TPTP Problem File: SLH0120^1.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_01202_049381__19443266_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1401 ( 504 unt; 289 typ; 0 def)
% Number of atoms : 2954 (1836 equ; 0 cnn)
% Maximal formula atoms : 18 ( 2 avg)
% Number of connectives : 12423 ( 573 ~; 79 |; 250 &;9840 @)
% ( 0 <=>;1681 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 49 ( 48 usr)
% Number of type conns : 1059 (1059 >; 0 *; 0 +; 0 <<)
% Number of symbols : 244 ( 241 usr; 25 con; 0-4 aty)
% Number of variables : 3821 ( 87 ^;3590 !; 144 ?;3821 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:37:26.069
%------------------------------------------------------------------------------
% Could-be-implicit typings (48)
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thf(ty_n_tf__a,type,
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% Explicit typings (241)
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thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Matrix__Omat_Itf__a_J,type,
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member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_B1____,type,
b1: mat_a ).
thf(sy_v_B2____,type,
b2: mat_a ).
thf(sy_v_B3____,type,
b3: mat_a ).
thf(sy_v_B4____,type,
b4: mat_a ).
thf(sy_v_Ba____,type,
ba: mat_a ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_l,type,
l: list_nat ).
thf(sy_v_la____,type,
la: list_nat ).
% Relevant facts (1100)
thf(fact_0__092_060open_062l_A_061_Ahd_Al_A_D_Atl_Al_092_060close_062,axiom,
( la
= ( cons_nat @ ( hd_nat @ la ) @ ( tl_nat @ la ) ) ) ).
% \<open>l = hd l # tl l\<close>
thf(fact_1__092_060open_062_Ihd_Al_A_D_Atl_Al_J_A_B_ASuc_Ai_A_061_Atl_Al_A_B_Ai_092_060close_062,axiom,
( ( nth_nat @ ( cons_nat @ ( hd_nat @ la ) @ ( tl_nat @ la ) ) @ ( suc @ ia ) )
= ( nth_nat @ ( tl_nat @ la ) @ ia ) ) ).
% \<open>(hd l # tl l) ! Suc i = tl l ! i\<close>
thf(fact_2__092_060open_062l_A_B_ASuc_Ai_A_061_Atl_Al_A_B_Ai_092_060close_062,axiom,
( ( nth_nat @ la @ ( suc @ ia ) )
= ( nth_nat @ ( tl_nat @ la ) @ ia ) ) ).
% \<open>l ! Suc i = tl l ! i\<close>
thf(fact_3_Suc_I2_J,axiom,
ord_less_nat @ ( suc @ ia ) @ ( size_size_list_nat @ la ) ).
% Suc(2)
thf(fact_4_calculation,axiom,
member_mat_a @ ( nth_mat_a @ ( commut2531942506349284476iags_a @ ba @ la ) @ ( suc @ ia ) ) @ ( carrier_mat_a @ ( nth_nat @ ( tl_nat @ la ) @ ia ) @ ( nth_nat @ ( tl_nat @ la ) @ ia ) ) ).
% calculation
thf(fact_5__092_060open_062extract__subdiags_AB4_A_Itl_Al_J_A_B_Ai_A_092_060in_062_Acarrier__mat_A_Itl_Al_A_B_Ai_J_A_Itl_Al_A_B_Ai_J_092_060close_062,axiom,
member_mat_a @ ( nth_mat_a @ ( commut2531942506349284476iags_a @ b4 @ ( tl_nat @ la ) ) @ ia ) @ ( carrier_mat_a @ ( nth_nat @ ( tl_nat @ la ) @ ia ) @ ( nth_nat @ ( tl_nat @ la ) @ ia ) ) ).
% \<open>extract_subdiags B4 (tl l) ! i \<in> carrier_mat (tl l ! i) (tl l ! i)\<close>
thf(fact_6_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_7_nth__Cons__Suc,axiom,
! [X: mat_a,Xs: list_mat_a,N: nat] :
( ( nth_mat_a @ ( cons_mat_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_mat_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_8_nth__Cons__Suc,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_9_nat_Osimps_I1_J,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.simps(1)
thf(fact_10_old_Onat_Osimps_I1_J,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.simps(1)
thf(fact_11_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_12_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
% Suc_n_not_n
thf(fact_13_assms,axiom,
ord_less_nat @ i @ ( size_size_list_nat @ l ) ).
% assms
thf(fact_14_Suc_Ohyps,axiom,
! [L: list_nat,B: mat_a] :
( ( ord_less_nat @ ia @ ( size_size_list_nat @ L ) )
=> ( member_mat_a @ ( nth_mat_a @ ( commut2531942506349284476iags_a @ B @ L ) @ ia ) @ ( carrier_mat_a @ ( nth_nat @ L @ ia ) @ ( nth_nat @ L @ ia ) ) ) ) ).
% Suc.hyps
thf(fact_15__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_a @ ( commut2531942506349284476iags_a @ ba @ la ) @ ( suc @ ia ) )
= ( nth_mat_a @ ( commut2531942506349284476iags_a @ b4 @ ( tl_nat @ la ) ) @ ia ) ) ).
% \<open>extract_subdiags B l ! Suc i = extract_subdiags B4 (tl l) ! i\<close>
thf(fact_16__092_060open_062extract__subdiags_AB_Al_A_061_AB1_A_D_Aextract__subdiags_AB4_A_Itl_Al_J_092_060close_062,axiom,
( ( commut2531942506349284476iags_a @ ba @ la )
= ( cons_mat_a @ b1 @ ( commut2531942506349284476iags_a @ b4 @ ( tl_nat @ la ) ) ) ) ).
% \<open>extract_subdiags B l = B1 # extract_subdiags B4 (tl l)\<close>
thf(fact_17_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_18_list_Osel_I1_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( ( hd_mat_a @ ( cons_mat_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_19_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_20_list_Osimps_I1_J,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_21_list_Osimps_I1_J,axiom,
! [X21: mat_a,X22: list_mat_a,Y21: mat_a,Y22: list_mat_a] :
( ( ( cons_mat_a @ X21 @ X22 )
= ( cons_mat_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_22_list_Osimps_I1_J,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.simps(1)
thf(fact_23_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_24_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_25_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_26_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_27_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_28_not__Cons__self,axiom,
! [Xs: list_nat,X: nat] :
( Xs
!= ( cons_nat @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_29_not__Cons__self,axiom,
! [Xs: list_mat_a,X: mat_a] :
( Xs
!= ( cons_mat_a @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_30_not__Cons__self,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( Xs
!= ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_31_nth__equalityI,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_mat_a @ Xs ) )
=> ( ( nth_mat_a @ Xs @ I )
= ( nth_mat_a @ Ys2 @ I ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_32_nth__equalityI,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs @ I )
= ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_33_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Ys2 @ I ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_34_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_35_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_36_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_37_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_38_Skolem__list__nth,axiom,
! [K: nat,P: nat > mat_a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: mat_a] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_mat_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_39_Skolem__list__nth,axiom,
! [K: nat,P: nat > product_prod_nat_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: product_prod_nat_nat] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_Pr7617993195940197384at_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_40_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: nat] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_41_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_42_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_43_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_44_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_45_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_mat_a,Z: list_mat_a] : ( Y3 = Z ) )
= ( ^ [Xs3: list_mat_a,Ys3: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Xs3 ) )
=> ( ( nth_mat_a @ Xs3 @ I2 )
= ( nth_mat_a @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_46_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_P6011104703257516679at_nat,Z: list_P6011104703257516679at_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs3 )
= ( size_s5460976970255530739at_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs3 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs3 @ I2 )
= ( nth_Pr7617993195940197384at_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_47_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_48_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_49_mem__Collect__eq,axiom,
! [A: mat_a,P: mat_a > $o] :
( ( member_mat_a @ A @ ( collect_mat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_mat_a] :
( ( collect_mat_a
@ ^ [X4: mat_a] : ( member_mat_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_length__Suc__conv,axiom,
! [Xs: list_mat_a,N: nat] :
( ( ( size_size_list_mat_a @ Xs )
= ( suc @ N ) )
= ( ? [Y4: mat_a,Ys3: list_mat_a] :
( ( Xs
= ( cons_mat_a @ Y4 @ Ys3 ) )
& ( ( size_size_list_mat_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_52_length__Suc__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ Y4 @ Ys3 ) )
& ( ( size_s5460976970255530739at_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_53_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_54_Suc__length__conv,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ( suc @ N )
= ( size_size_list_mat_a @ Xs ) )
= ( ? [Y4: mat_a,Ys3: list_mat_a] :
( ( Xs
= ( cons_mat_a @ Y4 @ Ys3 ) )
& ( ( size_size_list_mat_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_55_Suc__length__conv,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ( suc @ N )
= ( size_s5460976970255530739at_nat @ Xs ) )
= ( ? [Y4: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ Y4 @ Ys3 ) )
& ( ( size_s5460976970255530739at_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_56_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_57_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_58_strict__inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_59_less__Suc__induct,axiom,
! [I3: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_60_less__trans__Suc,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_61_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_62_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_63_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_64_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_65_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_66_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_67_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_68_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_69_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_70_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_71_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_72_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_73_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_74_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_75_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_76_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_77_nth__tl,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ ( tl_Pro4228036916689694320at_nat @ Xs ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( tl_Pro4228036916689694320at_nat @ Xs ) @ N )
= ( nth_Pr7617993195940197384at_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_78_nth__tl,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ ( tl_mat_a @ Xs ) ) )
=> ( ( nth_mat_a @ ( tl_mat_a @ Xs ) @ N )
= ( nth_mat_a @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_79_nth__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_80_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_81_list_Osel_I3_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( ( tl_mat_a @ ( cons_mat_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_82_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_83__092_060open_062tl_A_Iextract__subdiags_AB_Al_J_A_061_Aextract__subdiags_AB4_A_Itl_Al_J_092_060close_062,axiom,
( ( tl_mat_a @ ( commut2531942506349284476iags_a @ ba @ la ) )
= ( commut2531942506349284476iags_a @ b4 @ ( tl_nat @ la ) ) ) ).
% \<open>tl (extract_subdiags B l) = extract_subdiags B4 (tl l)\<close>
thf(fact_84_length__Cons,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( size_size_list_mat_a @ ( cons_mat_a @ X @ Xs ) )
= ( suc @ ( size_size_list_mat_a @ Xs ) ) ) ).
% length_Cons
thf(fact_85_length__Cons,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( size_s5460976970255530739at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( suc @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_86_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_87_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_88_gen__length__code_I2_J,axiom,
! [N: nat,X: mat_a,Xs: list_mat_a] :
( ( gen_length_mat_a @ N @ ( cons_mat_a @ X @ Xs ) )
= ( gen_length_mat_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_89_gen__length__code_I2_J,axiom,
! [N: nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( gen_le2383899666085517716at_nat @ N @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( gen_le2383899666085517716at_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_90_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_mat_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( cons_mat_a @ ( nth_mat_a @ Xs @ I3 ) @ ( drop_mat_a @ ( suc @ I3 ) @ Xs ) )
= ( drop_mat_a @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_91_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( cons_P6512896166579812791at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ ( drop_P8868858903918902087at_nat @ ( suc @ I3 ) @ Xs ) )
= ( drop_P8868858903918902087at_nat @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_92_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) )
= ( drop_nat @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_93_list__ex__length,axiom,
( list_ex_mat_a
= ( ^ [P2: mat_a > $o,Xs3: list_mat_a] :
? [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_mat_a @ Xs3 ) )
& ( P2 @ ( nth_mat_a @ Xs3 @ N4 ) ) ) ) ) ).
% list_ex_length
thf(fact_94_list__ex__length,axiom,
( list_e7689525607045846085at_nat
= ( ^ [P2: product_prod_nat_nat > $o,Xs3: list_P6011104703257516679at_nat] :
? [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_s5460976970255530739at_nat @ Xs3 ) )
& ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs3 @ N4 ) ) ) ) ) ).
% list_ex_length
thf(fact_95_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P2: nat > $o,Xs3: list_nat] :
? [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs3 ) )
& ( P2 @ ( nth_nat @ Xs3 @ N4 ) ) ) ) ) ).
% list_ex_length
thf(fact_96_remdups__adj__adjacent,axiom,
! [I3: nat,Xs: list_mat_a] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_mat_a @ ( remdups_adj_mat_a @ Xs ) ) )
=> ( ( nth_mat_a @ ( remdups_adj_mat_a @ Xs ) @ I3 )
!= ( nth_mat_a @ ( remdups_adj_mat_a @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_97_remdups__adj__adjacent,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_s5460976970255530739at_nat @ ( remdup844249387045036349at_nat @ Xs ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( remdup844249387045036349at_nat @ Xs ) @ I3 )
!= ( nth_Pr7617993195940197384at_nat @ ( remdup844249387045036349at_nat @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_98_remdups__adj__adjacent,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I3 )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_99_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ Xs ) )
=> ( ( hd_mat_a @ ( drop_mat_a @ N @ Xs ) )
= ( nth_mat_a @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_100_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( hd_Pro3460610213475200108at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs ) )
= ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_101_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_102_nth__butlast,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ ( butlast_mat_a @ Xs ) ) )
=> ( ( nth_mat_a @ ( butlast_mat_a @ Xs ) @ N )
= ( nth_mat_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_103_nth__butlast,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ ( butlas5569151904373679443at_nat @ Xs ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( butlas5569151904373679443at_nat @ Xs ) @ N )
= ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_104_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_105_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_106_hd__Cons__tl,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( cons_mat_a @ ( hd_mat_a @ Xs ) @ ( tl_mat_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_107_hd__Cons__tl,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ ( tl_Pro4228036916689694320at_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_108_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_109_list_Oexhaust__sel,axiom,
! [List: list_mat_a] :
( ( List != nil_mat_a )
=> ( List
= ( cons_mat_a @ ( hd_mat_a @ List ) @ ( tl_mat_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_110_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_111_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_112_list_Ocollapse,axiom,
! [List: list_mat_a] :
( ( List != nil_mat_a )
=> ( ( cons_mat_a @ ( hd_mat_a @ List ) @ ( tl_mat_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_113_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_114_drop_Osimps_I1_J,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop.simps(1)
thf(fact_115_drop_Osimps_I1_J,axiom,
! [N: nat] :
( ( drop_P8868858903918902087at_nat @ N @ nil_Pr5478986624290739719at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% drop.simps(1)
thf(fact_116_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_117_butlast_Osimps_I1_J,axiom,
( ( butlas5569151904373679443at_nat @ nil_Pr5478986624290739719at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% butlast.simps(1)
thf(fact_118_list__ex__simps_I2_J,axiom,
! [P: nat > $o] :
~ ( list_ex_nat @ P @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_119_list__ex__simps_I2_J,axiom,
! [P: product_prod_nat_nat > $o] :
~ ( list_e7689525607045846085at_nat @ P @ nil_Pr5478986624290739719at_nat ) ).
% list_ex_simps(2)
thf(fact_120_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_121_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le2383899666085517716at_nat @ N @ nil_Pr5478986624290739719at_nat )
= N ) ).
% gen_length_code(1)
thf(fact_122_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_123_remdups__adj_Osimps_I1_J,axiom,
( ( remdup844249387045036349at_nat @ nil_Pr5478986624290739719at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% remdups_adj.simps(1)
thf(fact_124_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_125_remdups__adj__Nil__iff,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( remdup844249387045036349at_nat @ Xs )
= nil_Pr5478986624290739719at_nat )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_126_remdups__adj_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X5: nat] :
( ( X
= ( cons_nat @ X5 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X5 @ nil_nat ) ) )
=> ~ ! [X5: nat,Y5: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Xs2 ) ) )
=> ~ ( ( ( X5 = Y5 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y5 )
=> ( Y
= ( cons_nat @ X5 @ ( remdups_adj_nat @ ( cons_nat @ Y5 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_127_remdups__adj_Oelims,axiom,
! [X: list_mat_a,Y: list_mat_a] :
( ( ( remdups_adj_mat_a @ X )
= Y )
=> ( ( ( X = nil_mat_a )
=> ( Y != nil_mat_a ) )
=> ( ! [X5: mat_a] :
( ( X
= ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ( Y
!= ( cons_mat_a @ X5 @ nil_mat_a ) ) )
=> ~ ! [X5: mat_a,Y5: mat_a,Xs2: list_mat_a] :
( ( X
= ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Xs2 ) ) )
=> ~ ( ( ( X5 = Y5 )
=> ( Y
= ( remdups_adj_mat_a @ ( cons_mat_a @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y5 )
=> ( Y
= ( cons_mat_a @ X5 @ ( remdups_adj_mat_a @ ( cons_mat_a @ Y5 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_128_remdups__adj_Oelims,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( remdup844249387045036349at_nat @ X )
= Y )
=> ( ( ( X = nil_Pr5478986624290739719at_nat )
=> ( Y != nil_Pr5478986624290739719at_nat ) )
=> ( ! [X5: product_prod_nat_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ( Y
!= ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) ) )
=> ~ ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) )
=> ~ ( ( ( X5 = Y5 )
=> ( Y
= ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y5 )
=> ( Y
= ( cons_P6512896166579812791at_nat @ X5 @ ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_129_remdups__adj_Osimps_I2_J,axiom,
! [X: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_130_remdups__adj_Osimps_I2_J,axiom,
! [X: mat_a] :
( ( remdups_adj_mat_a @ ( cons_mat_a @ X @ nil_mat_a ) )
= ( cons_mat_a @ X @ nil_mat_a ) ) ).
% remdups_adj.simps(2)
thf(fact_131_remdups__adj_Osimps_I2_J,axiom,
! [X: product_prod_nat_nat] :
( ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) )
= ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_132_List_Otranspose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X5: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_133_List_Otranspose_Ocases,axiom,
! [X: list_list_mat_a] :
( ( X != nil_list_mat_a )
=> ( ! [Xss: list_list_mat_a] :
( X
!= ( cons_list_mat_a @ nil_mat_a @ Xss ) )
=> ~ ! [X5: mat_a,Xs2: list_mat_a,Xss: list_list_mat_a] :
( X
!= ( cons_list_mat_a @ ( cons_mat_a @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_134_List_Otranspose_Ocases,axiom,
! [X: list_l3264859301627795341at_nat] :
( ( X != nil_li8973309667444810893at_nat )
=> ( ! [Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ Xss ) )
=> ~ ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_135_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_136_butlast_Osimps_I2_J,axiom,
! [Xs: list_mat_a,X: mat_a] :
( ( ( Xs = nil_mat_a )
=> ( ( butlast_mat_a @ ( cons_mat_a @ X @ Xs ) )
= nil_mat_a ) )
& ( ( Xs != nil_mat_a )
=> ( ( butlast_mat_a @ ( cons_mat_a @ X @ Xs ) )
= ( cons_mat_a @ X @ ( butlast_mat_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_137_butlast_Osimps_I2_J,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ( Xs = nil_Pr5478986624290739719at_nat )
=> ( ( butlas5569151904373679443at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= nil_Pr5478986624290739719at_nat ) )
& ( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( butlas5569151904373679443at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( cons_P6512896166579812791at_nat @ X @ ( butlas5569151904373679443at_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_138_extract__subdiags_Osimps_I1_J,axiom,
! [B: mat_a] :
( ( commut2531942506349284476iags_a @ B @ nil_nat )
= nil_mat_a ) ).
% extract_subdiags.simps(1)
thf(fact_139_drop__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_tl
thf(fact_140_drop__tl,axiom,
! [N: nat,Xs: list_mat_a] :
( ( drop_mat_a @ N @ ( tl_mat_a @ Xs ) )
= ( tl_mat_a @ ( drop_mat_a @ N @ Xs ) ) ) ).
% drop_tl
thf(fact_141_remdups__adj_Osimps_I3_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( cons_nat @ X @ ( remdups_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_142_remdups__adj_Osimps_I3_J,axiom,
! [X: mat_a,Y: mat_a,Xs: list_mat_a] :
( ( ( X = Y )
=> ( ( remdups_adj_mat_a @ ( cons_mat_a @ X @ ( cons_mat_a @ Y @ Xs ) ) )
= ( remdups_adj_mat_a @ ( cons_mat_a @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_mat_a @ ( cons_mat_a @ X @ ( cons_mat_a @ Y @ Xs ) ) )
= ( cons_mat_a @ X @ ( remdups_adj_mat_a @ ( cons_mat_a @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_143_remdups__adj_Osimps_I3_J,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( ( X = Y )
=> ( ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) )
= ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) )
= ( cons_P6512896166579812791at_nat @ X @ ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_144_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X5: nat] : ( P @ ( cons_nat @ X5 @ nil_nat ) )
=> ( ! [X5: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_145_list__nonempty__induct,axiom,
! [Xs: list_mat_a,P: list_mat_a > $o] :
( ( Xs != nil_mat_a )
=> ( ! [X5: mat_a] : ( P @ ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ( ! [X5: mat_a,Xs2: list_mat_a] :
( ( Xs2 != nil_mat_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_146_list__nonempty__induct,axiom,
! [Xs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_147_induct__list012,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X5: nat] : ( P @ ( cons_nat @ X5 @ nil_nat ) )
=> ( ! [X5: nat,Y5: nat,Zs: list_nat] :
( ( P @ Zs )
=> ( ( P @ ( cons_nat @ Y5 @ Zs ) )
=> ( P @ ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_148_induct__list012,axiom,
! [P: list_mat_a > $o,Xs: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X5: mat_a] : ( P @ ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ( ! [X5: mat_a,Y5: mat_a,Zs: list_mat_a] :
( ( P @ Zs )
=> ( ( P @ ( cons_mat_a @ Y5 @ Zs ) )
=> ( P @ ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_149_induct__list012,axiom,
! [P: list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat,Zs: list_P6011104703257516679at_nat] :
( ( P @ Zs )
=> ( ( P @ ( cons_P6512896166579812791at_nat @ Y5 @ Zs ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_150_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X5 @ Xs2 ) @ nil_nat )
=> ( ! [Y5: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y5 @ Ys4 ) )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_151_list__induct2_H,axiom,
! [P: list_nat > list_mat_a > $o,Xs: list_nat,Ys2: list_mat_a] :
( ( P @ nil_nat @ nil_mat_a )
=> ( ! [X5: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X5 @ Xs2 ) @ nil_mat_a )
=> ( ! [Y5: mat_a,Ys4: list_mat_a] : ( P @ nil_nat @ ( cons_mat_a @ Y5 @ Ys4 ) )
=> ( ! [X5: nat,Xs2: list_nat,Y5: mat_a,Ys4: list_mat_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_152_list__induct2_H,axiom,
! [P: list_nat > list_P6011104703257516679at_nat > $o,Xs: list_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X5 @ Xs2 ) @ nil_Pr5478986624290739719at_nat )
=> ( ! [Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] : ( P @ nil_nat @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) )
=> ( ! [X5: nat,Xs2: list_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_153_list__induct2_H,axiom,
! [P: list_mat_a > list_nat > $o,Xs: list_mat_a,Ys2: list_nat] :
( ( P @ nil_mat_a @ nil_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a] : ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ nil_nat )
=> ( ! [Y5: nat,Ys4: list_nat] : ( P @ nil_mat_a @ ( cons_nat @ Y5 @ Ys4 ) )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_154_list__induct2_H,axiom,
! [P: list_mat_a > list_mat_a > $o,Xs: list_mat_a,Ys2: list_mat_a] :
( ( P @ nil_mat_a @ nil_mat_a )
=> ( ! [X5: mat_a,Xs2: list_mat_a] : ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ nil_mat_a )
=> ( ! [Y5: mat_a,Ys4: list_mat_a] : ( P @ nil_mat_a @ ( cons_mat_a @ Y5 @ Ys4 ) )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_155_list__induct2_H,axiom,
! [P: list_mat_a > list_P6011104703257516679at_nat > $o,Xs: list_mat_a,Ys2: list_P6011104703257516679at_nat] :
( ( P @ nil_mat_a @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a] : ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ nil_Pr5478986624290739719at_nat )
=> ( ! [Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] : ( P @ nil_mat_a @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_156_list__induct2_H,axiom,
! [P: list_P6011104703257516679at_nat > list_nat > $o,Xs: list_P6011104703257516679at_nat,Ys2: list_nat] :
( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ nil_nat )
=> ( ! [Y5: nat,Ys4: list_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_nat @ Y5 @ Ys4 ) )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_157_list__induct2_H,axiom,
! [P: list_P6011104703257516679at_nat > list_mat_a > $o,Xs: list_P6011104703257516679at_nat,Ys2: list_mat_a] :
( ( P @ nil_Pr5478986624290739719at_nat @ nil_mat_a )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ nil_mat_a )
=> ( ! [Y5: mat_a,Ys4: list_mat_a] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_mat_a @ Y5 @ Ys4 ) )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: mat_a,Ys4: list_mat_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_158_list__induct2_H,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ nil_Pr5478986624290739719at_nat )
=> ( ! [Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_159_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_160_neq__Nil__conv,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
= ( ? [Y4: mat_a,Ys3: list_mat_a] :
( Xs
= ( cons_mat_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_161_neq__Nil__conv,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
= ( ? [Y4: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( Xs
= ( cons_P6512896166579812791at_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_162_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_163_map__tailrec__rev_Oinduct,axiom,
! [P: ( mat_a > nat ) > list_mat_a > list_nat > $o,A0: mat_a > nat,A1: list_mat_a,A22: list_nat] :
( ! [F2: mat_a > nat,X_1: list_nat] : ( P @ F2 @ nil_mat_a @ X_1 )
=> ( ! [F2: mat_a > nat,A3: mat_a,As: list_mat_a,Bs: list_nat] :
( ( P @ F2 @ As @ ( cons_nat @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_mat_a @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_164_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_165_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > mat_a ) > list_nat > list_mat_a > $o,A0: nat > mat_a,A1: list_nat,A22: list_mat_a] :
( ! [F2: nat > mat_a,X_1: list_mat_a] : ( P @ F2 @ nil_nat @ X_1 )
=> ( ! [F2: nat > mat_a,A3: nat,As: list_nat,Bs: list_mat_a] :
( ( P @ F2 @ As @ ( cons_mat_a @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_nat @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_166_map__tailrec__rev_Oinduct,axiom,
! [P: ( mat_a > mat_a ) > list_mat_a > list_mat_a > $o,A0: mat_a > mat_a,A1: list_mat_a,A22: list_mat_a] :
( ! [F2: mat_a > mat_a,X_1: list_mat_a] : ( P @ F2 @ nil_mat_a @ X_1 )
=> ( ! [F2: mat_a > mat_a,A3: mat_a,As: list_mat_a,Bs: list_mat_a] :
( ( P @ F2 @ As @ ( cons_mat_a @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_mat_a @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_167_map__tailrec__rev_Oinduct,axiom,
! [P: ( product_prod_nat_nat > mat_a ) > list_P6011104703257516679at_nat > list_mat_a > $o,A0: product_prod_nat_nat > mat_a,A1: list_P6011104703257516679at_nat,A22: list_mat_a] :
( ! [F2: product_prod_nat_nat > mat_a,X_1: list_mat_a] : ( P @ F2 @ nil_Pr5478986624290739719at_nat @ X_1 )
=> ( ! [F2: product_prod_nat_nat > mat_a,A3: product_prod_nat_nat,As: list_P6011104703257516679at_nat,Bs: list_mat_a] :
( ( P @ F2 @ As @ ( cons_mat_a @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_P6512896166579812791at_nat @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_168_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_169_map__tailrec__rev_Oinduct,axiom,
! [P: ( mat_a > product_prod_nat_nat ) > list_mat_a > list_P6011104703257516679at_nat > $o,A0: mat_a > product_prod_nat_nat,A1: list_mat_a,A22: list_P6011104703257516679at_nat] :
( ! [F2: mat_a > product_prod_nat_nat,X_1: list_P6011104703257516679at_nat] : ( P @ F2 @ nil_mat_a @ X_1 )
=> ( ! [F2: mat_a > product_prod_nat_nat,A3: mat_a,As: list_mat_a,Bs: list_P6011104703257516679at_nat] :
( ( P @ F2 @ As @ ( cons_P6512896166579812791at_nat @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons_mat_a @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_170_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_171_successively_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P3: nat > nat > $o] : ( P @ P3 @ nil_nat )
=> ( ! [P3: nat > nat > $o,X5: nat] : ( P @ P3 @ ( cons_nat @ X5 @ nil_nat ) )
=> ( ! [P3: nat > nat > $o,X5: nat,Y5: nat,Xs2: list_nat] :
( ( P @ P3 @ ( cons_nat @ Y5 @ Xs2 ) )
=> ( P @ P3 @ ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_172_successively_Oinduct,axiom,
! [P: ( mat_a > mat_a > $o ) > list_mat_a > $o,A0: mat_a > mat_a > $o,A1: list_mat_a] :
( ! [P3: mat_a > mat_a > $o] : ( P @ P3 @ nil_mat_a )
=> ( ! [P3: mat_a > mat_a > $o,X5: mat_a] : ( P @ P3 @ ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ( ! [P3: mat_a > mat_a > $o,X5: mat_a,Y5: mat_a,Xs2: list_mat_a] :
( ( P @ P3 @ ( cons_mat_a @ Y5 @ Xs2 ) )
=> ( P @ P3 @ ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_173_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] :
( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o] : ( P @ P3 @ nil_Pr5478986624290739719at_nat )
=> ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat] : ( P @ P3 @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( P @ P3 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) )
=> ( P @ P3 @ ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_174_remdups__adj_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X5: nat] : ( P @ ( cons_nat @ X5 @ nil_nat ) )
=> ( ! [X5: nat,Y5: nat,Xs2: list_nat] :
( ( ( X5 = Y5 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) ) )
=> ( ( ( X5 != Y5 )
=> ( P @ ( cons_nat @ Y5 @ Xs2 ) ) )
=> ( P @ ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_175_remdups__adj_Oinduct,axiom,
! [P: list_mat_a > $o,A0: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X5: mat_a] : ( P @ ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ( ! [X5: mat_a,Y5: mat_a,Xs2: list_mat_a] :
( ( ( X5 = Y5 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) ) )
=> ( ( ( X5 != Y5 )
=> ( P @ ( cons_mat_a @ Y5 @ Xs2 ) ) )
=> ( P @ ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_176_remdups__adj_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( ( X5 = Y5 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) ) )
=> ( ( ( X5 != Y5 )
=> ( P @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_177_sorted__wrt_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P3: nat > nat > $o] : ( P @ P3 @ nil_nat )
=> ( ! [P3: nat > nat > $o,X5: nat,Ys4: list_nat] :
( ( P @ P3 @ Ys4 )
=> ( P @ P3 @ ( cons_nat @ X5 @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_178_sorted__wrt_Oinduct,axiom,
! [P: ( mat_a > mat_a > $o ) > list_mat_a > $o,A0: mat_a > mat_a > $o,A1: list_mat_a] :
( ! [P3: mat_a > mat_a > $o] : ( P @ P3 @ nil_mat_a )
=> ( ! [P3: mat_a > mat_a > $o,X5: mat_a,Ys4: list_mat_a] :
( ( P @ P3 @ Ys4 )
=> ( P @ P3 @ ( cons_mat_a @ X5 @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_179_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] :
( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o] : ( P @ P3 @ nil_Pr5478986624290739719at_nat )
=> ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( P @ P3 @ Ys4 )
=> ( P @ P3 @ ( cons_P6512896166579812791at_nat @ X5 @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_180_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X5: nat] :
( X
!= ( cons_nat @ X5 @ nil_nat ) )
=> ~ ! [X5: nat,Y5: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_181_remdups__adj_Ocases,axiom,
! [X: list_mat_a] :
( ( X != nil_mat_a )
=> ( ! [X5: mat_a] :
( X
!= ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ~ ! [X5: mat_a,Y5: mat_a,Xs2: list_mat_a] :
( X
!= ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_182_remdups__adj_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ( X != nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_183_shuffles_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( ( P @ Xs2 @ ( cons_nat @ Y5 @ Ys4 ) )
=> ( ( P @ ( cons_nat @ X5 @ Xs2 ) @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_184_shuffles_Oinduct,axiom,
! [P: list_mat_a > list_mat_a > $o,A0: list_mat_a,A1: list_mat_a] :
( ! [X_1: list_mat_a] : ( P @ nil_mat_a @ X_1 )
=> ( ! [Xs2: list_mat_a] : ( P @ Xs2 @ nil_mat_a )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a] :
( ( P @ Xs2 @ ( cons_mat_a @ Y5 @ Ys4 ) )
=> ( ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ Ys4 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_185_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 )
=> ( ! [Xs2: list_P6011104703257516679at_nat] : ( P @ Xs2 @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( P @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) )
=> ( ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_186_min__list_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X5: nat,Xs2: list_nat] :
( ! [X212: nat,X222: list_nat] :
( ( Xs2
= ( cons_nat @ X212 @ X222 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_187_min__list_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ! [X212: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ X212 @ X222 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_188_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X5: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X5 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_189_min__list_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) )
=> ( X = nil_Pr5478986624290739719at_nat ) ) ).
% min_list.cases
thf(fact_190_splice_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [X5: nat,Xs2: list_nat,Ys4: list_nat] :
( ( P @ Ys4 @ Xs2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ Ys4 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_191_splice_Oinduct,axiom,
! [P: list_mat_a > list_mat_a > $o,A0: list_mat_a,A1: list_mat_a] :
( ! [X_1: list_mat_a] : ( P @ nil_mat_a @ X_1 )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Ys4: list_mat_a] :
( ( P @ Ys4 @ Xs2 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ Ys4 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_192_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 )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys4: list_P6011104703257516679at_nat] :
( ( P @ Ys4 @ Xs2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ Ys4 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_193_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_194_list_Oinducts,axiom,
! [P: list_mat_a > $o,List: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X1: mat_a,X23: list_mat_a] :
( ( P @ X23 )
=> ( P @ ( cons_mat_a @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_195_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_196_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X213: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_197_list_Oexhaust,axiom,
! [Y: list_mat_a] :
( ( Y != nil_mat_a )
=> ~ ! [X213: mat_a,X223: list_mat_a] :
( Y
!= ( cons_mat_a @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_198_list_Oexhaust,axiom,
! [Y: list_P6011104703257516679at_nat] :
( ( Y != nil_Pr5478986624290739719at_nat )
=> ~ ! [X213: product_prod_nat_nat,X223: list_P6011104703257516679at_nat] :
( Y
!= ( cons_P6512896166579812791at_nat @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_199_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_200_list_OdiscI,axiom,
! [List: list_mat_a,X21: mat_a,X22: list_mat_a] :
( ( List
= ( cons_mat_a @ X21 @ X22 ) )
=> ( List != nil_mat_a ) ) ).
% list.discI
thf(fact_201_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_202_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_203_list_Odistinct_I1_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( nil_mat_a
!= ( cons_mat_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_204_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_205_hd__remdups__adj,axiom,
! [Xs: list_nat] :
( ( hd_nat @ ( remdups_adj_nat @ Xs ) )
= ( hd_nat @ Xs ) ) ).
% hd_remdups_adj
thf(fact_206_list_Osel_I2_J,axiom,
( ( tl_mat_a @ nil_mat_a )
= nil_mat_a ) ).
% list.sel(2)
thf(fact_207_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_208_list_Osel_I2_J,axiom,
( ( tl_Pro4228036916689694320at_nat @ nil_Pr5478986624290739719at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% list.sel(2)
thf(fact_209_butlast__tl,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( butlast_nat @ Xs ) ) ) ).
% butlast_tl
thf(fact_210_butlast__tl,axiom,
! [Xs: list_mat_a] :
( ( butlast_mat_a @ ( tl_mat_a @ Xs ) )
= ( tl_mat_a @ ( butlast_mat_a @ Xs ) ) ) ).
% butlast_tl
thf(fact_211_list__ex__simps_I1_J,axiom,
! [P: nat > $o,X: nat,Xs: list_nat] :
( ( list_ex_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex_nat @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_212_list__ex__simps_I1_J,axiom,
! [P: mat_a > $o,X: mat_a,Xs: list_mat_a] :
( ( list_ex_mat_a @ P @ ( cons_mat_a @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex_mat_a @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_213_list__ex__simps_I1_J,axiom,
! [P: product_prod_nat_nat > $o,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( list_e7689525607045846085at_nat @ P @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( ( P @ X )
| ( list_e7689525607045846085at_nat @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_214_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_215_drop__Suc__Cons,axiom,
! [N: nat,X: mat_a,Xs: list_mat_a] :
( ( drop_mat_a @ ( suc @ N ) @ ( cons_mat_a @ X @ Xs ) )
= ( drop_mat_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_216_drop__Suc__Cons,axiom,
! [N: nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( drop_P8868858903918902087at_nat @ ( suc @ N ) @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( drop_P8868858903918902087at_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_217_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_218_nth__via__drop,axiom,
! [N: nat,Xs: list_mat_a,Y: mat_a,Ys2: list_mat_a] :
( ( ( drop_mat_a @ N @ Xs )
= ( cons_mat_a @ Y @ Ys2 ) )
=> ( ( nth_mat_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_219_nth__via__drop,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( drop_P8868858903918902087at_nat @ N @ Xs )
= ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_220_drop__Suc,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ Xs )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% drop_Suc
thf(fact_221_drop__Suc,axiom,
! [N: nat,Xs: list_mat_a] :
( ( drop_mat_a @ ( suc @ N ) @ Xs )
= ( drop_mat_a @ N @ ( tl_mat_a @ Xs ) ) ) ).
% drop_Suc
thf(fact_222_remdups__adj__Cons__alt,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_223_remdups__adj__Cons__alt,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( cons_mat_a @ X @ ( tl_mat_a @ ( remdups_adj_mat_a @ ( cons_mat_a @ X @ Xs ) ) ) )
= ( remdups_adj_mat_a @ ( cons_mat_a @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_224_remdups__adj__Cons__alt,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( cons_P6512896166579812791at_nat @ X @ ( tl_Pro4228036916689694320at_nat @ ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ) )
= ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_225_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs2: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( 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 @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_226_list__induct4,axiom,
! [Xs: list_mat_a,Ys2: list_nat,Zs2: list_nat,Ws: list_nat,P: list_mat_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_mat_a @ 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 @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_227_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_mat_a,Zs2: list_nat,Ws: list_nat,P: list_nat > list_mat_a > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_mat_a @ nil_nat @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: mat_a,Ys4: list_mat_a,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_228_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs2: list_mat_a,Ws: list_nat,P: list_nat > list_nat > list_mat_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_mat_a @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat,Z2: mat_a,Zs: list_mat_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_229_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs2: list_nat,Ws: list_mat_a,P: list_nat > list_nat > list_nat > list_mat_a > $o] :
( ( ( 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_mat_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_mat_a )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat,W: mat_a,Ws2: list_mat_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_mat_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_230_list__induct4,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,Zs2: list_nat,Ws: list_nat,P: list_mat_a > list_mat_a > list_nat > list_nat > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_nat @ nil_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_231_list__induct4,axiom,
! [Xs: list_mat_a,Ys2: list_nat,Zs2: list_mat_a,Ws: list_nat,P: list_mat_a > list_nat > list_mat_a > list_nat > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_nat @ nil_mat_a @ nil_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: nat,Ys4: list_nat,Z2: mat_a,Zs: list_mat_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_232_list__induct4,axiom,
! [Xs: list_mat_a,Ys2: list_nat,Zs2: list_nat,Ws: list_mat_a,P: list_mat_a > list_nat > list_nat > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_mat_a @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_nat @ nil_nat @ nil_mat_a )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat,W: mat_a,Ws2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_mat_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_233_list__induct4,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_nat,Zs2: list_nat,Ws: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > list_nat > $o] :
( ( ( 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 @ Ws ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_234_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_mat_a,Zs2: list_mat_a,Ws: list_nat,P: list_nat > list_mat_a > list_mat_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_mat_a @ nil_mat_a @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs: list_mat_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_235_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs2: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_236_list__induct3,axiom,
! [Xs: list_mat_a,Ys2: list_nat,Zs2: list_nat,P: list_mat_a > list_nat > list_nat > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_mat_a @ nil_nat @ nil_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_237_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_mat_a,Zs2: list_nat,P: list_nat > list_mat_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_mat_a @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: mat_a,Ys4: list_mat_a,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_238_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs2: list_mat_a,P: list_nat > list_nat > list_mat_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_mat_a )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat,Z2: mat_a,Zs: list_mat_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_239_list__induct3,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,Zs2: list_nat,P: list_mat_a > list_mat_a > list_nat > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_240_list__induct3,axiom,
! [Xs: list_mat_a,Ys2: list_nat,Zs2: list_mat_a,P: list_mat_a > list_nat > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ nil_mat_a @ nil_nat @ nil_mat_a )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: nat,Ys4: list_nat,Z2: mat_a,Zs: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_241_list__induct3,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_nat,Zs2: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: nat,Ys4: list_nat,Z2: nat,Zs: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_242_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_mat_a,Zs2: list_mat_a,P: list_nat > list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( size_size_list_mat_a @ Ys2 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_mat_a @ nil_mat_a )
=> ( ! [X5: nat,Xs2: list_nat,Y5: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs: list_mat_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_243_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_P6011104703257516679at_nat,Zs2: list_nat,P: list_nat > list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys4 ) )
=> ( ( ( size_s5460976970255530739at_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_244_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs2: list_P6011104703257516679at_nat,P: list_nat > list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5460976970255530739at_nat @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat,Z2: product_prod_nat_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s5460976970255530739at_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_P6512896166579812791at_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_245_list__induct2,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,P: list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( P @ nil_mat_a @ nil_mat_a )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_246_list__induct2,axiom,
! [Xs: list_mat_a,Ys2: list_P6011104703257516679at_nat,P: list_mat_a > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( P @ nil_mat_a @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_247_list__induct2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_mat_a,P: list_P6011104703257516679at_nat > list_mat_a > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_mat_a )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: mat_a,Ys4: list_mat_a] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_248_list__induct2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_249_list__induct2,axiom,
! [Xs: list_mat_a,Ys2: list_nat,P: list_mat_a > list_nat > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_mat_a @ nil_nat )
=> ( ! [X5: mat_a,Xs2: list_mat_a,Y5: nat,Ys4: list_nat] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_250_list__induct2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_nat,P: list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: nat,Ys4: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_251_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_mat_a,P: list_nat > list_mat_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_mat_a )
=> ( ! [X5: nat,Xs2: list_nat,Y5: mat_a,Ys4: list_mat_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_252_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_253_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_254_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X4: nat] :
( Xs
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_255_Nil__tl,axiom,
! [Xs: list_mat_a] :
( ( nil_mat_a
= ( tl_mat_a @ Xs ) )
= ( ( Xs = nil_mat_a )
| ? [X4: mat_a] :
( Xs
= ( cons_mat_a @ X4 @ nil_mat_a ) ) ) ) ).
% Nil_tl
thf(fact_256_Nil__tl,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( tl_Pro4228036916689694320at_nat @ Xs ) )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
| ? [X4: product_prod_nat_nat] :
( Xs
= ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% Nil_tl
thf(fact_257_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X4: nat] :
( Xs
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_258_tl__Nil,axiom,
! [Xs: list_mat_a] :
( ( ( tl_mat_a @ Xs )
= nil_mat_a )
= ( ( Xs = nil_mat_a )
| ? [X4: mat_a] :
( Xs
= ( cons_mat_a @ X4 @ nil_mat_a ) ) ) ) ).
% tl_Nil
thf(fact_259_tl__Nil,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( tl_Pro4228036916689694320at_nat @ Xs )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
| ? [X4: product_prod_nat_nat] :
( Xs
= ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% tl_Nil
thf(fact_260_list_Oexpand,axiom,
! [List: list_mat_a,List2: list_mat_a] :
( ( ( List = nil_mat_a )
= ( List2 = nil_mat_a ) )
=> ( ( ( List != nil_mat_a )
=> ( ( List2 != nil_mat_a )
=> ( ( ( hd_mat_a @ List )
= ( hd_mat_a @ List2 ) )
& ( ( tl_mat_a @ List )
= ( tl_mat_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_261_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_262_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_263_extract__subdiags__length,axiom,
! [B: mat_a,L: list_nat] :
( ( size_size_list_mat_a @ ( commut2531942506349284476iags_a @ B @ L ) )
= ( size_size_list_nat @ L ) ) ).
% extract_subdiags_length
thf(fact_264_extract__subdiags__neq__Nil,axiom,
! [B: mat_a,A: nat,L: list_nat] :
( ( commut2531942506349284476iags_a @ B @ ( cons_nat @ A @ L ) )
!= nil_mat_a ) ).
% extract_subdiags_neq_Nil
thf(fact_265_sorted__list__subset_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [A3: nat,As: list_nat,B2: nat,Bs: list_nat] :
( ( ( A3 = B2 )
=> ( P @ As @ ( cons_nat @ B2 @ Bs ) ) )
=> ( ( ( A3 != B2 )
=> ( ( ord_less_nat @ B2 @ A3 )
=> ( P @ ( cons_nat @ A3 @ As ) @ Bs ) ) )
=> ( P @ ( cons_nat @ A3 @ As ) @ ( cons_nat @ B2 @ 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_266_max__list__non__empty_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X5: nat] : ( P @ ( cons_nat @ X5 @ nil_nat ) )
=> ( ! [X5: nat,V: nat,Va: list_nat] :
( ( P @ ( cons_nat @ V @ Va ) )
=> ( P @ ( cons_nat @ X5 @ ( cons_nat @ V @ Va ) ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ) ).
% max_list_non_empty.induct
thf(fact_267_max__list__non__empty_Ocases,axiom,
! [X: list_nat] :
( ! [X5: nat] :
( X
!= ( cons_nat @ X5 @ nil_nat ) )
=> ( ! [X5: nat,V: nat,Va: list_nat] :
( X
!= ( cons_nat @ X5 @ ( cons_nat @ V @ Va ) ) )
=> ( X = nil_nat ) ) ) ).
% max_list_non_empty.cases
thf(fact_268_longest__common__prefix_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( ( ( X5 = Y5 )
=> ( P @ Xs2 @ Ys4 ) )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Uu: list_nat] : ( P @ Uu @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_269_longest__common__prefix_Oinduct,axiom,
! [P: list_mat_a > list_mat_a > $o,A0: list_mat_a,A1: list_mat_a] :
( ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a] :
( ( ( X5 = Y5 )
=> ( P @ Xs2 @ Ys4 ) )
=> ( P @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) )
=> ( ! [X_1: list_mat_a] : ( P @ nil_mat_a @ X_1 )
=> ( ! [Uu: list_mat_a] : ( P @ Uu @ nil_mat_a )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_270_longest__common__prefix_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat] :
( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( ( X5 = Y5 )
=> ( P @ Xs2 @ Ys4 ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) )
=> ( ! [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_271_plus__coeffs_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_nat )
=> ( ! [V: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V @ Va ) )
=> ( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_272_plus__coeffs_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat] :
( ! [Xs2: list_P6011104703257516679at_nat] : ( P @ Xs2 @ nil_Pr5478986624290739719at_nat )
=> ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_273_B1__def,axiom,
( b1
= ( produc7700291086614992977_mat_a @ ( split_block_a @ ba @ ( hd_nat @ la ) @ ( hd_nat @ la ) ) ) ) ).
% B1_def
thf(fact_274_max__list_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X5: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X5 @ Xs2 ) ) ) ).
% max_list.cases
thf(fact_275_max__list_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X5 @ Xs2 ) ) )
=> ( P @ A0 ) ) ) ).
% max_list.induct
thf(fact_276_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_277_sublists_Osimps_I1_J,axiom,
( ( sublis2272483767050531737at_nat @ nil_Pr5478986624290739719at_nat )
= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ nil_li8973309667444810893at_nat ) ) ).
% sublists.simps(1)
thf(fact_278_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_279_product__lists_Osimps_I1_J,axiom,
( ( produc8746550462604311920at_nat @ nil_li8973309667444810893at_nat )
= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ nil_li8973309667444810893at_nat ) ) ).
% product_lists.simps(1)
thf(fact_280_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_281_subseqs_Osimps_I1_J,axiom,
( ( subseq4535541509918465494at_nat @ nil_Pr5478986624290739719at_nat )
= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ nil_li8973309667444810893at_nat ) ) ).
% subseqs.simps(1)
thf(fact_282_take__Suc,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ ( suc @ N ) @ Xs )
= ( cons_nat @ ( hd_nat @ Xs ) @ ( take_nat @ N @ ( tl_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_283_take__Suc,axiom,
! [Xs: list_mat_a,N: nat] :
( ( Xs != nil_mat_a )
=> ( ( take_mat_a @ ( suc @ N ) @ Xs )
= ( cons_mat_a @ ( hd_mat_a @ Xs ) @ ( take_mat_a @ N @ ( tl_mat_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_284_take__Suc,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( take_P2173866234530122223at_nat @ ( suc @ N ) @ Xs )
= ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ ( take_P2173866234530122223at_nat @ N @ ( tl_Pro4228036916689694320at_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_285_take_Osimps_I1_J,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take.simps(1)
thf(fact_286_take_Osimps_I1_J,axiom,
! [N: nat] :
( ( take_P2173866234530122223at_nat @ N @ nil_Pr5478986624290739719at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% take.simps(1)
thf(fact_287_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_288_take__Suc__Cons,axiom,
! [N: nat,X: mat_a,Xs: list_mat_a] :
( ( take_mat_a @ ( suc @ N ) @ ( cons_mat_a @ X @ Xs ) )
= ( cons_mat_a @ X @ ( take_mat_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_289_take__Suc__Cons,axiom,
! [N: nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( take_P2173866234530122223at_nat @ ( suc @ N ) @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( cons_P6512896166579812791at_nat @ X @ ( take_P2173866234530122223at_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_290_nth__take,axiom,
! [I3: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I3 )
= ( nth_nat @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_291_nth__take,axiom,
! [I3: nat,N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_mat_a @ ( take_mat_a @ N @ Xs ) @ I3 )
= ( nth_mat_a @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_292_nth__take,axiom,
! [I3: nat,N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_Pr7617993195940197384at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs ) @ I3 )
= ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_293_take__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_294_take__tl,axiom,
! [N: nat,Xs: list_mat_a] :
( ( take_mat_a @ N @ ( tl_mat_a @ Xs ) )
= ( tl_mat_a @ ( take_mat_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_295_take__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_296_undef__vec__def,axiom,
( undef_vec_mat_a
= ( nth_mat_a @ nil_mat_a ) ) ).
% undef_vec_def
thf(fact_297_undef__vec__def,axiom,
( undef_vec_nat
= ( nth_nat @ nil_nat ) ) ).
% undef_vec_def
thf(fact_298_undef__vec__def,axiom,
( undef_7626143578040714507at_nat
= ( nth_Pr7617993195940197384at_nat @ nil_Pr5478986624290739719at_nat ) ) ).
% undef_vec_def
thf(fact_299_take__hd__drop,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ Xs ) )
=> ( ( append_mat_a @ ( take_mat_a @ N @ Xs ) @ ( cons_mat_a @ ( hd_mat_a @ ( drop_mat_a @ N @ Xs ) ) @ nil_mat_a ) )
= ( take_mat_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_300_take__hd__drop,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs ) @ ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs ) ) @ nil_Pr5478986624290739719at_nat ) )
= ( take_P2173866234530122223at_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_301_take__hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_302_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_mat_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( Xs
= ( append_mat_a @ ( take_mat_a @ I3 @ Xs ) @ ( cons_mat_a @ ( nth_mat_a @ Xs @ I3 ) @ ( drop_mat_a @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_303_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( Xs
= ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ I3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ ( drop_P8868858903918902087at_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_304_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_305_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_mat_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( take_mat_a @ ( suc @ I3 ) @ Xs )
= ( append_mat_a @ ( take_mat_a @ I3 @ Xs ) @ ( cons_mat_a @ ( nth_mat_a @ Xs @ I3 ) @ nil_mat_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_306_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( take_P2173866234530122223at_nat @ ( suc @ I3 ) @ Xs )
= ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ I3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_307_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I3 ) @ Xs )
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_308_Cons__in__shuffles__iff,axiom,
! [Z3: nat,Zs2: list_nat,Xs: list_nat,Ys2: list_nat] :
( ( member_list_nat @ ( cons_nat @ Z3 @ Zs2 ) @ ( shuffles_nat @ Xs @ Ys2 ) )
= ( ( ( Xs != nil_nat )
& ( ( hd_nat @ Xs )
= Z3 )
& ( member_list_nat @ Zs2 @ ( shuffles_nat @ ( tl_nat @ Xs ) @ Ys2 ) ) )
| ( ( Ys2 != nil_nat )
& ( ( hd_nat @ Ys2 )
= Z3 )
& ( member_list_nat @ Zs2 @ ( shuffles_nat @ Xs @ ( tl_nat @ Ys2 ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_309_Cons__in__shuffles__iff,axiom,
! [Z3: mat_a,Zs2: list_mat_a,Xs: list_mat_a,Ys2: list_mat_a] :
( ( member_list_mat_a @ ( cons_mat_a @ Z3 @ Zs2 ) @ ( shuffles_mat_a @ Xs @ Ys2 ) )
= ( ( ( Xs != nil_mat_a )
& ( ( hd_mat_a @ Xs )
= Z3 )
& ( member_list_mat_a @ Zs2 @ ( shuffles_mat_a @ ( tl_mat_a @ Xs ) @ Ys2 ) ) )
| ( ( Ys2 != nil_mat_a )
& ( ( hd_mat_a @ Ys2 )
= Z3 )
& ( member_list_mat_a @ Zs2 @ ( shuffles_mat_a @ Xs @ ( tl_mat_a @ Ys2 ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_310_Cons__in__shuffles__iff,axiom,
! [Z3: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ Z3 @ Zs2 ) @ ( shuffl5088453890617037852at_nat @ Xs @ Ys2 ) )
= ( ( ( Xs != nil_Pr5478986624290739719at_nat )
& ( ( hd_Pro3460610213475200108at_nat @ Xs )
= Z3 )
& ( member3067507820990806192at_nat @ Zs2 @ ( shuffl5088453890617037852at_nat @ ( tl_Pro4228036916689694320at_nat @ Xs ) @ Ys2 ) ) )
| ( ( Ys2 != nil_Pr5478986624290739719at_nat )
& ( ( hd_Pro3460610213475200108at_nat @ Ys2 )
= Z3 )
& ( member3067507820990806192at_nat @ Zs2 @ ( shuffl5088453890617037852at_nat @ Xs @ ( tl_Pro4228036916689694320at_nat @ Ys2 ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_311_nth__take__lemma,axiom,
! [K: nat,Xs: list_mat_a,Ys2: list_mat_a] :
( ( ord_less_eq_nat @ K @ ( size_size_list_mat_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_mat_a @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_mat_a @ Xs @ I )
= ( nth_mat_a @ Ys2 @ I ) ) )
=> ( ( take_mat_a @ K @ Xs )
= ( take_mat_a @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_312_nth__take__lemma,axiom,
! [K: nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ord_less_eq_nat @ K @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs @ I )
= ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) )
=> ( ( take_P2173866234530122223at_nat @ K @ Xs )
= ( take_P2173866234530122223at_nat @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_313_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Ys2 @ I ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_314_vec1__index,axiom,
! [J: nat,N: nat,I3: nat,Ze: nat,On: nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ( I3 = J )
=> ( ( nth_nat @ ( matrix_vec1I_nat @ Ze @ On @ N @ I3 ) @ J )
= On ) )
& ( ( I3 != J )
=> ( ( nth_nat @ ( matrix_vec1I_nat @ Ze @ On @ N @ I3 ) @ J )
= Ze ) ) ) ) ).
% vec1_index
thf(fact_315_vec1__index,axiom,
! [J: nat,N: nat,I3: nat,Ze: mat_a,On: mat_a] :
( ( ord_less_nat @ J @ N )
=> ( ( ( I3 = J )
=> ( ( nth_mat_a @ ( matrix_vec1I_mat_a @ Ze @ On @ N @ I3 ) @ J )
= On ) )
& ( ( I3 != J )
=> ( ( nth_mat_a @ ( matrix_vec1I_mat_a @ Ze @ On @ N @ I3 ) @ J )
= Ze ) ) ) ) ).
% vec1_index
thf(fact_316_vec1__index,axiom,
! [J: nat,N: nat,I3: nat,Ze: product_prod_nat_nat,On: product_prod_nat_nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ( I3 = J )
=> ( ( nth_Pr7617993195940197384at_nat @ ( matrix5788826640217845642at_nat @ Ze @ On @ N @ I3 ) @ J )
= On ) )
& ( ( I3 != J )
=> ( ( nth_Pr7617993195940197384at_nat @ ( matrix5788826640217845642at_nat @ Ze @ On @ N @ I3 ) @ J )
= Ze ) ) ) ) ).
% vec1_index
thf(fact_317_remdups__adj__append,axiom,
! [Xs_1: list_nat,X: nat,Xs_2: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X @ Xs_2 ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X @ nil_nat ) ) ) @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_318_remdups__adj__append,axiom,
! [Xs_1: list_mat_a,X: mat_a,Xs_2: list_mat_a] :
( ( remdups_adj_mat_a @ ( append_mat_a @ Xs_1 @ ( cons_mat_a @ X @ Xs_2 ) ) )
= ( append_mat_a @ ( remdups_adj_mat_a @ ( append_mat_a @ Xs_1 @ ( cons_mat_a @ X @ nil_mat_a ) ) ) @ ( tl_mat_a @ ( remdups_adj_mat_a @ ( cons_mat_a @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_319_remdups__adj__append,axiom,
! [Xs_1: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Xs_2: list_P6011104703257516679at_nat] :
( ( remdup844249387045036349at_nat @ ( append985823374593552924at_nat @ Xs_1 @ ( cons_P6512896166579812791at_nat @ X @ Xs_2 ) ) )
= ( append985823374593552924at_nat @ ( remdup844249387045036349at_nat @ ( append985823374593552924at_nat @ Xs_1 @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ) @ ( tl_Pro4228036916689694320at_nat @ ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_320_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B3 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_321_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_322_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_323_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_324_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_325_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_326_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_327_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_328_append_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys2 )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys2 ) ) ) ).
% append.simps(2)
thf(fact_329_append_Osimps_I2_J,axiom,
! [X: mat_a,Xs: list_mat_a,Ys2: list_mat_a] :
( ( append_mat_a @ ( cons_mat_a @ X @ Xs ) @ Ys2 )
= ( cons_mat_a @ X @ ( append_mat_a @ Xs @ Ys2 ) ) ) ).
% append.simps(2)
thf(fact_330_append_Osimps_I2_J,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ Ys2 )
= ( cons_P6512896166579812791at_nat @ X @ ( append985823374593552924at_nat @ Xs @ Ys2 ) ) ) ).
% append.simps(2)
thf(fact_331_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys2: list_nat,Xs: list_nat,Zs2: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs2 ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_332_Cons__eq__appendI,axiom,
! [X: mat_a,Xs1: list_mat_a,Ys2: list_mat_a,Xs: list_mat_a,Zs2: list_mat_a] :
( ( ( cons_mat_a @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_mat_a @ Xs1 @ Zs2 ) )
=> ( ( cons_mat_a @ X @ Xs )
= ( append_mat_a @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_333_Cons__eq__appendI,axiom,
! [X: product_prod_nat_nat,Xs1: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append985823374593552924at_nat @ Xs1 @ Zs2 ) )
=> ( ( cons_P6512896166579812791at_nat @ X @ Xs )
= ( append985823374593552924at_nat @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_334_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_335_append__is__Nil__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys2 )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
& ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_336_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_337_Nil__is__append__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
& ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_338_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_339_self__append__conv2,axiom,
! [Y: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat] :
( ( Y
= ( append985823374593552924at_nat @ Xs @ Y ) )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv2
thf(fact_340_append__self__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_341_append__self__conv2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv2
thf(fact_342_self__append__conv,axiom,
! [Y: list_nat,Ys2: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_343_self__append__conv,axiom,
! [Y: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Y
= ( append985823374593552924at_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv
thf(fact_344_append__self__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_345_append__self__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv
thf(fact_346_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_347_eq__Nil__appendI,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Xs = Ys2 )
=> ( Xs
= ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_348_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_349_append__Nil2,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ Xs @ nil_Pr5478986624290739719at_nat )
= Xs ) ).
% append_Nil2
thf(fact_350_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_351_append_Oright__neutral,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ A @ nil_Pr5478986624290739719at_nat )
= A ) ).
% append.right_neutral
thf(fact_352_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_353_append_Oleft__neutral,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_354_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_355_append__Nil,axiom,
! [Ys2: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_356_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_357_Cons__in__shuffles__leftI,axiom,
! [Zs2: list_nat,Xs: list_nat,Ys2: list_nat,Z3: nat] :
( ( member_list_nat @ Zs2 @ ( shuffles_nat @ Xs @ Ys2 ) )
=> ( member_list_nat @ ( cons_nat @ Z3 @ Zs2 ) @ ( shuffles_nat @ ( cons_nat @ Z3 @ Xs ) @ Ys2 ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_358_Cons__in__shuffles__leftI,axiom,
! [Zs2: list_mat_a,Xs: list_mat_a,Ys2: list_mat_a,Z3: mat_a] :
( ( member_list_mat_a @ Zs2 @ ( shuffles_mat_a @ Xs @ Ys2 ) )
=> ( member_list_mat_a @ ( cons_mat_a @ Z3 @ Zs2 ) @ ( shuffles_mat_a @ ( cons_mat_a @ Z3 @ Xs ) @ Ys2 ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_359_Cons__in__shuffles__leftI,axiom,
! [Zs2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Z3: product_prod_nat_nat] :
( ( member3067507820990806192at_nat @ Zs2 @ ( shuffl5088453890617037852at_nat @ Xs @ Ys2 ) )
=> ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ Z3 @ Zs2 ) @ ( shuffl5088453890617037852at_nat @ ( cons_P6512896166579812791at_nat @ Z3 @ Xs ) @ Ys2 ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_360_Cons__in__shuffles__rightI,axiom,
! [Zs2: list_nat,Xs: list_nat,Ys2: list_nat,Z3: nat] :
( ( member_list_nat @ Zs2 @ ( shuffles_nat @ Xs @ Ys2 ) )
=> ( member_list_nat @ ( cons_nat @ Z3 @ Zs2 ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Z3 @ Ys2 ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_361_Cons__in__shuffles__rightI,axiom,
! [Zs2: list_mat_a,Xs: list_mat_a,Ys2: list_mat_a,Z3: mat_a] :
( ( member_list_mat_a @ Zs2 @ ( shuffles_mat_a @ Xs @ Ys2 ) )
=> ( member_list_mat_a @ ( cons_mat_a @ Z3 @ Zs2 ) @ ( shuffles_mat_a @ Xs @ ( cons_mat_a @ Z3 @ Ys2 ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_362_Cons__in__shuffles__rightI,axiom,
! [Zs2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Z3: product_prod_nat_nat] :
( ( member3067507820990806192at_nat @ Zs2 @ ( shuffl5088453890617037852at_nat @ Xs @ Ys2 ) )
=> ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ Z3 @ Zs2 ) @ ( shuffl5088453890617037852at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Z3 @ Ys2 ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_363_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( F @ K2 @ I4 ) )
=> ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_364_le__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(1)
thf(fact_365_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_366_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_367_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_368_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_369_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_370_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X5: nat] : ( R @ X5 @ X5 )
=> ( ! [X5: nat,Y5: nat,Z2: nat] :
( ( R @ X5 @ Y5 )
=> ( ( R @ Y5 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_371_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_372_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_373_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_374_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_375_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_376_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_377_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_378_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_379_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_380_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_381_Nil__in__shufflesI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs = nil_nat )
=> ( ( Ys2 = nil_nat )
=> ( member_list_nat @ nil_nat @ ( shuffles_nat @ Xs @ Ys2 ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_382_Nil__in__shufflesI,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Xs = nil_Pr5478986624290739719at_nat )
=> ( ( Ys2 = nil_Pr5478986624290739719at_nat )
=> ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ ( shuffl5088453890617037852at_nat @ Xs @ Ys2 ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_383_Nil__in__shuffles,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( member_list_nat @ nil_nat @ ( shuffles_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_in_shuffles
thf(fact_384_Nil__in__shuffles,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ ( shuffl5088453890617037852at_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
& ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ) ).
% Nil_in_shuffles
thf(fact_385_append__eq__append__conv__if,axiom,
! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
( ( ( append_nat @ Xs_1 @ Xs_2 )
= ( append_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_386_rev__cases,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y5: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y5 @ nil_nat ) ) ) ) ).
% rev_cases
thf(fact_387_rev__cases,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ~ ! [Ys4: list_mat_a,Y5: mat_a] :
( Xs
!= ( append_mat_a @ Ys4 @ ( cons_mat_a @ Y5 @ nil_mat_a ) ) ) ) ).
% rev_cases
thf(fact_388_rev__cases,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ~ ! [Ys4: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat] :
( Xs
!= ( append985823374593552924at_nat @ Ys4 @ ( cons_P6512896166579812791at_nat @ Y5 @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% rev_cases
thf(fact_389_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X5: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X5 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_390_rev__induct,axiom,
! [P: list_mat_a > $o,Xs: list_mat_a] :
( ( P @ nil_mat_a )
=> ( ! [X5: mat_a,Xs2: list_mat_a] :
( ( P @ Xs2 )
=> ( P @ ( append_mat_a @ Xs2 @ ( cons_mat_a @ X5 @ nil_mat_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_391_rev__induct,axiom,
! [P: list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( P @ Xs2 )
=> ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_392_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_393_append1__eq__conv,axiom,
! [Xs: list_mat_a,X: mat_a,Ys2: list_mat_a,Y: mat_a] :
( ( ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) )
= ( append_mat_a @ Ys2 @ ( cons_mat_a @ Y @ nil_mat_a ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_394_append1__eq__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) )
= ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_395_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs2 ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X @ Ys5 )
= Ys2 )
& ( Xs
= ( append_nat @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_396_Cons__eq__append__conv,axiom,
! [X: mat_a,Xs: list_mat_a,Ys2: list_mat_a,Zs2: list_mat_a] :
( ( ( cons_mat_a @ X @ Xs )
= ( append_mat_a @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_mat_a )
& ( ( cons_mat_a @ X @ Xs )
= Zs2 ) )
| ? [Ys5: list_mat_a] :
( ( ( cons_mat_a @ X @ Ys5 )
= Ys2 )
& ( Xs
= ( append_mat_a @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_397_Cons__eq__append__conv,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X @ Xs )
= ( append985823374593552924at_nat @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_Pr5478986624290739719at_nat )
& ( ( cons_P6512896166579812791at_nat @ X @ Xs )
= Zs2 ) )
| ? [Ys5: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X @ Ys5 )
= Ys2 )
& ( Xs
= ( append985823374593552924at_nat @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_398_append__eq__Cons__conv,axiom,
! [Ys2: list_nat,Zs2: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys2 @ Zs2 )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys2 = nil_nat )
& ( Zs2
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys5: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_399_append__eq__Cons__conv,axiom,
! [Ys2: list_mat_a,Zs2: list_mat_a,X: mat_a,Xs: list_mat_a] :
( ( ( append_mat_a @ Ys2 @ Zs2 )
= ( cons_mat_a @ X @ Xs ) )
= ( ( ( Ys2 = nil_mat_a )
& ( Zs2
= ( cons_mat_a @ X @ Xs ) ) )
| ? [Ys5: list_mat_a] :
( ( Ys2
= ( cons_mat_a @ X @ Ys5 ) )
& ( ( append_mat_a @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_400_append__eq__Cons__conv,axiom,
! [Ys2: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Ys2 @ Zs2 )
= ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( ( ( Ys2 = nil_Pr5478986624290739719at_nat )
& ( Zs2
= ( cons_P6512896166579812791at_nat @ X @ Xs ) ) )
| ? [Ys5: list_P6011104703257516679at_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ X @ Ys5 ) )
& ( ( append985823374593552924at_nat @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_401_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X5: nat] : ( P @ ( cons_nat @ X5 @ nil_nat ) )
=> ( ! [X5: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X5 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_402_rev__nonempty__induct,axiom,
! [Xs: list_mat_a,P: list_mat_a > $o] :
( ( Xs != nil_mat_a )
=> ( ! [X5: mat_a] : ( P @ ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ( ! [X5: mat_a,Xs2: list_mat_a] :
( ( Xs2 != nil_mat_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_mat_a @ Xs2 @ ( cons_mat_a @ X5 @ nil_mat_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_403_rev__nonempty__induct,axiom,
! [Xs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ! [X5: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_404_hd__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Ys2 ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_405_hd__append,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( Xs = nil_Pr5478986624290739719at_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( hd_Pro3460610213475200108at_nat @ Ys2 ) ) )
& ( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( hd_Pro3460610213475200108at_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_406_hd__append2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_407_hd__append2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( hd_Pro3460610213475200108at_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_408_longest__common__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
? [Ps: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs4 ) )
& ( Ys2
= ( append_nat @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( hd_nat @ Xs4 )
!= ( hd_nat @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_409_longest__common__prefix,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
? [Ps: list_P6011104703257516679at_nat,Xs4: list_P6011104703257516679at_nat,Ys6: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Ps @ Xs4 ) )
& ( Ys2
= ( append985823374593552924at_nat @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_Pr5478986624290739719at_nat )
| ( Ys6 = nil_Pr5478986624290739719at_nat )
| ( ( hd_Pro3460610213475200108at_nat @ Xs4 )
!= ( hd_Pro3460610213475200108at_nat @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_410_tl__append2,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( tl_mat_a @ ( append_mat_a @ Xs @ Ys2 ) )
= ( append_mat_a @ ( tl_mat_a @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_411_tl__append2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_412_tl__append2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( tl_Pro4228036916689694320at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( append985823374593552924at_nat @ ( tl_Pro4228036916689694320at_nat @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_413_tl__append__if,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a] :
( ( ( Xs = nil_mat_a )
=> ( ( tl_mat_a @ ( append_mat_a @ Xs @ Ys2 ) )
= ( tl_mat_a @ Ys2 ) ) )
& ( ( Xs != nil_mat_a )
=> ( ( tl_mat_a @ ( append_mat_a @ Xs @ Ys2 ) )
= ( append_mat_a @ ( tl_mat_a @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_414_tl__append__if,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( tl_nat @ Ys2 ) ) )
& ( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_415_tl__append__if,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( Xs = nil_Pr5478986624290739719at_nat )
=> ( ( tl_Pro4228036916689694320at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( tl_Pro4228036916689694320at_nat @ Ys2 ) ) )
& ( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( tl_Pro4228036916689694320at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( append985823374593552924at_nat @ ( tl_Pro4228036916689694320at_nat @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_416_le__simps_I3_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% le_simps(3)
thf(fact_417_le__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(2)
thf(fact_418_not__less__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(2)
thf(fact_419_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_420_dec__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ I3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_421_inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_422_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_423_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_424_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_425_shufflesE,axiom,
! [Zs2: list_nat,Xs: list_nat,Ys2: list_nat] :
( ( member_list_nat @ Zs2 @ ( shuffles_nat @ Xs @ Ys2 ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys2 != nil_nat ) )
=> ( ( ( Zs2 = Ys2 )
=> ( Xs != nil_nat ) )
=> ( ! [X5: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X5 @ Xs4 ) )
=> ! [Z2: nat,Zs3: list_nat] :
( ( Zs2
= ( cons_nat @ Z2 @ Zs3 ) )
=> ( ( X5 = Z2 )
=> ~ ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs4 @ Ys2 ) ) ) ) )
=> ~ ! [Y5: nat,Ys6: list_nat] :
( ( Ys2
= ( cons_nat @ Y5 @ Ys6 ) )
=> ! [Z2: nat,Zs3: list_nat] :
( ( Zs2
= ( cons_nat @ Z2 @ Zs3 ) )
=> ( ( Y5 = Z2 )
=> ~ ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_426_shufflesE,axiom,
! [Zs2: list_mat_a,Xs: list_mat_a,Ys2: list_mat_a] :
( ( member_list_mat_a @ Zs2 @ ( shuffles_mat_a @ Xs @ Ys2 ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys2 != nil_mat_a ) )
=> ( ( ( Zs2 = Ys2 )
=> ( Xs != nil_mat_a ) )
=> ( ! [X5: mat_a,Xs4: list_mat_a] :
( ( Xs
= ( cons_mat_a @ X5 @ Xs4 ) )
=> ! [Z2: mat_a,Zs3: list_mat_a] :
( ( Zs2
= ( cons_mat_a @ Z2 @ Zs3 ) )
=> ( ( X5 = Z2 )
=> ~ ( member_list_mat_a @ Zs3 @ ( shuffles_mat_a @ Xs4 @ Ys2 ) ) ) ) )
=> ~ ! [Y5: mat_a,Ys6: list_mat_a] :
( ( Ys2
= ( cons_mat_a @ Y5 @ Ys6 ) )
=> ! [Z2: mat_a,Zs3: list_mat_a] :
( ( Zs2
= ( cons_mat_a @ Z2 @ Zs3 ) )
=> ( ( Y5 = Z2 )
=> ~ ( member_list_mat_a @ Zs3 @ ( shuffles_mat_a @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_427_shufflesE,axiom,
! [Zs2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member3067507820990806192at_nat @ Zs2 @ ( shuffl5088453890617037852at_nat @ Xs @ Ys2 ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys2 != nil_Pr5478986624290739719at_nat ) )
=> ( ( ( Zs2 = Ys2 )
=> ( Xs != nil_Pr5478986624290739719at_nat ) )
=> ( ! [X5: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ X5 @ Xs4 ) )
=> ! [Z2: product_prod_nat_nat,Zs3: list_P6011104703257516679at_nat] :
( ( Zs2
= ( cons_P6512896166579812791at_nat @ Z2 @ Zs3 ) )
=> ( ( X5 = Z2 )
=> ~ ( member3067507820990806192at_nat @ Zs3 @ ( shuffl5088453890617037852at_nat @ Xs4 @ Ys2 ) ) ) ) )
=> ~ ! [Y5: product_prod_nat_nat,Ys6: list_P6011104703257516679at_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ Y5 @ Ys6 ) )
=> ! [Z2: product_prod_nat_nat,Zs3: list_P6011104703257516679at_nat] :
( ( Zs2
= ( cons_P6512896166579812791at_nat @ Z2 @ Zs3 ) )
=> ( ( Y5 = Z2 )
=> ~ ( member3067507820990806192at_nat @ Zs3 @ ( shuffl5088453890617037852at_nat @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_428_impossible__Cons,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,X: mat_a] :
( ( ord_less_eq_nat @ ( size_size_list_mat_a @ Xs ) @ ( size_size_list_mat_a @ Ys2 ) )
=> ( Xs
!= ( cons_mat_a @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_429_impossible__Cons,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( Xs
!= ( cons_P6512896166579812791at_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_430_impossible__Cons,axiom,
! [Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs
!= ( cons_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_431_butlast__append,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_432_butlast__append,axiom,
! [Ys2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat] :
( ( ( Ys2 = nil_Pr5478986624290739719at_nat )
=> ( ( butlas5569151904373679443at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( butlas5569151904373679443at_nat @ Xs ) ) )
& ( ( Ys2 != nil_Pr5478986624290739719at_nat )
=> ( ( butlas5569151904373679443at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) )
= ( append985823374593552924at_nat @ Xs @ ( butlas5569151904373679443at_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_433_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_434_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_435_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_436_same__length__different,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys2 ) )
=> ? [Pre: list_mat_a,X5: mat_a,Xs4: list_mat_a,Y5: mat_a,Ys6: list_mat_a] :
( ( X5 != Y5 )
& ( Xs
= ( append_mat_a @ Pre @ ( append_mat_a @ ( cons_mat_a @ X5 @ nil_mat_a ) @ Xs4 ) ) )
& ( Ys2
= ( append_mat_a @ Pre @ ( append_mat_a @ ( cons_mat_a @ Y5 @ nil_mat_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_437_same__length__different,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Xs != Ys2 )
=> ( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ? [Pre: list_P6011104703257516679at_nat,X5: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys6: list_P6011104703257516679at_nat] :
( ( X5 != Y5 )
& ( Xs
= ( append985823374593552924at_nat @ Pre @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) @ Xs4 ) ) )
& ( Ys2
= ( append985823374593552924at_nat @ Pre @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ Y5 @ nil_Pr5478986624290739719at_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_438_same__length__different,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ? [Pre: list_nat,X5: nat,Xs4: list_nat,Y5: nat,Ys6: list_nat] :
( ( X5 != Y5 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X5 @ nil_nat ) @ Xs4 ) ) )
& ( Ys2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y5 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_439_nth__append__length,axiom,
! [Xs: list_mat_a,X: mat_a,Ys2: list_mat_a] :
( ( nth_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ Ys2 ) ) @ ( size_size_list_mat_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_440_nth__append__length,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( nth_Pr7617993195940197384at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ Ys2 ) ) @ ( size_s5460976970255530739at_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_441_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_442_remdups__adj__append__two,axiom,
! [Xs: list_nat,X: nat,Y: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) @ ( if_list_nat @ ( X = Y ) @ nil_nat @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_443_remdups__adj__append__two,axiom,
! [Xs: list_mat_a,X: mat_a,Y: mat_a] :
( ( remdups_adj_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ ( cons_mat_a @ Y @ nil_mat_a ) ) ) )
= ( append_mat_a @ ( remdups_adj_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) ) ) @ ( if_list_mat_a @ ( X = Y ) @ nil_mat_a @ ( cons_mat_a @ Y @ nil_mat_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_444_remdups__adj__append__two,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( remdup844249387045036349at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) ) )
= ( append985823374593552924at_nat @ ( remdup844249387045036349at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ) @ ( if_lis9186351972506106189at_nat @ ( X = Y ) @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_445_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_446_butlast__snoc,axiom,
! [Xs: list_mat_a,X: mat_a] :
( ( butlast_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_447_butlast__snoc,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( butlas5569151904373679443at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_448_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_mat_a @ Xs ) )
= ( ? [X4: mat_a,Ys3: list_mat_a] :
( ( Xs
= ( cons_mat_a @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_mat_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_449_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s5460976970255530739at_nat @ Xs ) )
= ( ? [X4: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_450_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_451_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Zs2 )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs2 ) )
& ( Ys2
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs2 ) ) ) ) ).
% append_eq_conv_conj
thf(fact_452_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( drop_P8868858903918902087at_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_453_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_454_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ( drop_P8868858903918902087at_nat @ N @ Xs )
= nil_Pr5478986624290739719at_nat )
= ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_455_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_456_drop__all,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ N )
=> ( ( drop_P8868858903918902087at_nat @ N @ Xs )
= nil_Pr5478986624290739719at_nat ) ) ).
% drop_all
thf(fact_457_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_458_length__Suc__conv__rev,axiom,
! [Xs: list_mat_a,N: nat] :
( ( ( size_size_list_mat_a @ Xs )
= ( suc @ N ) )
= ( ? [Y4: mat_a,Ys3: list_mat_a] :
( ( Xs
= ( append_mat_a @ Ys3 @ ( cons_mat_a @ Y4 @ nil_mat_a ) ) )
& ( ( size_size_list_mat_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_459_length__Suc__conv__rev,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) ) )
& ( ( size_s5460976970255530739at_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_460_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_461_length__append__singleton,axiom,
! [Xs: list_mat_a,X: mat_a] :
( ( size_size_list_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) ) )
= ( suc @ ( size_size_list_mat_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_462_length__append__singleton,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( size_s5460976970255530739at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) )
= ( suc @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_463_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_464_find__indices__snoc,axiom,
! [X: mat_a,Ys2: list_mat_a,Y: mat_a] :
( ( missin4242638364659709089_mat_a @ X @ ( append_mat_a @ Ys2 @ ( cons_mat_a @ Y @ nil_mat_a ) ) )
= ( append_nat @ ( missin4242638364659709089_mat_a @ X @ Ys2 ) @ ( if_list_nat @ ( X = Y ) @ ( cons_nat @ ( size_size_list_mat_a @ Ys2 ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_465_find__indices__snoc,axiom,
! [X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( missin7441370236764828603at_nat @ X @ ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
= ( append_nat @ ( missin7441370236764828603at_nat @ X @ Ys2 ) @ ( if_list_nat @ ( X = Y ) @ ( cons_nat @ ( size_s5460976970255530739at_nat @ Ys2 ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_466_find__indices__snoc,axiom,
! [X: nat,Ys2: list_nat,Y: nat] :
( ( missin5050847376130023830es_nat @ X @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( append_nat @ ( missin5050847376130023830es_nat @ X @ Ys2 ) @ ( if_list_nat @ ( X = Y ) @ ( cons_nat @ ( size_size_list_nat @ Ys2 ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_467_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_mat_a,A: mat_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( list_update_mat_a @ Xs @ I3 @ A )
= ( append_mat_a @ ( take_mat_a @ I3 @ Xs ) @ ( cons_mat_a @ A @ ( drop_mat_a @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_468_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat,A: product_prod_nat_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( list_u6180841689913720943at_nat @ Xs @ I3 @ A )
= ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ I3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ A @ ( drop_P8868858903918902087at_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_469_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I3 @ A )
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_470_rotate1__hd__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( rotate1_nat @ Xs )
= ( append_nat @ ( tl_nat @ Xs ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_471_rotate1__hd__tl,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( rotate1_mat_a @ Xs )
= ( append_mat_a @ ( tl_mat_a @ Xs ) @ ( cons_mat_a @ ( hd_mat_a @ Xs ) @ nil_mat_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_472_rotate1__hd__tl,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( rotate3092716997870431060at_nat @ Xs )
= ( append985823374593552924at_nat @ ( tl_Pro4228036916689694320at_nat @ Xs ) @ ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_473_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_474_SuccI,axiom,
! [Kl: list_mat_a,K: mat_a,Kl2: set_list_mat_a] :
( ( member_list_mat_a @ ( append_mat_a @ Kl @ ( cons_mat_a @ K @ nil_mat_a ) ) @ Kl2 )
=> ( member_mat_a @ K @ ( bNF_Gr1459196596068634368_mat_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_475_SuccI,axiom,
! [Kl: list_P6011104703257516679at_nat,K: product_prod_nat_nat,Kl2: set_li5450038453877631591at_nat] :
( ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl @ ( cons_P6512896166579812791at_nat @ K @ nil_Pr5478986624290739719at_nat ) ) @ Kl2 )
=> ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_476_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_477_SuccD,axiom,
! [K: mat_a,Kl2: set_list_mat_a,Kl: list_mat_a] :
( ( member_mat_a @ K @ ( bNF_Gr1459196596068634368_mat_a @ Kl2 @ Kl ) )
=> ( member_list_mat_a @ ( append_mat_a @ Kl @ ( cons_mat_a @ K @ nil_mat_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_478_SuccD,axiom,
! [K: product_prod_nat_nat,Kl2: set_li5450038453877631591at_nat,Kl: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl2 @ Kl ) )
=> ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl @ ( cons_P6512896166579812791at_nat @ K @ nil_Pr5478986624290739719at_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_479_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_480_list__update__code_I1_J,axiom,
! [I3: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I3 @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_481_list__update__code_I1_J,axiom,
! [I3: nat,Y: product_prod_nat_nat] :
( ( list_u6180841689913720943at_nat @ nil_Pr5478986624290739719at_nat @ I3 @ Y )
= nil_Pr5478986624290739719at_nat ) ).
% list_update_code(1)
thf(fact_482_list__update_Osimps_I1_J,axiom,
! [I3: nat,V2: nat] :
( ( list_update_nat @ nil_nat @ I3 @ V2 )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_483_list__update_Osimps_I1_J,axiom,
! [I3: nat,V2: product_prod_nat_nat] :
( ( list_u6180841689913720943at_nat @ nil_Pr5478986624290739719at_nat @ I3 @ V2 )
= nil_Pr5478986624290739719at_nat ) ).
% list_update.simps(1)
thf(fact_484_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs @ K @ X )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_485_list__update__nonempty,axiom,
! [Xs: list_P6011104703257516679at_nat,K: nat,X: product_prod_nat_nat] :
( ( ( list_u6180841689913720943at_nat @ Xs @ K @ X )
= nil_Pr5478986624290739719at_nat )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% list_update_nonempty
thf(fact_486_length__list__update,axiom,
! [Xs: list_nat,I3: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_487_list__update__id,axiom,
! [Xs: list_nat,I3: nat] :
( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_488_list__update__id,axiom,
! [Xs: list_mat_a,I3: nat] :
( ( list_update_mat_a @ Xs @ I3 @ ( nth_mat_a @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_489_list__update__id,axiom,
! [Xs: list_P6011104703257516679at_nat,I3: nat] :
( ( list_u6180841689913720943at_nat @ Xs @ I3 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_490_nth__list__update__neq,axiom,
! [I3: nat,J: nat,Xs: list_nat,X: nat] :
( ( I3 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_491_nth__list__update__neq,axiom,
! [I3: nat,J: nat,Xs: list_mat_a,X: mat_a] :
( ( I3 != J )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I3 @ X ) @ J )
= ( nth_mat_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_492_nth__list__update__neq,axiom,
! [I3: nat,J: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( I3 != J )
=> ( ( nth_Pr7617993195940197384at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I3 @ X ) @ J )
= ( nth_Pr7617993195940197384at_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_493_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_494_rotate1_Osimps_I1_J,axiom,
( ( rotate3092716997870431060at_nat @ nil_Pr5478986624290739719at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% rotate1.simps(1)
thf(fact_495_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_496_rotate1__is__Nil__conv,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( rotate3092716997870431060at_nat @ Xs )
= nil_Pr5478986624290739719at_nat )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_497_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_498_list__update__code_I3_J,axiom,
! [X: nat,Xs: list_nat,I3: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I3 ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ I3 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_499_list__update__code_I3_J,axiom,
! [X: mat_a,Xs: list_mat_a,I3: nat,Y: mat_a] :
( ( list_update_mat_a @ ( cons_mat_a @ X @ Xs ) @ ( suc @ I3 ) @ Y )
= ( cons_mat_a @ X @ ( list_update_mat_a @ Xs @ I3 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_500_list__update__code_I3_J,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,I3: nat,Y: product_prod_nat_nat] :
( ( list_u6180841689913720943at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( suc @ I3 ) @ Y )
= ( cons_P6512896166579812791at_nat @ X @ ( list_u6180841689913720943at_nat @ Xs @ I3 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_501_list__update__beyond,axiom,
! [Xs: list_nat,I3: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
=> ( ( list_update_nat @ Xs @ I3 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_502_find__indices__Nil,axiom,
! [X: nat] :
( ( missin5050847376130023830es_nat @ X @ nil_nat )
= nil_nat ) ).
% find_indices_Nil
thf(fact_503_find__indices__Nil,axiom,
! [X: product_prod_nat_nat] :
( ( missin7441370236764828603at_nat @ X @ nil_Pr5478986624290739719at_nat )
= nil_nat ) ).
% find_indices_Nil
thf(fact_504_list__update__append1,axiom,
! [I3: nat,Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ I3 @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I3 @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_505_list__update__length,axiom,
! [Xs: list_mat_a,X: mat_a,Ys2: list_mat_a,Y: mat_a] :
( ( list_update_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ Ys2 ) ) @ ( size_size_list_mat_a @ Xs ) @ Y )
= ( append_mat_a @ Xs @ ( cons_mat_a @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_506_list__update__length,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( list_u6180841689913720943at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ Ys2 ) ) @ ( size_s5460976970255530739at_nat @ Xs ) @ Y )
= ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_507_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_508_list__update__same__conv,axiom,
! [I3: nat,Xs: list_mat_a,X: mat_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( ( list_update_mat_a @ Xs @ I3 @ X )
= Xs )
= ( ( nth_mat_a @ Xs @ I3 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_509_list__update__same__conv,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ( list_u6180841689913720943at_nat @ Xs @ I3 @ X )
= Xs )
= ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_510_list__update__same__conv,axiom,
! [I3: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I3 @ X )
= Xs )
= ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_511_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_mat_a,X: mat_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I3 @ X ) @ I3 )
= X ) ) ).
% nth_list_update_eq
thf(fact_512_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I3 @ X ) @ I3 )
= X ) ) ).
% nth_list_update_eq
thf(fact_513_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X ) @ I3 )
= X ) ) ).
% nth_list_update_eq
thf(fact_514_nth__list__update,axiom,
! [I3: nat,Xs: list_mat_a,J: nat,X: mat_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( ( I3 = J )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I3 @ X ) @ J )
= X ) )
& ( ( I3 != J )
=> ( ( nth_mat_a @ ( list_update_mat_a @ Xs @ I3 @ X ) @ J )
= ( nth_mat_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_515_nth__list__update,axiom,
! [I3: nat,Xs: list_P6011104703257516679at_nat,J: nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ( I3 = J )
=> ( ( nth_Pr7617993195940197384at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I3 @ X ) @ J )
= X ) )
& ( ( I3 != J )
=> ( ( nth_Pr7617993195940197384at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I3 @ X ) @ J )
= ( nth_Pr7617993195940197384at_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_516_nth__list__update,axiom,
! [I3: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I3 = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X ) @ J )
= X ) )
& ( ( I3 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_517_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_518_rotate1_Osimps_I2_J,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( rotate1_mat_a @ ( cons_mat_a @ X @ Xs ) )
= ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) ) ) ).
% rotate1.simps(2)
thf(fact_519_rotate1_Osimps_I2_J,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( rotate3092716997870431060at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_520_B2__def,axiom,
( b2
= ( produc7340730364199978039_mat_a @ ( produc1482081755353976211_mat_a @ ( split_block_a @ ba @ ( hd_nat @ la ) @ ( hd_nat @ la ) ) ) ) ) ).
% B2_def
thf(fact_521_empty__Shift,axiom,
! [Kl2: set_list_mat_a,K: mat_a] :
( ( member_list_mat_a @ nil_mat_a @ Kl2 )
=> ( ( member_mat_a @ K @ ( bNF_Gr1459196596068634368_mat_a @ Kl2 @ nil_mat_a ) )
=> ( member_list_mat_a @ nil_mat_a @ ( bNF_Gr4483329336800378748_mat_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_522_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl2 )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_523_empty__Shift,axiom,
! [Kl2: set_li5450038453877631591at_nat,K: product_prod_nat_nat] :
( ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ Kl2 )
=> ( ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl2 @ nil_Pr5478986624290739719at_nat ) )
=> ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ ( bNF_Gr3130287167067265568at_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_524_Succ__Shift,axiom,
! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_525_Succ__Shift,axiom,
! [Kl2: set_list_mat_a,K: mat_a,Kl: list_mat_a] :
( ( bNF_Gr1459196596068634368_mat_a @ ( bNF_Gr4483329336800378748_mat_a @ Kl2 @ K ) @ Kl )
= ( bNF_Gr1459196596068634368_mat_a @ Kl2 @ ( cons_mat_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_526_Succ__Shift,axiom,
! [Kl2: set_li5450038453877631591at_nat,K: product_prod_nat_nat,Kl: list_P6011104703257516679at_nat] :
( ( bNF_Gr5363859321595349404at_nat @ ( bNF_Gr3130287167067265568at_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr5363859321595349404at_nat @ Kl2 @ ( cons_P6512896166579812791at_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_527_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_528_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_529_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_530_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_531_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_532_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_533_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_534_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_535_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_536_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_537_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_538_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_539_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_540_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_541_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_542_ShiftD,axiom,
! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_543_ShiftD,axiom,
! [Kl: list_mat_a,Kl2: set_list_mat_a,K: mat_a] :
( ( member_list_mat_a @ Kl @ ( bNF_Gr4483329336800378748_mat_a @ Kl2 @ K ) )
=> ( member_list_mat_a @ ( cons_mat_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_544_ShiftD,axiom,
! [Kl: list_P6011104703257516679at_nat,Kl2: set_li5450038453877631591at_nat,K: product_prod_nat_nat] :
( ( member3067507820990806192at_nat @ Kl @ ( bNF_Gr3130287167067265568at_nat @ Kl2 @ K ) )
=> ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_545_less__eq__prod__def,axiom,
( ord_le8460144461188290721at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) )
& ( ord_less_eq_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_546_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_547_B3__def,axiom,
( b3
= ( produc8618483072558553147_mat_a @ ( produc7508173349661082485_mat_a @ ( produc1482081755353976211_mat_a @ ( split_block_a @ ba @ ( hd_nat @ la ) @ ( hd_nat @ la ) ) ) ) ) ) ).
% B3_def
thf(fact_548_B4__def,axiom,
( b4
= ( produc3539460521124201597_mat_a @ ( produc7508173349661082485_mat_a @ ( produc1482081755353976211_mat_a @ ( split_block_a @ ba @ ( hd_nat @ la ) @ ( hd_nat @ la ) ) ) ) ) ) ).
% B4_def
thf(fact_549_prod_Oexpand,axiom,
! [Prod: produc4216251508294696237_mat_a,Prod2: produc4216251508294696237_mat_a] :
( ( ( ( produc7700291086614992977_mat_a @ Prod )
= ( produc7700291086614992977_mat_a @ Prod2 ) )
& ( ( produc1482081755353976211_mat_a @ Prod )
= ( produc1482081755353976211_mat_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_550_prod_Oexpand,axiom,
! [Prod: produc5452184871688341745_mat_a,Prod2: produc5452184871688341745_mat_a] :
( ( ( ( produc7340730364199978039_mat_a @ Prod )
= ( produc7340730364199978039_mat_a @ Prod2 ) )
& ( ( produc7508173349661082485_mat_a @ Prod )
= ( produc7508173349661082485_mat_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_551_prod_Oexpand,axiom,
! [Prod: produc5370362606830271383_mat_a,Prod2: produc5370362606830271383_mat_a] :
( ( ( ( produc8618483072558553147_mat_a @ Prod )
= ( produc8618483072558553147_mat_a @ Prod2 ) )
& ( ( produc3539460521124201597_mat_a @ Prod )
= ( produc3539460521124201597_mat_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_552_prod__eqI,axiom,
! [P5: produc4216251508294696237_mat_a,Q3: produc4216251508294696237_mat_a] :
( ( ( produc7700291086614992977_mat_a @ P5 )
= ( produc7700291086614992977_mat_a @ Q3 ) )
=> ( ( ( produc1482081755353976211_mat_a @ P5 )
= ( produc1482081755353976211_mat_a @ Q3 ) )
=> ( P5 = Q3 ) ) ) ).
% prod_eqI
thf(fact_553_prod__eqI,axiom,
! [P5: produc5452184871688341745_mat_a,Q3: produc5452184871688341745_mat_a] :
( ( ( produc7340730364199978039_mat_a @ P5 )
= ( produc7340730364199978039_mat_a @ Q3 ) )
=> ( ( ( produc7508173349661082485_mat_a @ P5 )
= ( produc7508173349661082485_mat_a @ Q3 ) )
=> ( P5 = Q3 ) ) ) ).
% prod_eqI
thf(fact_554_prod__eqI,axiom,
! [P5: produc5370362606830271383_mat_a,Q3: produc5370362606830271383_mat_a] :
( ( ( produc8618483072558553147_mat_a @ P5 )
= ( produc8618483072558553147_mat_a @ Q3 ) )
=> ( ( ( produc3539460521124201597_mat_a @ P5 )
= ( produc3539460521124201597_mat_a @ Q3 ) )
=> ( P5 = Q3 ) ) ) ).
% prod_eqI
thf(fact_555_exE__realizer_H,axiom,
! [P: produc5452184871688341745_mat_a > mat_a > $o,P5: produc4216251508294696237_mat_a] :
( ( P @ ( produc1482081755353976211_mat_a @ P5 ) @ ( produc7700291086614992977_mat_a @ P5 ) )
=> ~ ! [X5: mat_a,Y5: produc5452184871688341745_mat_a] :
~ ( P @ Y5 @ X5 ) ) ).
% exE_realizer'
thf(fact_556_exE__realizer_H,axiom,
! [P: produc5370362606830271383_mat_a > mat_a > $o,P5: produc5452184871688341745_mat_a] :
( ( P @ ( produc7508173349661082485_mat_a @ P5 ) @ ( produc7340730364199978039_mat_a @ P5 ) )
=> ~ ! [X5: mat_a,Y5: produc5370362606830271383_mat_a] :
~ ( P @ Y5 @ X5 ) ) ).
% exE_realizer'
thf(fact_557_exE__realizer_H,axiom,
! [P: mat_a > mat_a > $o,P5: produc5370362606830271383_mat_a] :
( ( P @ ( produc3539460521124201597_mat_a @ P5 ) @ ( produc8618483072558553147_mat_a @ P5 ) )
=> ~ ! [X5: mat_a,Y5: mat_a] :
~ ( P @ Y5 @ X5 ) ) ).
% exE_realizer'
thf(fact_558_prod__eq__iff,axiom,
( ( ^ [Y3: produc4216251508294696237_mat_a,Z: produc4216251508294696237_mat_a] : ( Y3 = Z ) )
= ( ^ [S2: produc4216251508294696237_mat_a,T2: produc4216251508294696237_mat_a] :
( ( ( produc7700291086614992977_mat_a @ S2 )
= ( produc7700291086614992977_mat_a @ T2 ) )
& ( ( produc1482081755353976211_mat_a @ S2 )
= ( produc1482081755353976211_mat_a @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_559_prod__eq__iff,axiom,
( ( ^ [Y3: produc5452184871688341745_mat_a,Z: produc5452184871688341745_mat_a] : ( Y3 = Z ) )
= ( ^ [S2: produc5452184871688341745_mat_a,T2: produc5452184871688341745_mat_a] :
( ( ( produc7340730364199978039_mat_a @ S2 )
= ( produc7340730364199978039_mat_a @ T2 ) )
& ( ( produc7508173349661082485_mat_a @ S2 )
= ( produc7508173349661082485_mat_a @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_560_prod__eq__iff,axiom,
( ( ^ [Y3: produc5370362606830271383_mat_a,Z: produc5370362606830271383_mat_a] : ( Y3 = Z ) )
= ( ^ [S2: produc5370362606830271383_mat_a,T2: produc5370362606830271383_mat_a] :
( ( ( produc8618483072558553147_mat_a @ S2 )
= ( produc8618483072558553147_mat_a @ T2 ) )
& ( ( produc3539460521124201597_mat_a @ S2 )
= ( produc3539460521124201597_mat_a @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_561_sp,axiom,
( ( split_block_a @ ba @ ( hd_nat @ la ) @ ( hd_nat @ la ) )
= ( produc5286753621172121189_mat_a @ b1 @ ( produc7602877900562455331_mat_a @ b2 @ ( produc3091253522927621199_mat_a @ b3 @ b4 ) ) ) ) ).
% sp
thf(fact_562_basic__trans__rules_I22_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% basic_trans_rules(22)
thf(fact_563_basic__trans__rules_I21_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% basic_trans_rules(21)
thf(fact_564_basic__trans__rules_I18_J,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% basic_trans_rules(18)
thf(fact_565_basic__trans__rules_I17_J,axiom,
! [A: nat,B3: nat] :
( ( A != B3 )
=> ( ( ord_less_eq_nat @ A @ B3 )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% basic_trans_rules(17)
thf(fact_566_basic__trans__rules_I6_J,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_567_split__block__diag__carrier_I1_J,axiom,
! [D: mat_a,N: nat,A: nat,D1: mat_a,D2: mat_a,D3: mat_a,D4: mat_a] :
( ( member_mat_a @ D @ ( carrier_mat_a @ N @ N ) )
=> ( ( ord_less_eq_nat @ A @ N )
=> ( ( ( split_block_a @ D @ A @ A )
= ( produc5286753621172121189_mat_a @ D1 @ ( produc7602877900562455331_mat_a @ D2 @ ( produc3091253522927621199_mat_a @ D3 @ D4 ) ) ) )
=> ( member_mat_a @ D1 @ ( carrier_mat_a @ A @ A ) ) ) ) ) ).
% split_block_diag_carrier(1)
thf(fact_568_extract__subdiags_Oinduct,axiom,
! [P: mat_a > list_nat > $o,A0: mat_a,A1: list_nat] :
( ! [B4: mat_a] : ( P @ B4 @ nil_nat )
=> ( ! [B4: mat_a,X5: nat,Xs2: list_nat] :
( ! [Xa: produc4216251508294696237_mat_a,Xb: mat_a,Y6: produc5452184871688341745_mat_a,Xc: mat_a,Ya: produc5370362606830271383_mat_a,Xd: mat_a,Yb: mat_a] :
( ( Xa
= ( split_block_a @ B4 @ X5 @ X5 ) )
=> ( ( ( produc5286753621172121189_mat_a @ Xb @ Y6 )
= Xa )
=> ( ( ( produc7602877900562455331_mat_a @ Xc @ Ya )
= Y6 )
=> ( ( ( produc3091253522927621199_mat_a @ Xd @ Yb )
= Ya )
=> ( P @ Yb @ Xs2 ) ) ) ) )
=> ( P @ B4 @ ( cons_nat @ X5 @ Xs2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% extract_subdiags.induct
thf(fact_569_extract__subdiags__not__emp_I1_J,axiom,
! [B1: mat_a,B22: mat_a,B32: mat_a,B42: mat_a,B: mat_a,X: nat,L: list_nat] :
( ( ( produc5286753621172121189_mat_a @ B1 @ ( produc7602877900562455331_mat_a @ B22 @ ( produc3091253522927621199_mat_a @ B32 @ B42 ) ) )
= ( split_block_a @ B @ X @ X ) )
=> ( ( hd_mat_a @ ( commut2531942506349284476iags_a @ B @ ( cons_nat @ X @ L ) ) )
= B1 ) ) ).
% extract_subdiags_not_emp(1)
thf(fact_570_fst__conv,axiom,
! [X12: nat,X2: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_571_fst__conv,axiom,
! [X12: nat > nat,X2: nat] :
( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_572_fst__conv,axiom,
! [X12: mat_a,X2: produc5452184871688341745_mat_a] :
( ( produc7700291086614992977_mat_a @ ( produc5286753621172121189_mat_a @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_573_fst__conv,axiom,
! [X12: mat_a,X2: produc5370362606830271383_mat_a] :
( ( produc7340730364199978039_mat_a @ ( produc7602877900562455331_mat_a @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_574_fst__conv,axiom,
! [X12: mat_a,X2: mat_a] :
( ( produc8618483072558553147_mat_a @ ( produc3091253522927621199_mat_a @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_575_fst__eqD,axiom,
! [X: nat,Y: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_576_fst__eqD,axiom,
! [X: nat > nat,Y: nat,A: nat > nat] :
( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_577_fst__eqD,axiom,
! [X: mat_a,Y: produc5452184871688341745_mat_a,A: mat_a] :
( ( ( produc7700291086614992977_mat_a @ ( produc5286753621172121189_mat_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_578_fst__eqD,axiom,
! [X: mat_a,Y: produc5370362606830271383_mat_a,A: mat_a] :
( ( ( produc7340730364199978039_mat_a @ ( produc7602877900562455331_mat_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_579_fst__eqD,axiom,
! [X: mat_a,Y: mat_a,A: mat_a] :
( ( ( produc8618483072558553147_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_580_extract__subdiags__not__emp_I2_J,axiom,
! [B1: mat_a,B22: mat_a,B32: mat_a,B42: mat_a,B: mat_a,X: nat,L: list_nat] :
( ( ( produc5286753621172121189_mat_a @ B1 @ ( produc7602877900562455331_mat_a @ B22 @ ( produc3091253522927621199_mat_a @ B32 @ B42 ) ) )
= ( split_block_a @ B @ X @ X ) )
=> ( ( tl_mat_a @ ( commut2531942506349284476iags_a @ B @ ( cons_nat @ X @ L ) ) )
= ( commut2531942506349284476iags_a @ B42 @ L ) ) ) ).
% extract_subdiags_not_emp(2)
thf(fact_581_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: nat > nat > $o,X: nat,Y: nat,A: product_prod_nat_nat] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_nat_nat @ X @ Y ) )
=> ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_582_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( nat > nat ) > nat > $o,X: nat > nat,Y: nat,A: produc8199716216217303280at_nat] :
( ( P @ X @ Y )
=> ( ( A
= ( produc72220940542539688at_nat @ X @ Y ) )
=> ( P @ ( produc6156676138143019412at_nat @ A ) @ ( produc1852801350702243542at_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_583_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: mat_a > produc5452184871688341745_mat_a > $o,X: mat_a,Y: produc5452184871688341745_mat_a,A: produc4216251508294696237_mat_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc5286753621172121189_mat_a @ X @ Y ) )
=> ( P @ ( produc7700291086614992977_mat_a @ A ) @ ( produc1482081755353976211_mat_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_584_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: mat_a > produc5370362606830271383_mat_a > $o,X: mat_a,Y: produc5370362606830271383_mat_a,A: produc5452184871688341745_mat_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc7602877900562455331_mat_a @ X @ Y ) )
=> ( P @ ( produc7340730364199978039_mat_a @ A ) @ ( produc7508173349661082485_mat_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_585_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: mat_a > mat_a > $o,X: mat_a,Y: mat_a,A: produc5370362606830271383_mat_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc3091253522927621199_mat_a @ X @ Y ) )
=> ( P @ ( produc8618483072558553147_mat_a @ A ) @ ( produc3539460521124201597_mat_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_586_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_587_prod_Ocollapse,axiom,
! [Prod: produc8199716216217303280at_nat] :
( ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_588_prod_Ocollapse,axiom,
! [Prod: produc4216251508294696237_mat_a] :
( ( produc5286753621172121189_mat_a @ ( produc7700291086614992977_mat_a @ Prod ) @ ( produc1482081755353976211_mat_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_589_prod_Ocollapse,axiom,
! [Prod: produc5452184871688341745_mat_a] :
( ( produc7602877900562455331_mat_a @ ( produc7340730364199978039_mat_a @ Prod ) @ ( produc7508173349661082485_mat_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_590_prod_Ocollapse,axiom,
! [Prod: produc5370362606830271383_mat_a] :
( ( produc3091253522927621199_mat_a @ ( produc8618483072558553147_mat_a @ Prod ) @ ( produc3539460521124201597_mat_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_591_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_592_prod_Oexhaust__sel,axiom,
! [Prod: produc8199716216217303280at_nat] :
( Prod
= ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_593_prod_Oexhaust__sel,axiom,
! [Prod: produc4216251508294696237_mat_a] :
( Prod
= ( produc5286753621172121189_mat_a @ ( produc7700291086614992977_mat_a @ Prod ) @ ( produc1482081755353976211_mat_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_594_prod_Oexhaust__sel,axiom,
! [Prod: produc5452184871688341745_mat_a] :
( Prod
= ( produc7602877900562455331_mat_a @ ( produc7340730364199978039_mat_a @ Prod ) @ ( produc7508173349661082485_mat_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_595_prod_Oexhaust__sel,axiom,
! [Prod: produc5370362606830271383_mat_a] :
( Prod
= ( produc3091253522927621199_mat_a @ ( produc8618483072558553147_mat_a @ Prod ) @ ( produc3539460521124201597_mat_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_596_exI__realizer,axiom,
! [P: nat > nat > $o,Y: nat,X: nat] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_597_exI__realizer,axiom,
! [P: nat > ( nat > nat ) > $o,Y: nat,X: nat > nat] :
( ( P @ Y @ X )
=> ( P @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X @ Y ) ) @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_598_exI__realizer,axiom,
! [P: produc5452184871688341745_mat_a > mat_a > $o,Y: produc5452184871688341745_mat_a,X: mat_a] :
( ( P @ Y @ X )
=> ( P @ ( produc1482081755353976211_mat_a @ ( produc5286753621172121189_mat_a @ X @ Y ) ) @ ( produc7700291086614992977_mat_a @ ( produc5286753621172121189_mat_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_599_exI__realizer,axiom,
! [P: produc5370362606830271383_mat_a > mat_a > $o,Y: produc5370362606830271383_mat_a,X: mat_a] :
( ( P @ Y @ X )
=> ( P @ ( produc7508173349661082485_mat_a @ ( produc7602877900562455331_mat_a @ X @ Y ) ) @ ( produc7340730364199978039_mat_a @ ( produc7602877900562455331_mat_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_600_exI__realizer,axiom,
! [P: mat_a > mat_a > $o,Y: mat_a,X: mat_a] :
( ( P @ Y @ X )
=> ( P @ ( produc3539460521124201597_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) ) @ ( produc8618483072558553147_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_601_conjI__realizer,axiom,
! [P: nat > $o,P5: nat,Q: nat > $o,Q3: nat] :
( ( P @ P5 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P5 @ Q3 ) ) )
& ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P5 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_602_conjI__realizer,axiom,
! [P: ( nat > nat ) > $o,P5: nat > nat,Q: nat > $o,Q3: nat] :
( ( P @ P5 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ P5 @ Q3 ) ) )
& ( Q @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ P5 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_603_conjI__realizer,axiom,
! [P: mat_a > $o,P5: mat_a,Q: produc5452184871688341745_mat_a > $o,Q3: produc5452184871688341745_mat_a] :
( ( P @ P5 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc7700291086614992977_mat_a @ ( produc5286753621172121189_mat_a @ P5 @ Q3 ) ) )
& ( Q @ ( produc1482081755353976211_mat_a @ ( produc5286753621172121189_mat_a @ P5 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_604_conjI__realizer,axiom,
! [P: mat_a > $o,P5: mat_a,Q: produc5370362606830271383_mat_a > $o,Q3: produc5370362606830271383_mat_a] :
( ( P @ P5 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc7340730364199978039_mat_a @ ( produc7602877900562455331_mat_a @ P5 @ Q3 ) ) )
& ( Q @ ( produc7508173349661082485_mat_a @ ( produc7602877900562455331_mat_a @ P5 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_605_conjI__realizer,axiom,
! [P: mat_a > $o,P5: mat_a,Q: mat_a > $o,Q3: mat_a] :
( ( P @ P5 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc8618483072558553147_mat_a @ ( produc3091253522927621199_mat_a @ P5 @ Q3 ) ) )
& ( Q @ ( produc3539460521124201597_mat_a @ ( produc3091253522927621199_mat_a @ P5 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_606_surjective__pairing,axiom,
! [T: product_prod_nat_nat] :
( T
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_607_surjective__pairing,axiom,
! [T: produc8199716216217303280at_nat] :
( T
= ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ T ) @ ( produc1852801350702243542at_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_608_surjective__pairing,axiom,
! [T: produc4216251508294696237_mat_a] :
( T
= ( produc5286753621172121189_mat_a @ ( produc7700291086614992977_mat_a @ T ) @ ( produc1482081755353976211_mat_a @ T ) ) ) ).
% surjective_pairing
thf(fact_609_surjective__pairing,axiom,
! [T: produc5452184871688341745_mat_a] :
( T
= ( produc7602877900562455331_mat_a @ ( produc7340730364199978039_mat_a @ T ) @ ( produc7508173349661082485_mat_a @ T ) ) ) ).
% surjective_pairing
thf(fact_610_surjective__pairing,axiom,
! [T: produc5370362606830271383_mat_a] :
( T
= ( produc3091253522927621199_mat_a @ ( produc8618483072558553147_mat_a @ T ) @ ( produc3539460521124201597_mat_a @ T ) ) ) ).
% surjective_pairing
thf(fact_611_Pair__mono,axiom,
! [X: nat > nat,X7: nat > nat,Y: nat,Y7: nat] :
( ( ord_less_eq_nat_nat @ X @ X7 )
=> ( ( ord_less_eq_nat @ Y @ Y7 )
=> ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ ( produc72220940542539688at_nat @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_612_Pair__mono,axiom,
! [X: nat,X7: nat,Y: nat,Y7: nat] :
( ( ord_less_eq_nat @ X @ X7 )
=> ( ( ord_less_eq_nat @ Y @ Y7 )
=> ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_613_Pair__le,axiom,
! [A: nat > nat,B3: nat,C: nat > nat,D5: nat] :
( ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ A @ B3 ) @ ( produc72220940542539688at_nat @ C @ D5 ) )
= ( ( ord_less_eq_nat_nat @ A @ C )
& ( ord_less_eq_nat @ B3 @ D5 ) ) ) ).
% Pair_le
thf(fact_614_Pair__le,axiom,
! [A: nat,B3: nat,C: nat,D5: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ ( product_Pair_nat_nat @ C @ D5 ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ B3 @ D5 ) ) ) ).
% Pair_le
thf(fact_615_basic__trans__rules_I26_J,axiom,
! [A: nat,B3: nat,C: nat] :
( ( A = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_616_basic__trans__rules_I25_J,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_617_basic__trans__rules_I24_J,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_618_basic__trans__rules_I23_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% basic_trans_rules(23)
thf(fact_619_basic__trans__rules_I10_J,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_620_basic__trans__rules_I9_J,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_621_basic__trans__rules_I8_J,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_622_basic__trans__rules_I7_J,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_623_basic__trans__rules_I28_J,axiom,
! [A: nat,B3: nat,C: nat] :
( ( A = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% basic_trans_rules(28)
thf(fact_624_basic__trans__rules_I27_J,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% basic_trans_rules(27)
thf(fact_625_basic__trans__rules_I20_J,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ~ ( ord_less_nat @ B3 @ A ) ) ).
% basic_trans_rules(20)
thf(fact_626_basic__trans__rules_I19_J,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% basic_trans_rules(19)
thf(fact_627_basic__trans__rules_I12_J,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(12)
thf(fact_628_basic__trans__rules_I11_J,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(11)
thf(fact_629_basic__trans__rules_I2_J,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(2)
thf(fact_630_basic__trans__rules_I1_J,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(1)
thf(fact_631_basic__trans__rules_I3_J,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_632_basic__trans__rules_I4_J,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_633_basic__trans__rules_I5_J,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_634_split__pairs,axiom,
! [A2: nat,B: nat,X8: product_prod_nat_nat] :
( ( ( product_Pair_nat_nat @ A2 @ B )
= X8 )
= ( ( ( product_fst_nat_nat @ X8 )
= A2 )
& ( ( product_snd_nat_nat @ X8 )
= B ) ) ) ).
% split_pairs
thf(fact_635_split__pairs,axiom,
! [A2: nat > nat,B: nat,X8: produc8199716216217303280at_nat] :
( ( ( produc72220940542539688at_nat @ A2 @ B )
= X8 )
= ( ( ( produc6156676138143019412at_nat @ X8 )
= A2 )
& ( ( produc1852801350702243542at_nat @ X8 )
= B ) ) ) ).
% split_pairs
thf(fact_636_split__pairs,axiom,
! [A2: mat_a,B: produc5452184871688341745_mat_a,X8: produc4216251508294696237_mat_a] :
( ( ( produc5286753621172121189_mat_a @ A2 @ B )
= X8 )
= ( ( ( produc7700291086614992977_mat_a @ X8 )
= A2 )
& ( ( produc1482081755353976211_mat_a @ X8 )
= B ) ) ) ).
% split_pairs
thf(fact_637_split__pairs,axiom,
! [A2: mat_a,B: produc5370362606830271383_mat_a,X8: produc5452184871688341745_mat_a] :
( ( ( produc7602877900562455331_mat_a @ A2 @ B )
= X8 )
= ( ( ( produc7340730364199978039_mat_a @ X8 )
= A2 )
& ( ( produc7508173349661082485_mat_a @ X8 )
= B ) ) ) ).
% split_pairs
thf(fact_638_split__pairs,axiom,
! [A2: mat_a,B: mat_a,X8: produc5370362606830271383_mat_a] :
( ( ( produc3091253522927621199_mat_a @ A2 @ B )
= X8 )
= ( ( ( produc8618483072558553147_mat_a @ X8 )
= A2 )
& ( ( produc3539460521124201597_mat_a @ X8 )
= B ) ) ) ).
% split_pairs
thf(fact_639_split__block__diag__carrier_I2_J,axiom,
! [D: mat_a,N: nat,A: nat,D1: mat_a,D2: mat_a,D3: mat_a,D4: mat_a] :
( ( member_mat_a @ D @ ( carrier_mat_a @ N @ N ) )
=> ( ( ord_less_eq_nat @ A @ N )
=> ( ( ( split_block_a @ D @ A @ A )
= ( produc5286753621172121189_mat_a @ D1 @ ( produc7602877900562455331_mat_a @ D2 @ ( produc3091253522927621199_mat_a @ D3 @ D4 ) ) ) )
=> ( member_mat_a @ D4 @ ( carrier_mat_a @ ( minus_minus_nat @ N @ A ) @ ( minus_minus_nat @ N @ A ) ) ) ) ) ) ).
% split_block_diag_carrier(2)
thf(fact_640_lex__take__index,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( take_P2173866234530122223at_nat @ I @ Xs )
= ( take_P2173866234530122223at_nat @ I @ Ys2 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_641_lex__take__index,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( lex_mat_a @ R2 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_mat_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_mat_a @ Ys2 ) )
=> ( ( ( take_mat_a @ I @ Xs )
= ( take_mat_a @ I @ Ys2 ) )
=> ~ ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ ( nth_mat_a @ Xs @ I ) @ ( nth_mat_a @ Ys2 @ I ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_642_lex__take__index,axiom,
! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R2 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( ( take_nat @ I @ Xs )
= ( take_nat @ I @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_643_fstI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z3: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z3 ) )
=> ( ( product_fst_nat_nat @ X )
= Y ) ) ).
% fstI
thf(fact_644_fstI,axiom,
! [X: produc8199716216217303280at_nat,Y: nat > nat,Z3: nat] :
( ( X
= ( produc72220940542539688at_nat @ Y @ Z3 ) )
=> ( ( produc6156676138143019412at_nat @ X )
= Y ) ) ).
% fstI
thf(fact_645_fstI,axiom,
! [X: produc4216251508294696237_mat_a,Y: mat_a,Z3: produc5452184871688341745_mat_a] :
( ( X
= ( produc5286753621172121189_mat_a @ Y @ Z3 ) )
=> ( ( produc7700291086614992977_mat_a @ X )
= Y ) ) ).
% fstI
thf(fact_646_fstI,axiom,
! [X: produc5452184871688341745_mat_a,Y: mat_a,Z3: produc5370362606830271383_mat_a] :
( ( X
= ( produc7602877900562455331_mat_a @ Y @ Z3 ) )
=> ( ( produc7340730364199978039_mat_a @ X )
= Y ) ) ).
% fstI
thf(fact_647_fstI,axiom,
! [X: produc5370362606830271383_mat_a,Y: mat_a,Z3: mat_a] :
( ( X
= ( produc3091253522927621199_mat_a @ Y @ Z3 ) )
=> ( ( produc8618483072558553147_mat_a @ X )
= Y ) ) ).
% fstI
thf(fact_648_Nil__notin__lex,axiom,
! [Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) @ ( lex_nat @ R2 ) ) ).
% Nil_notin_lex
thf(fact_649_Nil__notin__lex,axiom,
! [Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys2 ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ).
% Nil_notin_lex
thf(fact_650_Nil2__notin__lex,axiom,
! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_651_Nil2__notin__lex,axiom,
! [Xs: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ nil_Pr5478986624290739719at_nat ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_652_diff__commute,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% diff_commute
thf(fact_653_lex__append__rightI,axiom,
! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R2 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys2 @ Vs ) ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_654_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_655_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_656_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_657_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_658_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_659_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_660_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_661_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_662_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_663_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_664_le__diff__iff_H,axiom,
! [A: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_665_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_666_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_667_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ( ! [P3: nat > nat > $o,X5: nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X5 @ nil_nat ) ) )
=> ~ ! [P3: nat > nat > $o,X5: nat,Y5: nat,Xs2: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_668_successively_Ocases,axiom,
! [X: produc5176992444116710762_mat_a] :
( ! [P3: mat_a > mat_a > $o] :
( X
!= ( produc2635954048118976996_mat_a @ P3 @ nil_mat_a ) )
=> ( ! [P3: mat_a > mat_a > $o,X5: mat_a] :
( X
!= ( produc2635954048118976996_mat_a @ P3 @ ( cons_mat_a @ X5 @ nil_mat_a ) ) )
=> ~ ! [P3: mat_a > mat_a > $o,X5: mat_a,Y5: mat_a,Xs2: list_mat_a] :
( X
!= ( produc2635954048118976996_mat_a @ P3 @ ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_669_successively_Ocases,axiom,
! [X: produc2366258654402830848at_nat] :
( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o] :
( X
!= ( produc3352296309980913008at_nat @ P3 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat] :
( X
!= ( produc3352296309980913008at_nat @ P3 @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) ) )
=> ~ ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( X
!= ( produc3352296309980913008at_nat @ P3 @ ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_670_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ~ ! [P3: nat > nat > $o,X5: nat,Ys4: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_671_sorted__wrt_Ocases,axiom,
! [X: produc5176992444116710762_mat_a] :
( ! [P3: mat_a > mat_a > $o] :
( X
!= ( produc2635954048118976996_mat_a @ P3 @ nil_mat_a ) )
=> ~ ! [P3: mat_a > mat_a > $o,X5: mat_a,Ys4: list_mat_a] :
( X
!= ( produc2635954048118976996_mat_a @ P3 @ ( cons_mat_a @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_672_sorted__wrt_Ocases,axiom,
! [X: produc2366258654402830848at_nat] :
( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o] :
( X
!= ( produc3352296309980913008at_nat @ P3 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( X
!= ( produc3352296309980913008at_nat @ P3 @ ( cons_P6512896166579812791at_nat @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_673_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_674_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_675_pderiv__coeffs__code_Ocases,axiom,
! [X: produc4575160907756185873st_nat] :
( ! [F2: nat,X5: nat,Xs2: list_nat] :
( X
!= ( produc8282810413953273033st_nat @ F2 @ ( cons_nat @ X5 @ Xs2 ) ) )
=> ~ ! [F2: nat] :
( X
!= ( produc8282810413953273033st_nat @ F2 @ nil_nat ) ) ) ).
% pderiv_coeffs_code.cases
thf(fact_676_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ( ! [Xs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ~ ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_677_shuffles_Ocases,axiom,
! [X: produc3237720520546573239_mat_a] :
( ! [Ys4: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ nil_mat_a @ Ys4 ) )
=> ( ! [Xs2: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ Xs2 @ nil_mat_a ) )
=> ~ ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_678_shuffles_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [Ys4: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys4 ) )
=> ( ! [Xs2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ Xs2 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_679_splice_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ~ ! [X5: nat,Xs2: list_nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_680_splice_Ocases,axiom,
! [X: produc3237720520546573239_mat_a] :
( ! [Ys4: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ nil_mat_a @ Ys4 ) )
=> ~ ! [X5: mat_a,Xs2: list_mat_a,Ys4: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_681_splice_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [Ys4: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys4 ) )
=> ~ ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys4: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_682_plus__coeffs_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Xs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ( ! [V: nat,Va: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V @ Va ) ) )
=> ~ ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_683_plus__coeffs_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [Xs2: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ Xs2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) ) )
=> ~ ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_684_sorted__list__subset_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [A3: nat,As: list_nat,B2: nat,Bs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ ( cons_nat @ B2 @ 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_685_longest__common__prefix_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [X5: nat,Xs2: list_nat,Y5: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs2 ) @ ( cons_nat @ Y5 @ Ys4 ) ) )
=> ( ! [Uv: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
=> ~ ! [Uu: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Uu @ nil_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_686_longest__common__prefix_Ocases,axiom,
! [X: produc3237720520546573239_mat_a] :
( ! [X5: mat_a,Xs2: list_mat_a,Y5: mat_a,Ys4: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X5 @ Xs2 ) @ ( cons_mat_a @ Y5 @ Ys4 ) ) )
=> ( ! [Uv: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ nil_mat_a @ Uv ) )
=> ~ ! [Uu: list_mat_a] :
( X
!= ( produc4079638386681799535_mat_a @ Uu @ nil_mat_a ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_687_longest__common__prefix_Ocases,axiom,
! [X: produc6392793444374437607at_nat] :
( ! [X5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( X
!= ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys4 ) ) )
=> ( ! [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_688_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_689_diff__less__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_690_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_691_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K2: nat,M6: nat] :
( ( ~ ( ord_less_eq_nat @ M6 @ K2 )
=> ( P @ ( suc @ K2 ) @ ( minus_minus_nat @ M6 @ ( suc @ K2 ) ) ) )
=> ( P @ K2 @ M6 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_692_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_693_lex__append__leftD,axiom,
! [R2: set_Pr3154870478303372279_mat_a,Xs: list_mat_a,Ys2: list_mat_a,Zs2: list_mat_a] :
( ! [X5: mat_a] :
~ ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X5 @ X5 ) @ R2 )
=> ( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( append_mat_a @ Xs @ Ys2 ) @ ( append_mat_a @ Xs @ Zs2 ) ) @ ( lex_mat_a @ R2 ) )
=> ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Ys2 @ Zs2 ) @ ( lex_mat_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_694_lex__append__leftD,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ! [X5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs2 ) ) @ ( lex_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs2 ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_695_lex__append__left__iff,axiom,
! [R2: set_Pr3154870478303372279_mat_a,Xs: list_mat_a,Ys2: list_mat_a,Zs2: list_mat_a] :
( ! [X5: mat_a] :
~ ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X5 @ X5 ) @ R2 )
=> ( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( append_mat_a @ Xs @ Ys2 ) @ ( append_mat_a @ Xs @ Zs2 ) ) @ ( lex_mat_a @ R2 ) )
= ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Ys2 @ Zs2 ) @ ( lex_mat_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_696_lex__append__left__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ! [X5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs2 ) ) @ ( lex_nat @ R2 ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs2 ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_697_take__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_698_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_699_Cons__in__lex,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
= ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
& ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys2 ) ) )
| ( ( X = Y )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_700_Cons__in__lex,axiom,
! [X: mat_a,Xs: list_mat_a,Y: mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X @ Xs ) @ ( cons_mat_a @ Y @ Ys2 ) ) @ ( lex_mat_a @ R2 ) )
= ( ( ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) @ R2 )
& ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys2 ) ) )
| ( ( X = Y )
& ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( lex_mat_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_701_Cons__in__lex,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( lex_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_702_nth__append,axiom,
! [N: nat,Xs: list_mat_a,Ys2: list_mat_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_mat_a @ Xs ) )
=> ( ( nth_mat_a @ ( append_mat_a @ Xs @ Ys2 ) @ N )
= ( nth_mat_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_mat_a @ Xs ) )
=> ( ( nth_mat_a @ ( append_mat_a @ Xs @ Ys2 ) @ N )
= ( nth_mat_a @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_mat_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_703_nth__append,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) @ N )
= ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( append985823374593552924at_nat @ Xs @ Ys2 ) @ N )
= ( nth_Pr7617993195940197384at_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_704_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_705_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_706_minus__prod__def,axiom,
( minus_9067931446424981591at_nat
= ( ^ [X4: produc8199716216217303280at_nat,Y4: produc8199716216217303280at_nat] : ( produc72220940542539688at_nat @ ( minus_minus_nat_nat @ ( produc6156676138143019412at_nat @ X4 ) @ ( produc6156676138143019412at_nat @ Y4 ) ) @ ( minus_minus_nat @ ( produc1852801350702243542at_nat @ X4 ) @ ( produc1852801350702243542at_nat @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_707_minus__prod__def,axiom,
( minus_4365393887724441320at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( minus_minus_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) ) @ ( minus_minus_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_708_Cons__lenlex__iff,axiom,
! [M: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
| ( ( ( size_s5460976970255530739at_nat @ Ms )
= ( size_s5460976970255530739at_nat @ Ns ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_709_Cons__lenlex__iff,axiom,
! [M: mat_a,Ms: list_mat_a,N: mat_a,Ns: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( cons_mat_a @ M @ Ms ) @ ( cons_mat_a @ N @ Ns ) ) @ ( lenlex_mat_a @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_mat_a @ Ms ) @ ( size_size_list_mat_a @ Ns ) )
| ( ( ( size_size_list_mat_a @ Ms )
= ( size_size_list_mat_a @ Ns ) )
& ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Ms @ Ns ) @ ( lenlex_mat_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_710_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_711_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_712_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K2: nat,M6: nat] :
( X
!= ( product_Pair_nat_nat @ K2 @ M6 ) ) ).
% prod_decode_aux.cases
thf(fact_713_lenlex__irreflexive,axiom,
! [R2: set_Pr3154870478303372279_mat_a,Xs: list_mat_a] :
( ! [X5: mat_a] :
~ ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X5 @ X5 ) @ R2 )
=> ~ ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Xs ) @ ( lenlex_mat_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_714_lenlex__irreflexive,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_715_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_716_Nil__lenlex__iff2,axiom,
! [Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ns @ nil_Pr5478986624290739719at_nat ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_717_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R2 ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_718_Nil__lenlex__iff1,axiom,
! [Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ns ) @ ( lenlex325483962726685836at_nat @ R2 ) )
= ( Ns != nil_Pr5478986624290739719at_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_719_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_720_lenlex__append1,axiom,
! [Us: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys2 ) ) @ ( lenlex_nat @ R ) ) ) ) ).
% lenlex_append1
thf(fact_721_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,P6: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( ( ( ( product_fst_nat_nat @ P6 )
!= K2 )
=> ( P @ K2 @ F2 @ Ps ) )
=> ( P @ K2 @ F2 @ ( cons_P6512896166579812791at_nat @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_entry.induct
thf(fact_722_map__entry_Oinduct,axiom,
! [P: mat_a > ( produc5452184871688341745_mat_a > produc5452184871688341745_mat_a ) > list_P2872167576551266355_mat_a > $o,A0: mat_a,A1: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,A22: list_P2872167576551266355_mat_a] :
( ! [K2: mat_a,F2: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a] : ( P @ K2 @ F2 @ nil_Pr8081019204233271603_mat_a )
=> ( ! [K2: mat_a,F2: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,P6: produc4216251508294696237_mat_a,Ps: list_P2872167576551266355_mat_a] :
( ( ( ( produc7700291086614992977_mat_a @ P6 )
!= K2 )
=> ( P @ K2 @ F2 @ Ps ) )
=> ( P @ K2 @ F2 @ ( cons_P9119692492650804451_mat_a @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_entry.induct
thf(fact_723_map__entry_Oinduct,axiom,
! [P: mat_a > ( produc5370362606830271383_mat_a > produc5370362606830271383_mat_a ) > list_P798859136818506497_mat_a > $o,A0: mat_a,A1: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,A22: list_P798859136818506497_mat_a] :
( ! [K2: mat_a,F2: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a] : ( P @ K2 @ F2 @ nil_Pr3902087586535856747_mat_a )
=> ( ! [K2: mat_a,F2: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,P6: produc5452184871688341745_mat_a,Ps: list_P798859136818506497_mat_a] :
( ( ( ( produc7340730364199978039_mat_a @ P6 )
!= K2 )
=> ( P @ K2 @ F2 @ Ps ) )
=> ( P @ K2 @ F2 @ ( cons_P2417854964248693435_mat_a @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_entry.induct
thf(fact_724_map__entry_Oinduct,axiom,
! [P: mat_a > ( mat_a > mat_a ) > list_P5411175341357971485_mat_a > $o,A0: mat_a,A1: mat_a > mat_a,A22: list_P5411175341357971485_mat_a] :
( ! [K2: mat_a,F2: mat_a > mat_a] : ( P @ K2 @ F2 @ nil_Pr2784087112350407837_mat_a )
=> ( ! [K2: mat_a,F2: mat_a > mat_a,P6: produc5370362606830271383_mat_a,Ps: list_P5411175341357971485_mat_a] :
( ( ( ( produc8618483072558553147_mat_a @ P6 )
!= K2 )
=> ( P @ K2 @ F2 @ Ps ) )
=> ( P @ K2 @ F2 @ ( cons_P3230921977152692301_mat_a @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_entry.induct
thf(fact_725_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,P6: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( ( ( ( product_fst_nat_nat @ P6 )
!= K2 )
=> ( P @ K2 @ V @ F2 @ Ps ) )
=> ( P @ K2 @ V @ F2 @ ( cons_P6512896166579812791at_nat @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% map_default.induct
thf(fact_726_map__default_Oinduct,axiom,
! [P: mat_a > produc5452184871688341745_mat_a > ( produc5452184871688341745_mat_a > produc5452184871688341745_mat_a ) > list_P2872167576551266355_mat_a > $o,A0: mat_a,A1: produc5452184871688341745_mat_a,A22: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,A32: list_P2872167576551266355_mat_a] :
( ! [K2: mat_a,V: produc5452184871688341745_mat_a,F2: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a] : ( P @ K2 @ V @ F2 @ nil_Pr8081019204233271603_mat_a )
=> ( ! [K2: mat_a,V: produc5452184871688341745_mat_a,F2: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,P6: produc4216251508294696237_mat_a,Ps: list_P2872167576551266355_mat_a] :
( ( ( ( produc7700291086614992977_mat_a @ P6 )
!= K2 )
=> ( P @ K2 @ V @ F2 @ Ps ) )
=> ( P @ K2 @ V @ F2 @ ( cons_P9119692492650804451_mat_a @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% map_default.induct
thf(fact_727_map__default_Oinduct,axiom,
! [P: mat_a > produc5370362606830271383_mat_a > ( produc5370362606830271383_mat_a > produc5370362606830271383_mat_a ) > list_P798859136818506497_mat_a > $o,A0: mat_a,A1: produc5370362606830271383_mat_a,A22: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,A32: list_P798859136818506497_mat_a] :
( ! [K2: mat_a,V: produc5370362606830271383_mat_a,F2: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a] : ( P @ K2 @ V @ F2 @ nil_Pr3902087586535856747_mat_a )
=> ( ! [K2: mat_a,V: produc5370362606830271383_mat_a,F2: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,P6: produc5452184871688341745_mat_a,Ps: list_P798859136818506497_mat_a] :
( ( ( ( produc7340730364199978039_mat_a @ P6 )
!= K2 )
=> ( P @ K2 @ V @ F2 @ Ps ) )
=> ( P @ K2 @ V @ F2 @ ( cons_P2417854964248693435_mat_a @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% map_default.induct
thf(fact_728_map__default_Oinduct,axiom,
! [P: mat_a > mat_a > ( mat_a > mat_a ) > list_P5411175341357971485_mat_a > $o,A0: mat_a,A1: mat_a,A22: mat_a > mat_a,A32: list_P5411175341357971485_mat_a] :
( ! [K2: mat_a,V: mat_a,F2: mat_a > mat_a] : ( P @ K2 @ V @ F2 @ nil_Pr2784087112350407837_mat_a )
=> ( ! [K2: mat_a,V: mat_a,F2: mat_a > mat_a,P6: produc5370362606830271383_mat_a,Ps: list_P5411175341357971485_mat_a] :
( ( ( ( produc8618483072558553147_mat_a @ P6 )
!= K2 )
=> ( P @ K2 @ V @ F2 @ Ps ) )
=> ( P @ K2 @ V @ F2 @ ( cons_P3230921977152692301_mat_a @ P6 @ Ps ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% map_default.induct
thf(fact_729_map__entry_Oelims,axiom,
! [X: nat,Xa2: nat > nat,Xb2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( map_entry_nat_nat @ X @ Xa2 @ Xb2 )
= Y )
=> ( ( ( Xb2 = nil_Pr5478986624290739719at_nat )
=> ( Y != nil_Pr5478986624290739719at_nat ) )
=> ~ ! [P6: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( ( Xb2
= ( cons_P6512896166579812791at_nat @ P6 @ Ps ) )
=> ~ ( ( ( ( product_fst_nat_nat @ P6 )
= X )
=> ( Y
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ ( Xa2 @ ( product_snd_nat_nat @ P6 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_nat_nat @ P6 )
!= X )
=> ( Y
= ( cons_P6512896166579812791at_nat @ P6 @ ( map_entry_nat_nat @ X @ Xa2 @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_730_map__entry_Oelims,axiom,
! [X: nat > nat,Xa2: nat > nat,Xb2: list_P5366131564312172150at_nat,Y: list_P5366131564312172150at_nat] :
( ( ( map_en6171526509811521626at_nat @ X @ Xa2 @ Xb2 )
= Y )
=> ( ( ( Xb2 = nil_Pr2223394031645031670at_nat )
=> ( Y != nil_Pr2223394031645031670at_nat ) )
=> ~ ! [P6: produc8199716216217303280at_nat,Ps: list_P5366131564312172150at_nat] :
( ( Xb2
= ( cons_P4219629788700907686at_nat @ P6 @ Ps ) )
=> ~ ( ( ( ( produc6156676138143019412at_nat @ P6 )
= X )
=> ( Y
= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X @ ( Xa2 @ ( produc1852801350702243542at_nat @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc6156676138143019412at_nat @ P6 )
!= X )
=> ( Y
= ( cons_P4219629788700907686at_nat @ P6 @ ( map_en6171526509811521626at_nat @ X @ Xa2 @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_731_map__entry_Oelims,axiom,
! [X: mat_a,Xa2: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,Xb2: list_P2872167576551266355_mat_a,Y: list_P2872167576551266355_mat_a] :
( ( ( map_en5402142467459766039_mat_a @ X @ Xa2 @ Xb2 )
= Y )
=> ( ( ( Xb2 = nil_Pr8081019204233271603_mat_a )
=> ( Y != nil_Pr8081019204233271603_mat_a ) )
=> ~ ! [P6: produc4216251508294696237_mat_a,Ps: list_P2872167576551266355_mat_a] :
( ( Xb2
= ( cons_P9119692492650804451_mat_a @ P6 @ Ps ) )
=> ~ ( ( ( ( produc7700291086614992977_mat_a @ P6 )
= X )
=> ( Y
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ X @ ( Xa2 @ ( produc1482081755353976211_mat_a @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc7700291086614992977_mat_a @ P6 )
!= X )
=> ( Y
= ( cons_P9119692492650804451_mat_a @ P6 @ ( map_en5402142467459766039_mat_a @ X @ Xa2 @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_732_map__entry_Oelims,axiom,
! [X: mat_a,Xa2: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,Xb2: list_P798859136818506497_mat_a,Y: list_P798859136818506497_mat_a] :
( ( ( map_en2605797910578914033_mat_a @ X @ Xa2 @ Xb2 )
= Y )
=> ( ( ( Xb2 = nil_Pr3902087586535856747_mat_a )
=> ( Y != nil_Pr3902087586535856747_mat_a ) )
=> ~ ! [P6: produc5452184871688341745_mat_a,Ps: list_P798859136818506497_mat_a] :
( ( Xb2
= ( cons_P2417854964248693435_mat_a @ P6 @ Ps ) )
=> ~ ( ( ( ( produc7340730364199978039_mat_a @ P6 )
= X )
=> ( Y
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ X @ ( Xa2 @ ( produc7508173349661082485_mat_a @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc7340730364199978039_mat_a @ P6 )
!= X )
=> ( Y
= ( cons_P2417854964248693435_mat_a @ P6 @ ( map_en2605797910578914033_mat_a @ X @ Xa2 @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_733_map__entry_Oelims,axiom,
! [X: mat_a,Xa2: mat_a > mat_a,Xb2: list_P5411175341357971485_mat_a,Y: list_P5411175341357971485_mat_a] :
( ( ( map_en7478846251724979201_mat_a @ X @ Xa2 @ Xb2 )
= Y )
=> ( ( ( Xb2 = nil_Pr2784087112350407837_mat_a )
=> ( Y != nil_Pr2784087112350407837_mat_a ) )
=> ~ ! [P6: produc5370362606830271383_mat_a,Ps: list_P5411175341357971485_mat_a] :
( ( Xb2
= ( cons_P3230921977152692301_mat_a @ P6 @ Ps ) )
=> ~ ( ( ( ( produc8618483072558553147_mat_a @ P6 )
= X )
=> ( Y
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ X @ ( Xa2 @ ( produc3539460521124201597_mat_a @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc8618483072558553147_mat_a @ P6 )
!= X )
=> ( Y
= ( cons_P3230921977152692301_mat_a @ P6 @ ( map_en7478846251724979201_mat_a @ X @ Xa2 @ Ps ) ) ) ) ) ) ) ) ).
% map_entry.elims
thf(fact_734_map__default_Oelims,axiom,
! [X: nat,Xa2: nat,Xb2: nat > nat,Xc2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( map_default_nat_nat @ X @ Xa2 @ Xb2 @ Xc2 )
= Y )
=> ( ( ( Xc2 = nil_Pr5478986624290739719at_nat )
=> ( Y
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Xa2 ) @ nil_Pr5478986624290739719at_nat ) ) )
=> ~ ! [P6: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
( ( Xc2
= ( cons_P6512896166579812791at_nat @ P6 @ Ps ) )
=> ~ ( ( ( ( product_fst_nat_nat @ P6 )
= X )
=> ( Y
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ ( Xb2 @ ( product_snd_nat_nat @ P6 ) ) ) @ Ps ) ) )
& ( ( ( product_fst_nat_nat @ P6 )
!= X )
=> ( Y
= ( cons_P6512896166579812791at_nat @ P6 @ ( map_default_nat_nat @ X @ Xa2 @ Xb2 @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_735_map__default_Oelims,axiom,
! [X: nat > nat,Xa2: nat,Xb2: nat > nat,Xc2: list_P5366131564312172150at_nat,Y: list_P5366131564312172150at_nat] :
( ( ( map_de6768392647288954345at_nat @ X @ Xa2 @ Xb2 @ Xc2 )
= Y )
=> ( ( ( Xc2 = nil_Pr2223394031645031670at_nat )
=> ( Y
!= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X @ Xa2 ) @ nil_Pr2223394031645031670at_nat ) ) )
=> ~ ! [P6: produc8199716216217303280at_nat,Ps: list_P5366131564312172150at_nat] :
( ( Xc2
= ( cons_P4219629788700907686at_nat @ P6 @ Ps ) )
=> ~ ( ( ( ( produc6156676138143019412at_nat @ P6 )
= X )
=> ( Y
= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X @ ( Xb2 @ ( produc1852801350702243542at_nat @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc6156676138143019412at_nat @ P6 )
!= X )
=> ( Y
= ( cons_P4219629788700907686at_nat @ P6 @ ( map_de6768392647288954345at_nat @ X @ Xa2 @ Xb2 @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_736_map__default_Oelims,axiom,
! [X: mat_a,Xa2: produc5452184871688341745_mat_a,Xb2: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,Xc2: list_P2872167576551266355_mat_a,Y: list_P2872167576551266355_mat_a] :
( ( ( map_de3954425106173982886_mat_a @ X @ Xa2 @ Xb2 @ Xc2 )
= Y )
=> ( ( ( Xc2 = nil_Pr8081019204233271603_mat_a )
=> ( Y
!= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ X @ Xa2 ) @ nil_Pr8081019204233271603_mat_a ) ) )
=> ~ ! [P6: produc4216251508294696237_mat_a,Ps: list_P2872167576551266355_mat_a] :
( ( Xc2
= ( cons_P9119692492650804451_mat_a @ P6 @ Ps ) )
=> ~ ( ( ( ( produc7700291086614992977_mat_a @ P6 )
= X )
=> ( Y
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ X @ ( Xb2 @ ( produc1482081755353976211_mat_a @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc7700291086614992977_mat_a @ P6 )
!= X )
=> ( Y
= ( cons_P9119692492650804451_mat_a @ P6 @ ( map_de3954425106173982886_mat_a @ X @ Xa2 @ Xb2 @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_737_map__default_Oelims,axiom,
! [X: mat_a,Xa2: produc5370362606830271383_mat_a,Xb2: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,Xc2: list_P798859136818506497_mat_a,Y: list_P798859136818506497_mat_a] :
( ( ( map_de7291990965617922850_mat_a @ X @ Xa2 @ Xb2 @ Xc2 )
= Y )
=> ( ( ( Xc2 = nil_Pr3902087586535856747_mat_a )
=> ( Y
!= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ X @ Xa2 ) @ nil_Pr3902087586535856747_mat_a ) ) )
=> ~ ! [P6: produc5452184871688341745_mat_a,Ps: list_P798859136818506497_mat_a] :
( ( Xc2
= ( cons_P2417854964248693435_mat_a @ P6 @ Ps ) )
=> ~ ( ( ( ( produc7340730364199978039_mat_a @ P6 )
= X )
=> ( Y
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ X @ ( Xb2 @ ( produc7508173349661082485_mat_a @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc7340730364199978039_mat_a @ P6 )
!= X )
=> ( Y
= ( cons_P2417854964248693435_mat_a @ P6 @ ( map_de7291990965617922850_mat_a @ X @ Xa2 @ Xb2 @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_738_map__default_Oelims,axiom,
! [X: mat_a,Xa2: mat_a,Xb2: mat_a > mat_a,Xc2: list_P5411175341357971485_mat_a,Y: list_P5411175341357971485_mat_a] :
( ( ( map_de1790062285897181712_mat_a @ X @ Xa2 @ Xb2 @ Xc2 )
= Y )
=> ( ( ( Xc2 = nil_Pr2784087112350407837_mat_a )
=> ( Y
!= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ X @ Xa2 ) @ nil_Pr2784087112350407837_mat_a ) ) )
=> ~ ! [P6: produc5370362606830271383_mat_a,Ps: list_P5411175341357971485_mat_a] :
( ( Xc2
= ( cons_P3230921977152692301_mat_a @ P6 @ Ps ) )
=> ~ ( ( ( ( produc8618483072558553147_mat_a @ P6 )
= X )
=> ( Y
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ X @ ( Xb2 @ ( produc3539460521124201597_mat_a @ P6 ) ) ) @ Ps ) ) )
& ( ( ( produc8618483072558553147_mat_a @ P6 )
!= X )
=> ( Y
= ( cons_P3230921977152692301_mat_a @ P6 @ ( map_de1790062285897181712_mat_a @ X @ Xa2 @ Xb2 @ Ps ) ) ) ) ) ) ) ) ).
% map_default.elims
thf(fact_739_find__largest__block_Oelims,axiom,
! [X: product_prod_nat_nat,Xa2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( ( jordan1665469968453478129_block @ X @ Xa2 )
= Y )
=> ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
=> ( Y != X ) )
=> ~ ! [M_start: nat,M_end: nat] :
( ( X
= ( product_Pair_nat_nat @ M_start @ M_end ) )
=> ! [I_start: nat,I_end: nat,Blocks: list_P6011104703257516679at_nat] :
( ( Xa2
= ( cons_P6512896166579812791at_nat @ ( 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 ) )
=> ( Y
= ( jordan1665469968453478129_block @ ( 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 ) )
=> ( Y
= ( jordan1665469968453478129_block @ ( product_Pair_nat_nat @ M_start @ M_end ) @ Blocks ) ) ) ) ) ) ) ) ).
% find_largest_block.elims
thf(fact_740_update__with__aux_Osimps_I2_J,axiom,
! [P5: product_prod_nat_nat,K: nat,V2: nat,F: nat > nat,Ps2: list_P6011104703257516679at_nat] :
( ( ( ( product_fst_nat_nat @ P5 )
= K )
=> ( ( update528237659335440164at_nat @ V2 @ K @ F @ ( cons_P6512896166579812791at_nat @ P5 @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ ( product_snd_nat_nat @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_nat_nat @ P5 )
!= K )
=> ( ( update528237659335440164at_nat @ V2 @ K @ F @ ( cons_P6512896166579812791at_nat @ P5 @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ P5 @ ( update528237659335440164at_nat @ V2 @ K @ F @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_741_update__with__aux_Osimps_I2_J,axiom,
! [P5: produc8199716216217303280at_nat,K: nat > nat,V2: nat,F: nat > nat,Ps2: list_P5366131564312172150at_nat] :
( ( ( ( produc6156676138143019412at_nat @ P5 )
= K )
=> ( ( update516555593404511891at_nat @ V2 @ K @ F @ ( cons_P4219629788700907686at_nat @ P5 @ Ps2 ) )
= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K @ ( F @ ( produc1852801350702243542at_nat @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc6156676138143019412at_nat @ P5 )
!= K )
=> ( ( update516555593404511891at_nat @ V2 @ K @ F @ ( cons_P4219629788700907686at_nat @ P5 @ Ps2 ) )
= ( cons_P4219629788700907686at_nat @ P5 @ ( update516555593404511891at_nat @ V2 @ K @ F @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_742_update__with__aux_Osimps_I2_J,axiom,
! [P5: produc4216251508294696237_mat_a,K: mat_a,V2: produc5452184871688341745_mat_a,F: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,Ps2: list_P2872167576551266355_mat_a] :
( ( ( ( produc7700291086614992977_mat_a @ P5 )
= K )
=> ( ( update842895306872828624_mat_a @ V2 @ K @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ K @ ( F @ ( produc1482081755353976211_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc7700291086614992977_mat_a @ P5 )
!= K )
=> ( ( update842895306872828624_mat_a @ V2 @ K @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ P5 @ ( update842895306872828624_mat_a @ V2 @ K @ F @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_743_update__with__aux_Osimps_I2_J,axiom,
! [P5: produc5452184871688341745_mat_a,K: mat_a,V2: produc5370362606830271383_mat_a,F: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,Ps2: list_P798859136818506497_mat_a] :
( ( ( ( produc7340730364199978039_mat_a @ P5 )
= K )
=> ( ( update3285386300047510896_mat_a @ V2 @ K @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ K @ ( F @ ( produc7508173349661082485_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc7340730364199978039_mat_a @ P5 )
!= K )
=> ( ( update3285386300047510896_mat_a @ V2 @ K @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ P5 @ ( update3285386300047510896_mat_a @ V2 @ K @ F @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_744_update__with__aux_Osimps_I2_J,axiom,
! [P5: produc5370362606830271383_mat_a,K: mat_a,V2: mat_a,F: mat_a > mat_a,Ps2: list_P5411175341357971485_mat_a] :
( ( ( ( produc8618483072558553147_mat_a @ P5 )
= K )
=> ( ( update8196492349025996602_mat_a @ V2 @ K @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ K @ ( F @ ( produc3539460521124201597_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc8618483072558553147_mat_a @ P5 )
!= K )
=> ( ( update8196492349025996602_mat_a @ V2 @ K @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ P5 @ ( update8196492349025996602_mat_a @ V2 @ K @ F @ Ps2 ) ) ) ) ) ).
% update_with_aux.simps(2)
thf(fact_745_find__largest__block_Osimps_I2_J,axiom,
! [M_end2: nat,M_start2: nat,I_end2: nat,I_start2: nat,Blocks2: list_P6011104703257516679at_nat] :
( ( ( ord_less_eq_nat @ ( minus_minus_nat @ M_end2 @ M_start2 ) @ ( minus_minus_nat @ I_end2 @ I_start2 ) )
=> ( ( jordan1665469968453478129_block @ ( product_Pair_nat_nat @ M_start2 @ M_end2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ I_start2 @ I_end2 ) @ Blocks2 ) )
= ( jordan1665469968453478129_block @ ( product_Pair_nat_nat @ I_start2 @ I_end2 ) @ Blocks2 ) ) )
& ( ~ ( ord_less_eq_nat @ ( minus_minus_nat @ M_end2 @ M_start2 ) @ ( minus_minus_nat @ I_end2 @ I_start2 ) )
=> ( ( jordan1665469968453478129_block @ ( product_Pair_nat_nat @ M_start2 @ M_end2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ I_start2 @ I_end2 ) @ Blocks2 ) )
= ( jordan1665469968453478129_block @ ( product_Pair_nat_nat @ M_start2 @ M_end2 ) @ Blocks2 ) ) ) ) ).
% find_largest_block.simps(2)
thf(fact_746_map__default_Osimps_I2_J,axiom,
! [P5: product_prod_nat_nat,K: nat,V2: nat,F: nat > nat,Ps2: list_P6011104703257516679at_nat] :
( ( ( ( product_fst_nat_nat @ P5 )
= K )
=> ( ( map_default_nat_nat @ K @ V2 @ F @ ( cons_P6512896166579812791at_nat @ P5 @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ ( product_snd_nat_nat @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_nat_nat @ P5 )
!= K )
=> ( ( map_default_nat_nat @ K @ V2 @ F @ ( cons_P6512896166579812791at_nat @ P5 @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ P5 @ ( map_default_nat_nat @ K @ V2 @ F @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_747_map__default_Osimps_I2_J,axiom,
! [P5: produc8199716216217303280at_nat,K: nat > nat,V2: nat,F: nat > nat,Ps2: list_P5366131564312172150at_nat] :
( ( ( ( produc6156676138143019412at_nat @ P5 )
= K )
=> ( ( map_de6768392647288954345at_nat @ K @ V2 @ F @ ( cons_P4219629788700907686at_nat @ P5 @ Ps2 ) )
= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K @ ( F @ ( produc1852801350702243542at_nat @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc6156676138143019412at_nat @ P5 )
!= K )
=> ( ( map_de6768392647288954345at_nat @ K @ V2 @ F @ ( cons_P4219629788700907686at_nat @ P5 @ Ps2 ) )
= ( cons_P4219629788700907686at_nat @ P5 @ ( map_de6768392647288954345at_nat @ K @ V2 @ F @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_748_map__default_Osimps_I2_J,axiom,
! [P5: produc4216251508294696237_mat_a,K: mat_a,V2: produc5452184871688341745_mat_a,F: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,Ps2: list_P2872167576551266355_mat_a] :
( ( ( ( produc7700291086614992977_mat_a @ P5 )
= K )
=> ( ( map_de3954425106173982886_mat_a @ K @ V2 @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ K @ ( F @ ( produc1482081755353976211_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc7700291086614992977_mat_a @ P5 )
!= K )
=> ( ( map_de3954425106173982886_mat_a @ K @ V2 @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ P5 @ ( map_de3954425106173982886_mat_a @ K @ V2 @ F @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_749_map__default_Osimps_I2_J,axiom,
! [P5: produc5452184871688341745_mat_a,K: mat_a,V2: produc5370362606830271383_mat_a,F: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,Ps2: list_P798859136818506497_mat_a] :
( ( ( ( produc7340730364199978039_mat_a @ P5 )
= K )
=> ( ( map_de7291990965617922850_mat_a @ K @ V2 @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ K @ ( F @ ( produc7508173349661082485_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc7340730364199978039_mat_a @ P5 )
!= K )
=> ( ( map_de7291990965617922850_mat_a @ K @ V2 @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ P5 @ ( map_de7291990965617922850_mat_a @ K @ V2 @ F @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_750_map__default_Osimps_I2_J,axiom,
! [P5: produc5370362606830271383_mat_a,K: mat_a,V2: mat_a,F: mat_a > mat_a,Ps2: list_P5411175341357971485_mat_a] :
( ( ( ( produc8618483072558553147_mat_a @ P5 )
= K )
=> ( ( map_de1790062285897181712_mat_a @ K @ V2 @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ K @ ( F @ ( produc3539460521124201597_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc8618483072558553147_mat_a @ P5 )
!= K )
=> ( ( map_de1790062285897181712_mat_a @ K @ V2 @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ P5 @ ( map_de1790062285897181712_mat_a @ K @ V2 @ F @ Ps2 ) ) ) ) ) ).
% map_default.simps(2)
thf(fact_751_map__entry_Osimps_I2_J,axiom,
! [P5: product_prod_nat_nat,K: nat,F: nat > nat,Ps2: list_P6011104703257516679at_nat] :
( ( ( ( product_fst_nat_nat @ P5 )
= K )
=> ( ( map_entry_nat_nat @ K @ F @ ( cons_P6512896166579812791at_nat @ P5 @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ ( product_snd_nat_nat @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( product_fst_nat_nat @ P5 )
!= K )
=> ( ( map_entry_nat_nat @ K @ F @ ( cons_P6512896166579812791at_nat @ P5 @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ P5 @ ( map_entry_nat_nat @ K @ F @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_752_map__entry_Osimps_I2_J,axiom,
! [P5: produc8199716216217303280at_nat,K: nat > nat,F: nat > nat,Ps2: list_P5366131564312172150at_nat] :
( ( ( ( produc6156676138143019412at_nat @ P5 )
= K )
=> ( ( map_en6171526509811521626at_nat @ K @ F @ ( cons_P4219629788700907686at_nat @ P5 @ Ps2 ) )
= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K @ ( F @ ( produc1852801350702243542at_nat @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc6156676138143019412at_nat @ P5 )
!= K )
=> ( ( map_en6171526509811521626at_nat @ K @ F @ ( cons_P4219629788700907686at_nat @ P5 @ Ps2 ) )
= ( cons_P4219629788700907686at_nat @ P5 @ ( map_en6171526509811521626at_nat @ K @ F @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_753_map__entry_Osimps_I2_J,axiom,
! [P5: produc4216251508294696237_mat_a,K: mat_a,F: produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,Ps2: list_P2872167576551266355_mat_a] :
( ( ( ( produc7700291086614992977_mat_a @ P5 )
= K )
=> ( ( map_en5402142467459766039_mat_a @ K @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ K @ ( F @ ( produc1482081755353976211_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc7700291086614992977_mat_a @ P5 )
!= K )
=> ( ( map_en5402142467459766039_mat_a @ K @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ P5 @ ( map_en5402142467459766039_mat_a @ K @ F @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_754_map__entry_Osimps_I2_J,axiom,
! [P5: produc5452184871688341745_mat_a,K: mat_a,F: produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,Ps2: list_P798859136818506497_mat_a] :
( ( ( ( produc7340730364199978039_mat_a @ P5 )
= K )
=> ( ( map_en2605797910578914033_mat_a @ K @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ K @ ( F @ ( produc7508173349661082485_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc7340730364199978039_mat_a @ P5 )
!= K )
=> ( ( map_en2605797910578914033_mat_a @ K @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ P5 @ ( map_en2605797910578914033_mat_a @ K @ F @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_755_map__entry_Osimps_I2_J,axiom,
! [P5: produc5370362606830271383_mat_a,K: mat_a,F: mat_a > mat_a,Ps2: list_P5411175341357971485_mat_a] :
( ( ( ( produc8618483072558553147_mat_a @ P5 )
= K )
=> ( ( map_en7478846251724979201_mat_a @ K @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ K @ ( F @ ( produc3539460521124201597_mat_a @ P5 ) ) ) @ Ps2 ) ) )
& ( ( ( produc8618483072558553147_mat_a @ P5 )
!= K )
=> ( ( map_en7478846251724979201_mat_a @ K @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ P5 @ ( map_en7478846251724979201_mat_a @ K @ F @ Ps2 ) ) ) ) ) ).
% map_entry.simps(2)
thf(fact_756_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K5: nat,M4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M4 @ K5 ) @ ( product_Pair_nat_nat @ M4 @ ( minus_minus_nat @ K5 @ M4 ) ) @ ( nat_prod_decode_aux @ ( suc @ K5 ) @ ( minus_minus_nat @ M4 @ ( suc @ K5 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_757_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_758_listrel1__iff__update,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) )
= ( ? [Y4: product_prod_nat_nat,N4: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N4 ) @ Y4 ) @ R2 )
& ( ord_less_nat @ N4 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( Ys2
= ( list_u6180841689913720943at_nat @ Xs @ N4 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_759_listrel1__iff__update,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( listrel1_mat_a @ R2 ) )
= ( ? [Y4: mat_a,N4: nat] :
( ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ ( nth_mat_a @ Xs @ N4 ) @ Y4 ) @ R2 )
& ( ord_less_nat @ N4 @ ( size_size_list_mat_a @ Xs ) )
& ( Ys2
= ( list_update_mat_a @ Xs @ N4 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_760_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R2 ) )
= ( ? [Y4: nat,N4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N4 ) @ Y4 ) @ R2 )
& ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs ) )
& ( Ys2
= ( list_update_nat @ Xs @ N4 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_761_listrel1I2,axiom,
! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat,X: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ X @ Ys2 ) ) @ ( listrel1_nat @ R2 ) ) ) ).
% listrel1I2
thf(fact_762_listrel1I2,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a,X: mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( listrel1_mat_a @ R2 ) )
=> ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X @ Xs ) @ ( cons_mat_a @ X @ Ys2 ) ) @ ( listrel1_mat_a @ R2 ) ) ) ).
% listrel1I2
thf(fact_763_listrel1I2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( cons_P6512896166579812791at_nat @ X @ Ys2 ) ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ).
% listrel1I2
thf(fact_764_not__Nil__listrel1,axiom,
! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_765_not__Nil__listrel1,axiom,
! [Xs: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xs ) @ ( listre4828114922151135584at_nat @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_766_not__listrel1__Nil,axiom,
! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_767_not__listrel1__Nil,axiom,
! [Xs: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ nil_Pr5478986624290739719at_nat ) @ ( listre4828114922151135584at_nat @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_768_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_769_Cons__listrel1__Cons,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( listre4828114922151135584at_nat @ R2 ) )
= ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_770_Cons__listrel1__Cons,axiom,
! [X: mat_a,Xs: list_mat_a,Y: mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X @ Xs ) @ ( cons_mat_a @ Y @ Ys2 ) ) @ ( listrel1_mat_a @ R2 ) )
= ( ( ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) @ R2 )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( listrel1_mat_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_771_Cons__listrel1__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_772_Cons__listrel1E2,axiom,
! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( listre4828114922151135584at_nat @ R2 ) )
=> ( ! [X5: product_prod_nat_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ X5 @ Ys2 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ Y @ Zs ) )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Zs @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_773_Cons__listrel1E2,axiom,
! [Xs: list_mat_a,Y: mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ ( cons_mat_a @ Y @ Ys2 ) ) @ ( listrel1_mat_a @ R2 ) )
=> ( ! [X5: mat_a] :
( ( Xs
= ( cons_mat_a @ X5 @ Ys2 ) )
=> ~ ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X5 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_mat_a] :
( ( Xs
= ( cons_mat_a @ Y @ Zs ) )
=> ~ ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Zs @ Ys2 ) @ ( listrel1_mat_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_774_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R2 ) )
=> ( ! [X5: nat] :
( ( Xs
= ( cons_nat @ X5 @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs @ Ys2 ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_775_Cons__listrel1E1,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) )
=> ( ! [Y5: product_prod_nat_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ Y5 @ Xs ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y5 ) @ R2 ) )
=> ~ ! [Zs: list_P6011104703257516679at_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ X @ Zs ) )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Zs ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_776_Cons__listrel1E1,axiom,
! [X: mat_a,Xs: list_mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X @ Xs ) @ Ys2 ) @ ( listrel1_mat_a @ R2 ) )
=> ( ! [Y5: mat_a] :
( ( Ys2
= ( cons_mat_a @ Y5 @ Xs ) )
=> ~ ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y5 ) @ R2 ) )
=> ~ ! [Zs: list_mat_a] :
( ( Ys2
= ( cons_mat_a @ X @ Zs ) )
=> ~ ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Zs ) @ ( listrel1_mat_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_777_Cons__listrel1E1,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys2 ) @ ( listrel1_nat @ R2 ) )
=> ( ! [Y5: nat] :
( ( Ys2
= ( cons_nat @ Y5 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ R2 ) )
=> ~ ! [Zs: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Zs ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_778_listrel1I1,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ).
% listrel1I1
thf(fact_779_listrel1I1,axiom,
! [X: mat_a,Y: mat_a,R2: set_Pr3154870478303372279_mat_a,Xs: list_mat_a] :
( ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) @ R2 )
=> ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( cons_mat_a @ X @ Xs ) @ ( cons_mat_a @ Y @ Xs ) ) @ ( listrel1_mat_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_780_listrel1I1,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R2 ) ) ) ).
% listrel1I1
thf(fact_781_listrel1I,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ X @ Vs ) ) )
=> ( ( Ys2
= ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ Y @ Vs ) ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_782_listrel1I,axiom,
! [X: mat_a,Y: mat_a,R2: set_Pr3154870478303372279_mat_a,Xs: list_mat_a,Us: list_mat_a,Vs: list_mat_a,Ys2: list_mat_a] :
( ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_mat_a @ Us @ ( cons_mat_a @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_mat_a @ Us @ ( cons_mat_a @ Y @ Vs ) ) )
=> ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( listrel1_mat_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_783_listrel1I,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_784_listrel1E,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) )
=> ~ ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y5 ) @ R2 )
=> ! [Us2: list_P6011104703257516679at_nat,Vs2: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ X5 @ Vs2 ) ) )
=> ( Ys2
!= ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ Y5 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_785_listrel1E,axiom,
! [Xs: list_mat_a,Ys2: list_mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( listrel1_mat_a @ R2 ) )
=> ~ ! [X5: mat_a,Y5: mat_a] :
( ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X5 @ Y5 ) @ R2 )
=> ! [Us2: list_mat_a,Vs2: list_mat_a] :
( ( Xs
= ( append_mat_a @ Us2 @ ( cons_mat_a @ X5 @ Vs2 ) ) )
=> ( Ys2
!= ( append_mat_a @ Us2 @ ( cons_mat_a @ Y5 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_786_listrel1E,axiom,
! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R2 ) )
=> ~ ! [X5: nat,Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R2 )
=> ! [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ ( cons_nat @ X5 @ Vs2 ) ) )
=> ( Ys2
!= ( append_nat @ Us2 @ ( cons_nat @ Y5 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_787_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) @ ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) ) @ ( listre4828114922151135584at_nat @ R2 ) )
= ( ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre4828114922151135584at_nat @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_788_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_mat_a,X: mat_a,Ys2: list_mat_a,Y: mat_a,R2: set_Pr3154870478303372279_mat_a] :
( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) ) @ ( append_mat_a @ Ys2 @ ( cons_mat_a @ Y @ nil_mat_a ) ) ) @ ( listrel1_mat_a @ R2 ) )
= ( ( ( member8250092531365124960_mat_a @ ( produc4079638386681799535_mat_a @ Xs @ Ys2 ) @ ( listrel1_mat_a @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_789_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R2 ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_790_eq__fst__iff,axiom,
! [A: nat,P5: product_prod_nat_nat] :
( ( A
= ( product_fst_nat_nat @ P5 ) )
= ( ? [B5: nat] :
( P5
= ( product_Pair_nat_nat @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_791_eq__fst__iff,axiom,
! [A: nat > nat,P5: produc8199716216217303280at_nat] :
( ( A
= ( produc6156676138143019412at_nat @ P5 ) )
= ( ? [B5: nat] :
( P5
= ( produc72220940542539688at_nat @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_792_eq__fst__iff,axiom,
! [A: mat_a,P5: produc4216251508294696237_mat_a] :
( ( A
= ( produc7700291086614992977_mat_a @ P5 ) )
= ( ? [B5: produc5452184871688341745_mat_a] :
( P5
= ( produc5286753621172121189_mat_a @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_793_eq__fst__iff,axiom,
! [A: mat_a,P5: produc5452184871688341745_mat_a] :
( ( A
= ( produc7340730364199978039_mat_a @ P5 ) )
= ( ? [B5: produc5370362606830271383_mat_a] :
( P5
= ( produc7602877900562455331_mat_a @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_794_eq__fst__iff,axiom,
! [A: mat_a,P5: produc5370362606830271383_mat_a] :
( ( A
= ( produc8618483072558553147_mat_a @ P5 ) )
= ( ? [B5: mat_a] :
( P5
= ( produc3091253522927621199_mat_a @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_795_Abstract__Rewriting_Ochain__mono,axiom,
! [R3: set_Pr3154870478303372279_mat_a,R: set_Pr3154870478303372279_mat_a,Seq: nat > mat_a] :
( ( ord_le4146993573842611095_mat_a @ R3 @ R )
=> ( ! [I: nat] : ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R3 )
=> ! [I4: nat] : ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_796_Abstract__Rewriting_Ochain__mono,axiom,
! [R3: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ R )
=> ( ! [I: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R3 )
=> ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_797_subset__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A4: set_mat_a,B6: set_mat_a] :
! [T2: mat_a] :
( ( member_mat_a @ T2 @ A4 )
=> ( member_mat_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_798_subset__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A4: set_mat_a,B6: set_mat_a] :
! [X4: mat_a] :
( ( member_mat_a @ X4 @ A4 )
=> ( member_mat_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_799_subsetI,axiom,
! [A2: set_mat_a,B: set_mat_a] :
( ! [X5: mat_a] :
( ( member_mat_a @ X5 @ A2 )
=> ( member_mat_a @ X5 @ B ) )
=> ( ord_le3318621148231462513_mat_a @ A2 @ B ) ) ).
% subsetI
thf(fact_800_subsetD,axiom,
! [A2: set_mat_a,B: set_mat_a,C: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B )
=> ( ( member_mat_a @ C @ A2 )
=> ( member_mat_a @ C @ B ) ) ) ).
% subsetD
thf(fact_801_in__mono,axiom,
! [A2: set_mat_a,B: set_mat_a,X: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B )
=> ( ( member_mat_a @ X @ A2 )
=> ( member_mat_a @ X @ B ) ) ) ).
% in_mono
thf(fact_802_subrelI,axiom,
! [R2: set_Pr4108788433434999053_mat_a,S: set_Pr4108788433434999053_mat_a] :
( ! [X5: mat_a,Y5: produc5452184871688341745_mat_a] :
( ( member6160517978331616854_mat_a @ ( produc5286753621172121189_mat_a @ X5 @ Y5 ) @ R2 )
=> ( member6160517978331616854_mat_a @ ( produc5286753621172121189_mat_a @ X5 @ Y5 ) @ S ) )
=> ( ord_le6596545746210689197_mat_a @ R2 @ S ) ) ).
% subrelI
thf(fact_803_subrelI,axiom,
! [R2: set_Pr1606082691126482087_mat_a,S: set_Pr1606082691126482087_mat_a] :
( ! [X5: mat_a,Y5: produc5370362606830271383_mat_a] :
( ( member7270109072717380616_mat_a @ ( produc7602877900562455331_mat_a @ X5 @ Y5 ) @ R2 )
=> ( member7270109072717380616_mat_a @ ( produc7602877900562455331_mat_a @ X5 @ Y5 ) @ S ) )
=> ( ord_le4619910584120534279_mat_a @ R2 @ S ) ) ).
% subrelI
thf(fact_804_subrelI,axiom,
! [R2: set_Pr3154870478303372279_mat_a,S: set_Pr3154870478303372279_mat_a] :
( ! [X5: mat_a,Y5: mat_a] :
( ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X5 @ Y5 ) @ R2 )
=> ( member7323531280862645312_mat_a @ ( produc3091253522927621199_mat_a @ X5 @ Y5 ) @ S ) )
=> ( ord_le4146993573842611095_mat_a @ R2 @ S ) ) ).
% subrelI
thf(fact_805_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X5: nat,Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_806_subrelI,axiom,
! [R2: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ! [X5: nat > nat,Y5: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y5 ) @ R2 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y5 ) @ S ) )
=> ( ord_le3678578370064672496at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_807_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_808_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_809_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_810_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_811_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_812_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_813_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_814_dual__order_Otrans,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_815_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_816_dual__order_Oantisym,axiom,
! [B3: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_817_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_818_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B3: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_819_order_Otrans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_820_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_821_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_822_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_823_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_824_nle__le,axiom,
! [A: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_825_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_826_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_827_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_828_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_829_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_830_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_831_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_832_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_833_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_834_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_835_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( A != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_836_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( A != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_837_dual__order_Ostrict__trans,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_838_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_839_order_Ostrict__trans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_840_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B3: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_841_exists__least__iff,axiom,
( ( ^ [P7: nat > $o] :
? [X9: nat] : ( P7 @ X9 ) )
= ( ^ [P2: nat > $o] :
? [N4: nat] :
( ( P2 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P2 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_842_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_843_dual__order_Oasym,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ~ ( ord_less_nat @ A @ B3 ) ) ).
% dual_order.asym
thf(fact_844_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_845_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_846_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X5: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X5 )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_847_order_Oasym,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ~ ( ord_less_nat @ B3 @ A ) ) ).
% order.asym
thf(fact_848_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_849_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_850_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_851_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_852_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% le_less
thf(fact_853_less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% less_le
thf(fact_854_nless__le,axiom,
! [A: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A @ B3 )
| ( A = B3 ) ) ) ).
% nless_le
thf(fact_855_not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% not_le
thf(fact_856_not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% not_less
thf(fact_857_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_858_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_859_less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% less_imp_le
thf(fact_860_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_861_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_862_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_863_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_864_order_Ostrict__trans1,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_865_order_Ostrict__trans2,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_866_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_867_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_868_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_869_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_870_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_871_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_872_order_Ostrict__implies__order,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ord_less_eq_nat @ A @ B3 ) ) ).
% order.strict_implies_order
thf(fact_873_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ord_less_eq_nat @ B3 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_874_verit__comp__simplify_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify(3)
thf(fact_875_verit__comp__simplify_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_876_verit__la__disequality,axiom,
! [A: nat,B3: nat] :
( ( A = B3 )
| ~ ( ord_less_eq_nat @ A @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A ) ) ).
% verit_la_disequality
thf(fact_877_verit__comp__simplify_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_878_complete__interval,axiom,
! [A: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B3 )
=> ( ( P @ A )
=> ( ~ ( P @ B3 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B3 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D6: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ D6 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D6 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_879_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_880_prefixes__snoc,axiom,
! [Xs: list_mat_a,X: mat_a] :
( ( prefixes_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) ) )
= ( append_list_mat_a @ ( prefixes_mat_a @ Xs ) @ ( cons_list_mat_a @ ( append_mat_a @ Xs @ ( cons_mat_a @ X @ nil_mat_a ) ) @ nil_list_mat_a ) ) ) ).
% prefixes_snoc
thf(fact_881_prefixes__snoc,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( prefix1395342811948450574at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) )
= ( append1540555382121198114at_nat @ ( prefix1395342811948450574at_nat @ Xs ) @ ( cons_l7612840610449961021at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) @ nil_li8973309667444810893at_nat ) ) ) ).
% prefixes_snoc
thf(fact_882_hd__prefixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_883_hd__prefixes,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( hd_lis8853752595642532978at_nat @ ( prefix1395342811948450574at_nat @ Xs ) )
= nil_Pr5478986624290739719at_nat ) ).
% hd_prefixes
thf(fact_884_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_885_prefixes_Osimps_I1_J,axiom,
( ( prefix1395342811948450574at_nat @ nil_Pr5478986624290739719at_nat )
= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ nil_li8973309667444810893at_nat ) ) ).
% prefixes.simps(1)
thf(fact_886_prefixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys2 )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z5: nat,Zs4: list_nat] :
( ( Ys2
= ( append_nat @ Zs4 @ ( cons_nat @ Z5 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs4 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_887_prefixes__eq__snoc,axiom,
! [Ys2: list_mat_a,Xs: list_list_mat_a,X: list_mat_a] :
( ( ( prefixes_mat_a @ Ys2 )
= ( append_list_mat_a @ Xs @ ( cons_list_mat_a @ X @ nil_list_mat_a ) ) )
= ( ( ( ( Ys2 = nil_mat_a )
& ( Xs = nil_list_mat_a ) )
| ? [Z5: mat_a,Zs4: list_mat_a] :
( ( Ys2
= ( append_mat_a @ Zs4 @ ( cons_mat_a @ Z5 @ nil_mat_a ) ) )
& ( Xs
= ( prefixes_mat_a @ Zs4 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_888_prefixes__eq__snoc,axiom,
! [Ys2: list_P6011104703257516679at_nat,Xs: list_l3264859301627795341at_nat,X: list_P6011104703257516679at_nat] :
( ( ( prefix1395342811948450574at_nat @ Ys2 )
= ( append1540555382121198114at_nat @ Xs @ ( cons_l7612840610449961021at_nat @ X @ nil_li8973309667444810893at_nat ) ) )
= ( ( ( ( Ys2 = nil_Pr5478986624290739719at_nat )
& ( Xs = nil_li8973309667444810893at_nat ) )
| ? [Z5: product_prod_nat_nat,Zs4: list_P6011104703257516679at_nat] :
( ( Ys2
= ( append985823374593552924at_nat @ Zs4 @ ( cons_P6512896166579812791at_nat @ Z5 @ nil_Pr5478986624290739719at_nat ) ) )
& ( Xs
= ( prefix1395342811948450574at_nat @ Zs4 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_889_suffixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys2 )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z5: nat,Zs4: list_nat] :
( ( Ys2
= ( cons_nat @ Z5 @ Zs4 ) )
& ( Xs
= ( suffixes_nat @ Zs4 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_890_suffixes__eq__snoc,axiom,
! [Ys2: list_mat_a,Xs: list_list_mat_a,X: list_mat_a] :
( ( ( suffixes_mat_a @ Ys2 )
= ( append_list_mat_a @ Xs @ ( cons_list_mat_a @ X @ nil_list_mat_a ) ) )
= ( ( ( ( Ys2 = nil_mat_a )
& ( Xs = nil_list_mat_a ) )
| ? [Z5: mat_a,Zs4: list_mat_a] :
( ( Ys2
= ( cons_mat_a @ Z5 @ Zs4 ) )
& ( Xs
= ( suffixes_mat_a @ Zs4 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_891_suffixes__eq__snoc,axiom,
! [Ys2: list_P6011104703257516679at_nat,Xs: list_l3264859301627795341at_nat,X: list_P6011104703257516679at_nat] :
( ( ( suffix8103968943353111119at_nat @ Ys2 )
= ( append1540555382121198114at_nat @ Xs @ ( cons_l7612840610449961021at_nat @ X @ nil_li8973309667444810893at_nat ) ) )
= ( ( ( ( Ys2 = nil_Pr5478986624290739719at_nat )
& ( Xs = nil_li8973309667444810893at_nat ) )
| ? [Z5: product_prod_nat_nat,Zs4: list_P6011104703257516679at_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ Z5 @ Zs4 ) )
& ( Xs
= ( suffix8103968943353111119at_nat @ Zs4 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_892_map__ran__Cons__sel,axiom,
! [F: nat > nat > nat,P5: product_prod_nat_nat,Ps2: list_P6011104703257516679at_nat] :
( ( map_ran_nat_nat_nat @ F @ ( cons_P6512896166579812791at_nat @ P5 @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( product_fst_nat_nat @ P5 ) @ ( F @ ( product_fst_nat_nat @ P5 ) @ ( product_snd_nat_nat @ P5 ) ) ) @ ( map_ran_nat_nat_nat @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_893_map__ran__Cons__sel,axiom,
! [F: mat_a > produc5452184871688341745_mat_a > produc5452184871688341745_mat_a,P5: produc4216251508294696237_mat_a,Ps2: list_P2872167576551266355_mat_a] :
( ( map_ra8166749758470591350_mat_a @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ ( produc7700291086614992977_mat_a @ P5 ) @ ( F @ ( produc7700291086614992977_mat_a @ P5 ) @ ( produc1482081755353976211_mat_a @ P5 ) ) ) @ ( map_ra8166749758470591350_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_894_map__ran__Cons__sel,axiom,
! [F: mat_a > produc5452184871688341745_mat_a > produc5370362606830271383_mat_a,P5: produc4216251508294696237_mat_a,Ps2: list_P2872167576551266355_mat_a] :
( ( map_ra8397368593876798546_mat_a @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ ( produc7700291086614992977_mat_a @ P5 ) @ ( F @ ( produc7700291086614992977_mat_a @ P5 ) @ ( produc1482081755353976211_mat_a @ P5 ) ) ) @ ( map_ra8397368593876798546_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_895_map__ran__Cons__sel,axiom,
! [F: mat_a > produc5452184871688341745_mat_a > mat_a,P5: produc4216251508294696237_mat_a,Ps2: list_P2872167576551266355_mat_a] :
( ( map_ra5007938043284462816_mat_a @ F @ ( cons_P9119692492650804451_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ ( produc7700291086614992977_mat_a @ P5 ) @ ( F @ ( produc7700291086614992977_mat_a @ P5 ) @ ( produc1482081755353976211_mat_a @ P5 ) ) ) @ ( map_ra5007938043284462816_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_896_map__ran__Cons__sel,axiom,
! [F: mat_a > produc5370362606830271383_mat_a > produc5452184871688341745_mat_a,P5: produc5452184871688341745_mat_a,Ps2: list_P798859136818506497_mat_a] :
( ( map_ra3129878362297023990_mat_a @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ ( produc7340730364199978039_mat_a @ P5 ) @ ( F @ ( produc7340730364199978039_mat_a @ P5 ) @ ( produc7508173349661082485_mat_a @ P5 ) ) ) @ ( map_ra3129878362297023990_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_897_map__ran__Cons__sel,axiom,
! [F: mat_a > produc5370362606830271383_mat_a > produc5370362606830271383_mat_a,P5: produc5452184871688341745_mat_a,Ps2: list_P798859136818506497_mat_a] :
( ( map_ra711709222571885010_mat_a @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ ( produc7340730364199978039_mat_a @ P5 ) @ ( F @ ( produc7340730364199978039_mat_a @ P5 ) @ ( produc7508173349661082485_mat_a @ P5 ) ) ) @ ( map_ra711709222571885010_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_898_map__ran__Cons__sel,axiom,
! [F: mat_a > produc5370362606830271383_mat_a > mat_a,P5: produc5452184871688341745_mat_a,Ps2: list_P798859136818506497_mat_a] :
( ( map_ra5636908361876965216_mat_a @ F @ ( cons_P2417854964248693435_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ ( produc7340730364199978039_mat_a @ P5 ) @ ( F @ ( produc7340730364199978039_mat_a @ P5 ) @ ( produc7508173349661082485_mat_a @ P5 ) ) ) @ ( map_ra5636908361876965216_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_899_map__ran__Cons__sel,axiom,
! [F: mat_a > mat_a > produc5452184871688341745_mat_a,P5: produc5370362606830271383_mat_a,Ps2: list_P5411175341357971485_mat_a] :
( ( map_ra5711430780487591136_mat_a @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P9119692492650804451_mat_a @ ( produc5286753621172121189_mat_a @ ( produc8618483072558553147_mat_a @ P5 ) @ ( F @ ( produc8618483072558553147_mat_a @ P5 ) @ ( produc3539460521124201597_mat_a @ P5 ) ) ) @ ( map_ra5711430780487591136_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_900_map__ran__Cons__sel,axiom,
! [F: mat_a > mat_a > produc5370362606830271383_mat_a,P5: produc5370362606830271383_mat_a,Ps2: list_P5411175341357971485_mat_a] :
( ( map_ra6655213778065426792_mat_a @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P2417854964248693435_mat_a @ ( produc7602877900562455331_mat_a @ ( produc8618483072558553147_mat_a @ P5 ) @ ( F @ ( produc8618483072558553147_mat_a @ P5 ) @ ( produc3539460521124201597_mat_a @ P5 ) ) ) @ ( map_ra6655213778065426792_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_901_map__ran__Cons__sel,axiom,
! [F: mat_a > mat_a > mat_a,P5: produc5370362606830271383_mat_a,Ps2: list_P5411175341357971485_mat_a] :
( ( map_ra1029780840500392266_mat_a @ F @ ( cons_P3230921977152692301_mat_a @ P5 @ Ps2 ) )
= ( cons_P3230921977152692301_mat_a @ ( produc3091253522927621199_mat_a @ ( produc8618483072558553147_mat_a @ P5 ) @ ( F @ ( produc8618483072558553147_mat_a @ P5 ) @ ( produc3539460521124201597_mat_a @ P5 ) ) ) @ ( map_ra1029780840500392266_mat_a @ F @ Ps2 ) ) ) ).
% map_ran_Cons_sel
thf(fact_902_nat__distrib_I4_J,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% nat_distrib(4)
thf(fact_903_nat__distrib_I3_J,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% nat_distrib(3)
thf(fact_904_hd__suffixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_905_hd__suffixes,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( hd_lis8853752595642532978at_nat @ ( suffix8103968943353111119at_nat @ Xs ) )
= nil_Pr5478986624290739719at_nat ) ).
% hd_suffixes
thf(fact_906_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_907_mult__le__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_908_mult__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_909_mult__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_910_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_911_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_912_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_913_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_914_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_915_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_916_suffixes_Osimps_I1_J,axiom,
( ( suffix8103968943353111119at_nat @ nil_Pr5478986624290739719at_nat )
= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ nil_li8973309667444810893at_nat ) ) ).
% suffixes.simps(1)
thf(fact_917_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_918_suffixes_Osimps_I2_J,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( suffixes_mat_a @ ( cons_mat_a @ X @ Xs ) )
= ( append_list_mat_a @ ( suffixes_mat_a @ Xs ) @ ( cons_list_mat_a @ ( cons_mat_a @ X @ Xs ) @ nil_list_mat_a ) ) ) ).
% suffixes.simps(2)
thf(fact_919_suffixes_Osimps_I2_J,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( suffix8103968943353111119at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( append1540555382121198114at_nat @ ( suffix8103968943353111119at_nat @ Xs ) @ ( cons_l7612840610449961021at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ nil_li8973309667444810893at_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_920_elements__matI,axiom,
! [A2: mat_mat_a,Nr: nat,Nc: nat,I3: nat,J: nat,A: mat_a] :
( ( member_mat_mat_a @ A2 @ ( carrier_mat_mat_a @ Nr @ Nc ) )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( A
= ( index_mat_mat_a @ A2 @ ( product_Pair_nat_nat @ I3 @ J ) ) )
=> ( member_mat_a @ A @ ( elements_mat_mat_a @ A2 ) ) ) ) ) ) ).
% elements_matI
thf(fact_921_elements__matI,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,I3: nat,J: nat,A: a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( A
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I3 @ J ) ) )
=> ( member_a @ A @ ( elements_mat_a @ A2 ) ) ) ) ) ) ).
% elements_matI
thf(fact_922_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_923_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_924_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_925_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_926_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_927_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_928_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_929_zero__order_I1_J,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_order(1)
thf(fact_930_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_931_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_932_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_933_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_934_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_935_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X5: nat,Y5: nat] :
( ( P @ X5 @ Y5 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_936_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_937_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_938_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_939_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_940_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_941_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_942_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_943_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,N3: nat] :
( ( P @ F2 @ N3 )
=> ( P @ F2 @ ( suc @ N3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inf_concat_simple.induct
thf(fact_944_unit__vecs__last_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N3: nat] : ( P @ N3 @ zero_zero_nat )
=> ( ! [N3: nat,I: nat] :
( ( P @ N3 @ I )
=> ( P @ N3 @ ( suc @ I ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_last.induct
thf(fact_945_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_946_inf__concat_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [N3: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ N3 @ zero_zero_nat ) )
=> ~ ! [N3: nat > nat,K2: nat] :
( X
!= ( produc72220940542539688at_nat @ N3 @ ( suc @ K2 ) ) ) ) ).
% inf_concat.cases
thf(fact_947_minus__nat_Osimps_I1_J,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.simps(1)
thf(fact_948_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_949_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_950_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_951_zero__order_I5_J,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_952_zero__order_I4_J,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_order(4)
thf(fact_953_zero__order_I3_J,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% zero_order(3)
thf(fact_954_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_955_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_956_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_957_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_958_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_959_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_960_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_961_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_962_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_963_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_964_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_965_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_966_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_967_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_968_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_969_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_970_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_971_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_972_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_973_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_974_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_975_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_976_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_977_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_978_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_979_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_980_length__0__conv,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% length_0_conv
thf(fact_981_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_982_list_Osize_I3_J,axiom,
( ( size_s5460976970255530739at_nat @ nil_Pr5478986624290739719at_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_983_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_984_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_985_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_986_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_987_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_988_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel1
thf(fact_989_mult__less__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_990_mult__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_991_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_992_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_993_unit__vecs__first_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [N3: nat] :
( X
!= ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) )
=> ~ ! [N3: nat,I: nat] :
( X
!= ( product_Pair_nat_nat @ N3 @ ( suc @ I ) ) ) ) ).
% unit_vecs_first.cases
thf(fact_994_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_995_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_996_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_997_nth__Cons__0,axiom,
! [X: mat_a,Xs: list_mat_a] :
( ( nth_mat_a @ ( cons_mat_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_998_nth__Cons__0,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_999_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_1000_take0,axiom,
( ( take_P2173866234530122223at_nat @ zero_zero_nat )
= ( ^ [Xs3: list_P6011104703257516679at_nat] : nil_Pr5478986624290739719at_nat ) ) ).
% take0
thf(fact_1001_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_1002_take__0,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( take_P2173866234530122223at_nat @ zero_zero_nat @ Xs )
= nil_Pr5478986624290739719at_nat ) ).
% take_0
thf(fact_1003_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_1004_take__eq__Nil,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ( take_P2173866234530122223at_nat @ N @ Xs )
= nil_Pr5478986624290739719at_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Pr5478986624290739719at_nat ) ) ) ).
% take_eq_Nil
thf(fact_1005_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_1006_take__eq__Nil2,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( take_P2173866234530122223at_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Pr5478986624290739719at_nat ) ) ) ).
% take_eq_Nil2
thf(fact_1007_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1008_list__update__code_I2_J,axiom,
! [X: mat_a,Xs: list_mat_a,Y: mat_a] :
( ( list_update_mat_a @ ( cons_mat_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_mat_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1009_list__update__code_I2_J,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( list_u6180841689913720943at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1010_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_1011_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1012_length__greater__0__conv,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) )
= ( Xs != nil_Pr5478986624290739719at_nat ) ) ).
% length_greater_0_conv
thf(fact_1013_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_1014_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_1015_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1016_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1017_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1018_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1019_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1020_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel1
thf(fact_1021_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1022_hd__conv__nth,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( hd_mat_a @ Xs )
= ( nth_mat_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1023_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1024_hd__conv__nth,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ Xs )
= ( nth_Pr7617993195940197384at_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1025_hd__take,axiom,
! [J: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_nat @ ( take_nat @ J @ Xs ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_take
thf(fact_1026_index__zero__mat_I1_J,axiom,
! [I3: nat,Nr: nat,J: nat,Nc: nat] :
( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( index_mat_nat @ ( zero_mat_nat @ Nr @ Nc ) @ ( product_Pair_nat_nat @ I3 @ J ) )
= zero_zero_nat ) ) ) ).
% index_zero_mat(1)
thf(fact_1027_remdups__adj__length__ge1,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s5460976970255530739at_nat @ ( remdup844249387045036349at_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_1028_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_1029_expand__powers_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [N3: nat,A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N3 @ A3 ) @ Ps ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ 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_1030_expand__powers_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ( X != nil_Pr5478986624290739719at_nat )
=> ( ! [N3: nat,A3: nat,Ps: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ 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_1031_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_mat_a
= ( ^ [Xs3: list_mat_a] : ( if_nat @ ( Xs3 = nil_mat_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_mat_a @ ( tl_mat_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1032_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_1033_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_1034_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1035_semiring__norm_I113_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% semiring_norm(113)
thf(fact_1036_semiring__norm_I137_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% semiring_norm(137)
thf(fact_1037_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1038_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1039_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1040_mult__nonneg__nonpos2,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1041_mult__nonpos__nonneg,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1042_mult__nonneg__nonpos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1043_mult__nonneg__nonneg,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1044_split__mult__neg__le,axiom,
! [A: nat,B3: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1045_mult__right__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1046_mult__left__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_1047_mult__mono_H,axiom,
! [A: nat,B3: nat,C: nat,D5: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C @ D5 )
=> ( ( 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 @ B3 @ D5 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1048_mult__mono,axiom,
! [A: nat,B3: nat,C: nat,D5: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C @ D5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B3 @ D5 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1049_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1050_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1051_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1052_zero__less__mult__pos2,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1053_zero__less__mult__pos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_1054_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1055_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1056_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1057_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1058_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B3: nat,C: nat,D5: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C @ D5 )
=> ( ( 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 @ B3 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1059_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B3: nat,C: nat,D5: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_nat @ C @ D5 )
=> ( ( 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 @ B3 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1060_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1061_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1062_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B3: nat,C: nat,D5: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ C @ D5 )
=> ( ( 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 @ B3 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1063_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1064_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B3: nat,C: nat,D5: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ C @ D5 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B3 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1065_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1066_expand__powers_Oelims,axiom,
! [X: list_P2678535509005046162_mat_a,Y: list_mat_a] :
( ( ( missin2919694041448319826_mat_a @ X )
= Y )
=> ( ( ( X = nil_Pr4811644691399623058_mat_a )
=> ( Y != nil_mat_a ) )
=> ( ! [N3: nat,A3: mat_a,Ps: list_P2678535509005046162_mat_a] :
( ( X
= ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ ( suc @ N3 ) @ A3 ) @ Ps ) )
=> ( Y
!= ( cons_mat_a @ A3 @ ( missin2919694041448319826_mat_a @ ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ N3 @ A3 ) @ Ps ) ) ) ) )
=> ~ ! [A3: mat_a,Ps: list_P2678535509005046162_mat_a] :
( ( X
= ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( Y
!= ( missin2919694041448319826_mat_a @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_1067_expand__powers_Oelims,axiom,
! [X: list_P1909269847677398966at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( missin2748503833011120330at_nat @ X )
= Y )
=> ( ( ( X = nil_Pr5468900520374568608at_nat )
=> ( Y != nil_Pr5478986624290739719at_nat ) )
=> ( ! [N3: nat,A3: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N3 ) @ A3 ) @ Ps ) )
=> ( Y
!= ( cons_P6512896166579812791at_nat @ A3 @ ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N3 @ A3 ) @ Ps ) ) ) ) )
=> ~ ! [A3: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( Y
!= ( missin2748503833011120330at_nat @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_1068_expand__powers_Oelims,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_nat] :
( ( ( missin6482572040563731271rs_nat @ X )
= Y )
=> ( ( ( X = nil_Pr5478986624290739719at_nat )
=> ( Y != nil_nat ) )
=> ( ! [N3: nat,A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ A3 ) @ Ps ) )
=> ( Y
!= ( cons_nat @ A3 @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N3 @ A3 ) @ Ps ) ) ) ) )
=> ~ ! [A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( Y
!= ( missin6482572040563731271rs_nat @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_1069_expand__powers_Osimps_I1_J,axiom,
( ( missin2748503833011120330at_nat @ nil_Pr5468900520374568608at_nat )
= nil_Pr5478986624290739719at_nat ) ).
% expand_powers.simps(1)
thf(fact_1070_expand__powers_Osimps_I1_J,axiom,
( ( missin6482572040563731271rs_nat @ nil_Pr5478986624290739719at_nat )
= nil_nat ) ).
% expand_powers.simps(1)
thf(fact_1071_expand__powers_Osimps_I2_J,axiom,
! [N: nat,A: mat_a,Ps2: list_P2678535509005046162_mat_a] :
( ( missin2919694041448319826_mat_a @ ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ ( suc @ N ) @ A ) @ Ps2 ) )
= ( cons_mat_a @ A @ ( missin2919694041448319826_mat_a @ ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ N @ A ) @ Ps2 ) ) ) ) ).
% expand_powers.simps(2)
thf(fact_1072_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_1073_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_1074_n__sum_Oinduct,axiom,
! [P: nat > list_nat > $o,A0: nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ zero_zero_nat @ X_1 )
=> ( ! [N3: nat,L2: list_nat] :
( ( P @ N3 @ ( tl_nat @ L2 ) )
=> ( P @ ( suc @ N3 ) @ L2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% n_sum.induct
thf(fact_1075_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_1076_n__lists_Osimps_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( n_list5849586770479959567at_nat @ zero_zero_nat @ Xs )
= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ nil_li8973309667444810893at_nat ) ) ).
% n_lists.simps(1)
thf(fact_1077_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1078_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list5849586770479959567at_nat @ N @ nil_Pr5478986624290739719at_nat )
= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ nil_li8973309667444810893at_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list5849586770479959567at_nat @ N @ nil_Pr5478986624290739719at_nat )
= nil_li8973309667444810893at_nat ) ) ) ).
% n_lists_Nil
thf(fact_1079_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1080_sum__list__geq__tl,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J2 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( tl_nat @ L ) ) @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% sum_list_geq_tl
thf(fact_1081_expand__powers_Opelims,axiom,
! [X: list_P2678535509005046162_mat_a,Y: list_mat_a] :
( ( ( missin2919694041448319826_mat_a @ X )
= Y )
=> ( ( accp_l1588916554921591195_mat_a @ missin8112068078703955743_mat_a @ X )
=> ( ( ( X = nil_Pr4811644691399623058_mat_a )
=> ( ( Y = nil_mat_a )
=> ~ ( accp_l1588916554921591195_mat_a @ missin8112068078703955743_mat_a @ nil_Pr4811644691399623058_mat_a ) ) )
=> ( ! [N3: nat,A3: mat_a,Ps: list_P2678535509005046162_mat_a] :
( ( X
= ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ ( suc @ N3 ) @ A3 ) @ Ps ) )
=> ( ( Y
= ( cons_mat_a @ A3 @ ( missin2919694041448319826_mat_a @ ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ N3 @ A3 ) @ Ps ) ) ) )
=> ~ ( accp_l1588916554921591195_mat_a @ missin8112068078703955743_mat_a @ ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ ( suc @ N3 ) @ A3 ) @ Ps ) ) ) )
=> ~ ! [A3: mat_a,Ps: list_P2678535509005046162_mat_a] :
( ( X
= ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( ( Y
= ( missin2919694041448319826_mat_a @ Ps ) )
=> ~ ( accp_l1588916554921591195_mat_a @ missin8112068078703955743_mat_a @ ( cons_P8936595845287417154_mat_a @ ( produc966498310820561988_mat_a @ zero_zero_nat @ A3 ) @ Ps ) ) ) ) ) ) ) ) ).
% expand_powers.pelims
thf(fact_1082_expand__powers_Opelims,axiom,
! [X: list_P1909269847677398966at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( missin2748503833011120330at_nat @ X )
= Y )
=> ( ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ X )
=> ( ( ( X = nil_Pr5468900520374568608at_nat )
=> ( ( Y = nil_Pr5478986624290739719at_nat )
=> ~ ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ nil_Pr5468900520374568608at_nat ) ) )
=> ( ! [N3: nat,A3: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N3 ) @ A3 ) @ Ps ) )
=> ( ( Y
= ( cons_P6512896166579812791at_nat @ A3 @ ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N3 @ A3 ) @ Ps ) ) ) )
=> ~ ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N3 ) @ A3 ) @ Ps ) ) ) )
=> ~ ! [A3: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( ( Y
= ( missin2748503833011120330at_nat @ Ps ) )
=> ~ ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A3 ) @ Ps ) ) ) ) ) ) ) ) ).
% expand_powers.pelims
thf(fact_1083_expand__powers_Opelims,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_nat] :
( ( ( missin6482572040563731271rs_nat @ X )
= Y )
=> ( ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ X )
=> ( ( ( X = nil_Pr5478986624290739719at_nat )
=> ( ( Y = nil_nat )
=> ~ ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ nil_Pr5478986624290739719at_nat ) ) )
=> ( ! [N3: nat,A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ A3 ) @ Ps ) )
=> ( ( Y
= ( cons_nat @ A3 @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N3 @ A3 ) @ Ps ) ) ) )
=> ~ ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ A3 ) @ Ps ) ) ) )
=> ~ ! [A3: nat,Ps: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A3 ) @ Ps ) )
=> ( ( Y
= ( missin6482572040563731271rs_nat @ Ps ) )
=> ~ ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A3 ) @ Ps ) ) ) ) ) ) ) ) ).
% expand_powers.pelims
thf(fact_1084_sum__list__cong,axiom,
! [L: list_P6011104703257516679at_nat,M: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ L )
= ( size_s5460976970255530739at_nat @ M ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ L ) )
=> ( ( nth_Pr7617993195940197384at_nat @ L @ I )
= ( nth_Pr7617993195940197384at_nat @ M @ I ) ) )
=> ( ( groups4206474380581351322at_nat @ L )
= ( groups4206474380581351322at_nat @ M ) ) ) ) ).
% sum_list_cong
thf(fact_1085_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_1086_sum__list__geq__0,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% sum_list_geq_0
thf(fact_1087_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_1088_sum__list__mono2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_mono2
thf(fact_1089_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_1090_sum__list_ONil,axiom,
( ( groups4206474380581351322at_nat @ nil_Pr5478986624290739719at_nat )
= zero_z3979849011205770936at_nat ) ).
% sum_list.Nil
thf(fact_1091_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_1092_remdups__adj_Opelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X )
= Y )
=> ( ( accp_list_nat @ remdups_adj_rel_nat @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = nil_nat )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ nil_nat ) ) )
=> ( ! [X5: nat] :
( ( X
= ( cons_nat @ X5 @ nil_nat ) )
=> ( ( Y
= ( cons_nat @ X5 @ nil_nat ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X5 @ nil_nat ) ) ) )
=> ~ ! [X5: nat,Y5: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Xs2 ) ) )
=> ( ( ( ( X5 = Y5 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y5 )
=> ( Y
= ( cons_nat @ X5 @ ( remdups_adj_nat @ ( cons_nat @ Y5 @ Xs2 ) ) ) ) ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X5 @ ( cons_nat @ Y5 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_1093_remdups__adj_Opelims,axiom,
! [X: list_mat_a,Y: list_mat_a] :
( ( ( remdups_adj_mat_a @ X )
= Y )
=> ( ( accp_list_mat_a @ remdup4805512644640555794_mat_a @ X )
=> ( ( ( X = nil_mat_a )
=> ( ( Y = nil_mat_a )
=> ~ ( accp_list_mat_a @ remdup4805512644640555794_mat_a @ nil_mat_a ) ) )
=> ( ! [X5: mat_a] :
( ( X
= ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ( ( Y
= ( cons_mat_a @ X5 @ nil_mat_a ) )
=> ~ ( accp_list_mat_a @ remdup4805512644640555794_mat_a @ ( cons_mat_a @ X5 @ nil_mat_a ) ) ) )
=> ~ ! [X5: mat_a,Y5: mat_a,Xs2: list_mat_a] :
( ( X
= ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Xs2 ) ) )
=> ( ( ( ( X5 = Y5 )
=> ( Y
= ( remdups_adj_mat_a @ ( cons_mat_a @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y5 )
=> ( Y
= ( cons_mat_a @ X5 @ ( remdups_adj_mat_a @ ( cons_mat_a @ Y5 @ Xs2 ) ) ) ) ) )
=> ~ ( accp_list_mat_a @ remdup4805512644640555794_mat_a @ ( cons_mat_a @ X5 @ ( cons_mat_a @ Y5 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_1094_remdups__adj_Opelims,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( remdup844249387045036349at_nat @ X )
= Y )
=> ( ( accp_l244970489926305168at_nat @ remdup6488300026587562250at_nat @ X )
=> ( ( ( X = nil_Pr5478986624290739719at_nat )
=> ( ( Y = nil_Pr5478986624290739719at_nat )
=> ~ ( accp_l244970489926305168at_nat @ remdup6488300026587562250at_nat @ nil_Pr5478986624290739719at_nat ) ) )
=> ( ! [X5: product_prod_nat_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ( ( Y
= ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ( accp_l244970489926305168at_nat @ remdup6488300026587562250at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ nil_Pr5478986624290739719at_nat ) ) ) )
=> ~ ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( X
= ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) )
=> ( ( ( ( X5 = Y5 )
=> ( Y
= ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y5 )
=> ( Y
= ( cons_P6512896166579812791at_nat @ X5 @ ( remdup844249387045036349at_nat @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) ) ) ) )
=> ~ ( accp_l244970489926305168at_nat @ remdup6488300026587562250at_nat @ ( cons_P6512896166579812791at_nat @ X5 @ ( cons_P6512896166579812791at_nat @ Y5 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_1095_n__sum__sum__list,axiom,
! [I3: nat,L: list_nat] :
( ( ord_less_eq_nat @ I3 @ ( size_size_list_nat @ L ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J2 ) ) )
=> ( ord_less_eq_nat @ ( commut2019222099004354946um_nat @ I3 @ L ) @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% n_sum_sum_list
thf(fact_1096_n__sum_Oelims,axiom,
! [X: nat,Xa2: list_nat,Y: nat] :
( ( ( commut2019222099004354946um_nat @ X @ Xa2 )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( plus_plus_nat @ ( hd_nat @ Xa2 ) @ ( commut2019222099004354946um_nat @ N3 @ ( tl_nat @ Xa2 ) ) ) ) ) ) ) ).
% n_sum.elims
thf(fact_1097_n__sum__last__lt,axiom,
! [J: nat,L: list_nat,I3: nat] :
( ( ord_less_nat @ J @ ( nth_nat @ L @ I3 ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ ( commut2019222099004354946um_nat @ I3 @ L ) @ J ) @ ( commut2019222099004354946um_nat @ ( suc @ I3 ) @ L ) ) ) ) ).
% n_sum_last_lt
thf(fact_1098_plus__prod__def,axiom,
( plus_p9057090461656269880at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) ) @ ( plus_plus_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% plus_prod_def
thf(fact_1099_zero__compare__simps_I1_J,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% zero_compare_simps(1)
% Helper facts (11)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J_T,axiom,
! [X: list_mat_a,Y: list_mat_a] :
( ( if_list_mat_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J_T,axiom,
! [X: list_mat_a,Y: list_mat_a] :
( ( if_list_mat_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= 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,Y: list_P6011104703257516679at_nat] :
( ( if_lis9186351972506106189at_nat @ $false @ X @ Y )
= Y ) ).
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,Y: list_P6011104703257516679at_nat] :
( ( if_lis9186351972506106189at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( carrier_mat_a @ ( nth_nat @ ( tl_nat @ la ) @ ia ) @ ( nth_nat @ ( tl_nat @ la ) @ ia ) )
= ( carrier_mat_a @ ( nth_nat @ la @ ( suc @ ia ) ) @ ( nth_nat @ la @ ( suc @ ia ) ) ) ) ).
%------------------------------------------------------------------------------