thf(fact_158_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_159_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_160_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_161_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_162_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_163_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_164_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_165_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_166_list_Oinject,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,Y21: product_prod_nat_nat,Y22: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X21 @ X22 )
= ( cons_P6512896166579812791at_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_167_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_168_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_169_list_Osel_I1_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
( ( hd_Pro3460610213475200108at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_170_list_Osize_I4_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
( ( size_s5460976970255530739at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s5460976970255530739at_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_171_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_172_diff__Pair,axiom,
! [A: mat_complex,B: produc5677646155008957607omplex,C: mat_complex,D: produc5677646155008957607omplex] :
( ( minus_5093045068546291278omplex @ ( produc1901862033385395287omplex @ A @ B ) @ ( produc1901862033385395287omplex @ C @ D ) )
= ( produc1901862033385395287omplex @ ( minus_2412168080157227406omplex @ A @ C ) @ ( minus_4882995375997288846omplex @ B @ D ) ) ) ).
% diff_Pair
thf(fact_173_diff__Pair,axiom,
! [A: mat_complex,B: produc352478934956084711omplex,C: mat_complex,D: produc352478934956084711omplex] :
( ( minus_4882995375997288846omplex @ ( produc2861545499953221015omplex @ A @ B ) @ ( produc2861545499953221015omplex @ C @ D ) )
= ( produc2861545499953221015omplex @ ( minus_2412168080157227406omplex @ A @ C ) @ ( minus_2734116836287720782omplex @ B @ D ) ) ) ).
% diff_Pair
thf(fact_174_diff__Pair,axiom,
! [A: mat_complex,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( minus_2734116836287720782omplex @ ( produc3658446505030690647omplex @ A @ B ) @ ( produc3658446505030690647omplex @ C @ D ) )
= ( produc3658446505030690647omplex @ ( minus_2412168080157227406omplex @ A @ C ) @ ( minus_2412168080157227406omplex @ B @ D ) ) ) ).
% diff_Pair
thf(fact_175_diff__Pair,axiom,
! [A: nat > nat,B: nat,C: nat > nat,D: nat] :
( ( minus_9067931446424981591at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ ( produc72220940542539688at_nat @ C @ D ) )
= ( produc72220940542539688at_nat @ ( minus_minus_nat_nat @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).
% diff_Pair
thf(fact_176_diff__Pair,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).
% diff_Pair
thf(fact_177_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K2: nat,M3: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ K2 )
=> ( P @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) )
=> ( P @ K2 @ M3 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_178_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_179_zero__prod__def,axiom,
( zero_z738777567634093332t_real
= ( produc7837566107596912789t_real @ zero_zero_nat @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_180_zero__prod__def,axiom,
( zero_z5987101913011988884al_nat
= ( produc3181502643871035669al_nat @ zero_zero_real @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_181_zero__prod__def,axiom,
( zero_z1365759597461889520l_real
= ( produc4511245868158468465l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_182_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs: list_nat] :
( ! [Ys4: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_183_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_184_list__nonempty__induct,axiom,
! [Xs2: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_185_induct__list012,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,Zs2: list_nat] :
( ( P @ Zs2 )
=> ( ( P @ ( cons_nat @ Y @ Zs2 ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Zs2 ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% induct_list012
thf(fact_186_induct__list012,axiom,
! [P: list_P6011104703257516679at_nat > $o,Xs2: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
( ( P @ Zs2 )
=> ( ( P @ ( cons_P6512896166579812791at_nat @ Y @ Zs2 ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y @ Zs2 ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% induct_list012
thf(fact_187_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_188_list__induct2_H,axiom,
! [P: list_nat > list_P6011104703257516679at_nat > $o,Xs2: list_nat,Ys: list_P6011104703257516679at_nat] :
( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_Pr5478986624290739719at_nat )
=> ( ! [Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] : ( P @ nil_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_189_list__induct2_H,axiom,
! [P: list_P6011104703257516679at_nat > list_nat > $o,Xs2: list_P6011104703257516679at_nat,Ys: list_nat] :
( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_190_list__induct2_H,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ nil_Pr5478986624290739719at_nat )
=> ( ! [Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_191_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y2: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_192_neq__Nil__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
= ( ? [Y2: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( Xs2
= ( cons_P6512896166579812791at_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_193_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > nat ) > list_nat > list_nat > $o,A0: nat > nat,A1: list_nat,A22: list_nat] :
( ! [F2: nat > nat,X_1: list_nat] : ( P @ F2 @ nil_nat @ X_1 )
=> ( ! [F2: nat > nat,A3: nat,As: list_nat,Bs: list_nat] :
( ( P @ F2 @ As @ ( cons_nat @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_nat @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_194_map__tailrec__rev_Oinduct,axiom,
! [P: ( product_prod_nat_nat > nat ) > list_P6011104703257516679at_nat > list_nat > $o,A0: product_prod_nat_nat > nat,A1: list_P6011104703257516679at_nat,A22: list_nat] :
( ! [F2: product_prod_nat_nat > nat,X_1: list_nat] : ( P @ F2 @ nil_Pr5478986624290739719at_nat @ X_1 )
=> ( ! [F2: product_prod_nat_nat > nat,A3: product_prod_nat_nat,As: list_P6011104703257516679at_nat,Bs: list_nat] :
( ( P @ F2 @ As @ ( cons_nat @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_P6512896166579812791at_nat @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_195_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > product_prod_nat_nat ) > list_nat > list_P6011104703257516679at_nat > $o,A0: nat > product_prod_nat_nat,A1: list_nat,A22: list_P6011104703257516679at_nat] :
( ! [F2: nat > product_prod_nat_nat,X_1: list_P6011104703257516679at_nat] : ( P @ F2 @ nil_nat @ X_1 )
=> ( ! [F2: nat > product_prod_nat_nat,A3: nat,As: list_nat,Bs: list_P6011104703257516679at_nat] :
( ( P @ F2 @ As @ ( cons_P6512896166579812791at_nat @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_nat @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_196_map__tailrec__rev_Oinduct,axiom,
! [P: ( product_prod_nat_nat > product_prod_nat_nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,A0: product_prod_nat_nat > product_prod_nat_nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat] :
( ! [F2: product_prod_nat_nat > product_prod_nat_nat,X_1: list_P6011104703257516679at_nat] : ( P @ F2 @ nil_Pr5478986624290739719at_nat @ X_1 )
=> ( ! [F2: product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,As: list_P6011104703257516679at_nat,Bs: list_P6011104703257516679at_nat] :
( ( P @ F2 @ As @ ( cons_P6512896166579812791at_nat @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_P6512896166579812791at_nat @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_197_successively_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
=> ( ! [P2: nat > nat > $o,X2: nat] : ( P @ P2 @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [P2: nat > nat > $o,X2: nat,Y: nat,Xs: list_nat] :
( ( P @ P2 @ ( cons_nat @ Y @ Xs ) )
=> ( P @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_198_successively_Oinduct,axiom,
! [P: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > $o,A0: product_prod_nat_nat > product_prod_nat_nat > $o,A1: list_P6011104703257516679at_nat] :
( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o] : ( P @ P2 @ nil_Pr5478986624290739719at_nat )
=> ( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat] : ( P @ P2 @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( P @ P2 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) )
=> ( P @ P2 @ ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_199_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P2: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
=> ( ! [P2: nat > nat > $o,X2: nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ nil_nat ) ) )
=> ~ ! [P2: nat > nat > $o,X2: nat,Y: nat,Xs: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_200_successively_Ocases,axiom,
! [X: produc2366258654402830848at_nat] :
( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( X
!= ( produc3352296309980913008at_nat @ P2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
( X
!= ( produc3352296309980913008at_nat @ P2 @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) ) )
=> ~ ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( X
!= ( produc3352296309980913008at_nat @ P2 @ ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_201_remdups__adj_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,Xs: list_nat] :
( ( ( X2 = Y )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons_nat @ Y @ Xs ) ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_202_remdups__adj_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( ( X2 = Y )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_203_sorted__wrt_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
=> ( ! [P2: nat > nat > $o,X2: nat,Ys2: list_nat] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_nat @ X2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_204_sorted__wrt_Oinduct,axiom,
! [P: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > $o,A0: product_prod_nat_nat > product_prod_nat_nat > $o,A1: list_P6011104703257516679at_nat] :
( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o] : ( P @ P2 @ nil_Pr5478986624290739719at_nat )
=> ( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_P6512896166579812791at_nat @ X2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_205_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_206_remdups__adj_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ( X != nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_207_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P2: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
=> ~ ! [P2: nat > nat > $o,X2: nat,Ys2: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_208_sorted__wrt_Ocases,axiom,
! [X: produc2366258654402830848at_nat] :
( ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( X
!= ( produc3352296309980913008at_nat @ P2 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( X
!= ( produc3352296309980913008at_nat @ P2 @ ( cons_P6512896166579812791at_nat @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_209_shuffles_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Xs: list_nat] : ( P @ Xs @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ ( cons_nat @ Y @ Ys2 ) )
=> ( ( P @ ( cons_nat @ X2 @ Xs ) @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_210_shuffles_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat] :
( ! [X_1: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ X_1 )
=> ( ! [Xs: list_P6011104703257516679at_nat] : ( P @ Xs @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) )
=> ( ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ Ys2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_211_min__list_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( ! [X212: nat,X222: list_nat] :
( ( Xs
= ( cons_nat @ X212 @ X222 ) )
=> ( P @ Xs ) )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_212_min__list_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ! [X212: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ X212 @ X222 ) )
=> ( P @ Xs ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_213_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ( ! [Xs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
=> ~ ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_214_shuffles_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [Ys2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys2 ) )
=> ( ! [Xs: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ Xs @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_215_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_216_min__list_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
=> ( X = nil_Pr5478986624290739719at_nat ) ) ).
% min_list.cases
thf(fact_217_splice_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [X2: nat,Xs: list_nat,Ys2: list_nat] :
( ( P @ Ys2 @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_218_splice_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat] :
( ! [X_1: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ X_1 )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ Ys2 @ Xs )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_219_splice_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ~ ! [X2: nat,Xs: list_nat,Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_220_splice_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [Ys2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys2 ) )
=> ~ ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_221_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_222_list_Oinducts,axiom,
! [P: list_P6011104703257516679at_nat > $o,List: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X1: product_prod_nat_nat,X23: list_P6011104703257516679at_nat] :
( ( P @ X23 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_223_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X213: nat,X223: list_nat] :
( Y3
!= ( cons_nat @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_224_list_Oexhaust,axiom,
! [Y3: list_P6011104703257516679at_nat] :
( ( Y3 != nil_Pr5478986624290739719at_nat )
=> ~ ! [X213: product_prod_nat_nat,X223: list_P6011104703257516679at_nat] :
( Y3
!= ( cons_P6512896166579812791at_nat @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_225_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_226_list_OdiscI,axiom,
! [List: list_P6011104703257516679at_nat,X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
( ( List
= ( cons_P6512896166579812791at_nat @ X21 @ X22 ) )
=> ( List != nil_Pr5478986624290739719at_nat ) ) ).
% list.discI
thf(fact_227_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_228_list_Odistinct_I1_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
( nil_Pr5478986624290739719at_nat
!= ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_229_less__imp__diff__less,axiom,
! [J3: nat,K3: nat,N: nat] :
( ( ord_less_nat @ J3 @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_230_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_231_zero__induct__lemma,axiom,
! [P: nat > $o,K3: nat,I2: nat] :
( ( P @ K3 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K3 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_232_Suc__diff__diff,axiom,
! [M: nat,N: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K3 ) ) ).
% Suc_diff_diff
thf(fact_233_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_234_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J3: nat] :
( ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_235_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_236_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_237_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_238_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_239_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_240_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y: nat,Z: nat] :
( ( R @ X2 @ Y )
=> ( ( R @ Y @ Z )
=> ( R @ X2 @ Z ) ) )
=> ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_241_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_242_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_243_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_244_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_245_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_246_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_247_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M3: nat] :
( M5
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_248_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_249_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_250_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_251_Nat_Odiff__cancel,axiom,
! [K3: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_252_diff__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K3 ) @ ( plus_plus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_253_diff__diff__left,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J3 ) @ K3 )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J3 @ K3 ) ) ) ).
% diff_diff_left
thf(fact_254_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_255_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_256_nat__arith_Orule0,axiom,
! [A: product_prod_nat_nat] :
( A
= ( plus_p9057090461656269880at_nat @ A @ zero_z3979849011205770936at_nat ) ) ).
% nat_arith.rule0
thf(fact_257_nat__arith_Orule0,axiom,
! [A: nat] :
( A
= ( plus_plus_nat @ A @ zero_zero_nat ) ) ).
% nat_arith.rule0
thf(fact_258_nat__arith_Orule0,axiom,
! [A: real] :
( A
= ( plus_plus_real @ A @ zero_zero_real ) ) ).
% nat_arith.rule0
thf(fact_259_add__leE,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K3 @ N ) ) ) ).
% add_leE
thf(fact_260_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_261_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_262_add__leD1,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_263_add__leD2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ K3 @ N ) ) ).
% add_leD2
thf(fact_264_le__Suc__ex,axiom,
! [K3: nat,L: nat] :
( ( ord_less_eq_nat @ K3 @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K3 @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_265_add__le__mono,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_266_add__le__mono1,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ K3 ) ) ) ).
% add_le_mono1
thf(fact_267_trans__le__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_268_trans__le__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_269_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K: nat] :
( N2
= ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_270_nat__add__left__cancel__le,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_271_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_272_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_273_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_274_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_275_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_276_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_277_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_278_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_279_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_280_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_281_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_282_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_283_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_284_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_285_zero__induct,axiom,
! [P: nat > $o,K3: nat] :
( ( P @ K3 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_286_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X2: nat,Y: nat] :
( ( P @ X2 @ Y )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_287_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_288_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat2: nat] :
( ( P @ Nat2 )
=> ( P @ ( suc @ Nat2 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_289_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat2: nat] :
( Y3
!= ( suc @ Nat2 ) ) ) ).
% old.nat.exhaust
thf(fact_290_nat_OdiscI,axiom,
! [Nat: nat,X24: nat] :
( ( Nat
= ( suc @ X24 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_291_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_292_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_293_nat_Odistinct_I1_J,axiom,
! [X24: nat] :
( zero_zero_nat
!= ( suc @ X24 ) ) ).
% nat.distinct(1)
thf(fact_294_plus__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.simps(1)
thf(fact_295_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_296_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_297_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_298_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_299_list_Osel_I3_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
( ( tl_Pro4228036916689694320at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_300_extract__subdiags__neq__Nil,axiom,
! [B2: mat_complex,A: nat,L: list_nat] :
( ( commut6900707758132580272omplex @ B2 @ ( cons_nat @ A @ L ) )
!= nil_mat_complex ) ).
% extract_subdiags_neq_Nil
thf(fact_301_hd__Cons__tl,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs2 ) @ ( tl_nat @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_302_hd__Cons__tl,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs2 ) @ ( tl_Pro4228036916689694320at_nat @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_303_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_304_list_Oexhaust__sel,axiom,
! [List: list_P6011104703257516679at_nat] :
( ( List != nil_Pr5478986624290739719at_nat )
=> ( List
= ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ List ) @ ( tl_Pro4228036916689694320at_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_305_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_306_list_Ocollapse,axiom,
! [List: list_P6011104703257516679at_nat] :
( ( List != nil_Pr5478986624290739719at_nat )
=> ( ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ List ) @ ( tl_Pro4228036916689694320at_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_307_hd__conv__nth,axiom,
! [Xs2: list_complex] :
( ( Xs2 != nil_complex )
=> ( ( hd_complex @ Xs2 )
= ( nth_complex @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_308_hd__conv__nth,axiom,
! [Xs2: list_mat_complex] :
( ( Xs2 != nil_mat_complex )
=> ( ( hd_mat_complex @ Xs2 )
= ( nth_mat_complex @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_309_hd__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ Xs2 )
= ( nth_nat @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_310_hd__conv__nth,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ Xs2 )
= ( nth_Pr7617993195940197384at_nat @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_311_less__diff__conv2,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K3 ) @ I2 )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I2 @ K3 ) ) ) ) ).
% less_diff_conv2
thf(fact_312_diff__Suc__diff__eq2,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J3 @ K3 ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J3 ) @ ( plus_plus_nat @ K3 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_313_diff__Suc__diff__eq1,axiom,
! [K3: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J3 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J3 @ K3 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( suc @ J3 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_314_length__greater__0__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) )
= ( Xs2 != nil_Pr5478986624290739719at_nat ) ) ).
% length_greater_0_conv
thf(fact_315_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_316_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_317_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_318_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_319_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_320_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_321_extract__subdiags__carrier,axiom,
! [I2: nat,L: list_nat,B2: mat_complex] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
=> ( member_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ B2 @ L ) @ I2 ) @ ( carrier_mat_complex @ ( nth_nat @ L @ I2 ) @ ( nth_nat @ L @ I2 ) ) ) ) ).
% extract_subdiags_carrier
thf(fact_322_list_Oexpand,axiom,
! [List: list_nat,List2: list_nat] :
( ( ( List = nil_nat )
= ( List2 = nil_nat ) )
=> ( ( ( List != nil_nat )
=> ( ( List2 != nil_nat )
=> ( ( ( hd_nat @ List )
= ( hd_nat @ List2 ) )
& ( ( tl_nat @ List )
= ( tl_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_323_list_Oexpand,axiom,
! [List: list_P6011104703257516679at_nat,List2: list_P6011104703257516679at_nat] :
( ( ( List = nil_Pr5478986624290739719at_nat )
= ( List2 = nil_Pr5478986624290739719at_nat ) )
=> ( ( ( List != nil_Pr5478986624290739719at_nat )
=> ( ( List2 != nil_Pr5478986624290739719at_nat )
=> ( ( ( hd_Pro3460610213475200108at_nat @ List )
= ( hd_Pro3460610213475200108at_nat @ List2 ) )
& ( ( tl_Pro4228036916689694320at_nat @ List )
= ( tl_Pro4228036916689694320at_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_324_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_325_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_326_list__encode_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] :
( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) )
=> ( P @ A0 ) ) ) ).
% list_encode.induct
thf(fact_327_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) ) ) ).
% list_encode.cases
thf(fact_328_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_complex,Z2: list_complex] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_complex,Ys3: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs3 ) )
=> ( ( nth_complex @ Xs3 @ I4 )
= ( nth_complex @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_329_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_mat_complex,Z2: list_mat_complex] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_mat_complex,Ys3: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5969786470865220249omplex @ Xs3 ) )
=> ( ( nth_mat_complex @ Xs3 @ I4 )
= ( nth_mat_complex @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_330_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z2: list_nat] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_331_Skolem__list__nth,axiom,
! [K3: nat,P: nat > complex > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ? [X4: complex] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= K3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth_complex @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_332_Skolem__list__nth,axiom,
! [K3: nat,P: nat > mat_complex > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ? [X4: mat_complex] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= K3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth_mat_complex @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_333_Skolem__list__nth,axiom,
! [K3: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ? [X4: nat] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_334_nth__equalityI,axiom,
! [Xs2: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( ( nth_complex @ Xs2 @ I )
= ( nth_complex @ Ys @ I ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_335_nth__equalityI,axiom,
! [Xs2: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs2 )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( ( nth_mat_complex @ Xs2 @ I )
= ( nth_mat_complex @ Ys @ I ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_336_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys @ I ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_337_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_338_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_339_nth__Cons__Suc,axiom,
! [X: complex,Xs2: list_complex,N: nat] :
( ( nth_complex @ ( cons_complex @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_complex @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_340_nth__Cons__Suc,axiom,
! [X: mat_complex,Xs2: list_mat_complex,N: nat] :
( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_mat_complex @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_341_nth__Cons__Suc,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_342_nth__Cons__Suc,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,N: nat] :
( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_343_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_344_less__diff__conv,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J3 @ K3 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ J3 ) ) ).
% less_diff_conv
thf(fact_345_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_346_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_347_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_348_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_349_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_350_inc__induct,axiom,
! [I2: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( P @ J3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J3 )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_351_dec__induct,axiom,
! [I2: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J3 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_352_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_353_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_354_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K3: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K3 ) @ ( F @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_355_tl__Nil,axiom,
! [Xs2: list_nat] :
( ( ( tl_nat @ Xs2 )
= nil_nat )
= ( ( Xs2 = nil_nat )
| ? [X3: nat] :
( Xs2
= ( cons_nat @ X3 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_356_tl__Nil,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( ( tl_Pro4228036916689694320at_nat @ Xs2 )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs2 = nil_Pr5478986624290739719at_nat )
| ? [X3: product_prod_nat_nat] :
( Xs2
= ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% tl_Nil
thf(fact_357_Nil__tl,axiom,
! [Xs2: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs2 ) )
= ( ( Xs2 = nil_nat )
| ? [X3: nat] :
( Xs2
= ( cons_nat @ X3 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_358_Nil__tl,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( tl_Pro4228036916689694320at_nat @ Xs2 ) )
= ( ( Xs2 = nil_Pr5478986624290739719at_nat )
| ? [X3: product_prod_nat_nat] :
( Xs2
= ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% Nil_tl
thf(fact_359_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J4: nat] :
( ( M
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_360_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_361_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_362_All__less__Suc2,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_less_Suc2
thf(fact_363_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_364_Ex__less__Suc2,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_less_Suc2
thf(fact_365_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_366_less__imp__add__positive,axiom,
! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_367_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_368_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_369_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_370_Nitpick_Osize__list__simp_I2_J,axiom,
( size_s5460976970255530739at_nat
= ( ^ [Xs3: list_P6011104703257516679at_nat] : ( if_nat @ ( Xs3 = nil_Pr5478986624290739719at_nat ) @ zero_zero_nat @ ( suc @ ( size_s5460976970255530739at_nat @ ( tl_Pro4228036916689694320at_nat @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_371_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs3: list_nat] : ( if_nat @ ( Xs3 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_372_extract__subdiags_Osimps_I1_J,axiom,
! [B2: mat_complex] :
( ( commut6900707758132580272omplex @ B2 @ nil_nat )
= nil_mat_complex ) ).
% extract_subdiags.simps(1)
thf(fact_373_nth__tl,axiom,
! [N: nat,Xs2: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ ( tl_complex @ Xs2 ) ) )
=> ( ( nth_complex @ ( tl_complex @ Xs2 ) @ N )
= ( nth_complex @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_374_nth__tl,axiom,
! [N: nat,Xs2: list_mat_complex] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ ( tl_mat_complex @ Xs2 ) ) )
=> ( ( nth_mat_complex @ ( tl_mat_complex @ Xs2 ) @ N )
= ( nth_mat_complex @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_375_nth__tl,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs2 ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_376_nat__arith_Oadd1,axiom,
! [A2: nat,K3: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K3 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% nat_arith.add1
thf(fact_377_nat__arith_Oadd1,axiom,
! [A2: product_prod_nat_nat,K3: product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( A2
= ( plus_p9057090461656269880at_nat @ K3 @ A ) )
=> ( ( plus_p9057090461656269880at_nat @ A2 @ B )
= ( plus_p9057090461656269880at_nat @ K3 @ ( plus_p9057090461656269880at_nat @ A @ B ) ) ) ) ).
% nat_arith.add1
thf(fact_378_nat__arith_Oadd2,axiom,
! [B2: nat,K3: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K3 @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_379_nat__arith_Oadd2,axiom,
! [B2: product_prod_nat_nat,K3: product_prod_nat_nat,B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( B2
= ( plus_p9057090461656269880at_nat @ K3 @ B ) )
=> ( ( plus_p9057090461656269880at_nat @ A @ B2 )
= ( plus_p9057090461656269880at_nat @ K3 @ ( plus_p9057090461656269880at_nat @ A @ B ) ) ) ) ).
% nat_arith.add2
thf(fact_380_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_381_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_382_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_383_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_384_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_385_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_386_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_387_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_388_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_389_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_390_old_Onat_Oinject,axiom,
! [Nat: nat,Nat3: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat3 ) )
= ( Nat = Nat3 ) ) ).
% old.nat.inject
thf(fact_391_nat_Oinject,axiom,
! [X24: nat,Y23: nat] :
( ( ( suc @ X24 )
= ( suc @ Y23 ) )
= ( X24 = Y23 ) ) ).
% nat.inject
thf(fact_392_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_393_strict__inc__induct,axiom,
! [I2: nat,J3: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ! [I: nat] :
( ( J3
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_394_less__Suc__induct,axiom,
! [I2: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I2 @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_395_less__trans__Suc,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K3 ) ) ) ).
% less_trans_Suc
thf(fact_396_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_397_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_398_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_399_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_400_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_401_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_402_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_403_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_404_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_405_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_406_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_407_Suc__lessE,axiom,
! [I2: nat,K3: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K3 )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( K3
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_408_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_409_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_410_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_411_Nat_OlessE,axiom,
! [I2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ( ( K3
!= ( suc @ I2 ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( K3
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_412_add__lessD1,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J3 ) @ K3 )
=> ( ord_less_nat @ I2 @ K3 ) ) ).
% add_lessD1
thf(fact_413_add__less__mono,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ K3 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_414_not__add__less1,axiom,
! [I2: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J3 ) @ I2 ) ).
% not_add_less1
thf(fact_415_not__add__less2,axiom,
! [J3: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_416_add__less__mono1,axiom,
! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ K3 ) ) ) ).
% add_less_mono1
thf(fact_417_trans__less__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_418_trans__less__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_419_less__add__eq__less,axiom,
! [K3: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K3 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_420_nat__add__left__cancel__less,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_421_plus__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% plus_nat.simps(2)
thf(fact_422_nat__arith_Osuc1,axiom,
! [A2: nat,K3: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K3 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K3 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_423_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_424_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_425_Suc_I1_J,axiom,
! [B2: mat_complex,N: nat,L: list_nat] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( L != nil_nat )
=> ( ( ord_less_nat @ ia @ ( size_size_list_nat @ L ) )
=> ( ( ord_less_nat @ j @ ( nth_nat @ L @ ia ) )
=> ( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ L ) @ N )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
=> ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ L @ J ) ) )
=> ( ( index_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ B2 @ L ) @ ia ) @ ( product_Pair_nat_nat @ j @ j ) )
= ( nth_complex @ ( diag_mat_complex @ B2 ) @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ia @ L ) @ j ) ) ) ) ) ) ) ) ) ) ).
% Suc(1)
thf(fact_426_n__sum_Oelims,axiom,
! [X: nat,Xa: list_P6011104703257516679at_nat,Y3: product_prod_nat_nat] :
( ( ( commut2293368841785035983at_nat @ X @ Xa )
= Y3 )
=> ( ( ( X = zero_zero_nat )
=> ( Y3 != zero_z3979849011205770936at_nat ) )
=> ~ ! [N4: nat] :
( ( X
= ( suc @ N4 ) )
=> ( Y3
!= ( plus_p9057090461656269880at_nat @ ( hd_Pro3460610213475200108at_nat @ Xa ) @ ( commut2293368841785035983at_nat @ N4 @ ( tl_Pro4228036916689694320at_nat @ Xa ) ) ) ) ) ) ) ).
% n_sum.elims
thf(fact_427_n__sum_Oelims,axiom,
! [X: nat,Xa: list_real,Y3: real] :
( ( ( commut5569088596779277150m_real @ X @ Xa )
= Y3 )
=> ( ( ( X = zero_zero_nat )
=> ( Y3 != zero_zero_real ) )
=> ~ ! [N4: nat] :
( ( X
= ( suc @ N4 ) )
=> ( Y3
!= ( plus_plus_real @ ( hd_real @ Xa ) @ ( commut5569088596779277150m_real @ N4 @ ( tl_real @ Xa ) ) ) ) ) ) ) ).
% n_sum.elims
thf(fact_428_n__sum_Oelims,axiom,
! [X: nat,Xa: list_nat,Y3: nat] :
( ( ( commut2019222099004354946um_nat @ X @ Xa )
= Y3 )
=> ( ( ( X = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) )
=> ~ ! [N4: nat] :
( ( X
= ( suc @ N4 ) )
=> ( Y3
!= ( plus_plus_nat @ ( hd_nat @ Xa ) @ ( commut2019222099004354946um_nat @ N4 @ ( tl_nat @ Xa ) ) ) ) ) ) ) ).
% n_sum.elims
thf(fact_429_n__sum__last__lt,axiom,
! [J3: complex,L: list_complex,I2: nat] :
( ( ord_less_complex @ J3 @ ( nth_complex @ L @ I2 ) )
=> ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ ( commut6323218633641605728omplex @ I2 @ L ) @ J3 ) @ ( commut6323218633641605728omplex @ ( suc @ I2 ) @ L ) ) ) ) ).
% n_sum_last_lt
thf(fact_430_n__sum__last__lt,axiom,
! [J3: nat,L: list_nat,I2: nat] :
( ( ord_less_nat @ J3 @ ( nth_nat @ L @ I2 ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I2 @ L ) @ J3 ) @ ( commut2019222099004354946um_nat @ ( suc @ I2 ) @ L ) ) ) ) ).
% n_sum_last_lt
thf(fact_431_n__sum__last,axiom,
! [I2: nat,L: list_complex] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ( commut6323218633641605728omplex @ ( suc @ I2 ) @ L )
= ( plus_plus_complex @ ( commut6323218633641605728omplex @ I2 @ L ) @ ( nth_complex @ L @ I2 ) ) ) ) ).
% n_sum_last
thf(fact_432_n__sum__last,axiom,
! [I2: nat,L: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ L ) )
=> ( ( commut2293368841785035983at_nat @ ( suc @ I2 ) @ L )
= ( plus_p9057090461656269880at_nat @ ( commut2293368841785035983at_nat @ I2 @ L ) @ ( nth_Pr7617993195940197384at_nat @ L @ I2 ) ) ) ) ).
% n_sum_last
thf(fact_433_n__sum__last,axiom,
! [I2: nat,L: list_nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
=> ( ( commut2019222099004354946um_nat @ ( suc @ I2 ) @ L )
= ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I2 @ L ) @ ( nth_nat @ L @ I2 ) ) ) ) ).
% n_sum_last
thf(fact_434_n__sum_Osimps_I2_J,axiom,
! [N: nat,L: list_P6011104703257516679at_nat] :
( ( commut2293368841785035983at_nat @ ( suc @ N ) @ L )
= ( plus_p9057090461656269880at_nat @ ( hd_Pro3460610213475200108at_nat @ L ) @ ( commut2293368841785035983at_nat @ N @ ( tl_Pro4228036916689694320at_nat @ L ) ) ) ) ).
% n_sum.simps(2)
thf(fact_435_n__sum_Osimps_I2_J,axiom,
! [N: nat,L: list_nat] :
( ( commut2019222099004354946um_nat @ ( suc @ N ) @ L )
= ( plus_plus_nat @ ( hd_nat @ L ) @ ( commut2019222099004354946um_nat @ N @ ( tl_nat @ L ) ) ) ) ).
% n_sum.simps(2)
thf(fact_436__092_060open_062n__sum_A_ISuc_A_ISuc_Ai_J_J_Al_A_092_060le_062_Asum__list_Al_092_060close_062,axiom,
ord_less_eq_nat @ ( commut2019222099004354946um_nat @ ( suc @ ( suc @ ia ) ) @ la ) @ ( groups4561878855575611511st_nat @ la ) ).
% \<open>n_sum (Suc (Suc i)) l \<le> sum_list l\<close>
thf(fact_437__092_060open_0621_A_060_Alength_Al_092_060close_062,axiom,
ord_less_nat @ one_one_nat @ ( size_size_list_nat @ la ) ).
% \<open>1 < length l\<close>
thf(fact_438_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,I: nat,A4: mat_complex] :
( ( ( ( index_mat_complex @ A4 @ ( product_Pair_nat_nat @ I @ I ) )
!= Ev )
=> ( P @ Ev @ I @ A4 ) )
=> ( P @ Ev @ ( suc @ I ) @ A4 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_ev.induct
thf(fact_439_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,I: nat,A4: mat_complex] :
( ( ( ( index_mat_complex @ A4 @ ( product_Pair_nat_nat @ I @ I ) )
= Ev )
=> ( P @ Ev @ I @ A4 ) )
=> ( P @ Ev @ ( suc @ I ) @ A4 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_other_ev.induct
thf(fact_440_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_441_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_442_assms_I6_J,axiom,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ l ) @ n ).
% assms(6)
thf(fact_443_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_444_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_445_find__largest__block_Ocases,axiom,
! [X: produc7489448085829838189at_nat] :
( ! [Block: product_prod_nat_nat] :
( X
!= ( produc1593612501639298397at_nat @ Block @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [M_start: nat,M_end: nat,I_start: nat,I_end: nat,Blocks: list_P6011104703257516679at_nat] :
( X
!= ( produc1593612501639298397at_nat @ ( product_Pair_nat_nat @ M_start @ M_end ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ I_start @ I_end ) @ Blocks ) ) ) ) ).
% find_largest_block.cases
thf(fact_446_sum__list__cong,axiom,
! [L: list_complex,M: list_complex] :
( ( ( size_s3451745648224563538omplex @ L )
= ( size_s3451745648224563538omplex @ M ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L ) )
=> ( ( nth_complex @ L @ I )
= ( nth_complex @ M @ I ) ) )
=> ( ( groups486868518411355989omplex @ L )
= ( groups486868518411355989omplex @ M ) ) ) ) ).
% sum_list_cong
thf(fact_447_sum__list__cong,axiom,
! [L: list_nat,M: list_nat] :
( ( ( size_size_list_nat @ L )
= ( size_size_list_nat @ M ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
=> ( ( nth_nat @ L @ I )
= ( nth_nat @ M @ I ) ) )
=> ( ( groups4561878855575611511st_nat @ L )
= ( groups4561878855575611511st_nat @ M ) ) ) ) ).
% sum_list_cong
thf(fact_448_find__largest__block_Oinduct,axiom,
! [P: product_prod_nat_nat > list_P6011104703257516679at_nat > $o,A0: product_prod_nat_nat,A1: list_P6011104703257516679at_nat] :
( ! [Block: product_prod_nat_nat] : ( P @ Block @ nil_Pr5478986624290739719at_nat )
=> ( ! [M_start: nat,M_end: nat,I_start: nat,I_end: nat,Blocks: list_P6011104703257516679at_nat] :
( ( ( ord_less_eq_nat @ ( minus_minus_nat @ M_end @ M_start ) @ ( minus_minus_nat @ I_end @ I_start ) )
=> ( P @ ( product_Pair_nat_nat @ I_start @ I_end ) @ Blocks ) )
=> ( ( ~ ( ord_less_eq_nat @ ( minus_minus_nat @ M_end @ M_start ) @ ( minus_minus_nat @ I_end @ I_start ) )
=> ( P @ ( product_Pair_nat_nat @ M_start @ M_end ) @ Blocks ) )
=> ( P @ ( product_Pair_nat_nat @ M_start @ M_end ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ I_start @ I_end ) @ Blocks ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% find_largest_block.induct
thf(fact_449_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_450_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_451_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_452_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_453_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_454_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_455_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_456_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_457_identify__block_Oinduct,axiom,
! [P: mat_complex > nat > $o,A0: mat_complex,A1: nat] :
( ! [A4: mat_complex] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: mat_complex,I: nat] :
( ( ( ( index_mat_complex @ A4 @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
= one_one_complex )
=> ( P @ A4 @ I ) )
=> ( P @ A4 @ ( suc @ I ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_458_identify__block_Oinduct,axiom,
! [P: mat_nat > nat > $o,A0: mat_nat,A1: nat] :
( ! [A4: mat_nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: mat_nat,I: nat] :
( ( ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
= one_one_nat )
=> ( P @ A4 @ I ) )
=> ( P @ A4 @ ( suc @ I ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_459_identify__block_Oinduct,axiom,
! [P: mat_real > nat > $o,A0: mat_real,A1: nat] :
( ! [A4: mat_real] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: mat_real,I: nat] :
( ( ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
= one_one_real )
=> ( P @ A4 @ I ) )
=> ( P @ A4 @ ( suc @ I ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_460_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_461_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_462_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_463_add__right__imp__eq,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ B @ A )
= ( plus_p9057090461656269880at_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_464_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_465_add__right__cancel,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ B @ A )
= ( plus_p9057090461656269880at_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_466_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_467_add__left__imp__eq,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ A @ B )
= ( plus_p9057090461656269880at_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_468_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_469_add__left__cancel,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ A @ B )
= ( plus_p9057090461656269880at_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_470_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_471_add_Oleft__commute,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( plus_p9057090461656269880at_nat @ B @ ( plus_p9057090461656269880at_nat @ A @ C ) )
= ( plus_p9057090461656269880at_nat @ A @ ( plus_p9057090461656269880at_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_472_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_473_add_Ocommute,axiom,
( plus_p9057090461656269880at_nat
= ( ^ [A5: product_prod_nat_nat,B4: product_prod_nat_nat] : ( plus_p9057090461656269880at_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_474_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_475_add_Oassoc,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( plus_p9057090461656269880at_nat @ ( plus_p9057090461656269880at_nat @ A @ B ) @ C )
= ( plus_p9057090461656269880at_nat @ A @ ( plus_p9057090461656269880at_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_476_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( I2 = J3 )
& ( K3 = L ) )
=> ( ( plus_plus_nat @ I2 @ K3 )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_477_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_478_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( plus_p9057090461656269880at_nat @ ( plus_p9057090461656269880at_nat @ A @ B ) @ C )
= ( plus_p9057090461656269880at_nat @ A @ ( plus_p9057090461656269880at_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_479_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_480_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_481_length__tl,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_482_sum__list__geq__0,axiom,
! [L: list_complex] :
( ( L != nil_complex )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3451745648224563538omplex @ L ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( nth_complex @ L @ J ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( groups486868518411355989omplex @ L ) ) ) ) ).
% sum_list_geq_0
thf(fact_483_sum__list__geq__0,axiom,
! [L: list_real] :
( ( L != nil_real )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_real @ L ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( nth_real @ L @ J ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6723090944982001619t_real @ L ) ) ) ) ).
% sum_list_geq_0
thf(fact_484_sum__list__geq__0,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% sum_list_geq_0
thf(fact_485_sum__list__tl__leq,axiom,
! [L: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ L ) @ N )
=> ( ( L != nil_nat )
=> ( ( ord_less_eq_nat @ ( hd_nat @ L ) @ N )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( tl_nat @ L ) ) @ ( minus_minus_nat @ N @ ( hd_nat @ L ) ) ) ) ) ) ).
% sum_list_tl_leq
thf(fact_486_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_487_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_488_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_489_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_490_nth__Cons_H,axiom,
! [N: nat,X: complex,Xs2: list_complex] :
( ( ( N = zero_zero_nat )
=> ( ( nth_complex @ ( cons_complex @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_complex @ ( cons_complex @ X @ Xs2 ) @ N )
= ( nth_complex @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_491_nth__Cons_H,axiom,
! [N: nat,X: mat_complex,Xs2: list_mat_complex] :
( ( ( N = zero_zero_nat )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs2 ) @ N )
= ( nth_mat_complex @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_492_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_493_nth__Cons_H,axiom,
! [N: nat,X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ N )
= ( nth_Pr7617993195940197384at_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_494_sum__list__geq__tl,axiom,
! [L: list_complex] :
( ( L != nil_complex )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3451745648224563538omplex @ L ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( nth_complex @ L @ J ) ) )
=> ( ord_less_eq_complex @ ( groups486868518411355989omplex @ ( tl_complex @ L ) ) @ ( groups486868518411355989omplex @ L ) ) ) ) ).
% sum_list_geq_tl
thf(fact_495_sum__list__geq__tl,axiom,
! [L: list_real] :
( ( L != nil_real )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_real @ L ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( nth_real @ L @ J ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ ( tl_real @ L ) ) @ ( groups6723090944982001619t_real @ L ) ) ) ) ).
% sum_list_geq_tl
thf(fact_496_sum__list__geq__tl,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( tl_nat @ L ) ) @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% sum_list_geq_tl
thf(fact_497_n__sum__sum__list,axiom,
! [I2: nat,L: list_complex] :
( ( ord_less_eq_nat @ I2 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3451745648224563538omplex @ L ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( nth_complex @ L @ J ) ) )
=> ( ord_less_eq_complex @ ( commut6323218633641605728omplex @ I2 @ L ) @ ( groups486868518411355989omplex @ L ) ) ) ) ).
% n_sum_sum_list
thf(fact_498_n__sum__sum__list,axiom,
! [I2: nat,L: list_real] :
( ( ord_less_eq_nat @ I2 @ ( size_size_list_real @ L ) )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_real @ L ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( nth_real @ L @ J ) ) )
=> ( ord_less_eq_real @ ( commut5569088596779277150m_real @ I2 @ L ) @ ( groups6723090944982001619t_real @ L ) ) ) ) ).
% n_sum_sum_list
thf(fact_499_n__sum__sum__list,axiom,
! [I2: nat,L: list_nat] :
( ( ord_less_eq_nat @ I2 @ ( size_size_list_nat @ L ) )
=> ( ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J ) ) )
=> ( ord_less_eq_nat @ ( commut2019222099004354946um_nat @ I2 @ L ) @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% n_sum_sum_list
thf(fact_500_nth__non__equal__first__eq,axiom,
! [X: complex,Y3: complex,Xs2: list_complex,N: nat] :
( ( X != Y3 )
=> ( ( ( nth_complex @ ( cons_complex @ X @ Xs2 ) @ N )
= Y3 )
= ( ( ( nth_complex @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_501_nth__non__equal__first__eq,axiom,
! [X: mat_complex,Y3: mat_complex,Xs2: list_mat_complex,N: nat] :
( ( X != Y3 )
=> ( ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs2 ) @ N )
= Y3 )
= ( ( ( nth_mat_complex @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_502_nth__non__equal__first__eq,axiom,
! [X: nat,Y3: nat,Xs2: list_nat,N: nat] :
( ( X != Y3 )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= Y3 )
= ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_503_nth__non__equal__first__eq,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,N: nat] :
( ( X != Y3 )
=> ( ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ N )
= Y3 )
= ( ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_504_nth__Cons__pos,axiom,
! [N: nat,X: complex,Xs2: list_complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_complex @ ( cons_complex @ X @ Xs2 ) @ N )
= ( nth_complex @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_505_nth__Cons__pos,axiom,
! [N: nat,X: mat_complex,Xs2: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs2 ) @ N )
= ( nth_mat_complex @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_506_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_507_nth__Cons__pos,axiom,
! [N: nat,X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ N )
= ( nth_Pr7617993195940197384at_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_508_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_509_zero__order_I1_J,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_order(1)
thf(fact_510_zero__order_I5_J,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_511_zero__order_I4_J,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_order(4)
thf(fact_512_zero__order_I3_J,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% zero_order(3)
thf(fact_513_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_514_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_515_add__0,axiom,
! [A: product_prod_nat_nat] :
( ( plus_p9057090461656269880at_nat @ zero_z3979849011205770936at_nat @ A )
= A ) ).
% add_0
thf(fact_516_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_517_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_518_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_519_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_520_add__cancel__right__right,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( A
= ( plus_p9057090461656269880at_nat @ A @ B ) )
= ( B = zero_z3979849011205770936at_nat ) ) ).
% add_cancel_right_right
thf(fact_521_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_522_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_523_add__cancel__right__left,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( A
= ( plus_p9057090461656269880at_nat @ B @ A ) )
= ( B = zero_z3979849011205770936at_nat ) ) ).
% add_cancel_right_left
thf(fact_524_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_525_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_526_add__cancel__left__right,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ A @ B )
= A )
= ( B = zero_z3979849011205770936at_nat ) ) ).
% add_cancel_left_right
thf(fact_527_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_528_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_529_add__cancel__left__left,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ B @ A )
= A )
= ( B = zero_z3979849011205770936at_nat ) ) ).
% add_cancel_left_left
thf(fact_530_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_531_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_532_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_533_double__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_zero
thf(fact_534_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_535_add_Oright__neutral,axiom,
! [A: product_prod_nat_nat] :
( ( plus_p9057090461656269880at_nat @ A @ zero_z3979849011205770936at_nat )
= A ) ).
% add.right_neutral
thf(fact_536_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_537_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_538_comm__monoid__add__class_Oadd__0,axiom,
! [A: product_prod_nat_nat] :
( ( plus_p9057090461656269880at_nat @ zero_z3979849011205770936at_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_539_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_540_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_541_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_542_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_543_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_544_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_545_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_546_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_547_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_548_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_549_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C2: nat] :
( B4
= ( plus_plus_nat @ A5 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_550_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_551_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_552_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_553_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_554_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_555_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_556_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_557_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_558_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J3: real,K3: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J3 )
& ( ord_less_eq_real @ K3 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K3 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_559_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( I2 = J3 )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_560_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J3: real,K3: real,L: real] :
( ( ( I2 = J3 )
& ( ord_less_eq_real @ K3 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K3 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_561_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( K3 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_562_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J3: real,K3: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J3 )
& ( K3 = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K3 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_563_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_564_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_565_right__minus__eq,axiom,
! [A: real,B: real] :
( ( ( minus_minus_real @ A @ B )
= zero_zero_real )
= ( A = B ) ) ).
% right_minus_eq
thf(fact_566_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_567_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_568_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_569_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_570_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_571_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_572_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_573_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_574_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_575_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_576_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_577_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_578_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_579_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_580_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_581_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_582_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J3 )
& ( K3 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_583_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( I2 = J3 )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_584_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J3 )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_585_diff__diff__eq,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
( ( minus_4365393887724441320at_nat @ ( minus_4365393887724441320at_nat @ A @ B ) @ C )
= ( minus_4365393887724441320at_nat @ A @ ( plus_p9057090461656269880at_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_586_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_587_add__diff__cancel__right_H,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( minus_4365393887724441320at_nat @ ( plus_p9057090461656269880at_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_588_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_589_add__diff__cancel__right,axiom,
! [A: product_prod_nat_nat,C: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( minus_4365393887724441320at_nat @ ( plus_p9057090461656269880at_nat @ A @ C ) @ ( plus_p9057090461656269880at_nat @ B @ C ) )
= ( minus_4365393887724441320at_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_590_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_591_add__diff__cancel__left_H,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( minus_4365393887724441320at_nat @ ( plus_p9057090461656269880at_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_592_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_593_add__diff__cancel__left,axiom,
! [C: product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( minus_4365393887724441320at_nat @ ( plus_p9057090461656269880at_nat @ C @ A ) @ ( plus_p9057090461656269880at_nat @ C @ B ) )
= ( minus_4365393887724441320at_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_594_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_595_add__implies__diff,axiom,
! [C: product_prod_nat_nat,B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ C @ B )
= A )
=> ( C
= ( minus_4365393887724441320at_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_596_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_597_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_598_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_599_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_600_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_601_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_602_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_603_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_604_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_605_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_606_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_607_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_608_add__nonpos__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_609_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_610_add__nonneg__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_611_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_612_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_613_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_614_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_615_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_616_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_617_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_618_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_619_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_620_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_621_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_622_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_623_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_624_diff__le__0__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ B ) ) ).
% diff_le_0_iff_le
thf(fact_625_add__sign__intros_I6_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(6)
thf(fact_626_add__sign__intros_I6_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(6)
thf(fact_627_add__sign__intros_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_sign_intros(2)
thf(fact_628_add__sign__intros_I2_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_sign_intros(2)
thf(fact_629_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_630_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_631_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_632_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_633_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_634_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_635_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_636_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_637_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_638_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_639_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_640_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_641_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_642_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_643_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J3: real,K3: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J3 )
& ( ord_less_real @ K3 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K3 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_644_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J3: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J3 )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_645_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J3: real,K3: real,L: real] :
( ( ( ord_less_real @ I2 @ J3 )
& ( ord_less_eq_real @ K3 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K3 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_646_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_647_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_648_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_649_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_650_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_651_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_652_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_653_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_654_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_655_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_656_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_657_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_658_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_659_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_660_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_661_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_662_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_663_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_664_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_665_n__sum_Oinduct,axiom,
! [P: nat > list_nat > $o,A0: nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ zero_zero_nat @ X_1 )
=> ( ! [N4: nat,L2: list_nat] :
( ( P @ N4 @ ( tl_nat @ L2 ) )
=> ( P @ ( suc @ N4 ) @ L2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% n_sum.induct
thf(fact_666_n__sum_Osimps_I1_J,axiom,
! [L: list_real] :
( ( commut5569088596779277150m_real @ zero_zero_nat @ L )
= zero_zero_real ) ).
% n_sum.simps(1)
thf(fact_667_n__sum_Osimps_I1_J,axiom,
! [L: list_nat] :
( ( commut2019222099004354946um_nat @ zero_zero_nat @ L )
= zero_zero_nat ) ).
% n_sum.simps(1)
thf(fact_668_add__sign__intros_I7_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(7)
thf(fact_669_add__sign__intros_I7_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(7)
thf(fact_670_add__sign__intros_I5_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(5)
thf(fact_671_add__sign__intros_I5_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(5)
thf(fact_672_add__sign__intros_I3_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_sign_intros(3)
thf(fact_673_add__sign__intros_I3_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_sign_intros(3)
thf(fact_674_add__sign__intros_I1_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_sign_intros(1)
thf(fact_675_add__sign__intros_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_sign_intros(1)
thf(fact_676_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_677_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_678_elem__le__sum__list,axiom,
! [K3: nat,Ns: list_nat] :
( ( ord_less_nat @ K3 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K3 ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_679_sum__list__mono2,axiom,
! [Xs2: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( ord_less_eq_complex @ ( nth_complex @ Xs2 @ I ) @ ( nth_complex @ Ys @ I ) ) )
=> ( ord_less_eq_complex @ ( groups486868518411355989omplex @ Xs2 ) @ ( groups486868518411355989omplex @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_680_sum__list__mono2,axiom,
! [Xs2: list_real,Ys: list_real] :
( ( ( size_size_list_real @ Xs2 )
= ( size_size_list_real @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs2 @ I ) @ ( nth_real @ Ys @ I ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ Xs2 ) @ ( groups6723090944982001619t_real @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_681_sum__list__mono2,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_682_kuhn__lemma,axiom,
! [P3: nat,N: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P3 )
=> ( ! [X2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_nat @ ( X2 @ I3 ) @ P3 ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( Label @ X2 @ I )
= zero_zero_nat )
| ( ( Label @ X2 @ I )
= one_one_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_nat @ ( X2 @ I3 ) @ P3 ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( X2 @ I )
= zero_zero_nat )
=> ( ( Label @ X2 @ I )
= zero_zero_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_nat @ ( X2 @ I3 ) @ P3 ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ( X2 @ I )
= P3 )
=> ( ( Label @ X2 @ I )
= one_one_nat ) ) ) )
=> ~ ! [Q: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_nat @ ( Q @ I3 ) @ P3 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [R2: nat > nat] :
( ! [J2: nat] :
( ( ord_less_nat @ J2 @ N )
=> ( ( ord_less_eq_nat @ ( Q @ J2 ) @ ( R2 @ J2 ) )
& ( ord_less_eq_nat @ ( R2 @ J2 ) @ ( plus_plus_nat @ ( Q @ J2 ) @ one_one_nat ) ) ) )
& ? [S2: nat > nat] :
( ! [J2: nat] :
( ( ord_less_nat @ J2 @ N )
=> ( ( ord_less_eq_nat @ ( Q @ J2 ) @ ( S2 @ J2 ) )
& ( ord_less_eq_nat @ ( S2 @ J2 ) @ ( plus_plus_nat @ ( Q @ J2 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I3 )
!= ( Label @ S2 @ I3 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_683_pderiv__coeffs__code_Oinduct,axiom,
! [P: real > list_real > $o,A0: real,A1: list_real] :
( ! [F2: real,X2: real,Xs: list_real] :
( ( P @ ( plus_plus_real @ F2 @ one_one_real ) @ Xs )
=> ( P @ F2 @ ( cons_real @ X2 @ Xs ) ) )
=> ( ! [F2: real] : ( P @ F2 @ nil_real )
=> ( P @ A0 @ A1 ) ) ) ).
% pderiv_coeffs_code.induct
thf(fact_684_pderiv__coeffs__code_Oinduct,axiom,
! [P: nat > list_nat > $o,A0: nat,A1: list_nat] :
( ! [F2: nat,X2: nat,Xs: list_nat] :
( ( P @ ( plus_plus_nat @ F2 @ one_one_nat ) @ Xs )
=> ( P @ F2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ! [F2: nat] : ( P @ F2 @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ).
% pderiv_coeffs_code.induct
thf(fact_685_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I: nat] :
( ( Q2 @ I )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I ) )
& ( ord_less_eq_real @ ( X2 @ I ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I3 ) @ one_one_nat )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I3 )
= zero_zero_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= one_one_real ) )
=> ( ( L2 @ X5 @ I3 )
= one_one_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L2 @ X5 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I3 ) @ ( F @ X5 @ I3 ) ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L2 @ X5 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I3 ) @ ( X5 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_686_plus__coeffs_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [Xs: list_nat] : ( P @ Xs @ nil_nat )
=> ( ! [V: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V @ Va ) )
=> ( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_687_plus__coeffs_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat] :
( ! [Xs: list_P6011104703257516679at_nat] : ( P @ Xs @ nil_Pr5478986624290739719at_nat )
=> ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_688_sum__list__simps_I1_J,axiom,
( ( groups4206474380581351322at_nat @ nil_Pr5478986624290739719at_nat )
= zero_z3979849011205770936at_nat ) ).
% sum_list_simps(1)
thf(fact_689_sum__list__simps_I1_J,axiom,
( ( groups6723090944982001619t_real @ nil_real )
= zero_zero_real ) ).
% sum_list_simps(1)
thf(fact_690_sum__list__simps_I1_J,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list_simps(1)
thf(fact_691_pderiv__coeffs__code_Ocases,axiom,
! [X: produc4575160907756185873st_nat] :
( ! [F2: nat,X2: nat,Xs: list_nat] :
( X
!= ( produc8282810413953273033st_nat @ F2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ~ ! [F2: nat] :
( X
!= ( produc8282810413953273033st_nat @ F2 @ nil_nat ) ) ) ).
% pderiv_coeffs_code.cases
thf(fact_692_sum__list__simps_I2_J,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( groups4206474380581351322at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) )
= ( plus_p9057090461656269880at_nat @ X @ ( groups4206474380581351322at_nat @ Xs2 ) ) ) ).
% sum_list_simps(2)
thf(fact_693_sum__list__simps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( groups4561878855575611511st_nat @ ( cons_nat @ X @ Xs2 ) )
= ( plus_plus_nat @ X @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ).
% sum_list_simps(2)
thf(fact_694_plus__coeffs_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Xs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
=> ( ! [V: nat,Va: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V @ Va ) ) )
=> ~ ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_695_plus__coeffs_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [Xs: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ Xs @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) ) )
=> ~ ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_696_expand__powers_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [N4: nat,A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N4 @ A3 ) @ Ps ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N4 ) @ A3 ) @ Ps ) ) )
=> ( ! [A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( P @ Ps )
=> ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A3 ) @ Ps ) ) )
=> ( P @ A0 ) ) ) ) ).
% expand_powers.induct
thf(fact_697_expand__powers_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ( X != nil_Pr5478986624290739719at_nat )
=> ( ! [N4: nat,A3: nat,Ps: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N4 ) @ A3 ) @ Ps ) )
=> ~ ! [A3: nat,Ps: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A3 ) @ Ps ) ) ) ) ).
% expand_powers.cases
thf(fact_698_inverse__permutation__of__list_Ocases,axiom,
! [X: produc4244290832573747233omplex] :
( ! [X2: mat_complex] :
( X
!= ( produc25957052209774481omplex @ nil_Pr795171647102378477omplex @ X2 ) )
=> ~ ! [Y: mat_complex,X6: mat_complex,Xs: list_P2712776532842825709omplex,X2: mat_complex] :
( X
!= ( produc25957052209774481omplex @ ( cons_P6830038620677733789omplex @ ( produc3658446505030690647omplex @ Y @ X6 ) @ Xs ) @ X2 ) ) ) ).
% inverse_permutation_of_list.cases
thf(fact_699_inverse__permutation__of__list_Ocases,axiom,
! [X: produc4008378413191047942at_nat] :
( ! [X2: nat] :
( X
!= ( produc8424349340415155968at_nat @ nil_Pr5478986624290739719at_nat @ X2 ) )
=> ~ ! [Y: nat,X6: nat,Xs: list_P6011104703257516679at_nat,X2: nat] :
( X
!= ( produc8424349340415155968at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y @ X6 ) @ Xs ) @ X2 ) ) ) ).
% inverse_permutation_of_list.cases
thf(fact_700_delete__aux_Ocases,axiom,
! [X: produc1686404485030744877omplex] :
( ! [K2: mat_complex] :
( X
!= ( produc240115804040334493omplex @ K2 @ nil_Pr8537966556735835629omplex ) )
=> ~ ! [K2: mat_complex,K4: mat_complex,V: produc5677646155008957607omplex,Xs: list_P6834526116611618029omplex] :
( X
!= ( produc240115804040334493omplex @ K2 @ ( cons_P2742075696876001693omplex @ ( produc1901862033385395287omplex @ K4 @ V ) @ Xs ) ) ) ) ).
% delete_aux.cases
thf(fact_701_delete__aux_Ocases,axiom,
! [X: produc4940938296835214317omplex] :
( ! [K2: mat_complex] :
( X
!= ( produc5287803926896192989omplex @ K2 @ nil_Pr9007294345934890925omplex ) )
=> ~ ! [K2: mat_complex,K4: mat_complex,V: produc352478934956084711omplex,Xs: list_P7556462501187456557omplex] :
( X
!= ( produc5287803926896192989omplex @ K2 @ ( cons_P5134161419496323933omplex @ ( produc2861545499953221015omplex @ K4 @ V ) @ Xs ) ) ) ) ).
% delete_aux.cases
thf(fact_702_delete__aux_Ocases,axiom,
! [X: produc6632627751064020269omplex] :
( ! [K2: mat_complex] :
( X
!= ( produc1895390367874281373omplex @ K2 @ nil_Pr795171647102378477omplex ) )
=> ~ ! [K2: mat_complex,K4: mat_complex,V: mat_complex,Xs: list_P2712776532842825709omplex] :
( X
!= ( produc1895390367874281373omplex @ K2 @ ( cons_P6830038620677733789omplex @ ( produc3658446505030690647omplex @ K4 @ V ) @ Xs ) ) ) ) ).
% delete_aux.cases
thf(fact_703_delete__aux_Ocases,axiom,
! [X: produc8472197452120411308at_nat] :
( ! [K2: nat] :
( X
!= ( produc6109913384486294878at_nat @ K2 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [K2: nat,K4: nat,V: nat,Xs: list_P6011104703257516679at_nat] :
( X
!= ( produc6109913384486294878at_nat @ K2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ Xs ) ) ) ) ).
% delete_aux.cases
thf(fact_704_delete__aux_Ocases,axiom,
! [X: produc1458239053189343596at_nat] :
( ! [K2: nat > nat] :
( X
!= ( produc17425388850148510at_nat @ K2 @ nil_Pr2223394031645031670at_nat ) )
=> ~ ! [K2: nat > nat,K4: nat > nat,V: nat,Xs: list_P5366131564312172150at_nat] :
( X
!= ( produc17425388850148510at_nat @ K2 @ ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K4 @ V ) @ Xs ) ) ) ) ).
% delete_aux.cases
thf(fact_705_map__default_Ocases,axiom,
! [X: produc5405368317271509971at_nat] :
( ! [K2: nat,V: nat,F2: nat > nat] :
( X
!= ( produc2291548248119593221at_nat @ K2 @ ( produc1709345877921393766at_nat @ V @ ( produc1236331799044183215at_nat @ F2 @ nil_Pr5478986624290739719at_nat ) ) ) )
=> ~ ! [K2: nat,V: nat,F2: nat > nat,P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( X
!= ( produc2291548248119593221at_nat @ K2 @ ( produc1709345877921393766at_nat @ V @ ( produc1236331799044183215at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) ) ) ) ) ).
% map_default.cases
thf(fact_706_map__entry_Ocases,axiom,
! [X: produc6121082497140218670at_nat] :
( ! [K2: nat,F2: nat > nat] :
( X
!= ( produc1709345877921393766at_nat @ K2 @ ( produc1236331799044183215at_nat @ F2 @ nil_Pr5478986624290739719at_nat ) ) )
=> ~ ! [K2: nat,F2: nat > nat,P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( X
!= ( produc1709345877921393766at_nat @ K2 @ ( produc1236331799044183215at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) ) ) ) ).
% map_entry.cases
thf(fact_707_clearjunk_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ( X != nil_Pr5478986624290739719at_nat )
=> ~ ! [P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) ) ).
% clearjunk.cases
thf(fact_708_inverse__permutation__of__list_Oinduct,axiom,
! [P: list_P2712776532842825709omplex > mat_complex > $o,A0: list_P2712776532842825709omplex,A1: mat_complex] :
( ! [X_1: mat_complex] : ( P @ nil_Pr795171647102378477omplex @ X_1 )
=> ( ! [Y: mat_complex,X6: mat_complex,Xs: list_P2712776532842825709omplex,X2: mat_complex] :
( ( ( X2 != X6 )
=> ( P @ Xs @ X2 ) )
=> ( P @ ( cons_P6830038620677733789omplex @ ( produc3658446505030690647omplex @ Y @ X6 ) @ Xs ) @ X2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inverse_permutation_of_list.induct
thf(fact_709_inverse__permutation__of__list_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > nat > $o,A0: list_P6011104703257516679at_nat,A1: nat] :
( ! [X_1: nat] : ( P @ nil_Pr5478986624290739719at_nat @ X_1 )
=> ( ! [Y: nat,X6: nat,Xs: list_P6011104703257516679at_nat,X2: nat] :
( ( ( X2 != X6 )
=> ( P @ Xs @ X2 ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y @ X6 ) @ Xs ) @ X2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inverse_permutation_of_list.induct
thf(fact_710_delete__aux_Oinduct,axiom,
! [P: mat_complex > list_P6834526116611618029omplex > $o,A0: mat_complex,A1: list_P6834526116611618029omplex] :
( ! [K2: mat_complex] : ( P @ K2 @ nil_Pr8537966556735835629omplex )
=> ( ! [K2: mat_complex,K4: mat_complex,V: produc5677646155008957607omplex,Xs: list_P6834526116611618029omplex] :
( ( ( K2 != K4 )
=> ( P @ K2 @ Xs ) )
=> ( P @ K2 @ ( cons_P2742075696876001693omplex @ ( produc1901862033385395287omplex @ K4 @ V ) @ Xs ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% delete_aux.induct
thf(fact_711_delete__aux_Oinduct,axiom,
! [P: mat_complex > list_P7556462501187456557omplex > $o,A0: mat_complex,A1: list_P7556462501187456557omplex] :
( ! [K2: mat_complex] : ( P @ K2 @ nil_Pr9007294345934890925omplex )
=> ( ! [K2: mat_complex,K4: mat_complex,V: produc352478934956084711omplex,Xs: list_P7556462501187456557omplex] :
( ( ( K2 != K4 )
=> ( P @ K2 @ Xs ) )
=> ( P @ K2 @ ( cons_P5134161419496323933omplex @ ( produc2861545499953221015omplex @ K4 @ V ) @ Xs ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% delete_aux.induct
thf(fact_712_delete__aux_Oinduct,axiom,
! [P: mat_complex > list_P2712776532842825709omplex > $o,A0: mat_complex,A1: list_P2712776532842825709omplex] :
( ! [K2: mat_complex] : ( P @ K2 @ nil_Pr795171647102378477omplex )
=> ( ! [K2: mat_complex,K4: mat_complex,V: mat_complex,Xs: list_P2712776532842825709omplex] :
( ( ( K2 != K4 )
=> ( P @ K2 @ Xs ) )
=> ( P @ K2 @ ( cons_P6830038620677733789omplex @ ( produc3658446505030690647omplex @ K4 @ V ) @ Xs ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% delete_aux.induct
thf(fact_713_delete__aux_Oinduct,axiom,
! [P: nat > list_P6011104703257516679at_nat > $o,A0: nat,A1: list_P6011104703257516679at_nat] :
( ! [K2: nat] : ( P @ K2 @ nil_Pr5478986624290739719at_nat )
=> ( ! [K2: nat,K4: nat,V: nat,Xs: list_P6011104703257516679at_nat] :
( ( ( K2 != K4 )
=> ( P @ K2 @ Xs ) )
=> ( P @ K2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ Xs ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% delete_aux.induct
thf(fact_714_delete__aux_Oinduct,axiom,
! [P: ( nat > nat ) > list_P5366131564312172150at_nat > $o,A0: nat > nat,A1: list_P5366131564312172150at_nat] :
( ! [K2: nat > nat] : ( P @ K2 @ nil_Pr2223394031645031670at_nat )
=> ( ! [K2: nat > nat,K4: nat > nat,V: nat,Xs: list_P5366131564312172150at_nat] :
( ( ( K2 != K4 )
=> ( P @ K2 @ Xs ) )
=> ( P @ K2 @ ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K4 @ V ) @ Xs ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% delete_aux.induct
thf(fact_715_prod_Osimps_I1_J,axiom,
! [X12: mat_complex,X24: produc5677646155008957607omplex,Y1: mat_complex,Y23: produc5677646155008957607omplex] :
( ( ( produc1901862033385395287omplex @ X12 @ X24 )
= ( produc1901862033385395287omplex @ Y1 @ Y23 ) )
= ( ( X12 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.simps(1)
thf(fact_716_prod_Osimps_I1_J,axiom,
! [X12: mat_complex,X24: produc352478934956084711omplex,Y1: mat_complex,Y23: produc352478934956084711omplex] :
( ( ( produc2861545499953221015omplex @ X12 @ X24 )
= ( produc2861545499953221015omplex @ Y1 @ Y23 ) )
= ( ( X12 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.simps(1)
thf(fact_717_prod_Osimps_I1_J,axiom,
! [X12: mat_complex,X24: mat_complex,Y1: mat_complex,Y23: mat_complex] :
( ( ( produc3658446505030690647omplex @ X12 @ X24 )
= ( produc3658446505030690647omplex @ Y1 @ Y23 ) )
= ( ( X12 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.simps(1)
thf(fact_718_prod_Osimps_I1_J,axiom,
! [X12: nat,X24: nat,Y1: nat,Y23: nat] :
( ( ( product_Pair_nat_nat @ X12 @ X24 )
= ( product_Pair_nat_nat @ Y1 @ Y23 ) )
= ( ( X12 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.simps(1)
thf(fact_719_prod_Osimps_I1_J,axiom,
! [X12: nat > nat,X24: nat,Y1: nat > nat,Y23: nat] :
( ( ( produc72220940542539688at_nat @ X12 @ X24 )
= ( produc72220940542539688at_nat @ Y1 @ Y23 ) )
= ( ( X12 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.simps(1)
thf(fact_720_old_Oprod_Osimps_I1_J,axiom,
! [A: mat_complex,B: produc5677646155008957607omplex,A6: mat_complex,B5: produc5677646155008957607omplex] :
( ( ( produc1901862033385395287omplex @ A @ B )
= ( produc1901862033385395287omplex @ A6 @ B5 ) )
= ( ( A = A6 )
& ( B = B5 ) ) ) ).
% old.prod.simps(1)
thf(fact_721_old_Oprod_Osimps_I1_J,axiom,
! [A: mat_complex,B: produc352478934956084711omplex,A6: mat_complex,B5: produc352478934956084711omplex] :
( ( ( produc2861545499953221015omplex @ A @ B )
= ( produc2861545499953221015omplex @ A6 @ B5 ) )
= ( ( A = A6 )
& ( B = B5 ) ) ) ).
% old.prod.simps(1)
thf(fact_722_old_Oprod_Osimps_I1_J,axiom,
! [A: mat_complex,B: mat_complex,A6: mat_complex,B5: mat_complex] :
( ( ( produc3658446505030690647omplex @ A @ B )
= ( produc3658446505030690647omplex @ A6 @ B5 ) )
= ( ( A = A6 )
& ( B = B5 ) ) ) ).
% old.prod.simps(1)
thf(fact_723_old_Oprod_Osimps_I1_J,axiom,
! [A: nat,B: nat,A6: nat,B5: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A6 @ B5 ) )
= ( ( A = A6 )
& ( B = B5 ) ) ) ).
% old.prod.simps(1)
thf(fact_724_old_Oprod_Osimps_I1_J,axiom,
! [A: nat > nat,B: nat,A6: nat > nat,B5: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A6 @ B5 ) )
= ( ( A = A6 )
& ( B = B5 ) ) ) ).
% old.prod.simps(1)
thf(fact_725_prod_Oinduct,axiom,
! [P: produc1634985270395358183omplex > $o,Prod: produc1634985270395358183omplex] :
( ! [A3: mat_complex,B3: produc5677646155008957607omplex] : ( P @ ( produc1901862033385395287omplex @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_726_prod_Oinduct,axiom,
! [P: produc5677646155008957607omplex > $o,Prod: produc5677646155008957607omplex] :
( ! [A3: mat_complex,B3: produc352478934956084711omplex] : ( P @ ( produc2861545499953221015omplex @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_727_prod_Oinduct,axiom,
! [P: produc352478934956084711omplex > $o,Prod: produc352478934956084711omplex] :
( ! [A3: mat_complex,B3: mat_complex] : ( P @ ( produc3658446505030690647omplex @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_728_prod_Oinduct,axiom,
! [P: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_729_prod_Oinduct,axiom,
! [P: produc8199716216217303280at_nat > $o,Prod: produc8199716216217303280at_nat] :
( ! [A3: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_730_prod_Oexhaust,axiom,
! [Y3: produc1634985270395358183omplex] :
~ ! [X1: mat_complex,X23: produc5677646155008957607omplex] :
( Y3
!= ( produc1901862033385395287omplex @ X1 @ X23 ) ) ).
% prod.exhaust
thf(fact_731_prod_Oexhaust,axiom,
! [Y3: produc5677646155008957607omplex] :
~ ! [X1: mat_complex,X23: produc352478934956084711omplex] :
( Y3
!= ( produc2861545499953221015omplex @ X1 @ X23 ) ) ).
% prod.exhaust
thf(fact_732_prod_Oexhaust,axiom,
! [Y3: produc352478934956084711omplex] :
~ ! [X1: mat_complex,X23: mat_complex] :
( Y3
!= ( produc3658446505030690647omplex @ X1 @ X23 ) ) ).
% prod.exhaust
thf(fact_733_prod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [X1: nat,X23: nat] :
( Y3
!= ( product_Pair_nat_nat @ X1 @ X23 ) ) ).
% prod.exhaust
thf(fact_734_prod_Oexhaust,axiom,
! [Y3: produc8199716216217303280at_nat] :
~ ! [X1: nat > nat,X23: nat] :
( Y3
!= ( produc72220940542539688at_nat @ X1 @ X23 ) ) ).
% prod.exhaust
thf(fact_735_surj__pair,axiom,
! [P3: produc1634985270395358183omplex] :
? [X2: mat_complex,Y: produc5677646155008957607omplex] :
( P3
= ( produc1901862033385395287omplex @ X2 @ Y ) ) ).
% surj_pair
thf(fact_736_surj__pair,axiom,
! [P3: produc5677646155008957607omplex] :
? [X2: mat_complex,Y: produc352478934956084711omplex] :
( P3
= ( produc2861545499953221015omplex @ X2 @ Y ) ) ).
% surj_pair
thf(fact_737_surj__pair,axiom,
! [P3: produc352478934956084711omplex] :
? [X2: mat_complex,Y: mat_complex] :
( P3
= ( produc3658446505030690647omplex @ X2 @ Y ) ) ).
% surj_pair
thf(fact_738_surj__pair,axiom,
! [P3: product_prod_nat_nat] :
? [X2: nat,Y: nat] :
( P3
= ( product_Pair_nat_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_739_surj__pair,axiom,
! [P3: produc8199716216217303280at_nat] :
? [X2: nat > nat,Y: nat] :
( P3
= ( produc72220940542539688at_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_740_prod__cases,axiom,
! [P: produc1634985270395358183omplex > $o,P3: produc1634985270395358183omplex] :
( ! [A3: mat_complex,B3: produc5677646155008957607omplex] : ( P @ ( produc1901862033385395287omplex @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_741_prod__cases,axiom,
! [P: produc5677646155008957607omplex > $o,P3: produc5677646155008957607omplex] :
( ! [A3: mat_complex,B3: produc352478934956084711omplex] : ( P @ ( produc2861545499953221015omplex @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_742_prod__cases,axiom,
! [P: produc352478934956084711omplex > $o,P3: produc352478934956084711omplex] :
( ! [A3: mat_complex,B3: mat_complex] : ( P @ ( produc3658446505030690647omplex @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_743_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_744_prod__cases,axiom,
! [P: produc8199716216217303280at_nat > $o,P3: produc8199716216217303280at_nat] :
( ! [A3: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_745_Pair__inject,axiom,
! [A: mat_complex,B: produc5677646155008957607omplex,A6: mat_complex,B5: produc5677646155008957607omplex] :
( ( ( produc1901862033385395287omplex @ A @ B )
= ( produc1901862033385395287omplex @ A6 @ B5 ) )
=> ~ ( ( A = A6 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_746_Pair__inject,axiom,
! [A: mat_complex,B: produc352478934956084711omplex,A6: mat_complex,B5: produc352478934956084711omplex] :
( ( ( produc2861545499953221015omplex @ A @ B )
= ( produc2861545499953221015omplex @ A6 @ B5 ) )
=> ~ ( ( A = A6 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_747_Pair__inject,axiom,
! [A: mat_complex,B: mat_complex,A6: mat_complex,B5: mat_complex] :
( ( ( produc3658446505030690647omplex @ A @ B )
= ( produc3658446505030690647omplex @ A6 @ B5 ) )
=> ~ ( ( A = A6 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_748_Pair__inject,axiom,
! [A: nat,B: nat,A6: nat,B5: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A6 @ B5 ) )
=> ~ ( ( A = A6 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_749_Pair__inject,axiom,
! [A: nat > nat,B: nat,A6: nat > nat,B5: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A6 @ B5 ) )
=> ~ ( ( A = A6 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_750_prod__cases3,axiom,
! [Y3: produc1634985270395358183omplex] :
~ ! [A3: mat_complex,B3: mat_complex,C3: produc352478934956084711omplex] :
( Y3
!= ( produc1901862033385395287omplex @ A3 @ ( produc2861545499953221015omplex @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_751_prod__cases3,axiom,
! [Y3: produc5677646155008957607omplex] :
~ ! [A3: mat_complex,B3: mat_complex,C3: mat_complex] :
( Y3
!= ( produc2861545499953221015omplex @ A3 @ ( produc3658446505030690647omplex @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_752_prod__induct3,axiom,
! [P: produc1634985270395358183omplex > $o,X: produc1634985270395358183omplex] :
( ! [A3: mat_complex,B3: mat_complex,C3: produc352478934956084711omplex] : ( P @ ( produc1901862033385395287omplex @ A3 @ ( produc2861545499953221015omplex @ B3 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_753_prod__induct3,axiom,
! [P: produc5677646155008957607omplex > $o,X: produc5677646155008957607omplex] :
( ! [A3: mat_complex,B3: mat_complex,C3: mat_complex] : ( P @ ( produc2861545499953221015omplex @ A3 @ ( produc3658446505030690647omplex @ B3 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_754_prod__cases4,axiom,
! [Y3: produc1634985270395358183omplex] :
~ ! [A3: mat_complex,B3: mat_complex,C3: mat_complex,D3: mat_complex] :
( Y3
!= ( produc1901862033385395287omplex @ A3 @ ( produc2861545499953221015omplex @ B3 @ ( produc3658446505030690647omplex @ C3 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_755_prod__induct4,axiom,
! [P: produc1634985270395358183omplex > $o,X: produc1634985270395358183omplex] :
( ! [A3: mat_complex,B3: mat_complex,C3: mat_complex,D3: mat_complex] : ( P @ ( produc1901862033385395287omplex @ A3 @ ( produc2861545499953221015omplex @ B3 @ ( produc3658446505030690647omplex @ C3 @ D3 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_756_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_757_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_758_field__le__epsilon,axiom,
! [X: real,Y3: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y3 @ E ) ) )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% field_le_epsilon
thf(fact_759_gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( X
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% gcd.cases
thf(fact_760_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
% one_neq_zero
thf(fact_761_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
% one_neq_zero
thf(fact_762_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_763_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_764_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_765_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_766_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_767_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_768_add__less__zeroD,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_769_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_770_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_771_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_772_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_773_add__le__add__imp__diff__le,axiom,
! [I2: nat,K3: nat,N: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K3 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K3 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K3 ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_774_add__le__add__imp__diff__le,axiom,
! [I2: real,K3: real,N: real,J3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K3 ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J3 @ K3 ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K3 ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J3 @ K3 ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K3 ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_775_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_776_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_777_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_778_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_779_add__le__imp__le__diff,axiom,
! [I2: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K3 ) ) ) ).
% add_le_imp_le_diff
thf(fact_780_add__le__imp__le__diff,axiom,
! [I2: real,K3: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K3 ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K3 ) ) ) ).
% add_le_imp_le_diff
thf(fact_781_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_782_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_783_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_784_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_785_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_786_ev__blocks__part__def,axiom,
( jordan4637981584770492064omplex
= ( ^ [M2: nat,A7: mat_complex] :
! [I4: nat,J4: nat,K: nat] :
( ( ord_less_nat @ I4 @ J4 )
=> ( ( ord_less_nat @ J4 @ K )
=> ( ( ord_less_nat @ K @ M2 )
=> ( ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ K @ K ) )
= ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) ) )
=> ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ J4 @ J4 ) )
= ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) ) ) ) ) ) ) ) ) ).
% ev_blocks_part_def
thf(fact_787_expand__powers_Oelims,axiom,
! [X: list_P1909269847677398966at_nat,Y3: list_P6011104703257516679at_nat] :
( ( ( missin2748503833011120330at_nat @ X )
= Y3 )
=> ( ( ( X = nil_Pr5468900520374568608at_nat )
=> ( Y3 != nil_Pr5478986624290739719at_nat ) )
=> ( ! [N4: nat,A3: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N4 ) @ A3 ) @ Ps ) )
=> ( Y3
!= ( cons_P6512896166579812791at_nat @ A3 @ ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N4 @ A3 ) @ Ps ) ) ) ) )
=> ~ ! [A3: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( Y3
!= ( missin2748503833011120330at_nat @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_788_expand__powers_Oelims,axiom,
! [X: list_P6011104703257516679at_nat,Y3: list_nat] :
( ( ( missin6482572040563731271rs_nat @ X )
= Y3 )
=> ( ( ( X = nil_Pr5478986624290739719at_nat )
=> ( Y3 != nil_nat ) )
=> ( ! [N4: nat,A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N4 ) @ A3 ) @ Ps ) )
=> ( Y3
!= ( cons_nat @ A3 @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N4 @ A3 ) @ Ps ) ) ) ) )
=> ~ ! [A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( Y3
!= ( missin6482572040563731271rs_nat @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_789_identify__block__main,axiom,
! [A2: mat_complex,J3: nat,I2: nat] :
( ( ( jordan3525277539992963945omplex @ A2 @ J3 )
= I2 )
=> ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ( I2 = zero_zero_nat )
| ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ I2 ) )
!= one_one_complex ) )
& ! [K5: nat] :
( ( ord_less_eq_nat @ I2 @ K5 )
=> ( ( ord_less_nat @ K5 @ J3 )
=> ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ K5 @ ( suc @ K5 ) ) )
= one_one_complex ) ) ) ) ) ).
% identify_block_main
thf(fact_790_identify__block__main,axiom,
! [A2: mat_nat,J3: nat,I2: nat] :
( ( ( jordan8923406848002823307ck_nat @ A2 @ J3 )
= I2 )
=> ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ( I2 = zero_zero_nat )
| ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ I2 ) )
!= one_one_nat ) )
& ! [K5: nat] :
( ( ord_less_eq_nat @ I2 @ K5 )
=> ( ( ord_less_nat @ K5 @ J3 )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ K5 @ ( suc @ K5 ) ) )
= one_one_nat ) ) ) ) ) ).
% identify_block_main
thf(fact_791_identify__block__main,axiom,
! [A2: mat_real,J3: nat,I2: nat] :
( ( ( jordan6672758942465739239k_real @ A2 @ J3 )
= I2 )
=> ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ( I2 = zero_zero_nat )
| ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ I2 ) )
!= one_one_real ) )
& ! [K5: nat] :
( ( ord_less_eq_nat @ I2 @ K5 )
=> ( ( ord_less_nat @ K5 @ J3 )
=> ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ K5 @ ( suc @ K5 ) ) )
= one_one_real ) ) ) ) ) ).
% identify_block_main
thf(fact_792_expand__powers_Osimps_I1_J,axiom,
( ( missin2748503833011120330at_nat @ nil_Pr5468900520374568608at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% expand_powers.simps(1)
thf(fact_793_expand__powers_Osimps_I1_J,axiom,
( ( missin6482572040563731271rs_nat @ nil_Pr5478986624290739719at_nat )
= nil_nat ) ).
% expand_powers.simps(1)
thf(fact_794_identify__block_Osimps_I2_J,axiom,
! [A2: mat_complex,I2: nat] :
( ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
= one_one_complex )
=> ( ( jordan3525277539992963945omplex @ A2 @ ( suc @ I2 ) )
= ( jordan3525277539992963945omplex @ A2 @ I2 ) ) )
& ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
!= one_one_complex )
=> ( ( jordan3525277539992963945omplex @ A2 @ ( suc @ I2 ) )
= ( suc @ I2 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_795_identify__block_Osimps_I2_J,axiom,
! [A2: mat_nat,I2: nat] :
( ( ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
= one_one_nat )
=> ( ( jordan8923406848002823307ck_nat @ A2 @ ( suc @ I2 ) )
= ( jordan8923406848002823307ck_nat @ A2 @ I2 ) ) )
& ( ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
!= one_one_nat )
=> ( ( jordan8923406848002823307ck_nat @ A2 @ ( suc @ I2 ) )
= ( suc @ I2 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_796_identify__block_Osimps_I2_J,axiom,
! [A2: mat_real,I2: nat] :
( ( ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
= one_one_real )
=> ( ( jordan6672758942465739239k_real @ A2 @ ( suc @ I2 ) )
= ( jordan6672758942465739239k_real @ A2 @ I2 ) ) )
& ( ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
!= one_one_real )
=> ( ( jordan6672758942465739239k_real @ A2 @ ( suc @ I2 ) )
= ( suc @ I2 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_797_expand__powers_Osimps_I3_J,axiom,
! [A: nat,Ps2: list_P6011104703257516679at_nat] :
( ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A ) @ Ps2 ) )
= ( missin6482572040563731271rs_nat @ Ps2 ) ) ).
% expand_powers.simps(3)
thf(fact_798_identify__block_Oelims,axiom,
! [X: mat_complex,Xa: nat,Y3: nat] :
( ( ( jordan3525277539992963945omplex @ X @ Xa )
= Y3 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) )
=> ~ ! [I: nat] :
( ( Xa
= ( suc @ I ) )
=> ~ ( ( ( ( index_mat_complex @ X @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
= one_one_complex )
=> ( Y3
= ( jordan3525277539992963945omplex @ X @ I ) ) )
& ( ( ( index_mat_complex @ X @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
!= one_one_complex )
=> ( Y3
= ( suc @ I ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_799_identify__block_Oelims,axiom,
! [X: mat_nat,Xa: nat,Y3: nat] :
( ( ( jordan8923406848002823307ck_nat @ X @ Xa )
= Y3 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) )
=> ~ ! [I: nat] :
( ( Xa
= ( suc @ I ) )
=> ~ ( ( ( ( index_mat_nat @ X @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
= one_one_nat )
=> ( Y3
= ( jordan8923406848002823307ck_nat @ X @ I ) ) )
& ( ( ( index_mat_nat @ X @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
!= one_one_nat )
=> ( Y3
= ( suc @ I ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_800_identify__block_Oelims,axiom,
! [X: mat_real,Xa: nat,Y3: nat] :
( ( ( jordan6672758942465739239k_real @ X @ Xa )
= Y3 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) )
=> ~ ! [I: nat] :
( ( Xa
= ( suc @ I ) )
=> ~ ( ( ( ( index_mat_real @ X @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
= one_one_real )
=> ( Y3
= ( jordan6672758942465739239k_real @ X @ I ) ) )
& ( ( ( index_mat_real @ X @ ( product_Pair_nat_nat @ I @ ( suc @ I ) ) )
!= one_one_real )
=> ( Y3
= ( suc @ I ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_801_expand__powers_Osimps_I2_J,axiom,
! [N: nat,A: product_prod_nat_nat,Ps2: list_P1909269847677398966at_nat] :
( ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N ) @ A ) @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ A @ ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N @ A ) @ Ps2 ) ) ) ) ).
% expand_powers.simps(2)
thf(fact_802_expand__powers_Osimps_I2_J,axiom,
! [N: nat,A: nat,Ps2: list_P6011104703257516679at_nat] :
( ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N ) @ A ) @ Ps2 ) )
= ( cons_nat @ A @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ A ) @ Ps2 ) ) ) ) ).
% expand_powers.simps(2)
thf(fact_803_identify__block_I3_J,axiom,
! [A2: mat_complex,J3: nat,I2: nat,K3: nat] :
( ( ( jordan3525277539992963945omplex @ A2 @ J3 )
= I2 )
=> ( ( ord_less_eq_nat @ I2 @ K3 )
=> ( ( ord_less_nat @ K3 @ J3 )
=> ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ K3 @ ( suc @ K3 ) ) )
= one_one_complex ) ) ) ) ).
% identify_block(3)
thf(fact_804_identify__block_I3_J,axiom,
! [A2: mat_nat,J3: nat,I2: nat,K3: nat] :
( ( ( jordan8923406848002823307ck_nat @ A2 @ J3 )
= I2 )
=> ( ( ord_less_eq_nat @ I2 @ K3 )
=> ( ( ord_less_nat @ K3 @ J3 )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ K3 @ ( suc @ K3 ) ) )
= one_one_nat ) ) ) ) ).
% identify_block(3)
thf(fact_805_identify__block_I3_J,axiom,
! [A2: mat_real,J3: nat,I2: nat,K3: nat] :
( ( ( jordan6672758942465739239k_real @ A2 @ J3 )
= I2 )
=> ( ( ord_less_eq_nat @ I2 @ K3 )
=> ( ( ord_less_nat @ K3 @ J3 )
=> ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ K3 @ ( suc @ K3 ) ) )
= one_one_real ) ) ) ) ).
% identify_block(3)
thf(fact_806_identify__block_I2_J,axiom,
! [A2: mat_complex,J3: nat,I2: nat] :
( ( ( jordan3525277539992963945omplex @ A2 @ J3 )
= I2 )
=> ( ( I2 = zero_zero_nat )
| ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ I2 ) )
!= one_one_complex ) ) ) ).
% identify_block(2)
thf(fact_807_identify__block_I2_J,axiom,
! [A2: mat_nat,J3: nat,I2: nat] :
( ( ( jordan8923406848002823307ck_nat @ A2 @ J3 )
= I2 )
=> ( ( I2 = zero_zero_nat )
| ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ I2 ) )
!= one_one_nat ) ) ) ).
% identify_block(2)
thf(fact_808_identify__block_I2_J,axiom,
! [A2: mat_real,J3: nat,I2: nat] :
( ( ( jordan6672758942465739239k_real @ A2 @ J3 )
= I2 )
=> ( ( I2 = zero_zero_nat )
| ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ I2 ) )
!= one_one_real ) ) ) ).
% identify_block(2)
thf(fact_809_comm__add__mat,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ A2 @ B2 )
= ( plus_p8323303612493835998omplex @ B2 @ A2 ) ) ) ) ).
% comm_add_mat
thf(fact_810_assoc__add__mat,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,C4: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A2 @ B2 ) @ C4 )
= ( plus_p8323303612493835998omplex @ A2 @ ( plus_p8323303612493835998omplex @ B2 @ C4 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_811_add__carrier__mat,axiom,
! [B2: mat_complex,Nr: nat,Nc: nat,A2: mat_complex] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( plus_p8323303612493835998omplex @ A2 @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_812_minus__add__minus__mat,axiom,
! [U: mat_complex,Nr: nat,Nc: nat,V2: mat_complex,W2: mat_complex] :
( ( member_mat_complex @ U @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ V2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ W2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( minus_2412168080157227406omplex @ U @ ( plus_p8323303612493835998omplex @ V2 @ W2 ) )
= ( minus_2412168080157227406omplex @ ( minus_2412168080157227406omplex @ U @ V2 ) @ W2 ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_813_minus__carrier__mat,axiom,
! [B2: mat_complex,Nr: nat,Nc: nat,A2: mat_complex] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A2 @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_814_unit__vecs__first_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N4: nat] : ( P @ N4 @ zero_zero_nat )
=> ( ! [N4: nat,I: nat] :
( ( P @ N4 @ I )
=> ( P @ N4 @ ( suc @ I ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_first.induct
thf(fact_815_unit__vecs__first_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [N4: nat] :
( X
!= ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) )
=> ~ ! [N4: nat,I: nat] :
( X
!= ( product_Pair_nat_nat @ N4 @ ( suc @ I ) ) ) ) ).
% unit_vecs_first.cases
thf(fact_816_longest__common__prefix_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( ! [Uv: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
=> ~ ! [Uu: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Uu @ nil_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_817_longest__common__prefix_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) )
=> ( ! [Uv: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Uv ) )
=> ~ ! [Uu: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ Uu @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_818_sorted__list__subset_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [A3: nat,As: list_nat,B3: nat,Bs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ ( cons_nat @ B3 @ Bs ) ) )
=> ( ! [Uu: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Uu ) )
=> ~ ! [A3: nat,Uv: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ Uv ) @ nil_nat ) ) ) ) ).
% sorted_list_subset.cases
thf(fact_819_longest__common__prefix_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X2: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( X2 = Y )
=> ( P @ Xs @ Ys2 ) )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( ! [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_820_longest__common__prefix_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( X2 = Y )
=> ( P @ Xs @ Ys2 ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) )
=> ( ! [X_1: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ X_1 )
=> ( ! [Uu: list_P6011104703257516679at_nat] : ( P @ Uu @ nil_Pr5478986624290739719at_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_821_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( F @ K2 @ I3 ) )
=> ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ! [K5: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K5 @ K4 )
& ( F @ K4 @ I ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_822_sorted__list__subset_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [A3: nat,As: list_nat,B3: nat,Bs: list_nat] :
( ( ( A3 = B3 )
=> ( P @ As @ ( cons_nat @ B3 @ Bs ) ) )
=> ( ( ( A3 != B3 )
=> ( ( ord_less_nat @ B3 @ A3 )
=> ( P @ ( cons_nat @ A3 @ As ) @ Bs ) ) )
=> ( P @ ( cons_nat @ A3 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [A3: nat,Uv: list_nat] : ( P @ ( cons_nat @ A3 @ Uv ) @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sorted_list_subset.induct
thf(fact_823_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_824_mat__minus__minus,axiom,
! [A2: mat_complex,N: nat,M: nat,B2: mat_complex,C4: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ M ) )
=> ( ( minus_2412168080157227406omplex @ A2 @ ( minus_2412168080157227406omplex @ B2 @ C4 ) )
= ( plus_p8323303612493835998omplex @ ( minus_2412168080157227406omplex @ A2 @ B2 ) @ C4 ) ) ) ) ) ).
% mat_minus_minus
thf(fact_825_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_826_semiring__norm_I113_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% semiring_norm(113)
thf(fact_827_semiring__norm_I113_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% semiring_norm(113)
thf(fact_828_semiring__norm_I137_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% semiring_norm(137)
thf(fact_829_semiring__norm_I137_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% semiring_norm(137)
thf(fact_830_semiring__norm_I114_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% semiring_norm(114)
thf(fact_831_semiring__norm_I114_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% semiring_norm(114)
thf(fact_832_semiring__norm_I138_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% semiring_norm(138)
thf(fact_833_semiring__norm_I138_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% semiring_norm(138)
thf(fact_834_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_835_semiring__norm_I135_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% semiring_norm(135)
thf(fact_836_semiring__norm_I135_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% semiring_norm(135)
thf(fact_837_length__Cons,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( size_s5460976970255530739at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) )
= ( suc @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).
% length_Cons
thf(fact_838_length__Cons,axiom,
! [X: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_Cons
thf(fact_839_inf__concat__simple_Oinduct,axiom,
! [P: ( nat > nat ) > nat > $o,A0: nat > nat,A1: nat] :
( ! [F2: nat > nat] : ( P @ F2 @ zero_zero_nat )
=> ( ! [F2: nat > nat,N4: nat] :
( ( P @ F2 @ N4 )
=> ( P @ F2 @ ( suc @ N4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inf_concat_simple.induct
thf(fact_840_inf__concat__simple_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [F2: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
=> ~ ! [F2: nat > nat,N4: nat] :
( X
!= ( produc72220940542539688at_nat @ F2 @ ( suc @ N4 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_841_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_842_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_843_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_844_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_845_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_846_verit__eq__simplify_I6_J,axiom,
! [X: nat,Y3: nat] :
( ( X = Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% verit_eq_simplify(6)
thf(fact_847_verit__eq__simplify_I6_J,axiom,
! [X: real,Y3: real] :
( ( X = Y3 )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% verit_eq_simplify(6)
thf(fact_848_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_849_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A6 ) )
= ( ord_less_nat @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_850_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A6 ) )
= ( ord_less_real @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_851_verit__sum__simplify,axiom,
! [A: product_prod_nat_nat] :
( ( plus_p9057090461656269880at_nat @ A @ zero_z3979849011205770936at_nat )
= A ) ).
% verit_sum_simplify
thf(fact_852_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_853_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_854_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A5 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_855_same__diag__def,axiom,
( jordan2620430285385836103omplex
= ( ^ [N2: nat,A7: mat_complex,B6: mat_complex] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ( 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_856_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_857_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_858_diff__ev__def,axiom,
( jordan8650160714669549932omplex
= ( ^ [A7: mat_complex,I4: nat,J4: nat] :
( ( ord_less_nat @ I4 @ J4 )
=> ( ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) )
!= ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ J4 @ J4 ) ) )
=> ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= zero_zero_complex ) ) ) ) ) ).
% diff_ev_def
thf(fact_859_diff__ev__def,axiom,
( jordan8934236962569034858v_real
= ( ^ [A7: mat_real,I4: nat,J4: nat] :
( ( ord_less_nat @ I4 @ J4 )
=> ( ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) )
!= ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ J4 @ J4 ) ) )
=> ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= zero_zero_real ) ) ) ) ) ).
% diff_ev_def
thf(fact_860_extract__subdiags__diagonal,axiom,
! [B2: mat_complex,N: nat,L: list_nat,I2: 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 @ I2 @ ( size_size_list_nat @ L ) )
=> ( diagonal_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ B2 @ L ) @ I2 ) ) ) ) ) ) ) ).
% extract_subdiags_diagonal
thf(fact_861_pivot__positions__main__gen_Osimps,axiom,
( gauss_3265172588521606572omplex
= ( ^ [Zero2: complex,A7: mat_complex,Nr2: nat,Nc2: nat,I4: nat,J4: nat] :
( if_lis9186351972506106189at_nat @ ( ord_less_nat @ I4 @ Nr2 )
@ ( if_lis9186351972506106189at_nat @ ( ord_less_nat @ J4 @ Nc2 )
@ ( if_lis9186351972506106189at_nat
@ ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= Zero2 )
@ ( gauss_3265172588521606572omplex @ Zero2 @ A7 @ Nr2 @ Nc2 @ I4 @ ( suc @ J4 ) )
@ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ I4 @ J4 ) @ ( gauss_3265172588521606572omplex @ Zero2 @ A7 @ Nr2 @ Nc2 @ ( suc @ I4 ) @ ( suc @ J4 ) ) ) )
@ nil_Pr5478986624290739719at_nat )
@ nil_Pr5478986624290739719at_nat ) ) ) ).
% pivot_positions_main_gen.simps
thf(fact_862_pivot__positions__main__gen_Oelims,axiom,
! [Zero: complex,A2: mat_complex,Nr: nat,Nc: nat,X: nat,Xa: nat,Y3: list_P6011104703257516679at_nat] :
( ( ( gauss_3265172588521606572omplex @ Zero @ A2 @ Nr @ Nc @ X @ Xa )
= Y3 )
=> ( ( ( ord_less_nat @ X @ Nr )
=> ( ( ( ord_less_nat @ Xa @ Nc )
=> ( ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ X @ Xa ) )
= Zero )
=> ( Y3
= ( gauss_3265172588521606572omplex @ Zero @ A2 @ Nr @ Nc @ X @ ( suc @ Xa ) ) ) )
& ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ X @ Xa ) )
!= Zero )
=> ( Y3
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Xa ) @ ( gauss_3265172588521606572omplex @ Zero @ A2 @ Nr @ Nc @ ( suc @ X ) @ ( suc @ Xa ) ) ) ) ) ) )
& ( ~ ( ord_less_nat @ Xa @ Nc )
=> ( Y3 = nil_Pr5478986624290739719at_nat ) ) ) )
& ( ~ ( ord_less_nat @ X @ Nr )
=> ( Y3 = nil_Pr5478986624290739719at_nat ) ) ) ) ).
% pivot_positions_main_gen.elims
thf(fact_863_undef__vec__def,axiom,
( undef_vec_complex
= ( nth_complex @ nil_complex ) ) ).
% undef_vec_def
thf(fact_864_undef__vec__def,axiom,
( undef_2495355514574404529omplex
= ( nth_mat_complex @ nil_mat_complex ) ) ).
% undef_vec_def
thf(fact_865_undef__vec__def,axiom,
( undef_vec_nat
= ( nth_nat @ nil_nat ) ) ).
% undef_vec_def
thf(fact_866_undef__vec__def,axiom,
( undef_7626143578040714507at_nat
= ( nth_Pr7617993195940197384at_nat @ nil_Pr5478986624290739719at_nat ) ) ).
% undef_vec_def
thf(fact_867_sp,axiom,
( ( split_block_complex @ ba @ a @ a )
= ( produc1901862033385395287omplex @ b1 @ ( produc2861545499953221015omplex @ b2 @ ( produc3658446505030690647omplex @ b3 @ b4 ) ) ) ) ).
% sp
thf(fact_868_split__block__diag__carrier_I1_J,axiom,
! [D4: mat_complex,N: nat,A: nat,D1: mat_complex,D22: mat_complex,D32: mat_complex,D42: mat_complex] :
( ( member_mat_complex @ D4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A @ N )
=> ( ( ( split_block_complex @ D4 @ A @ A )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D32 @ D42 ) ) ) )
=> ( member_mat_complex @ D1 @ ( carrier_mat_complex @ A @ A ) ) ) ) ) ).
% split_block_diag_carrier(1)
thf(fact_869_extract__subdiags_Oinduct,axiom,
! [P: mat_complex > list_nat > $o,A0: mat_complex,A1: list_nat] :
( ! [B7: mat_complex] : ( P @ B7 @ nil_nat )
=> ( ! [B7: mat_complex,X2: nat,Xs: list_nat] :
( ! [Xa2: produc1634985270395358183omplex,Xb: mat_complex,Y4: produc5677646155008957607omplex,Xc: mat_complex,Ya: produc352478934956084711omplex,Xd: mat_complex,Yb: mat_complex] :
( ( Xa2
= ( split_block_complex @ B7 @ X2 @ X2 ) )
=> ( ( ( produc1901862033385395287omplex @ Xb @ Y4 )
= Xa2 )
=> ( ( ( produc2861545499953221015omplex @ Xc @ Ya )
= Y4 )
=> ( ( ( produc3658446505030690647omplex @ Xd @ Yb )
= Ya )
=> ( P @ Yb @ Xs ) ) ) ) )
=> ( P @ B7 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% extract_subdiags.induct
thf(fact_870_extract__subdiags__not__emp_I2_J,axiom,
! [B1: mat_complex,B22: mat_complex,B32: mat_complex,B42: mat_complex,B2: mat_complex,X: nat,L: list_nat] :
( ( ( produc1901862033385395287omplex @ B1 @ ( produc2861545499953221015omplex @ B22 @ ( produc3658446505030690647omplex @ B32 @ B42 ) ) )
= ( split_block_complex @ B2 @ X @ X ) )
=> ( ( tl_mat_complex @ ( commut6900707758132580272omplex @ B2 @ ( cons_nat @ X @ L ) ) )
= ( commut6900707758132580272omplex @ B42 @ L ) ) ) ).
% extract_subdiags_not_emp(2)
thf(fact_871_extract__subdiags__not__emp_I1_J,axiom,
! [B1: mat_complex,B22: mat_complex,B32: mat_complex,B42: mat_complex,B2: mat_complex,X: nat,L: list_nat] :
( ( ( produc1901862033385395287omplex @ B1 @ ( produc2861545499953221015omplex @ B22 @ ( produc3658446505030690647omplex @ B32 @ B42 ) ) )
= ( split_block_complex @ B2 @ X @ X ) )
=> ( ( hd_mat_complex @ ( commut6900707758132580272omplex @ B2 @ ( cons_nat @ X @ L ) ) )
= B1 ) ) ).
% extract_subdiags_not_emp(1)
thf(fact_872_split__block__diag__carrier_I2_J,axiom,
! [D4: mat_complex,N: nat,A: nat,D1: mat_complex,D22: mat_complex,D32: mat_complex,D42: mat_complex] :
( ( member_mat_complex @ D4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A @ N )
=> ( ( ( split_block_complex @ D4 @ A @ A )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D32 @ D42 ) ) ) )
=> ( member_mat_complex @ D42 @ ( carrier_mat_complex @ ( minus_minus_nat @ N @ A ) @ ( minus_minus_nat @ N @ A ) ) ) ) ) ) ).
% split_block_diag_carrier(2)
thf(fact_873_split__block__diagonal,axiom,
! [D4: mat_complex,N: nat,A: nat,D1: mat_complex,D22: mat_complex,D32: mat_complex,D42: mat_complex] :
( ( diagonal_mat_complex @ D4 )
=> ( ( member_mat_complex @ D4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A @ N )
=> ( ( ( split_block_complex @ D4 @ A @ A )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D32 @ D42 ) ) ) )
=> ( ( diagonal_mat_complex @ D1 )
& ( diagonal_mat_complex @ D42 ) ) ) ) ) ) ).
% split_block_diagonal
thf(fact_874_B1__def,axiom,
( b1
= ( produc8911724726559533635omplex @ ( split_block_complex @ ba @ a @ a ) ) ) ).
% B1_def
thf(fact_875_split__block__commute__subblock,axiom,
! [D4: mat_complex,N: nat,B2: mat_complex,A: nat,B1: mat_complex,B22: mat_complex,B32: mat_complex,B42: mat_complex,D1: mat_complex,D22: mat_complex,D32: mat_complex,D42: mat_complex] :
( ( diagonal_mat_complex @ D4 )
=> ( ( member_mat_complex @ D4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A @ N )
=> ( ( ( split_block_complex @ B2 @ A @ A )
= ( produc1901862033385395287omplex @ B1 @ ( produc2861545499953221015omplex @ B22 @ ( produc3658446505030690647omplex @ B32 @ B42 ) ) ) )
=> ( ( ( split_block_complex @ D4 @ A @ A )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D32 @ D42 ) ) ) )
=> ( ( ( times_8009071140041733218omplex @ B2 @ D4 )
= ( times_8009071140041733218omplex @ D4 @ B2 ) )
=> ( ( times_8009071140041733218omplex @ B42 @ D42 )
= ( times_8009071140041733218omplex @ D42 @ B42 ) ) ) ) ) ) ) ) ) ).
% split_block_commute_subblock
thf(fact_876_prod_Osel_I1_J,axiom,
! [X12: mat_complex,X24: produc352478934956084711omplex] :
( ( produc2697000228617323907omplex @ ( produc2861545499953221015omplex @ X12 @ X24 ) )
= X12 ) ).
% prod.sel(1)
thf(fact_877_prod_Osel_I1_J,axiom,
! [X12: mat_complex,X24: mat_complex] :
( ( produc9163778666669654339omplex @ ( produc3658446505030690647omplex @ X12 @ X24 ) )
= X12 ) ).
% prod.sel(1)
thf(fact_878_prod_Osel_I1_J,axiom,
! [X12: nat,X24: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X12 @ X24 ) )
= X12 ) ).
% prod.sel(1)
thf(fact_879_prod_Osel_I1_J,axiom,
! [X12: nat > nat,X24: nat] :
( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X12 @ X24 ) )
= X12 ) ).
% prod.sel(1)
thf(fact_880_prod_Osel_I1_J,axiom,
! [X12: mat_complex,X24: produc5677646155008957607omplex] :
( ( produc8911724726559533635omplex @ ( produc1901862033385395287omplex @ X12 @ X24 ) )
= X12 ) ).
% prod.sel(1)
thf(fact_881_fst__eqD,axiom,
! [X: mat_complex,Y3: produc352478934956084711omplex,A: mat_complex] :
( ( ( produc2697000228617323907omplex @ ( produc2861545499953221015omplex @ X @ Y3 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_882_fst__eqD,axiom,
! [X: mat_complex,Y3: mat_complex,A: mat_complex] :
( ( ( produc9163778666669654339omplex @ ( produc3658446505030690647omplex @ X @ Y3 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_883_fst__eqD,axiom,
! [X: nat,Y3: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_884_fst__eqD,axiom,
! [X: nat > nat,Y3: nat,A: nat > nat] :
( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_885_fst__eqD,axiom,
! [X: mat_complex,Y3: produc5677646155008957607omplex,A: mat_complex] :
( ( ( produc8911724726559533635omplex @ ( produc1901862033385395287omplex @ X @ Y3 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_886_verit__prod__simplify_I2_J,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% verit_prod_simplify(2)
thf(fact_887_verit__prod__simplify_I2_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% verit_prod_simplify(2)
thf(fact_888_verit__prod__simplify_I1_J,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% verit_prod_simplify(1)
thf(fact_889_verit__prod__simplify_I1_J,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% verit_prod_simplify(1)
thf(fact_890_minus__mult__distrib__mat,axiom,
! [A2: mat_complex,Nr: nat,N: nat,B2: mat_complex,C4: mat_complex,Nc: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( minus_2412168080157227406omplex @ A2 @ B2 ) @ C4 )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A2 @ C4 ) @ ( times_8009071140041733218omplex @ B2 @ C4 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_891_mult__minus__distrib__mat,axiom,
! [A2: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat,C4: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A2 @ ( minus_2412168080157227406omplex @ B2 @ C4 ) )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( times_8009071140041733218omplex @ A2 @ C4 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_892_arith__simps_I63_J,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% arith_simps(63)
thf(fact_893_arith__simps_I63_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% arith_simps(63)
thf(fact_894_arith__simps_I62_J,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% arith_simps(62)
thf(fact_895_arith__simps_I62_J,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% arith_simps(62)
thf(fact_896_nat__distrib_I2_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% nat_distrib(2)
thf(fact_897_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_898_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_899_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_900_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_901_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_902_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_903_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_904_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_905_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_906_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_907_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_908_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_909_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_910_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_911_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_912_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_913_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_914_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_915_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_916_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_917_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_918_assoc__mult__mat,axiom,
! [A2: mat_complex,N_1: nat,N_2: nat,B2: mat_complex,N_3: nat,C4: mat_complex,N_4: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ C4 )
= ( times_8009071140041733218omplex @ A2 @ ( times_8009071140041733218omplex @ B2 @ C4 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_919_mult__carrier__mat,axiom,
! [A2: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_920_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_921_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_922_semigroup__mult__class_Omult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% semigroup_mult_class.mult.assoc
thf(fact_923_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_924_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_925_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_926_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_927_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_928_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_929_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_930_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_931_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_932_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_933_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_934_sum__squares__eq__zero__iff,axiom,
! [X: real,Y3: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_935_zero__compare__simps_I10_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% zero_compare_simps(10)
thf(fact_936_zero__compare__simps_I6_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(6)
thf(fact_937_mult__sign__intros_I8_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_938_mult__sign__intros_I7_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(7)
thf(fact_939_mult__sign__intros_I7_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_940_mult__sign__intros_I6_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(6)
thf(fact_941_mult__sign__intros_I6_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_942_mult__sign__intros_I5_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_943_mult__sign__intros_I5_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_944_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_945_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_946_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_947_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_948_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_949_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_950_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_951_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_952_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_953_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_954_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_955_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_956_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_957_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_958_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_959_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_960_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_961_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_962_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_963_zero__compare__simps_I4_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(4)
thf(fact_964_zero__compare__simps_I8_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% zero_compare_simps(8)
thf(fact_965_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_966_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_967_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_968_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_969_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_970_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_971_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_972_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_973_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_974_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_975_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_976_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_977_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_978_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_979_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_980_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_981_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_982_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_983_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_984_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_985_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_986_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_987_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_988_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_989_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_990_mult__delta__left,axiom,
! [B: $o,X: real,Y3: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y3 )
= ( times_times_real @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y3 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_991_mult__delta__left,axiom,
! [B: $o,X: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y3 )
= ( times_times_nat @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y3 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_992_mult__delta__right,axiom,
! [B: $o,X: real,Y3: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= ( times_times_real @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_993_mult__delta__right,axiom,
! [B: $o,X: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_994_add__mult__distrib__mat,axiom,
! [A2: mat_complex,Nr: nat,N: nat,B2: mat_complex,C4: mat_complex,Nc: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A2 @ B2 ) @ C4 )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A2 @ C4 ) @ ( times_8009071140041733218omplex @ B2 @ C4 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_995_mult__add__distrib__mat,axiom,
! [A2: mat_complex,Nr: nat,N: nat,B2: mat_complex,Nc: nat,C4: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A2 @ ( plus_p8323303612493835998omplex @ B2 @ C4 ) )
= ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( times_8009071140041733218omplex @ A2 @ C4 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_996_map__default_Oinduct,axiom,
! [P: nat > nat > ( nat > nat ) > list_P6011104703257516679at_nat > $o,A0: nat,A1: nat,A22: nat > nat,A32: list_P6011104703257516679at_nat] :
( ! [K2: nat,V: nat,F2: nat > nat] : ( P @ K2 @ V @ F2 @ nil_Pr5478986624290739719at_nat )
=> ( ! [K2: nat,V: nat,F2: nat > nat,P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( ( ( ( product_fst_nat_nat @ P4 )
!= K2 )
=> ( P @ K2 @ V @ F2 @ Ps ) )
=> ( P @ K2 @ V @ F2 @ ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% map_default.induct
thf(fact_997_map__default_Oinduct,axiom,
! [P: mat_complex > produc5677646155008957607omplex > ( produc5677646155008957607omplex > produc5677646155008957607omplex ) > list_P6834526116611618029omplex > $o,A0: mat_complex,A1: produc5677646155008957607omplex,A22: produc5677646155008957607omplex > produc5677646155008957607omplex,A32: list_P6834526116611618029omplex] :
( ! [K2: mat_complex,V: produc5677646155008957607omplex,F2: produc5677646155008957607omplex > produc5677646155008957607omplex] : ( P @ K2 @ V @ F2 @ nil_Pr8537966556735835629omplex )
=> ( ! [K2: mat_complex,V: produc5677646155008957607omplex,F2: produc5677646155008957607omplex > produc5677646155008957607omplex,P4: produc1634985270395358183omplex,Ps: list_P6834526116611618029omplex] :
( ( ( ( produc8911724726559533635omplex @ P4 )
!= K2 )
=> ( P @ K2 @ V @ F2 @ Ps ) )
=> ( P @ K2 @ V @ F2 @ ( cons_P2742075696876001693omplex @ P4 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% map_default.induct
thf(fact_998_map__entry_Oinduct,axiom,
! [P: nat > ( nat > nat ) > list_P6011104703257516679at_nat > $o,A0: nat,A1: nat > nat,A22: list_P6011104703257516679at_nat] :
( ! [K2: nat,F2: nat > nat] : ( P @ K2 @ F2 @ nil_Pr5478986624290739719at_nat )
=> ( ! [K2: nat,F2: nat > nat,P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( ( ( ( product_fst_nat_nat @ P4 )
!= K2 )
=> ( P @ K2 @ F2 @ Ps ) )
=> ( P @ K2 @ F2 @ ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_entry.induct
thf(fact_999_map__entry_Oinduct,axiom,
! [P: mat_complex > ( produc5677646155008957607omplex > produc5677646155008957607omplex ) > list_P6834526116611618029omplex > $o,A0: mat_complex,A1: produc5677646155008957607omplex > produc5677646155008957607omplex,A22: list_P6834526116611618029omplex] :
( ! [K2: mat_complex,F2: produc5677646155008957607omplex > produc5677646155008957607omplex] : ( P @ K2 @ F2 @ nil_Pr8537966556735835629omplex )
=> ( ! [K2: mat_complex,F2: produc5677646155008957607omplex > produc5677646155008957607omplex,P4: produc1634985270395358183omplex,Ps: list_P6834526116611618029omplex] :
( ( ( ( produc8911724726559533635omplex @ P4 )
!= K2 )
=> ( P @ K2 @ F2 @ Ps ) )
=> ( P @ K2 @ F2 @ ( cons_P2742075696876001693omplex @ P4 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_entry.induct
thf(fact_1000_fst__add,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( product_fst_nat_nat @ ( plus_p9057090461656269880at_nat @ X @ Y3 ) )
= ( plus_plus_nat @ ( product_fst_nat_nat @ X ) @ ( product_fst_nat_nat @ Y3 ) ) ) ).
% fst_add
thf(fact_1001_fst__add,axiom,
! [X: produc352478934956084711omplex,Y3: produc352478934956084711omplex] :
( ( produc9163778666669654339omplex @ ( plus_p6104634242915576478omplex @ X @ Y3 ) )
= ( plus_p8323303612493835998omplex @ ( produc9163778666669654339omplex @ X ) @ ( produc9163778666669654339omplex @ Y3 ) ) ) ).
% fst_add
thf(fact_1002_fst__add,axiom,
! [X: produc5677646155008957607omplex,Y3: produc5677646155008957607omplex] :
( ( produc2697000228617323907omplex @ ( plus_p4451486566768871134omplex @ X @ Y3 ) )
= ( plus_p8323303612493835998omplex @ ( produc2697000228617323907omplex @ X ) @ ( produc2697000228617323907omplex @ Y3 ) ) ) ).
% fst_add
thf(fact_1003_fst__add,axiom,
! [X: produc1634985270395358183omplex,Y3: produc1634985270395358183omplex] :
( ( produc8911724726559533635omplex @ ( plus_p1405069618600264606omplex @ X @ Y3 ) )
= ( plus_p8323303612493835998omplex @ ( produc8911724726559533635omplex @ X ) @ ( produc8911724726559533635omplex @ Y3 ) ) ) ).
% fst_add
thf(fact_1004_fst__diff,axiom,
! [X: produc1634985270395358183omplex,Y3: produc1634985270395358183omplex] :
( ( produc8911724726559533635omplex @ ( minus_5093045068546291278omplex @ X @ Y3 ) )
= ( minus_2412168080157227406omplex @ ( produc8911724726559533635omplex @ X ) @ ( produc8911724726559533635omplex @ Y3 ) ) ) ).
% fst_diff
thf(fact_1005_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1006_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1007_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1008_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1009_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1010_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1011_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1012_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1013_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1014_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1015_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1016_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1017_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1018_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1019_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1020_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1021_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1022_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1023_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1024_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1025_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1026_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1027_sum__squares__ge__zero,axiom,
! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% sum_squares_ge_zero
thf(fact_1028_sum__squares__le__zero__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1029_mult__left__le__one__le,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y3 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1030_mult__right__le__one__le,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y3 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1031_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1032_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1033_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1034_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1035_not__sum__squares__lt__zero,axiom,
! [X: real,Y3: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_1036_sum__squares__gt__zero__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) )
= ( ( X != zero_zero_real )
| ( Y3 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1037_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1038_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1039_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_1040_field__le__mult__one__interval,axiom,
! [X: real,Y3: real] :
( ! [Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ Z @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ Y3 ) ) )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% field_le_mult_one_interval
thf(fact_1041_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1042_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1043_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1044_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1045_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1046_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1047_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1048_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1049_convex__bound__le,axiom,
! [X: real,A: real,Y3: real,U: real,V2: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y3 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1050_convex__bound__lt,axiom,
! [X: real,A: real,Y3: real,U: real,V2: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y3 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1051_commute__diag__mat__zero__comp,axiom,
! [D4: mat_complex,N: nat,B2: mat_complex,I2: nat,J3: nat] :
( ( diagonal_mat_complex @ D4 )
=> ( ( member_mat_complex @ D4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( times_8009071140041733218omplex @ B2 @ D4 )
= ( times_8009071140041733218omplex @ D4 @ B2 ) )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ( ( index_mat_complex @ D4 @ ( product_Pair_nat_nat @ I2 @ I2 ) )
!= ( index_mat_complex @ D4 @ ( product_Pair_nat_nat @ J3 @ J3 ) ) )
=> ( ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I2 @ J3 ) )
= zero_zero_complex ) ) ) ) ) ) ) ) ).
% commute_diag_mat_zero_comp
thf(fact_1052_commute__diag__mat__zero__comp,axiom,
! [D4: mat_real,N: nat,B2: mat_real,I2: nat,J3: nat] :
( ( diagonal_mat_real @ D4 )
=> ( ( member_mat_real @ D4 @ ( carrier_mat_real @ N @ N ) )
=> ( ( member_mat_real @ B2 @ ( carrier_mat_real @ N @ N ) )
=> ( ( ( times_times_mat_real @ B2 @ D4 )
= ( times_times_mat_real @ D4 @ B2 ) )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ( ( index_mat_real @ D4 @ ( product_Pair_nat_nat @ I2 @ I2 ) )
!= ( index_mat_real @ D4 @ ( product_Pair_nat_nat @ J3 @ J3 ) ) )
=> ( ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ I2 @ J3 ) )
= zero_zero_real ) ) ) ) ) ) ) ) ).
% commute_diag_mat_zero_comp
thf(fact_1053_diagonal__mat__mult__index_H,axiom,
! [A2: mat_complex,N: nat,B2: mat_complex,J3: nat,I2: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ B2 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I2 @ J3 ) )
= ( times_times_complex @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ J3 @ J3 ) ) @ ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I2 @ J3 ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index'
thf(fact_1054_diagonal__mat__mult__index,axiom,
! [A2: mat_complex,N: nat,B2: mat_complex,I2: nat,J3: nat] :
( ( diagonal_mat_complex @ A2 )
=> ( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I2 @ J3 ) )
= ( times_times_complex @ ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I2 @ I2 ) ) @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I2 @ J3 ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index
thf(fact_1055_Suc__mult__le__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1056_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1057_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_1058_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1059_nat__distrib_I1_J,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K3 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% nat_distrib(1)
thf(fact_1060_add__mult__distrib2,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1061_diff__mult__distrib2,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1062_diff__mult__distrib,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_1063_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J3: nat,K3: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ K3 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J3 ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib
thf(fact_1064_mult__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ( times_times_nat @ M @ K3 )
= ( times_times_nat @ N @ K3 ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1065_mult__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1066_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1067_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
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_P6011104703257516679at_nat,Y3: list_P6011104703257516679at_nat] :
( ( if_lis9186351972506106189at_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_P6011104703257516679at_nat,Y3: list_P6011104703257516679at_nat] :
( ( if_lis9186351972506106189at_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( nth_complex @ ( diag_mat_complex @ b4 ) @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) @ j ) )
= ( index_mat_complex @ b4 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) @ j ) @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ ia @ ( tl_nat @ la ) ) @ j ) ) ) ) ).
%------------------------------------------------------------------------------