TPTP Problem File: SLH0665^1.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 : FO_Theory_Rewriting/0049_Utils/prob_00239_007735__18281194_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1615 ( 468 unt; 343 typ; 0 def)
% Number of atoms : 3839 (1664 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 15445 ( 526 ~; 83 |; 377 &;12589 @)
% ( 0 <=>;1870 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 71 ( 70 usr)
% Number of type conns : 867 ( 867 >; 0 *; 0 +; 0 <<)
% Number of symbols : 276 ( 273 usr; 17 con; 0-3 aty)
% Number of variables : 4520 ( 125 ^;4075 !; 320 ?;4520 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:00:45.845
%------------------------------------------------------------------------------
% Could-be-implicit typings (70)
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% Explicit typings (273)
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thf(sy_c_Utils_Otrancl__list_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
trancl5658469110673436434list_a: set_Pr5382606609415531783list_a > set_Pr6368357167367666183list_a ).
thf(sy_c_Utils_Otrancl__list_001t__List__Olist_It__Nat__Onat_J,type,
trancl_list_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).
thf(sy_c_Utils_Otrancl__list_001t__List__Olist_Itf__a_J,type,
trancl_list_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_Utils_Otrancl__list_001t__Nat__Onat,type,
trancl_list_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Utils_Otrancl__list_001tf__a,type,
trancl_list_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_Utils_Otrancl__listp_001t__List__Olist_Itf__a_J,type,
trancl_listp_list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_Utils_Otrancl__listp_001t__Nat__Onat,type,
trancl_listp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_Utils_Otrancl__listp_001tf__a,type,
trancl_listp_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member8667732009277439056list_a: produc3789376428941379879list_a > set_Pr6368357167367666183list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
member8680655010358287850st_nat: produc4326814125627636033st_nat > set_Pr1190453367779242145st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member6137020595371076400st_nat: produc484362233401163801st_nat > set_Pr5218233386278408143st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member4371779931761811402list_a: produc7489172080673977121list_a > set_Pr1060768173594829441list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member7437635676025943222list_a: produc1784977314056030623list_a > set_Pr6168524219156002389list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__a_J_J,type,
member5932150393272073264list_a: produc1513410750981052825list_a > set_Pr7423161166939974351list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member3917598494194944214list_a: produc7034990643107109933list_a > set_Pr4412185308373534093list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member4851138774834033962st_nat: produc432399132543013523st_nat > set_Pr5046312416420021961st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
member873448109036076442_a_nat: produc424395135190311811_a_nat > set_Pr6248017591093387577_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member8006451231845903178st_a_a: produc2579390645249093025st_a_a > set_Pr8962057229576493569st_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member8189971380475638336list_a: produc7740918406629873705list_a > set_Pr3870785875473343199list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member4889668945541975382list_a: produc8685980395799941037list_a > set_Pr2070066670564046349list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060R_062,type,
r: set_Product_prod_a_a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xsa____,type,
xsa: list_a ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_ysa____,type,
ysa: list_a ).
% Relevant facts (1254)
thf(fact_0_base_Ohyps_I1_J,axiom,
( ( size_size_list_a @ xsa )
= ( size_size_list_a @ ysa ) ) ).
% base.hyps(1)
thf(fact_1_base_Oprems,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ x @ y ) @ r ).
% base.prems
thf(fact_2_base_Ohyps_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ysa ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ xsa @ I ) @ ( nth_a @ ysa @ I ) ) @ r ) ) ).
% base.hyps(2)
thf(fact_3_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_4_prod_Oinject,axiom,
! [X1: list_nat,X2: list_nat,Y1: list_nat,Y2: list_nat] :
( ( ( produc2694037385005941721st_nat @ X1 @ X2 )
= ( produc2694037385005941721st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_prod_Oinject,axiom,
! [X1: list_list_a,X2: list_list_a,Y1: list_list_a,Y2: list_list_a] :
( ( ( produc8696003437204565271list_a @ X1 @ X2 )
= ( produc8696003437204565271list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_prod_Oinject,axiom,
! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X2 )
= ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_7_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
( ( ( produc2694037385005941721st_nat @ A @ B )
= ( produc2694037385005941721st_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A: list_list_a,B: list_list_a,A2: list_list_a,B2: list_list_a] :
( ( ( produc8696003437204565271list_a @ A @ B )
= ( produc8696003437204565271list_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_14_trancl__list_Obase,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ I2 ) @ ( nth_list_nat @ Ys @ I2 ) ) @ R ) )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( trancl_list_list_nat @ R ) ) ) ) ).
% trancl_list.base
thf(fact_15_trancl__list_Obase,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a] :
( ( ( size_s2403821588304063868list_a @ Xs )
= ( size_s2403821588304063868list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2403821588304063868list_a @ Ys ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ Xs @ I2 ) @ ( nth_list_list_a @ Ys @ I2 ) ) @ R ) )
=> ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( trancl5658469110673436434list_a @ R ) ) ) ) ).
% trancl_list.base
thf(fact_16_trancl__list_Obase,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ I2 ) @ ( nth_list_a @ Ys @ I2 ) ) @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( trancl_list_list_a @ R ) ) ) ) ).
% trancl_list.base
thf(fact_17_trancl__list_Obase,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( trancl_list_a @ R ) ) ) ) ).
% trancl_list.base
thf(fact_18_trancl__list_Obase,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) ) ) ) ).
% trancl_list.base
thf(fact_19_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_20_old_Oprod_Oexhaust,axiom,
! [Y: produc1828647624359046049st_nat] :
~ ! [A3: list_nat,B3: list_nat] :
( Y
!= ( produc2694037385005941721st_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_21_old_Oprod_Oexhaust,axiom,
! [Y: produc7709606177366032167list_a] :
~ ! [A3: list_list_a,B3: list_list_a] :
( Y
!= ( produc8696003437204565271list_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_22_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A3: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_23_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A3: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_24_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_25_surj__pair,axiom,
! [P: produc1828647624359046049st_nat] :
? [X3: list_nat,Y3: list_nat] :
( P
= ( produc2694037385005941721st_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_26_surj__pair,axiom,
! [P: produc7709606177366032167list_a] :
? [X3: list_list_a,Y3: list_list_a] :
( P
= ( produc8696003437204565271list_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_27_surj__pair,axiom,
! [P: produc9164743771328383783list_a] :
? [X3: list_a,Y3: list_a] :
( P
= ( produc6837034575241423639list_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_28_surj__pair,axiom,
! [P: product_prod_a_a] :
? [X3: a,Y3: a] :
( P
= ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_29_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_30_prod__cases,axiom,
! [P2: produc9164743771328383783list_a > $o,P: produc9164743771328383783list_a] :
( ! [A3: list_a,B3: list_a] : ( P2 @ ( produc6837034575241423639list_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_31_prod__cases,axiom,
! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_32_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_33_prod__cases,axiom,
! [P2: produc1828647624359046049st_nat > $o,P: produc1828647624359046049st_nat] :
( ! [A3: list_nat,B3: list_nat] : ( P2 @ ( produc2694037385005941721st_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_34_prod__cases,axiom,
! [P2: produc7709606177366032167list_a > $o,P: produc7709606177366032167list_a] :
( ! [A3: list_list_a,B3: list_list_a] : ( P2 @ ( produc8696003437204565271list_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_35_Pair__inject,axiom,
! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_36_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_37_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_38_Pair__inject,axiom,
! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
( ( ( produc2694037385005941721st_nat @ A @ B )
= ( produc2694037385005941721st_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_39_Pair__inject,axiom,
! [A: list_list_a,B: list_list_a,A2: list_list_a,B2: list_list_a] :
( ( ( produc8696003437204565271list_a @ A @ B )
= ( produc8696003437204565271list_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_40_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_41_length__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_42_length__induct,axiom,
! [P2: list_list_a > $o,Xs: list_list_a] :
( ! [Xs2: list_list_a] :
( ! [Ys2: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys2 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_43_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_44_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_45_nth__equalityI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I2 )
= ( nth_list_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_46_nth__equalityE,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ~ ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I )
= ( nth_a @ Ys @ I ) ) ) ) ) ).
% nth_equalityE
thf(fact_47_nth__equalityE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ~ ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_equalityE
thf(fact_48_nth__equalityE,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ~ ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I )
= ( nth_list_a @ Ys @ I ) ) ) ) ) ).
% nth_equalityE
thf(fact_49_Skolem__list__nth,axiom,
! [K: nat,P2: nat > a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: a] : ( P2 @ I3 @ X4 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_a @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_50_Skolem__list__nth,axiom,
! [K: nat,P2: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: nat] : ( P2 @ I3 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_51_Skolem__list__nth,axiom,
! [K: nat,P2: nat > list_a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: list_a] : ( P2 @ I3 @ X4 ) ) )
= ( ? [Xs3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_list_a @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_52_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_53_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_54_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_55_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_56_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_57_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_58_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I3 )
= ( nth_a @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_59_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_60_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_a,Z: list_list_a] : ( Y4 = Z ) )
= ( ^ [Xs3: list_list_a,Ys3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs3 )
= ( size_s349497388124573686list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ( ( nth_list_a @ Xs3 @ I3 )
= ( nth_list_a @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_61_Cons__less__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X @ Y ) ) ) ) ).
% Cons_less_Cons
thf(fact_62_Ex__list__of__length__P,axiom,
! [N: nat,P2: a > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X5: a] : ( P2 @ X5 @ I2 ) )
=> ? [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_a @ Xs2 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_63_Ex__list__of__length__P,axiom,
! [N: nat,P2: nat > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X5: nat] : ( P2 @ X5 @ I2 ) )
=> ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_nat @ Xs2 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_64_Ex__list__of__length__P,axiom,
! [N: nat,P2: list_a > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X5: list_a] : ( P2 @ X5 @ I2 ) )
=> ? [Xs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_list_a @ Xs2 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_65_Cons__lenlex__iff,axiom,
! [M: list_nat,Ms: list_list_nat,N: list_nat,Ns: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ M @ Ms ) @ ( cons_list_nat @ N @ Ns ) ) @ ( lenlex_list_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ms ) @ ( size_s3023201423986296836st_nat @ Ns ) )
| ( ( ( size_s3023201423986296836st_nat @ Ms )
= ( size_s3023201423986296836st_nat @ Ns ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ms @ Ns ) @ ( lenlex_list_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_66_Cons__lenlex__iff,axiom,
! [M: list_list_a,Ms: list_list_list_a,N: list_list_a,Ns: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ M @ Ms ) @ ( cons_list_list_a @ N @ Ns ) ) @ ( lenlex_list_list_a @ R2 ) )
= ( ( ord_less_nat @ ( size_s2403821588304063868list_a @ Ms ) @ ( size_s2403821588304063868list_a @ Ns ) )
| ( ( ( size_s2403821588304063868list_a @ Ms )
= ( size_s2403821588304063868list_a @ Ns ) )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Ms @ Ns ) @ ( lenlex_list_list_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_67_Cons__lenlex__iff,axiom,
! [M: a,Ms: list_a,N: a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_68_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_69_Cons__lenlex__iff,axiom,
! [M: list_a,Ms: list_list_a,N: list_a,Ns: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ M @ Ms ) @ ( cons_list_a @ N @ Ns ) ) @ ( lenlex_list_a @ R2 ) )
= ( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ms ) @ ( size_s349497388124573686list_a @ Ns ) )
| ( ( ( size_s349497388124573686list_a @ Ms )
= ( size_s349497388124573686list_a @ Ns ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ms @ Ns ) @ ( lenlex_list_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_70_Cons__in__lex,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( lex_list_nat @ R2 ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
& ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) )
| ( ( X = Y )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( lex_list_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_71_Cons__in__lex,axiom,
! [X: list_list_a,Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ Y @ Ys ) ) @ ( lex_list_list_a @ R2 ) )
= ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R2 )
& ( ( size_s2403821588304063868list_a @ Xs )
= ( size_s2403821588304063868list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( lex_list_list_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_72_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_73_Cons__in__lex,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_74_Cons__in__lex,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( lex_list_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R2 )
& ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_75_less__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_list_nat @ Xs @ Ys ) ) ) ) ).
% less_list_code(3)
thf(fact_76_trancl__list_Olist__trancl,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat,I4: nat,Z2: list_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( trancl_list_list_nat @ R ) )
=> ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ys @ I4 ) @ Z2 ) @ R )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ ( list_update_list_nat @ Ys @ I4 @ Z2 ) ) @ ( trancl_list_list_nat @ R ) ) ) ) ) ).
% trancl_list.list_trancl
thf(fact_77_trancl__list_Olist__trancl,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a,I4: nat,Z2: list_list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( trancl5658469110673436434list_a @ R ) )
=> ( ( ord_less_nat @ I4 @ ( size_s2403821588304063868list_a @ Ys ) )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ Ys @ I4 ) @ Z2 ) @ R )
=> ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ ( list_u84128978910470136list_a @ Ys @ I4 @ Z2 ) ) @ ( trancl5658469110673436434list_a @ R ) ) ) ) ) ).
% trancl_list.list_trancl
thf(fact_78_trancl__list_Olist__trancl,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,I4: nat,Z2: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( trancl_list_a @ R ) )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Ys @ I4 ) @ Z2 ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( list_update_a @ Ys @ I4 @ Z2 ) ) @ ( trancl_list_a @ R ) ) ) ) ) ).
% trancl_list.list_trancl
thf(fact_79_trancl__list_Olist__trancl,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,I4: nat,Z2: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys @ I4 ) @ Z2 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( list_update_nat @ Ys @ I4 @ Z2 ) ) @ ( trancl_list_nat @ R ) ) ) ) ) ).
% trancl_list.list_trancl
thf(fact_80_trancl__list_Olist__trancl,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a,I4: nat,Z2: list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( trancl_list_list_a @ R ) )
=> ( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Ys @ I4 ) @ Z2 ) @ R )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( list_update_list_a @ Ys @ I4 @ Z2 ) ) @ ( trancl_list_list_a @ R ) ) ) ) ) ).
% trancl_list.list_trancl
thf(fact_81_trancl__list_Osimps,axiom,
! [A1: list_list_nat,A22: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ A22 ) @ ( trancl_list_list_nat @ R ) )
= ( ? [Xs3: list_list_nat,Ys3: list_list_nat] :
( ( A1 = Xs3 )
& ( A22 = Ys3 )
& ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs3 @ I3 ) @ ( nth_list_nat @ Ys3 @ I3 ) ) @ R ) ) )
| ? [Xs3: list_list_nat,Ys3: list_list_nat,I3: nat,Z3: list_nat] :
( ( A1 = Xs3 )
& ( A22
= ( list_update_list_nat @ Ys3 @ I3 @ Z3 ) )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs3 @ Ys3 ) @ ( trancl_list_list_nat @ R ) )
& ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).
% trancl_list.simps
thf(fact_82_trancl__list_Osimps,axiom,
! [A1: list_list_list_a,A22: list_list_list_a,R: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ A1 @ A22 ) @ ( trancl5658469110673436434list_a @ R ) )
= ( ? [Xs3: list_list_list_a,Ys3: list_list_list_a] :
( ( A1 = Xs3 )
& ( A22 = Ys3 )
& ( ( size_s2403821588304063868list_a @ Xs3 )
= ( size_s2403821588304063868list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s2403821588304063868list_a @ Ys3 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ Xs3 @ I3 ) @ ( nth_list_list_a @ Ys3 @ I3 ) ) @ R ) ) )
| ? [Xs3: list_list_list_a,Ys3: list_list_list_a,I3: nat,Z3: list_list_a] :
( ( A1 = Xs3 )
& ( A22
= ( list_u84128978910470136list_a @ Ys3 @ I3 @ Z3 ) )
& ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs3 @ Ys3 ) @ ( trancl5658469110673436434list_a @ R ) )
& ( ord_less_nat @ I3 @ ( size_s2403821588304063868list_a @ Ys3 ) )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).
% trancl_list.simps
thf(fact_83_trancl__list_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( trancl_list_a @ R ) )
= ( ? [Xs3: list_a,Ys3: list_a] :
( ( A1 = Xs3 )
& ( A22 = Ys3 )
& ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs3 @ I3 ) @ ( nth_a @ Ys3 @ I3 ) ) @ R ) ) )
| ? [Xs3: list_a,Ys3: list_a,I3: nat,Z3: a] :
( ( A1 = Xs3 )
& ( A22
= ( list_update_a @ Ys3 @ I3 @ Z3 ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( trancl_list_a @ R ) )
& ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).
% trancl_list.simps
thf(fact_84_trancl__list_Osimps,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R ) )
= ( ? [Xs3: list_nat,Ys3: list_nat] :
( ( A1 = Xs3 )
& ( A22 = Ys3 )
& ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) @ R ) ) )
| ? [Xs3: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
( ( A1 = Xs3 )
& ( A22
= ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( trancl_list_nat @ R ) )
& ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).
% trancl_list.simps
thf(fact_85_trancl__list_Osimps,axiom,
! [A1: list_list_a,A22: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A1 @ A22 ) @ ( trancl_list_list_a @ R ) )
= ( ? [Xs3: list_list_a,Ys3: list_list_a] :
( ( A1 = Xs3 )
& ( A22 = Ys3 )
& ( ( size_s349497388124573686list_a @ Xs3 )
= ( size_s349497388124573686list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ys3 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs3 @ I3 ) @ ( nth_list_a @ Ys3 @ I3 ) ) @ R ) ) )
| ? [Xs3: list_list_a,Ys3: list_list_a,I3: nat,Z3: list_a] :
( ( A1 = Xs3 )
& ( A22
= ( list_update_list_a @ Ys3 @ I3 @ Z3 ) )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs3 @ Ys3 ) @ ( trancl_list_list_a @ R ) )
& ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ys3 ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).
% trancl_list.simps
thf(fact_86_trancl__list_Ocases,axiom,
! [A1: list_list_nat,A22: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ A22 ) @ ( trancl_list_list_nat @ R ) )
=> ( ( ( ( size_s3023201423986296836st_nat @ A1 )
= ( size_s3023201423986296836st_nat @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ A22 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ A1 @ I ) @ ( nth_list_nat @ A22 @ I ) ) @ R ) ) )
=> ~ ! [Ys4: list_list_nat,I2: nat,Z4: list_nat] :
( ( A22
= ( list_update_list_nat @ Ys4 @ I2 @ Z4 ) )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ Ys4 ) @ ( trancl_list_list_nat @ R ) )
=> ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys4 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ys4 @ I2 ) @ Z4 ) @ R ) ) ) ) ) ) ).
% trancl_list.cases
thf(fact_87_trancl__list_Ocases,axiom,
! [A1: list_list_list_a,A22: list_list_list_a,R: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ A1 @ A22 ) @ ( trancl5658469110673436434list_a @ R ) )
=> ( ( ( ( size_s2403821588304063868list_a @ A1 )
= ( size_s2403821588304063868list_a @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s2403821588304063868list_a @ A22 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ A1 @ I ) @ ( nth_list_list_a @ A22 @ I ) ) @ R ) ) )
=> ~ ! [Ys4: list_list_list_a,I2: nat,Z4: list_list_a] :
( ( A22
= ( list_u84128978910470136list_a @ Ys4 @ I2 @ Z4 ) )
=> ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ A1 @ Ys4 ) @ ( trancl5658469110673436434list_a @ R ) )
=> ( ( ord_less_nat @ I2 @ ( size_s2403821588304063868list_a @ Ys4 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ Ys4 @ I2 ) @ Z4 ) @ R ) ) ) ) ) ) ).
% trancl_list.cases
thf(fact_88_trancl__list_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( trancl_list_a @ R ) )
=> ( ( ( ( size_size_list_a @ A1 )
= ( size_size_list_a @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ A22 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ A1 @ I ) @ ( nth_a @ A22 @ I ) ) @ R ) ) )
=> ~ ! [Ys4: list_a,I2: nat,Z4: a] :
( ( A22
= ( list_update_a @ Ys4 @ I2 @ Z4 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ Ys4 ) @ ( trancl_list_a @ R ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys4 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Ys4 @ I2 ) @ Z4 ) @ R ) ) ) ) ) ) ).
% trancl_list.cases
thf(fact_89_trancl__list_Ocases,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R ) )
=> ( ( ( ( size_size_list_nat @ A1 )
= ( size_size_list_nat @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ A22 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ A1 @ I ) @ ( nth_nat @ A22 @ I ) ) @ R ) ) )
=> ~ ! [Ys4: list_nat,I2: nat,Z4: nat] :
( ( A22
= ( list_update_nat @ Ys4 @ I2 @ Z4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ Ys4 ) @ ( trancl_list_nat @ R ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I2 ) @ Z4 ) @ R ) ) ) ) ) ) ).
% trancl_list.cases
thf(fact_90_trancl__list_Ocases,axiom,
! [A1: list_list_a,A22: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A1 @ A22 ) @ ( trancl_list_list_a @ R ) )
=> ( ( ( ( size_s349497388124573686list_a @ A1 )
= ( size_s349497388124573686list_a @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ A22 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ A1 @ I ) @ ( nth_list_a @ A22 @ I ) ) @ R ) ) )
=> ~ ! [Ys4: list_list_a,I2: nat,Z4: list_a] :
( ( A22
= ( list_update_list_a @ Ys4 @ I2 @ Z4 ) )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A1 @ Ys4 ) @ ( trancl_list_list_a @ R ) )
=> ( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys4 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Ys4 @ I2 ) @ Z4 ) @ R ) ) ) ) ) ) ).
% trancl_list.cases
thf(fact_91_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N2 ) @ ( nth_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_92_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R2 ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ ( nth_a @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_93_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_a,R2: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ Ys ) @ ( listrel_nat_a @ R2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ ( nth_nat @ Xs @ N2 ) @ ( nth_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_94_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_95_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_list_a,R2: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs @ Ys ) @ ( listrel_a_list_a @ R2 ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ ( nth_a @ Xs @ N2 ) @ ( nth_list_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_96_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_list_a,R2: set_Pr3870785875473343199list_a] :
( ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Xs @ Ys ) @ ( listrel_nat_list_a @ R2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ ( nth_nat @ Xs @ N2 ) @ ( nth_list_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_97_listrel__iff__nth,axiom,
! [Xs: list_list_a,Ys: list_a,R2: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs @ Ys ) @ ( listrel_list_a_a @ R2 ) )
= ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ ( nth_list_a @ Xs @ N2 ) @ ( nth_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_98_listrel__iff__nth,axiom,
! [Xs: list_list_a,Ys: list_nat,R2: set_Pr6248017591093387577_a_nat] :
( ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Xs @ Ys ) @ ( listrel_list_a_nat @ R2 ) )
= ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member873448109036076442_a_nat @ ( produc4010430043144778749_a_nat @ ( nth_list_a @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_99_listrel__iff__nth,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listre6091228620945859379st_nat @ R2 ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ N2 ) @ ( nth_list_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_100_listrel__iff__nth,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listre6772471554020304241list_a @ R2 ) )
= ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ N2 ) @ ( nth_list_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_101_in__measures_I2_J,axiom,
! [X: list_a,Y: list_a,F: list_a > nat,Fs: list_list_a_nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_102_in__measures_I2_J,axiom,
! [X: a,Y: a,F: a > nat,Fs: list_a_nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_103_in__measures_I2_J,axiom,
! [X: nat,Y: nat,F: nat > nat,Fs: list_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_104_in__measures_I2_J,axiom,
! [X: list_nat,Y: list_nat,F: list_nat > nat,Fs: list_list_nat_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ ( cons_list_nat_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_105_in__measures_I2_J,axiom,
! [X: list_list_a,Y: list_list_a,F: list_list_a > nat,Fs: list_list_list_a_nat] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ ( measures_list_list_a @ ( cons_list_list_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ ( measures_list_list_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_106_length__list__update,axiom,
! [Xs: list_a,I4: nat,X: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I4 @ X ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_107_length__list__update,axiom,
! [Xs: list_nat,I4: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I4 @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_108_length__list__update,axiom,
! [Xs: list_list_a,I4: nat,X: list_a] :
( ( size_s349497388124573686list_a @ ( list_update_list_a @ Xs @ I4 @ X ) )
= ( size_s349497388124573686list_a @ Xs ) ) ).
% length_list_update
thf(fact_109_mem__Collect__eq,axiom,
! [A: produc9164743771328383783list_a,P2: produc9164743771328383783list_a > $o] :
( ( member8191768239178080336list_a @ A @ ( collec943055143889122450list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
! [A: produc1828647624359046049st_nat,P2: produc1828647624359046049st_nat > $o] :
( ( member7340969449405702474st_nat @ A @ ( collec1570431334306492044st_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_113_mem__Collect__eq,axiom,
! [A: produc7709606177366032167list_a,P2: produc7709606177366032167list_a > $o] :
( ( member1318342207407915856list_a @ A @ ( collec4221375464948680082list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A4: set_Pr4048851178543822343list_a] :
( ( collec943055143889122450list_a
@ ^ [X6: produc9164743771328383783list_a] : ( member8191768239178080336list_a @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A4: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X6: product_prod_a_a] : ( member1426531477525435216od_a_a @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X6: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ( collec1570431334306492044st_nat
@ ^ [X6: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_118_Collect__mem__eq,axiom,
! [A4: set_Pr5382606609415531783list_a] :
( ( collec4221375464948680082list_a
@ ^ [X6: produc7709606177366032167list_a] : ( member1318342207407915856list_a @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_119_nth__list__update__neq,axiom,
! [I4: nat,J: nat,Xs: list_a,X: a] :
( ( I4 != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_120_nth__list__update__neq,axiom,
! [I4: nat,J: nat,Xs: list_nat,X: nat] :
( ( I4 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_121_nth__list__update__neq,axiom,
! [I4: nat,J: nat,Xs: list_list_a,X: list_a] :
( ( I4 != J )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I4 @ X ) @ J )
= ( nth_list_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_122_list__update__id,axiom,
! [Xs: list_a,I4: nat] :
( ( list_update_a @ Xs @ I4 @ ( nth_a @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_123_list__update__id,axiom,
! [Xs: list_nat,I4: nat] :
( ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_124_list__update__id,axiom,
! [Xs: list_list_a,I4: nat] :
( ( list_update_list_a @ Xs @ I4 @ ( nth_list_a @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_125_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X ) @ I4 )
= X ) ) ).
% nth_list_update_eq
thf(fact_126_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ I4 )
= X ) ) ).
% nth_list_update_eq
thf(fact_127_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_list_a,X: list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I4 @ X ) @ I4 )
= X ) ) ).
% nth_list_update_eq
thf(fact_128_parallel__list__update,axiom,
! [N: nat,R2: a > a > $o,P: list_a > $o,Xs: list_a,Ys: list_a] :
( ! [Xs2: list_a,I2: nat,Y3: a] :
( ( ( size_size_list_a @ Xs2 )
= N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( R2 @ ( nth_a @ Xs2 @ I2 ) @ Y3 )
=> ( ( P @ Xs2 )
=> ( P @ ( list_update_a @ Xs2 @ I2 @ Y3 ) ) ) ) ) )
=> ( ( ( size_size_list_a @ Xs )
= N )
=> ( ( P @ Xs )
=> ( ( ( size_size_list_a @ Ys )
= N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( R2 @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) )
=> ( P @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_129_parallel__list__update,axiom,
! [N: nat,R2: nat > nat > $o,P: list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ! [Xs2: list_nat,I2: nat,Y3: nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( R2 @ ( nth_nat @ Xs2 @ I2 ) @ Y3 )
=> ( ( P @ Xs2 )
=> ( P @ ( list_update_nat @ Xs2 @ I2 @ Y3 ) ) ) ) ) )
=> ( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( P @ Xs )
=> ( ( ( size_size_list_nat @ Ys )
= N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( R2 @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( P @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_130_parallel__list__update,axiom,
! [N: nat,R2: list_a > list_a > $o,P: list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ! [Xs2: list_list_a,I2: nat,Y3: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( R2 @ ( nth_list_a @ Xs2 @ I2 ) @ Y3 )
=> ( ( P @ Xs2 )
=> ( P @ ( list_update_list_a @ Xs2 @ I2 @ Y3 ) ) ) ) ) )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= N )
=> ( ( P @ Xs )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( R2 @ ( nth_list_a @ Xs @ I2 ) @ ( nth_list_a @ Ys @ I2 ) ) )
=> ( P @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_131_listrel__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_132_listrel__eq__len,axiom,
! [Xs: list_a,Ys: list_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_133_listrel__eq__len,axiom,
! [Xs: list_a,Ys: list_list_a,R2: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs @ Ys ) @ ( listrel_a_list_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_134_listrel__eq__len,axiom,
! [Xs: list_nat,Ys: list_a,R2: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ Ys ) @ ( listrel_nat_a @ R2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_135_listrel__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_136_listrel__eq__len,axiom,
! [Xs: list_nat,Ys: list_list_a,R2: set_Pr3870785875473343199list_a] :
( ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Xs @ Ys ) @ ( listrel_nat_list_a @ R2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_137_listrel__eq__len,axiom,
! [Xs: list_list_a,Ys: list_a,R2: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs @ Ys ) @ ( listrel_list_a_a @ R2 ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_138_listrel__eq__len,axiom,
! [Xs: list_list_a,Ys: list_nat,R2: set_Pr6248017591093387577_a_nat] :
( ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Xs @ Ys ) @ ( listrel_list_a_nat @ R2 ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_139_listrel__eq__len,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listre6772471554020304241list_a @ R2 ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_140_list__update__same__conv,axiom,
! [I4: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I4 @ X )
= Xs )
= ( ( nth_a @ Xs @ I4 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_141_list__update__same__conv,axiom,
! [I4: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I4 @ X )
= Xs )
= ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_142_list__update__same__conv,axiom,
! [I4: nat,Xs: list_list_a,X: list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( list_update_list_a @ Xs @ I4 @ X )
= Xs )
= ( ( nth_list_a @ Xs @ I4 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_143_nth__list__update,axiom,
! [I4: nat,Xs: list_a,J: nat,X: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ( I4 = J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X ) @ J )
= X ) )
& ( ( I4 != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_144_nth__list__update,axiom,
! [I4: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I4 = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ J )
= X ) )
& ( ( I4 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_145_nth__list__update,axiom,
! [I4: nat,Xs: list_list_a,J: nat,X: list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( I4 = J )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I4 @ X ) @ J )
= X ) )
& ( ( I4 != J )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I4 @ X ) @ J )
= ( nth_list_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_146_lenlex__irreflexive,axiom,
! [R2: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ! [X3: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ X3 ) @ R2 )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Xs ) @ ( lenlex_list_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_147_lenlex__irreflexive,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a] :
( ! [X3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Xs ) @ ( lenlex_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_148_lenlex__irreflexive,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_149_lenlex__irreflexive,axiom,
! [R2: set_Pr3451248702717554689st_nat,Xs: list_list_nat] :
( ! [X3: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ R2 )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Xs ) @ ( lenlex_list_nat @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_150_lenlex__irreflexive,axiom,
! [R2: set_Pr5382606609415531783list_a,Xs: list_list_list_a] :
( ! [X3: list_list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ X3 ) @ R2 )
=> ~ ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Xs ) @ ( lenlex_list_list_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_151_listrel_OCons,axiom,
! [X: list_a,Y: list_a,R2: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R2 )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listre6772471554020304241list_a @ R2 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listre6772471554020304241list_a @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_152_listrel_OCons,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_153_listrel_OCons,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_154_listrel_OCons,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listre6091228620945859379st_nat @ R2 ) )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( listre6091228620945859379st_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_155_listrel_OCons,axiom,
! [X: list_list_a,Y: list_list_a,R2: set_Pr5382606609415531783list_a,Xs: list_list_list_a,Ys: list_list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R2 )
=> ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listre7520310791835696497list_a @ R2 ) )
=> ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ Y @ Ys ) ) @ ( listre7520310791835696497list_a @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_156_listrel__Cons1,axiom,
! [Y: list_nat,Ys: list_list_nat,Xs: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ Y @ Ys ) @ Xs ) @ ( listre6091228620945859379st_nat @ R2 ) )
=> ~ ! [Y3: list_nat,Ys4: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y3 @ Ys4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Ys4 ) @ ( listre6091228620945859379st_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_157_listrel__Cons1,axiom,
! [Y: list_list_a,Ys: list_list_list_a,Xs: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ Y @ Ys ) @ Xs ) @ ( listre7520310791835696497list_a @ R2 ) )
=> ~ ! [Y3: list_list_a,Ys4: list_list_list_a] :
( ( Xs
= ( cons_list_list_a @ Y3 @ Ys4 ) )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Y @ Y3 ) @ R2 )
=> ~ ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Ys @ Ys4 ) @ ( listre7520310791835696497list_a @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_158_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_a @ R2 ) )
=> ~ ! [Y3: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys4 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y3 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Ys4 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_159_listrel__Cons1,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [Y3: nat,Ys4: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys4 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_160_listrel__Cons1,axiom,
! [Y: list_a,Ys: list_list_a,Xs: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ Y @ Ys ) @ Xs ) @ ( listre6772471554020304241list_a @ R2 ) )
=> ~ ! [Y3: list_a,Ys4: list_list_a] :
( ( Xs
= ( cons_list_a @ Y3 @ Ys4 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ Y3 ) @ R2 )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Ys4 ) @ ( listre6772471554020304241list_a @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_161_listrel__Cons2,axiom,
! [Xs: list_list_nat,Y: list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ ( cons_list_nat @ Y @ Ys ) ) @ ( listre6091228620945859379st_nat @ R2 ) )
=> ~ ! [X3: list_nat,Xs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ X3 @ Xs2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ R2 )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( listre6091228620945859379st_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_162_listrel__Cons2,axiom,
! [Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ ( cons_list_list_a @ Y @ Ys ) ) @ ( listre7520310791835696497list_a @ R2 ) )
=> ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
( ( Xs
= ( cons_list_list_a @ X3 @ Xs2 ) )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ Y ) @ R2 )
=> ~ ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs2 @ Ys ) @ ( listre7520310791835696497list_a @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_163_listrel__Cons2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R2 ) )
=> ~ ! [X3: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_164_listrel__Cons2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_165_listrel__Cons2,axiom,
! [Xs: list_list_a,Y: list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listre6772471554020304241list_a @ R2 ) )
=> ~ ! [X3: list_a,Xs2: list_list_a] :
( ( Xs
= ( cons_list_a @ X3 @ Xs2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y ) @ R2 )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys ) @ ( listre6772471554020304241list_a @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_166_measures__less,axiom,
! [F: list_a > nat,X: list_a,Y: list_a,Fs: list_list_a_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_167_measures__less,axiom,
! [F: a > nat,X: a,Y: a,Fs: list_a_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_168_measures__less,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_169_measures__less,axiom,
! [F: list_nat > nat,X: list_nat,Y: list_nat,Fs: list_list_nat_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ ( cons_list_nat_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_170_measures__less,axiom,
! [F: list_list_a > nat,X: list_list_a,Y: list_list_a,Fs: list_list_list_a_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ ( measures_list_list_a @ ( cons_list_list_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_171_permut__sound,axiom,
! [I4: nat,As: list_a,F: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ As ) )
=> ( ( nth_a @ ( missing_permut_a @ As @ F ) @ I4 )
= ( nth_a @ As @ ( F @ I4 ) ) ) ) ).
% permut_sound
thf(fact_172_permut__sound,axiom,
! [I4: nat,As: list_nat,F: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missing_permut_nat @ As @ F ) @ I4 )
= ( nth_nat @ As @ ( F @ I4 ) ) ) ) ).
% permut_sound
thf(fact_173_permut__sound,axiom,
! [I4: nat,As: list_list_a,F: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ As ) )
=> ( ( nth_list_a @ ( missin8258838934412359004list_a @ As @ F ) @ I4 )
= ( nth_list_a @ As @ ( F @ I4 ) ) ) ) ).
% permut_sound
thf(fact_174_trancl__listp_Olist__trancl,axiom,
! [R: a > a > $o,Xs: list_a,Ys: list_a,I4: nat,Z2: a] :
( ( trancl_listp_a @ R @ Xs @ Ys )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
=> ( ( R @ ( nth_a @ Ys @ I4 ) @ Z2 )
=> ( trancl_listp_a @ R @ Xs @ ( list_update_a @ Ys @ I4 @ Z2 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_175_trancl__listp_Olist__trancl,axiom,
! [R: nat > nat > $o,Xs: list_nat,Ys: list_nat,I4: nat,Z2: nat] :
( ( trancl_listp_nat @ R @ Xs @ Ys )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( R @ ( nth_nat @ Ys @ I4 ) @ Z2 )
=> ( trancl_listp_nat @ R @ Xs @ ( list_update_nat @ Ys @ I4 @ Z2 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_176_trancl__listp_Olist__trancl,axiom,
! [R: list_a > list_a > $o,Xs: list_list_a,Ys: list_list_a,I4: nat,Z2: list_a] :
( ( trancl_listp_list_a @ R @ Xs @ Ys )
=> ( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( R @ ( nth_list_a @ Ys @ I4 ) @ Z2 )
=> ( trancl_listp_list_a @ R @ Xs @ ( list_update_list_a @ Ys @ I4 @ Z2 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_177_trancl__listp_Osimps,axiom,
( trancl_listp_a
= ( ^ [R3: a > a > $o,A12: list_a,A23: list_a] :
( ? [Xs3: list_a,Ys3: list_a] :
( ( A12 = Xs3 )
& ( A23 = Ys3 )
& ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys3 ) )
=> ( R3 @ ( nth_a @ Xs3 @ I3 ) @ ( nth_a @ Ys3 @ I3 ) ) ) )
| ? [Xs3: list_a,Ys3: list_a,I3: nat,Z3: a] :
( ( A12 = Xs3 )
& ( A23
= ( list_update_a @ Ys3 @ I3 @ Z3 ) )
& ( trancl_listp_a @ R3 @ Xs3 @ Ys3 )
& ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys3 ) )
& ( R3 @ ( nth_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_178_trancl__listp_Osimps,axiom,
( trancl_listp_nat
= ( ^ [R3: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ? [Xs3: list_nat,Ys3: list_nat] :
( ( A12 = Xs3 )
& ( A23 = Ys3 )
& ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
=> ( R3 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) ) )
| ? [Xs3: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
( ( A12 = Xs3 )
& ( A23
= ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
& ( trancl_listp_nat @ R3 @ Xs3 @ Ys3 )
& ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
& ( R3 @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_179_trancl__listp_Osimps,axiom,
( trancl_listp_list_a
= ( ^ [R3: list_a > list_a > $o,A12: list_list_a,A23: list_list_a] :
( ? [Xs3: list_list_a,Ys3: list_list_a] :
( ( A12 = Xs3 )
& ( A23 = Ys3 )
& ( ( size_s349497388124573686list_a @ Xs3 )
= ( size_s349497388124573686list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ys3 ) )
=> ( R3 @ ( nth_list_a @ Xs3 @ I3 ) @ ( nth_list_a @ Ys3 @ I3 ) ) ) )
| ? [Xs3: list_list_a,Ys3: list_list_a,I3: nat,Z3: list_a] :
( ( A12 = Xs3 )
& ( A23
= ( list_update_list_a @ Ys3 @ I3 @ Z3 ) )
& ( trancl_listp_list_a @ R3 @ Xs3 @ Ys3 )
& ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ys3 ) )
& ( R3 @ ( nth_list_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_180_trancl__listp_Ocases,axiom,
! [R: a > a > $o,A1: list_a,A22: list_a] :
( ( trancl_listp_a @ R @ A1 @ A22 )
=> ( ( ( ( size_size_list_a @ A1 )
= ( size_size_list_a @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ A22 ) )
=> ( R @ ( nth_a @ A1 @ I ) @ ( nth_a @ A22 @ I ) ) ) )
=> ~ ! [Ys4: list_a,I2: nat,Z4: a] :
( ( A22
= ( list_update_a @ Ys4 @ I2 @ Z4 ) )
=> ( ( trancl_listp_a @ R @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys4 ) )
=> ~ ( R @ ( nth_a @ Ys4 @ I2 ) @ Z4 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_181_trancl__listp_Ocases,axiom,
! [R: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( trancl_listp_nat @ R @ A1 @ A22 )
=> ( ( ( ( size_size_list_nat @ A1 )
= ( size_size_list_nat @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ A22 ) )
=> ( R @ ( nth_nat @ A1 @ I ) @ ( nth_nat @ A22 @ I ) ) ) )
=> ~ ! [Ys4: list_nat,I2: nat,Z4: nat] :
( ( A22
= ( list_update_nat @ Ys4 @ I2 @ Z4 ) )
=> ( ( trancl_listp_nat @ R @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
=> ~ ( R @ ( nth_nat @ Ys4 @ I2 ) @ Z4 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_182_trancl__listp_Ocases,axiom,
! [R: list_a > list_a > $o,A1: list_list_a,A22: list_list_a] :
( ( trancl_listp_list_a @ R @ A1 @ A22 )
=> ( ( ( ( size_s349497388124573686list_a @ A1 )
= ( size_s349497388124573686list_a @ A22 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ A22 ) )
=> ( R @ ( nth_list_a @ A1 @ I ) @ ( nth_list_a @ A22 @ I ) ) ) )
=> ~ ! [Ys4: list_list_a,I2: nat,Z4: list_a] :
( ( A22
= ( list_update_list_a @ Ys4 @ I2 @ Z4 ) )
=> ( ( trancl_listp_list_a @ R @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys4 ) )
=> ~ ( R @ ( nth_list_a @ Ys4 @ I2 ) @ Z4 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_183_listrel1__iff__update,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R2 ) )
= ( ? [Y5: list_nat,N2: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( Ys
= ( list_update_list_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_184_listrel1__iff__update,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R2 ) )
= ( ? [Y5: list_list_a,N2: nat] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_s2403821588304063868list_a @ Xs ) )
& ( Ys
= ( list_u84128978910470136list_a @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_185_listrel1__iff__update,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
= ( ? [Y5: a,N2: nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
& ( Ys
= ( list_update_a @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_186_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
= ( ? [Y5: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_187_listrel1__iff__update,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) )
= ( ? [Y5: list_a,N2: nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( Ys
= ( list_update_list_a @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_188_permut__aux__sound,axiom,
! [I4: nat,As: list_a,F: nat > nat,Bs: list_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ As ) )
=> ( ( nth_a @ ( missing_permut_aux_a @ As @ F @ Bs ) @ I4 )
= ( nth_a @ Bs @ ( F @ I4 ) ) ) ) ).
% permut_aux_sound
thf(fact_189_permut__aux__sound,axiom,
! [I4: nat,As: list_nat,F: nat > nat,Bs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As @ F @ Bs ) @ I4 )
= ( nth_nat @ Bs @ ( F @ I4 ) ) ) ) ).
% permut_aux_sound
thf(fact_190_permut__aux__sound,axiom,
! [I4: nat,As: list_list_a,F: nat > nat,Bs: list_list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ As ) )
=> ( ( nth_list_a @ ( missin5389789484201179846list_a @ As @ F @ Bs ) @ I4 )
= ( nth_list_a @ Bs @ ( F @ I4 ) ) ) ) ).
% permut_aux_sound
thf(fact_191_lex__take__index,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( lex_list_nat @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( take_list_nat @ I2 @ Xs )
= ( take_list_nat @ I2 @ Ys ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ I2 ) @ ( nth_list_nat @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_192_lex__take__index,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( lex_list_list_a @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2403821588304063868list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s2403821588304063868list_a @ Ys ) )
=> ( ( ( take_list_list_a @ I2 @ Xs )
= ( take_list_list_a @ I2 @ Ys ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( nth_list_list_a @ Xs @ I2 ) @ ( nth_list_list_a @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_193_lex__take__index,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( ( take_a @ I2 @ Xs )
= ( take_a @ I2 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_194_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_195_lex__take__index,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( take_list_a @ I2 @ Xs )
= ( take_list_a @ I2 @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ I2 ) @ ( nth_list_a @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_196_trancl__listp_Obase,axiom,
! [Xs: list_a,Ys: list_a,R: a > a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( R @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) )
=> ( trancl_listp_a @ R @ Xs @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_197_trancl__listp_Obase,axiom,
! [Xs: list_nat,Ys: list_nat,R: nat > nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( R @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( trancl_listp_nat @ R @ Xs @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_198_trancl__listp_Obase,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( R @ ( nth_list_a @ Xs @ I2 ) @ ( nth_list_a @ Ys @ I2 ) ) )
=> ( trancl_listp_list_a @ R @ Xs @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_199_set__swap,axiom,
! [I4: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I4 @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I4 ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_200_set__swap,axiom,
! [I4: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I4 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_201_set__swap,axiom,
! [I4: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( set_list_a2 @ ( list_update_list_a @ ( list_update_list_a @ Xs @ I4 @ ( nth_list_a @ Xs @ J ) ) @ J @ ( nth_list_a @ Xs @ I4 ) ) )
= ( set_list_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_202_remove__nth__P__compat,axiom,
! [As: list_a,Bs: list_a,P2: a > a > $o,P: nat] :
( ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ As ) )
=> ( P2 @ ( nth_a @ As @ I2 ) @ ( nth_a @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ( missing_remove_nth_a @ P @ As ) ) )
=> ( P2 @ ( nth_a @ ( missing_remove_nth_a @ P @ As ) @ I ) @ ( nth_a @ ( missing_remove_nth_a @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_203_remove__nth__P__compat,axiom,
! [As: list_a,Bs: list_nat,P2: a > nat > $o,P: nat] :
( ( ( size_size_list_a @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ As ) )
=> ( P2 @ ( nth_a @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ( missing_remove_nth_a @ P @ As ) ) )
=> ( P2 @ ( nth_a @ ( missing_remove_nth_a @ P @ As ) @ I ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_204_remove__nth__P__compat,axiom,
! [As: list_a,Bs: list_list_a,P2: a > list_a > $o,P: nat] :
( ( ( size_size_list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ As ) )
=> ( P2 @ ( nth_a @ As @ I2 ) @ ( nth_list_a @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ( missing_remove_nth_a @ P @ As ) ) )
=> ( P2 @ ( nth_a @ ( missing_remove_nth_a @ P @ As ) @ I ) @ ( nth_list_a @ ( missin8041159176150063613list_a @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_205_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_a,P2: nat > a > $o,P: nat] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As ) )
=> ( P2 @ ( nth_nat @ As @ I2 ) @ ( nth_a @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P @ As ) ) )
=> ( P2 @ ( nth_nat @ ( missin7175274867594579095th_nat @ P @ As ) @ I ) @ ( nth_a @ ( missing_remove_nth_a @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_206_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_nat,P2: nat > nat > $o,P: nat] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As ) )
=> ( P2 @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P @ As ) ) )
=> ( P2 @ ( nth_nat @ ( missin7175274867594579095th_nat @ P @ As ) @ I ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_207_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_list_a,P2: nat > list_a > $o,P: nat] :
( ( ( size_size_list_nat @ As )
= ( size_s349497388124573686list_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As ) )
=> ( P2 @ ( nth_nat @ As @ I2 ) @ ( nth_list_a @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P @ As ) ) )
=> ( P2 @ ( nth_nat @ ( missin7175274867594579095th_nat @ P @ As ) @ I ) @ ( nth_list_a @ ( missin8041159176150063613list_a @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_208_remove__nth__P__compat,axiom,
! [As: list_list_a,Bs: list_a,P2: list_a > a > $o,P: nat] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ As ) )
=> ( P2 @ ( nth_list_a @ As @ I2 ) @ ( nth_a @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ P @ As ) ) )
=> ( P2 @ ( nth_list_a @ ( missin8041159176150063613list_a @ P @ As ) @ I ) @ ( nth_a @ ( missing_remove_nth_a @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_209_remove__nth__P__compat,axiom,
! [As: list_list_a,Bs: list_nat,P2: list_a > nat > $o,P: nat] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ As ) )
=> ( P2 @ ( nth_list_a @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ P @ As ) ) )
=> ( P2 @ ( nth_list_a @ ( missin8041159176150063613list_a @ P @ As ) @ I ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_210_remove__nth__P__compat,axiom,
! [As: list_list_a,Bs: list_list_a,P2: list_a > list_a > $o,P: nat] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ As ) )
=> ( P2 @ ( nth_list_a @ As @ I2 ) @ ( nth_list_a @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ P @ As ) ) )
=> ( P2 @ ( nth_list_a @ ( missin8041159176150063613list_a @ P @ As ) @ I ) @ ( nth_list_a @ ( missin8041159176150063613list_a @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_211_nth__take,axiom,
! [I4: nat,N: nat,Xs: list_a] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_a @ ( take_a @ N @ Xs ) @ I4 )
= ( nth_a @ Xs @ I4 ) ) ) ).
% nth_take
thf(fact_212_nth__take,axiom,
! [I4: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ).
% nth_take
thf(fact_213_nth__take,axiom,
! [I4: nat,N: nat,Xs: list_list_a] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_list_a @ ( take_list_a @ N @ Xs ) @ I4 )
= ( nth_list_a @ Xs @ I4 ) ) ) ).
% nth_take
thf(fact_214_Cons__listrel1__Cons,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( listrel1_list_nat @ R2 ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_215_Cons__listrel1__Cons,axiom,
! [X: list_list_a,Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ Y @ Ys ) ) @ ( listrel1_list_list_a @ R2 ) )
= ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_216_Cons__listrel1__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_217_Cons__listrel1__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_218_Cons__listrel1__Cons,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_219_in__set__takeD,axiom,
! [X: produc9164743771328383783list_a,N: nat,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ ( take_P7365604256116377365list_a @ N @ Xs ) ) )
=> ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) ) ) ).
% in_set_takeD
thf(fact_220_in__set__takeD,axiom,
! [X: product_prod_a_a,N: nat,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( take_P1986783995523548949od_a_a @ N @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_221_in__set__takeD,axiom,
! [X: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% in_set_takeD
thf(fact_222_in__set__takeD,axiom,
! [X: produc1828647624359046049st_nat,N: nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ ( take_P9169595709903062671st_nat @ N @ Xs ) ) )
=> ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) ) ) ).
% in_set_takeD
thf(fact_223_in__set__takeD,axiom,
! [X: produc7709606177366032167list_a,N: nat,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ ( take_P8397501012876893461list_a @ N @ Xs ) ) )
=> ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) ) ) ).
% in_set_takeD
thf(fact_224_set__ConsD,axiom,
! [Y: produc9164743771328383783list_a,X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ Y @ ( set_Pr2906193453920572092list_a @ ( cons_P5184657343811988189list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member8191768239178080336list_a @ Y @ ( set_Pr2906193453920572092list_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_225_set__ConsD,axiom,
! [Y: product_prod_a_a,X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_226_set__ConsD,axiom,
! [Y: product_prod_nat_nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_227_set__ConsD,axiom,
! [Y: produc1828647624359046049st_nat,X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ Y @ ( set_Pr3842133991353686454st_nat @ ( cons_P5007559046487125591st_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member7340969449405702474st_nat @ Y @ ( set_Pr3842133991353686454st_nat @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_228_set__ConsD,axiom,
! [Y: produc7709606177366032167list_a,X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ Y @ ( set_Pr1480943573297899964list_a @ ( cons_P4475838960125360349list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member1318342207407915856list_a @ Y @ ( set_Pr1480943573297899964list_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_229_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_230_list_Oset__cases,axiom,
! [E: produc9164743771328383783list_a,A: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ E @ ( set_Pr2906193453920572092list_a @ A ) )
=> ( ! [Z22: list_P321204300973800749list_a] :
( A
!= ( cons_P5184657343811988189list_a @ E @ Z22 ) )
=> ~ ! [Z1: produc9164743771328383783list_a,Z22: list_P321204300973800749list_a] :
( ( A
= ( cons_P5184657343811988189list_a @ Z1 @ Z22 ) )
=> ~ ( member8191768239178080336list_a @ E @ ( set_Pr2906193453920572092list_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_231_list_Oset__cases,axiom,
! [E: product_prod_a_a,A: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A ) )
=> ( ! [Z22: list_P1396940483166286381od_a_a] :
( A
!= ( cons_P7316939126706565853od_a_a @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_a_a,Z22: list_P1396940483166286381od_a_a] :
( ( A
= ( cons_P7316939126706565853od_a_a @ Z1 @ Z22 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_232_list_Oset__cases,axiom,
! [E: product_prod_nat_nat,A: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ A ) )
=> ( ! [Z22: list_P6011104703257516679at_nat] :
( A
!= ( cons_P6512896166579812791at_nat @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_nat_nat,Z22: list_P6011104703257516679at_nat] :
( ( A
= ( cons_P6512896166579812791at_nat @ Z1 @ Z22 ) )
=> ~ ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_233_list_Oset__cases,axiom,
! [E: produc1828647624359046049st_nat,A: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ E @ ( set_Pr3842133991353686454st_nat @ A ) )
=> ( ! [Z22: list_P7940050157051400743st_nat] :
( A
!= ( cons_P5007559046487125591st_nat @ E @ Z22 ) )
=> ~ ! [Z1: produc1828647624359046049st_nat,Z22: list_P7940050157051400743st_nat] :
( ( A
= ( cons_P5007559046487125591st_nat @ Z1 @ Z22 ) )
=> ~ ( member7340969449405702474st_nat @ E @ ( set_Pr3842133991353686454st_nat @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_234_list_Oset__cases,axiom,
! [E: produc7709606177366032167list_a,A: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ E @ ( set_Pr1480943573297899964list_a @ A ) )
=> ( ! [Z22: list_P3352141576573427245list_a] :
( A
!= ( cons_P4475838960125360349list_a @ E @ Z22 ) )
=> ~ ! [Z1: produc7709606177366032167list_a,Z22: list_P3352141576573427245list_a] :
( ( A
= ( cons_P4475838960125360349list_a @ Z1 @ Z22 ) )
=> ~ ( member1318342207407915856list_a @ E @ ( set_Pr1480943573297899964list_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_235_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_236_list_Oset__intros_I1_J,axiom,
! [X21: produc9164743771328383783list_a,X22: list_P321204300973800749list_a] : ( member8191768239178080336list_a @ X21 @ ( set_Pr2906193453920572092list_a @ ( cons_P5184657343811988189list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_237_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_238_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_239_list_Oset__intros_I1_J,axiom,
! [X21: produc1828647624359046049st_nat,X22: list_P7940050157051400743st_nat] : ( member7340969449405702474st_nat @ X21 @ ( set_Pr3842133991353686454st_nat @ ( cons_P5007559046487125591st_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_240_list_Oset__intros_I1_J,axiom,
! [X21: produc7709606177366032167list_a,X22: list_P3352141576573427245list_a] : ( member1318342207407915856list_a @ X21 @ ( set_Pr1480943573297899964list_a @ ( cons_P4475838960125360349list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_241_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_242_list_Oset__intros_I2_J,axiom,
! [Y: produc9164743771328383783list_a,X22: list_P321204300973800749list_a,X21: produc9164743771328383783list_a] :
( ( member8191768239178080336list_a @ Y @ ( set_Pr2906193453920572092list_a @ X22 ) )
=> ( member8191768239178080336list_a @ Y @ ( set_Pr2906193453920572092list_a @ ( cons_P5184657343811988189list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_243_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_a_a,X22: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ X22 ) )
=> ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_244_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ X22 ) )
=> ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_245_list_Oset__intros_I2_J,axiom,
! [Y: produc1828647624359046049st_nat,X22: list_P7940050157051400743st_nat,X21: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ Y @ ( set_Pr3842133991353686454st_nat @ X22 ) )
=> ( member7340969449405702474st_nat @ Y @ ( set_Pr3842133991353686454st_nat @ ( cons_P5007559046487125591st_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_246_list_Oset__intros_I2_J,axiom,
! [Y: produc7709606177366032167list_a,X22: list_P3352141576573427245list_a,X21: produc7709606177366032167list_a] :
( ( member1318342207407915856list_a @ Y @ ( set_Pr1480943573297899964list_a @ X22 ) )
=> ( member1318342207407915856list_a @ Y @ ( set_Pr1480943573297899964list_a @ ( cons_P4475838960125360349list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_247_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_248_set__take__nth,axiom,
! [X: produc9164743771328383783list_a,I4: nat,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ ( take_P7365604256116377365list_a @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s4944079540699745177list_a @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_Pr5917933638979213230list_a @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_249_set__take__nth,axiom,
! [X: product_prod_a_a,I4: nat,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( take_P1986783995523548949od_a_a @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_Product_prod_a_a @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_250_set__take__nth,axiom,
! [X: product_prod_nat_nat,I4: nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( take_P2173866234530122223at_nat @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_251_set__take__nth,axiom,
! [X: produc1828647624359046049st_nat,I4: nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ ( take_P9169595709903062671st_nat @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s7664791237847770771st_nat @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_Pr2569852650362941352st_nat @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_252_set__take__nth,axiom,
! [X: produc7709606177366032167list_a,I4: nat,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ ( take_P8397501012876893461list_a @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s411350714811752601list_a @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_Pr590429048667532462list_a @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_253_set__take__nth,axiom,
! [X: a,I4: nat,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( take_a @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_a @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_254_set__take__nth,axiom,
! [X: nat,I4: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_nat @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_255_set__take__nth,axiom,
! [X: list_a,I4: nat,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( take_list_a @ I4 @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ord_less_nat @ J2 @ I4 )
& ( ( nth_list_a @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_256_listrel1I2,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,X: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ X @ Ys ) ) @ ( listrel1_a @ R2 ) ) ) ).
% listrel1I2
thf(fact_257_listrel1I2,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,X: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ X @ Ys ) ) @ ( listrel1_nat @ R2 ) ) ) ).
% listrel1I2
thf(fact_258_listrel1I2,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a,X: list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ X @ Ys ) ) @ ( listrel1_list_a @ R2 ) ) ) ).
% listrel1I2
thf(fact_259_listrel1__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_260_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_261_listrel1__eq__len,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_262_nth__mem,axiom,
! [N: nat,Xs: list_P321204300973800749list_a] :
( ( ord_less_nat @ N @ ( size_s4944079540699745177list_a @ Xs ) )
=> ( member8191768239178080336list_a @ ( nth_Pr5917933638979213230list_a @ Xs @ N ) @ ( set_Pr2906193453920572092list_a @ Xs ) ) ) ).
% nth_mem
thf(fact_263_nth__mem,axiom,
! [N: nat,Xs: list_P1396940483166286381od_a_a] :
( ( ord_less_nat @ N @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Xs @ N ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_264_nth__mem,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_265_nth__mem,axiom,
! [N: nat,Xs: list_P7940050157051400743st_nat] :
( ( ord_less_nat @ N @ ( size_s7664791237847770771st_nat @ Xs ) )
=> ( member7340969449405702474st_nat @ ( nth_Pr2569852650362941352st_nat @ Xs @ N ) @ ( set_Pr3842133991353686454st_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_266_nth__mem,axiom,
! [N: nat,Xs: list_P3352141576573427245list_a] :
( ( ord_less_nat @ N @ ( size_s411350714811752601list_a @ Xs ) )
=> ( member1318342207407915856list_a @ ( nth_Pr590429048667532462list_a @ Xs @ N ) @ ( set_Pr1480943573297899964list_a @ Xs ) ) ) ).
% nth_mem
thf(fact_267_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_268_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_269_nth__mem,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( set_list_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_270_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P2: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_271_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P2: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_272_list__ball__nth,axiom,
! [N: nat,Xs: list_list_a,P2: list_a > $o] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_list_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_273_in__set__conv__nth,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s4944079540699745177list_a @ Xs ) )
& ( ( nth_Pr5917933638979213230list_a @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_274_in__set__conv__nth,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_275_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_276_in__set__conv__nth,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7664791237847770771st_nat @ Xs ) )
& ( ( nth_Pr2569852650362941352st_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_277_in__set__conv__nth,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s411350714811752601list_a @ Xs ) )
& ( ( nth_Pr590429048667532462list_a @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_278_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_279_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_280_in__set__conv__nth,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_281_all__nth__imp__all__set,axiom,
! [Xs: list_P321204300973800749list_a,P2: produc9164743771328383783list_a > $o,X: produc9164743771328383783list_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s4944079540699745177list_a @ Xs ) )
=> ( P2 @ ( nth_Pr5917933638979213230list_a @ Xs @ I2 ) ) )
=> ( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_282_all__nth__imp__all__set,axiom,
! [Xs: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o,X: product_prod_a_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( P2 @ ( nth_Product_prod_a_a @ Xs @ I2 ) ) )
=> ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_283_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_284_all__nth__imp__all__set,axiom,
! [Xs: list_P7940050157051400743st_nat,P2: produc1828647624359046049st_nat > $o,X: produc1828647624359046049st_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7664791237847770771st_nat @ Xs ) )
=> ( P2 @ ( nth_Pr2569852650362941352st_nat @ Xs @ I2 ) ) )
=> ( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_285_all__nth__imp__all__set,axiom,
! [Xs: list_P3352141576573427245list_a,P2: produc7709606177366032167list_a > $o,X: produc7709606177366032167list_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s411350714811752601list_a @ Xs ) )
=> ( P2 @ ( nth_Pr590429048667532462list_a @ Xs @ I2 ) ) )
=> ( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_286_all__nth__imp__all__set,axiom,
! [Xs: list_a,P2: a > $o,X: a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I2 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_287_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P2: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_288_all__nth__imp__all__set,axiom,
! [Xs: list_list_a,P2: list_a > $o,X: list_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P2 @ ( nth_list_a @ Xs @ I2 ) ) )
=> ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_289_all__set__conv__all__nth,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
=> ( P2 @ X6 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_290_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ! [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X6 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_291_all__set__conv__all__nth,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ( ! [X6: list_a] :
( ( member_list_a @ X6 @ ( set_list_a2 @ Xs ) )
=> ( P2 @ X6 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P2 @ ( nth_list_a @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_292_ex__set__conv__ex__nth,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( P2 @ ( nth_a @ Xs @ I3 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_293_ex__set__conv__ex__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P2 @ X6 ) ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( P2 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_294_ex__set__conv__ex__nth,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ( ? [X6: list_a] :
( ( member_list_a @ X6 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X6 ) ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
& ( P2 @ ( nth_list_a @ Xs @ I3 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_295_set__update__memI,axiom,
! [N: nat,Xs: list_P321204300973800749list_a,X: produc9164743771328383783list_a] :
( ( ord_less_nat @ N @ ( size_s4944079540699745177list_a @ Xs ) )
=> ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ ( list_u6458906768619699605list_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_296_set__update__memI,axiom,
! [N: nat,Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
( ( ord_less_nat @ N @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( list_u4967751623001292181od_a_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_297_set__update__memI,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_298_set__update__memI,axiom,
! [N: nat,Xs: list_P7940050157051400743st_nat,X: produc1828647624359046049st_nat] :
( ( ord_less_nat @ N @ ( size_s7664791237847770771st_nat @ Xs ) )
=> ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ ( list_u5682793034032774415st_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_299_set__update__memI,axiom,
! [N: nat,Xs: list_P3352141576573427245list_a,X: produc7709606177366032167list_a] :
( ( ord_less_nat @ N @ ( size_s411350714811752601list_a @ Xs ) )
=> ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ ( list_u7340727631749157781list_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_300_set__update__memI,axiom,
! [N: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_301_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_302_set__update__memI,axiom,
! [N: nat,Xs: list_list_a,X: list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ X @ ( set_list_a2 @ ( list_update_list_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_303_remove__nth__sound__l,axiom,
! [P: nat,N: nat,Xs: list_a] :
( ( ord_less_nat @ P @ N )
=> ( ( nth_a @ ( missing_remove_nth_a @ N @ Xs ) @ P )
= ( nth_a @ Xs @ P ) ) ) ).
% remove_nth_sound_l
thf(fact_304_remove__nth__sound__l,axiom,
! [P: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ P @ N )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ P )
= ( nth_nat @ Xs @ P ) ) ) ).
% remove_nth_sound_l
thf(fact_305_remove__nth__sound__l,axiom,
! [P: nat,N: nat,Xs: list_list_a] :
( ( ord_less_nat @ P @ N )
=> ( ( nth_list_a @ ( missin8041159176150063613list_a @ N @ Xs ) @ P )
= ( nth_list_a @ Xs @ P ) ) ) ).
% remove_nth_sound_l
thf(fact_306_Cons__listrel1E2,axiom,
! [Xs: list_list_nat,Y: list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ ( cons_list_nat @ Y @ Ys ) ) @ ( listrel1_list_nat @ R2 ) )
=> ( ! [X3: list_nat] :
( ( Xs
= ( cons_list_nat @ X3 @ Ys ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y @ Zs ) )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Zs @ Ys ) @ ( listrel1_list_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_307_Cons__listrel1E2,axiom,
! [Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ ( cons_list_list_a @ Y @ Ys ) ) @ ( listrel1_list_list_a @ R2 ) )
=> ( ! [X3: list_list_a] :
( ( Xs
= ( cons_list_list_a @ X3 @ Ys ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_list_list_a] :
( ( Xs
= ( cons_list_list_a @ Y @ Zs ) )
=> ~ ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Zs @ Ys ) @ ( listrel1_list_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_308_Cons__listrel1E2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R2 ) )
=> ( ! [X3: a] :
( ( Xs
= ( cons_a @ X3 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_a] :
( ( Xs
= ( cons_a @ Y @ Zs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_309_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
=> ( ! [X3: nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_310_Cons__listrel1E2,axiom,
! [Xs: list_list_a,Y: list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R2 ) )
=> ( ! [X3: list_a] :
( ( Xs
= ( cons_list_a @ X3 @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs: list_list_a] :
( ( Xs
= ( cons_list_a @ Y @ Zs ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Zs @ Ys ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_311_Cons__listrel1E1,axiom,
! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ Ys ) @ ( listrel1_list_nat @ R2 ) )
=> ( ! [Y3: list_nat] :
( ( Ys
= ( cons_list_nat @ Y3 @ Xs ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs: list_list_nat] :
( ( Ys
= ( cons_list_nat @ X @ Zs ) )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Zs ) @ ( listrel1_list_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_312_Cons__listrel1E1,axiom,
! [X: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ Ys ) @ ( listrel1_list_list_a @ R2 ) )
=> ( ! [Y3: list_list_a] :
( ( Ys
= ( cons_list_list_a @ Y3 @ Xs ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs: list_list_list_a] :
( ( Ys
= ( cons_list_list_a @ X @ Zs ) )
=> ~ ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Zs ) @ ( listrel1_list_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_313_Cons__listrel1E1,axiom,
! [X: a,Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ! [Y3: a] :
( ( Ys
= ( cons_a @ Y3 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs: list_a] :
( ( Ys
= ( cons_a @ X @ Zs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_314_Cons__listrel1E1,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ( ! [Y3: nat] :
( ( Ys
= ( cons_nat @ Y3 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs: list_nat] :
( ( Ys
= ( cons_nat @ X @ Zs ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_315_Cons__listrel1E1,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ Ys ) @ ( listrel1_list_a @ R2 ) )
=> ( ! [Y3: list_a] :
( ( Ys
= ( cons_list_a @ Y3 @ Xs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Zs ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_316_listrel1I1,axiom,
! [X: list_a,Y: list_a,R2: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R2 )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Xs ) ) @ ( listrel1_list_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_317_listrel1I1,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Xs ) ) @ ( listrel1_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_318_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_319_listrel1I1,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr3451248702717554689st_nat,Xs: list_list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Xs ) ) @ ( listrel1_list_nat @ R2 ) ) ) ).
% listrel1I1
thf(fact_320_listrel1I1,axiom,
! [X: list_list_a,Y: list_list_a,R2: set_Pr5382606609415531783list_a,Xs: list_list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R2 )
=> ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ Y @ Xs ) ) @ ( listrel1_list_list_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_321_in__set__idx,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s4944079540699745177list_a @ Xs ) )
& ( ( nth_Pr5917933638979213230list_a @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_322_in__set__idx,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_323_in__set__idx,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_324_in__set__idx,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7664791237847770771st_nat @ Xs ) )
& ( ( nth_Pr2569852650362941352st_nat @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_325_in__set__idx,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s411350714811752601list_a @ Xs ) )
& ( ( nth_Pr590429048667532462list_a @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_326_in__set__idx,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_327_in__set__idx,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_328_in__set__idx,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_329_adjust__idx__rev__nth,axiom,
! [I4: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( J != I4 )
=> ( ( nth_a @ ( missing_remove_nth_a @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
= ( nth_a @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_330_adjust__idx__rev__nth,axiom,
! [I4: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( J != I4 )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
= ( nth_nat @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_331_adjust__idx__rev__nth,axiom,
! [I4: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( J != I4 )
=> ( ( nth_list_a @ ( missin8041159176150063613list_a @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
= ( nth_list_a @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_332_adjust__idx__nth,axiom,
! [I4: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( missing_remove_nth_a @ I4 @ Xs ) @ J )
= ( nth_a @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_333_adjust__idx__nth,axiom,
! [I4: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) @ J )
= ( nth_nat @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_334_adjust__idx__nth,axiom,
! [I4: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( missin8041159176150063613list_a @ I4 @ Xs ) @ J )
= ( nth_list_a @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_335_adjust__idx__rev__length,axiom,
! [I4: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( J != I4 )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_size_list_a @ ( missing_remove_nth_a @ I4 @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_336_adjust__idx__rev__length,axiom,
! [I4: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( J != I4 )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_337_adjust__idx__rev__length,axiom,
! [I4: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( J != I4 )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ I4 @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_338_adjust__idx__length,axiom,
! [I4: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ ( missing_remove_nth_a @ I4 @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_size_list_a @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_339_adjust__idx__length,axiom,
! [I4: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_340_adjust__idx__length,axiom,
! [I4: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ I4 @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_s349497388124573686list_a @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_341_list__ex__length,axiom,
( list_ex_a
= ( ^ [P3: a > $o,Xs3: list_a] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs3 ) )
& ( P3 @ ( nth_a @ Xs3 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_342_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P3: nat > $o,Xs3: list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs3 ) )
& ( P3 @ ( nth_nat @ Xs3 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_343_list__ex__length,axiom,
( list_ex_list_a
= ( ^ [P3: list_a > $o,Xs3: list_list_a] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs3 ) )
& ( P3 @ ( nth_list_a @ Xs3 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_344_distinct__swap,axiom,
! [I4: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( distinct_a @ ( list_update_a @ ( list_update_a @ Xs @ I4 @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I4 ) ) )
= ( distinct_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_345_distinct__swap,axiom,
! [I4: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I4 ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_346_distinct__swap,axiom,
! [I4: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( distinct_list_a @ ( list_update_list_a @ ( list_update_list_a @ Xs @ I4 @ ( nth_list_a @ Xs @ J ) ) @ J @ ( nth_list_a @ Xs @ I4 ) ) )
= ( distinct_list_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_347_not__in__set__insert,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ( ( insert1218221085542900803list_a @ X @ Xs )
= ( cons_P5184657343811988189list_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_348_not__in__set__insert,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( insert7736115120964043331od_a_a @ X @ Xs )
= ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_349_not__in__set__insert,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( insert8944034826898310173at_nat @ X @ Xs )
= ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_350_not__in__set__insert,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ( ( insert3602830346519061181st_nat @ X @ Xs )
= ( cons_P5007559046487125591st_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_351_not__in__set__insert,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ( ( insert6700045326551096387list_a @ X @ Xs )
= ( cons_P4475838960125360349list_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_352_not__in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= ( cons_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_353_measures__lesseq,axiom,
! [F: list_a > nat,X: list_a,Y: list_a,Fs: list_list_a_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ Fs ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_354_measures__lesseq,axiom,
! [F: a > nat,X: a,Y: a,Fs: list_a_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ Fs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_355_measures__lesseq,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_356_measures__lesseq,axiom,
! [F: list_nat > nat,X: list_nat,Y: list_nat,Fs: list_list_nat_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ Fs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ ( cons_list_nat_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_357_measures__lesseq,axiom,
! [F: list_list_a > nat,X: list_list_a,Y: list_list_a,Fs: list_list_list_a_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ ( measures_list_list_a @ Fs ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ ( measures_list_list_a @ ( cons_list_list_a_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_358_list__ex__simps_I1_J,axiom,
! [P2: a > $o,X: a,Xs: list_a] :
( ( list_ex_a @ P2 @ ( cons_a @ X @ Xs ) )
= ( ( P2 @ X )
| ( list_ex_a @ P2 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_359_in__set__insert,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ( ( insert1218221085542900803list_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_360_in__set__insert,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( insert7736115120964043331od_a_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_361_in__set__insert,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( insert8944034826898310173at_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_362_in__set__insert,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ( ( insert3602830346519061181st_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_363_in__set__insert,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ( ( insert6700045326551096387list_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_364_take__all__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_365_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_366_take__all__iff,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( take_list_a @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_367_take__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( take_a @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_368_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_369_take__all,axiom,
! [Xs: list_list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N )
=> ( ( take_list_a @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_370_list__update__beyond,axiom,
! [Xs: list_a,I4: nat,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I4 )
=> ( ( list_update_a @ Xs @ I4 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_371_list__update__beyond,axiom,
! [Xs: list_nat,I4: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I4 )
=> ( ( list_update_nat @ Xs @ I4 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_372_list__update__beyond,axiom,
! [Xs: list_list_a,I4: nat,X: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ I4 )
=> ( ( list_update_list_a @ Xs @ I4 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_373_adjust__idx__rev1,axiom,
! [I4: nat,J: nat] :
( ( missin3815256168798769645dx_rev @ I4 @ ( missing_adjust_idx @ I4 @ J ) )
= J ) ).
% adjust_idx_rev1
thf(fact_374_adjust__idx__rev2,axiom,
! [J: nat,I4: nat] :
( ( J != I4 )
=> ( ( missing_adjust_idx @ I4 @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
= J ) ) ).
% adjust_idx_rev2
thf(fact_375_adjust__idx__i,axiom,
! [I4: nat,J: nat] :
( ( missing_adjust_idx @ I4 @ J )
!= I4 ) ).
% adjust_idx_i
thf(fact_376_distinct__length__2__or__more,axiom,
! [A: a,B: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ A @ ( cons_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_a @ ( cons_a @ A @ Xs ) )
& ( distinct_a @ ( cons_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_377_list__ex__cong,axiom,
! [Xs: list_P321204300973800749list_a,Ys: list_P321204300973800749list_a,F: produc9164743771328383783list_a > $o,G: produc9164743771328383783list_a > $o] :
( ( Xs = Ys )
=> ( ! [X3: produc9164743771328383783list_a] :
( ( member8191768239178080336list_a @ X3 @ ( set_Pr2906193453920572092list_a @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_e1784073413959309419list_a @ F @ Xs )
= ( list_e1784073413959309419list_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_378_list__ex__cong,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,F: product_prod_a_a > $o,G: product_prod_a_a > $o] :
( ( Xs = Ys )
=> ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_e6552556518106193515od_a_a @ F @ Xs )
= ( list_e6552556518106193515od_a_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_379_list__ex__cong,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,F: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_e7689525607045846085at_nat @ F @ Xs )
= ( list_e7689525607045846085at_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_380_list__ex__cong,axiom,
! [Xs: list_P7940050157051400743st_nat,Ys: list_P7940050157051400743st_nat,F: produc1828647624359046049st_nat > $o,G: produc1828647624359046049st_nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X3 @ ( set_Pr3842133991353686454st_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_e3351411598239959781st_nat @ F @ Xs )
= ( list_e3351411598239959781st_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_381_list__ex__cong,axiom,
! [Xs: list_P3352141576573427245list_a,Ys: list_P3352141576573427245list_a,F: produc7709606177366032167list_a > $o,G: produc7709606177366032167list_a > $o] :
( ( Xs = Ys )
=> ( ! [X3: produc7709606177366032167list_a] :
( ( member1318342207407915856list_a @ X3 @ ( set_Pr1480943573297899964list_a @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_e4120312986338338923list_a @ F @ Xs )
= ( list_e4120312986338338923list_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_382_distinct_Osimps_I2_J,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( distin9160064362609808800list_a @ ( cons_P5184657343811988189list_a @ X @ Xs ) )
= ( ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
& ( distin9160064362609808800list_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_383_distinct_Osimps_I2_J,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) )
= ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
& ( distin132333870042060960od_a_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_384_distinct_Osimps_I2_J,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
& ( distin6923225563576452346at_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_385_distinct_Osimps_I2_J,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( distin347524420504911514st_nat @ ( cons_P5007559046487125591st_nat @ X @ Xs ) )
= ( ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
& ( distin347524420504911514st_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_386_distinct_Osimps_I2_J,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( distin5851120023645153440list_a @ ( cons_P4475838960125360349list_a @ X @ Xs ) )
= ( ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
& ( distin5851120023645153440list_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_387_distinct_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X @ Xs ) )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_388_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_389_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_390_impossible__Cons,axiom,
! [Xs: list_list_a,Ys: list_list_a,X: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs
!= ( cons_list_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_391_remove__nth__id,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( missing_remove_nth_a @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_392_remove__nth__id,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( missin7175274867594579095th_nat @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_393_remove__nth__id,axiom,
! [Xs: list_list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N )
=> ( ( missin8041159176150063613list_a @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_394_nth__eq__iff__index__eq,axiom,
! [Xs: list_a,I4: nat,J: nat] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ Xs @ I4 )
= ( nth_a @ Xs @ J ) )
= ( I4 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_395_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I4: nat,J: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Xs @ J ) )
= ( I4 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_396_nth__eq__iff__index__eq,axiom,
! [Xs: list_list_a,I4: nat,J: nat] :
( ( distinct_list_a @ Xs )
=> ( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( nth_list_a @ Xs @ I4 )
= ( nth_list_a @ Xs @ J ) )
= ( I4 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_397_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs3: list_a] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs3 ) )
=> ( ( I3 != J3 )
=> ( ( nth_a @ Xs3 @ I3 )
!= ( nth_a @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_398_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I3 != J3 )
=> ( ( nth_nat @ Xs3 @ I3 )
!= ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_399_distinct__conv__nth,axiom,
( distinct_list_a
= ( ^ [Xs3: list_list_a] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ( ( I3 != J3 )
=> ( ( nth_list_a @ Xs3 @ I3 )
!= ( nth_list_a @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_400_List_Oinsert__def,axiom,
( insert1218221085542900803list_a
= ( ^ [X6: produc9164743771328383783list_a,Xs3: list_P321204300973800749list_a] : ( if_lis2255084897382415091list_a @ ( member8191768239178080336list_a @ X6 @ ( set_Pr2906193453920572092list_a @ Xs3 ) ) @ Xs3 @ ( cons_P5184657343811988189list_a @ X6 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_401_List_Oinsert__def,axiom,
( insert7736115120964043331od_a_a
= ( ^ [X6: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] : ( if_lis931442767461590515od_a_a @ ( member1426531477525435216od_a_a @ X6 @ ( set_Product_prod_a_a2 @ Xs3 ) ) @ Xs3 @ ( cons_P7316939126706565853od_a_a @ X6 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_402_List_Oinsert__def,axiom,
( insert8944034826898310173at_nat
= ( ^ [X6: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] : ( if_lis9186351972506106189at_nat @ ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P6512896166579812791at_nat @ X6 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_403_List_Oinsert__def,axiom,
( insert3602830346519061181st_nat
= ( ^ [X6: produc1828647624359046049st_nat,Xs3: list_P7940050157051400743st_nat] : ( if_lis5496265267951696621st_nat @ ( member7340969449405702474st_nat @ X6 @ ( set_Pr3842133991353686454st_nat @ Xs3 ) ) @ Xs3 @ ( cons_P5007559046487125591st_nat @ X6 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_404_List_Oinsert__def,axiom,
( insert6700045326551096387list_a
= ( ^ [X6: produc7709606177366032167list_a,Xs3: list_P3352141576573427245list_a] : ( if_lis8043406425296152051list_a @ ( member1318342207407915856list_a @ X6 @ ( set_Pr1480943573297899964list_a @ Xs3 ) ) @ Xs3 @ ( cons_P4475838960125360349list_a @ X6 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_405_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X6: a,Xs3: list_a] : ( if_list_a @ ( member_a @ X6 @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X6 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_406_lenlex__length,axiom,
! [Ms: list_a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_407_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_408_lenlex__length,axiom,
! [Ms: list_list_a,Ns: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ms @ Ns ) @ ( lenlex_list_a @ R2 ) )
=> ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Ms ) @ ( size_s349497388124573686list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_409_distinct__Ex1,axiom,
! [Xs: list_P321204300973800749list_a,X: produc9164743771328383783list_a] :
( ( distin9160064362609808800list_a @ Xs )
=> ( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s4944079540699745177list_a @ Xs ) )
& ( ( nth_Pr5917933638979213230list_a @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_s4944079540699745177list_a @ Xs ) )
& ( ( nth_Pr5917933638979213230list_a @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_410_distinct__Ex1,axiom,
! [Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
( ( distin132333870042060960od_a_a @ Xs )
=> ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_411_distinct__Ex1,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( distin6923225563576452346at_nat @ Xs )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_412_distinct__Ex1,axiom,
! [Xs: list_P7940050157051400743st_nat,X: produc1828647624359046049st_nat] :
( ( distin347524420504911514st_nat @ Xs )
=> ( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s7664791237847770771st_nat @ Xs ) )
& ( ( nth_Pr2569852650362941352st_nat @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_s7664791237847770771st_nat @ Xs ) )
& ( ( nth_Pr2569852650362941352st_nat @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_413_distinct__Ex1,axiom,
! [Xs: list_P3352141576573427245list_a,X: produc7709606177366032167list_a] :
( ( distin5851120023645153440list_a @ Xs )
=> ( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s411350714811752601list_a @ Xs ) )
& ( ( nth_Pr590429048667532462list_a @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_s411350714811752601list_a @ Xs ) )
& ( ( nth_Pr590429048667532462list_a @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_414_distinct__Ex1,axiom,
! [Xs: list_a,X: a] :
( ( distinct_a @ Xs )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_415_distinct__Ex1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_416_distinct__Ex1,axiom,
! [Xs: list_list_a,X: list_a] :
( ( distinct_list_a @ Xs )
=> ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_417_nth__take__prefix,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Ys ) @ ( size_size_list_a @ Xs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( ( take_a @ ( size_size_list_a @ Ys ) @ Xs )
= Ys ) ) ) ).
% nth_take_prefix
thf(fact_418_nth__take__prefix,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs )
= Ys ) ) ) ).
% nth_take_prefix
thf(fact_419_nth__take__prefix,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Ys ) @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( nth_list_a @ Xs @ I2 )
= ( nth_list_a @ Ys @ I2 ) ) )
=> ( ( take_list_a @ ( size_s349497388124573686list_a @ Ys ) @ Xs )
= Ys ) ) ) ).
% nth_take_prefix
thf(fact_420_nth__take__lemma,axiom,
! [K: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( ( take_a @ K @ Xs )
= ( take_a @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_421_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_422_nth__take__lemma,axiom,
! [K: nat,Xs: list_list_a,Ys: list_list_a] :
( ( ord_less_eq_nat @ K @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_list_a @ Xs @ I2 )
= ( nth_list_a @ Ys @ I2 ) ) )
=> ( ( take_list_a @ K @ Xs )
= ( take_list_a @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_423_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_424_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_425_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( F @ K2 @ I ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_426_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P2 @ I ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_427_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_428_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_429_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_430_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_431_Cons__le__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ).
% Cons_le_Cons
thf(fact_432_subset__code_I1_J,axiom,
! [Xs: list_P321204300973800749list_a,B4: set_Pr4048851178543822343list_a] :
( ( ord_le7857023143581076903list_a @ ( set_Pr2906193453920572092list_a @ Xs ) @ B4 )
= ( ! [X6: produc9164743771328383783list_a] :
( ( member8191768239178080336list_a @ X6 @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ( member8191768239178080336list_a @ X6 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_433_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B4 )
= ( ! [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X6 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_434_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B4 )
= ( ! [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X6 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_435_subset__code_I1_J,axiom,
! [Xs: list_P7940050157051400743st_nat,B4: set_Pr3451248702717554689st_nat] :
( ( ord_le8406513867147106209st_nat @ ( set_Pr3842133991353686454st_nat @ Xs ) @ B4 )
= ( ! [X6: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X6 @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ( member7340969449405702474st_nat @ X6 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_436_subset__code_I1_J,axiom,
! [Xs: list_P3352141576573427245list_a,B4: set_Pr5382606609415531783list_a] :
( ( ord_le251320724540548263list_a @ ( set_Pr1480943573297899964list_a @ Xs ) @ B4 )
= ( ! [X6: produc7709606177366032167list_a] :
( ( member1318342207407915856list_a @ X6 @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ( member1318342207407915856list_a @ X6 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_437_less__eq__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_eq_list_nat @ Xs @ Ys ) ) ) ) ).
% less_eq_list_code(3)
thf(fact_438_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_439_set__update__subsetI,axiom,
! [Xs: list_P321204300973800749list_a,A4: set_Pr4048851178543822343list_a,X: produc9164743771328383783list_a,I4: nat] :
( ( ord_le7857023143581076903list_a @ ( set_Pr2906193453920572092list_a @ Xs ) @ A4 )
=> ( ( member8191768239178080336list_a @ X @ A4 )
=> ( ord_le7857023143581076903list_a @ ( set_Pr2906193453920572092list_a @ ( list_u6458906768619699605list_a @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_440_set__update__subsetI,axiom,
! [Xs: list_P1396940483166286381od_a_a,A4: set_Product_prod_a_a,X: product_prod_a_a,I4: nat] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ A4 )
=> ( ( member1426531477525435216od_a_a @ X @ A4 )
=> ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ ( list_u4967751623001292181od_a_a @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_441_set__update__subsetI,axiom,
! [Xs: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I4: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A4 )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_442_set__update__subsetI,axiom,
! [Xs: list_P7940050157051400743st_nat,A4: set_Pr3451248702717554689st_nat,X: produc1828647624359046049st_nat,I4: nat] :
( ( ord_le8406513867147106209st_nat @ ( set_Pr3842133991353686454st_nat @ Xs ) @ A4 )
=> ( ( member7340969449405702474st_nat @ X @ A4 )
=> ( ord_le8406513867147106209st_nat @ ( set_Pr3842133991353686454st_nat @ ( list_u5682793034032774415st_nat @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_443_set__update__subsetI,axiom,
! [Xs: list_P3352141576573427245list_a,A4: set_Pr5382606609415531783list_a,X: produc7709606177366032167list_a,I4: nat] :
( ( ord_le251320724540548263list_a @ ( set_Pr1480943573297899964list_a @ Xs ) @ A4 )
=> ( ( member1318342207407915856list_a @ X @ A4 )
=> ( ord_le251320724540548263list_a @ ( set_Pr1480943573297899964list_a @ ( list_u7340727631749157781list_a @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_444_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_445_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_446_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_447_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_448_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_449_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_450_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_451_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_452_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_453_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_454_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_455_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_456_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_457_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_458_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_459_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_460_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_461_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_462_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_463_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_464_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_465_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_466_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_467_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_468_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_469_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_470_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_471_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_472_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_473_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_474_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_475_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_476_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_477_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_478_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_479_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_480_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_481_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_482_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_483_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_484_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_485_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_486_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_487_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_488_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_489_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_490_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_491_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_492_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_493_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_494_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_495_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_496_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_497_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_498_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_499_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_500_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_501_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_502_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_503_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_504_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_505_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_506_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_507_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_508_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_509_size__neq__size__imp__neq,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_510_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_511_le__trans,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_512_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_513_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_514_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_515_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_516_subseteq__set__conv__nth,axiom,
! [Ss: list_P321204300973800749list_a,T2: set_Pr4048851178543822343list_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s4944079540699745177list_a @ Ss ) )
=> ( member8191768239178080336list_a @ ( nth_Pr5917933638979213230list_a @ Ss @ I3 ) @ T2 ) ) )
= ( ord_le7857023143581076903list_a @ ( set_Pr2906193453920572092list_a @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_517_subseteq__set__conv__nth,axiom,
! [Ss: list_P1396940483166286381od_a_a,T2: set_Product_prod_a_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ss ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Ss @ I3 ) @ T2 ) ) )
= ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_518_subseteq__set__conv__nth,axiom,
! [Ss: list_P6011104703257516679at_nat,T2: set_Pr1261947904930325089at_nat] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Ss ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Ss @ I3 ) @ T2 ) ) )
= ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_519_subseteq__set__conv__nth,axiom,
! [Ss: list_P7940050157051400743st_nat,T2: set_Pr3451248702717554689st_nat] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7664791237847770771st_nat @ Ss ) )
=> ( member7340969449405702474st_nat @ ( nth_Pr2569852650362941352st_nat @ Ss @ I3 ) @ T2 ) ) )
= ( ord_le8406513867147106209st_nat @ ( set_Pr3842133991353686454st_nat @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_520_subseteq__set__conv__nth,axiom,
! [Ss: list_P3352141576573427245list_a,T2: set_Pr5382606609415531783list_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s411350714811752601list_a @ Ss ) )
=> ( member1318342207407915856list_a @ ( nth_Pr590429048667532462list_a @ Ss @ I3 ) @ T2 ) ) )
= ( ord_le251320724540548263list_a @ ( set_Pr1480943573297899964list_a @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_521_subseteq__set__conv__nth,axiom,
! [Ss: list_a,T2: set_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ss ) )
=> ( member_a @ ( nth_a @ Ss @ I3 ) @ T2 ) ) )
= ( ord_less_eq_set_a @ ( set_a2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_522_subseteq__set__conv__nth,axiom,
! [Ss: list_nat,T2: set_nat] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ss ) )
=> ( member_nat @ ( nth_nat @ Ss @ I3 ) @ T2 ) ) )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_523_subseteq__set__conv__nth,axiom,
! [Ss: list_list_a,T2: set_list_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ss ) )
=> ( member_list_a @ ( nth_list_a @ Ss @ I3 ) @ T2 ) ) )
= ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_524_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P321204300973800749list_a,A4: set_Pr4048851178543822343list_a] :
( ( ord_le7857023143581076903list_a @ ( set_Pr2906193453920572092list_a @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s4944079540699745177list_a @ Xs ) )
=> ( member8191768239178080336list_a @ ( nth_Pr5917933638979213230list_a @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_525_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P1396940483166286381od_a_a,A4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_526_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_527_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P7940050157051400743st_nat,A4: set_Pr3451248702717554689st_nat] :
( ( ord_le8406513867147106209st_nat @ ( set_Pr3842133991353686454st_nat @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7664791237847770771st_nat @ Xs ) )
=> ( member7340969449405702474st_nat @ ( nth_Pr2569852650362941352st_nat @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_528_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P3352141576573427245list_a,A4: set_Pr5382606609415531783list_a] :
( ( ord_le251320724540548263list_a @ ( set_Pr1480943573297899964list_a @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s411350714811752601list_a @ Xs ) )
=> ( member1318342207407915856list_a @ ( nth_Pr590429048667532462list_a @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_529_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_a,A4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_530_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_531_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_list_a,A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A4 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ I3 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_532_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_533_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_534_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_535_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_536_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_537_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_538_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_539_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_540_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_541_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_542_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_543_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_544_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_545_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_546_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_547_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_548_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_549_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_550_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_551_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_552_order__less__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ( X6 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_553_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_554_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_555_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_556_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_557_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_558_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_559_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_560_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_561_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_562_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_563_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_564_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_565_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_566_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_567_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_568_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_569_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_570_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_571_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_572_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_573_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P2 @ X5 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_574_min__list__nth,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_nth
thf(fact_575_nth__equal__first__eq,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a,N: nat] :
( ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s4944079540699745177list_a @ Xs ) )
=> ( ( ( nth_Pr5917933638979213230list_a @ ( cons_P5184657343811988189list_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_576_nth__equal__first__eq,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,N: nat] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( ( ( nth_Product_prod_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_577_nth__equal__first__eq,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_578_nth__equal__first__eq,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat,N: nat] :
( ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s7664791237847770771st_nat @ Xs ) )
=> ( ( ( nth_Pr2569852650362941352st_nat @ ( cons_P5007559046487125591st_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_579_nth__equal__first__eq,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a,N: nat] :
( ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s411350714811752601list_a @ Xs ) )
=> ( ( ( nth_Pr590429048667532462list_a @ ( cons_P4475838960125360349list_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_580_nth__equal__first__eq,axiom,
! [X: a,Xs: list_a,N: nat] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_581_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_582_nth__equal__first__eq,axiom,
! [X: list_a,Xs: list_list_a,N: nat] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_583_remove__nth__sound__r,axiom,
! [N: nat,P: nat,Xs: list_a] :
( ( ord_less_eq_nat @ N @ P )
=> ( ( ord_less_nat @ P @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( missing_remove_nth_a @ N @ Xs ) @ P )
= ( nth_a @ Xs @ ( suc @ P ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_584_remove__nth__sound__r,axiom,
! [N: nat,P: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ P )
=> ( ( ord_less_nat @ P @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ P )
= ( nth_nat @ Xs @ ( suc @ P ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_585_remove__nth__sound__r,axiom,
! [N: nat,P: nat,Xs: list_list_a] :
( ( ord_less_eq_nat @ N @ P )
=> ( ( ord_less_nat @ P @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( missin8041159176150063613list_a @ N @ Xs ) @ P )
= ( nth_list_a @ Xs @ ( suc @ P ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_586_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_587_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_588_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_589_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_590_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_591_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_592_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_593_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_594_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_595_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_596_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_597_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_598_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_599_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_600_nth__Cons__Suc,axiom,
! [X: list_a,Xs: list_list_a,N: nat] :
( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_list_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_601_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_602_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_603_nth__Cons__0,axiom,
! [X: list_a,Xs: list_list_a] :
( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_604_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_605_take__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( take_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_606_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_607_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_608_ex__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_609_all__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_610_all__less__two,axiom,
! [P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ( P2 @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_611_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_612_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_613_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_614_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_615_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_616_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_617_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_618_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_619_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_620_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_621_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_622_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_623_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_624_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_625_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_626_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K2 )
=> ~ ( P2 @ I ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_627_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_628_strict__inc__induct,axiom,
! [I4: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_629_less__Suc__induct,axiom,
! [I4: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I2 @ K2 ) ) ) ) )
=> ( P2 @ I4 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_630_less__trans__Suc,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_631_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_632_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_633_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_634_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_635_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_636_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_637_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_638_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_639_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_640_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_641_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_642_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_643_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_644_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R4 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z4: nat] :
( ( R4 @ X3 @ Y3 )
=> ( ( R4 @ Y3 @ Z4 )
=> ( R4 @ X3 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
=> ( R4 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_645_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_646_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_647_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_648_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_649_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_650_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_651_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_652_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_653_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_654_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_655_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_656_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_657_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_658_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_659_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_660_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_661_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_662_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_663_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_664_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_665_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_666_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_667_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_668_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_669_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y5: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_670_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_671_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( suc @ N )
= ( size_s349497388124573686list_a @ Xs ) )
= ( ? [Y5: list_a,Ys3: list_list_a] :
( ( Xs
= ( cons_list_a @ Y5 @ Ys3 ) )
& ( ( size_s349497388124573686list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_672_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_673_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_674_length__Suc__conv,axiom,
! [Xs: list_list_a,N: nat] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: list_a,Ys3: list_list_a] :
( ( Xs
= ( cons_list_a @ Y5 @ Ys3 ) )
& ( ( size_s349497388124573686list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_675_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_676_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_677_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_678_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_679_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_680_inc__induct,axiom,
! [I4: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% inc_induct
thf(fact_681_dec__induct,axiom,
! [I4: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P2 @ I4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_682_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_683_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_684_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I4: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I4 ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I4 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_685_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P2 @ I ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_686_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_687_adjust__idx__def,axiom,
( missing_adjust_idx
= ( ^ [I3: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) ) ).
% adjust_idx_def
thf(fact_688_min__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ X ) ) ).
% min_list
thf(fact_689_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X6: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ X6 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_690_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X6: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X6 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_691_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s349497388124573686list_a @ Xs ) )
= ( ? [X6: list_a,Ys3: list_list_a] :
( ( Xs
= ( cons_list_a @ X6 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s349497388124573686list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_692_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_693_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_694_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_695_pinf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P2 @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_696_pinf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P2 @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_697_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_698_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_699_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_700_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_701_minf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P2 @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_702_minf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P2 @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_703_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_704_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_705_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_706_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_707_length__pos__if__in__set,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s4944079540699745177list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_708_length__pos__if__in__set,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3885678630836030617od_a_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_709_length__pos__if__in__set,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_710_length__pos__if__in__set,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s7664791237847770771st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_711_length__pos__if__in__set,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s411350714811752601list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_712_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_713_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_714_length__pos__if__in__set,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_715_remove__nth__len,axiom,
! [I4: nat,Xs: list_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( size_size_list_a @ Xs )
= ( suc @ ( size_size_list_a @ ( missing_remove_nth_a @ I4 @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_716_remove__nth__len,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ Xs )
= ( suc @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I4 @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_717_remove__nth__len,axiom,
! [I4: nat,Xs: list_list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( suc @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ I4 @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_718_min__list__Cons,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ ( cons_nat @ X @ Xs ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys ) ) ) ) ) ) ).
% min_list_Cons
thf(fact_719_nth__remove__nth__conv,axiom,
! [I4: nat,N: nat,Xs: list_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ ( missing_remove_nth_a @ N @ Xs ) ) )
=> ( ( nth_a @ ( missing_remove_nth_a @ N @ Xs ) @ I4 )
= ( nth_a @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_720_nth__remove__nth__conv,axiom,
! [I4: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ I4 )
= ( nth_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_721_nth__remove__nth__conv,axiom,
! [I4: nat,N: nat,Xs: list_list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ N @ Xs ) ) )
=> ( ( nth_list_a @ ( missin8041159176150063613list_a @ N @ Xs ) @ I4 )
= ( nth_list_a @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_722_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_723_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_724_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr4048851178543822343list_a,R4: set_Pr4048851178543822343list_a,Seq: nat > list_a] :
( ( ord_le7857023143581076903list_a @ R5 @ R4 )
=> ( ! [I2: nat] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R5 )
=> ! [I: nat] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_725_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Product_prod_a_a,R4: set_Product_prod_a_a,Seq: nat > a] :
( ( ord_le746702958409616551od_a_a @ R5 @ R4 )
=> ( ! [I2: nat] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R5 )
=> ! [I: nat] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_726_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
( ( ord_le3146513528884898305at_nat @ R5 @ R4 )
=> ( ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R5 )
=> ! [I: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_727_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr3451248702717554689st_nat,R4: set_Pr3451248702717554689st_nat,Seq: nat > list_nat] :
( ( ord_le8406513867147106209st_nat @ R5 @ R4 )
=> ( ! [I2: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R5 )
=> ! [I: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_728_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr5382606609415531783list_a,R4: set_Pr5382606609415531783list_a,Seq: nat > list_list_a] :
( ( ord_le251320724540548263list_a @ R5 @ R4 )
=> ( ! [I2: nat] : ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R5 )
=> ! [I: nat] : ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_729_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_730_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_731_length__Cons,axiom,
! [X: list_a,Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( cons_list_a @ X @ Xs ) )
= ( suc @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_732_subsetI,axiom,
! [A4: set_Pr4048851178543822343list_a,B4: set_Pr4048851178543822343list_a] :
( ! [X3: produc9164743771328383783list_a] :
( ( member8191768239178080336list_a @ X3 @ A4 )
=> ( member8191768239178080336list_a @ X3 @ B4 ) )
=> ( ord_le7857023143581076903list_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_733_subsetI,axiom,
! [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A4 )
=> ( member1426531477525435216od_a_a @ X3 @ B4 ) )
=> ( ord_le746702958409616551od_a_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_734_subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ X3 @ B4 ) )
=> ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_735_subsetI,axiom,
! [A4: set_Pr3451248702717554689st_nat,B4: set_Pr3451248702717554689st_nat] :
( ! [X3: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X3 @ A4 )
=> ( member7340969449405702474st_nat @ X3 @ B4 ) )
=> ( ord_le8406513867147106209st_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_736_subsetI,axiom,
! [A4: set_Pr5382606609415531783list_a,B4: set_Pr5382606609415531783list_a] :
( ! [X3: produc7709606177366032167list_a] :
( ( member1318342207407915856list_a @ X3 @ A4 )
=> ( member1318342207407915856list_a @ X3 @ B4 ) )
=> ( ord_le251320724540548263list_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_737_in__mono,axiom,
! [A4: set_Pr4048851178543822343list_a,B4: set_Pr4048851178543822343list_a,X: produc9164743771328383783list_a] :
( ( ord_le7857023143581076903list_a @ A4 @ B4 )
=> ( ( member8191768239178080336list_a @ X @ A4 )
=> ( member8191768239178080336list_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_738_in__mono,axiom,
! [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B4 )
=> ( ( member1426531477525435216od_a_a @ X @ A4 )
=> ( member1426531477525435216od_a_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_739_in__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_740_in__mono,axiom,
! [A4: set_Pr3451248702717554689st_nat,B4: set_Pr3451248702717554689st_nat,X: produc1828647624359046049st_nat] :
( ( ord_le8406513867147106209st_nat @ A4 @ B4 )
=> ( ( member7340969449405702474st_nat @ X @ A4 )
=> ( member7340969449405702474st_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_741_in__mono,axiom,
! [A4: set_Pr5382606609415531783list_a,B4: set_Pr5382606609415531783list_a,X: produc7709606177366032167list_a] :
( ( ord_le251320724540548263list_a @ A4 @ B4 )
=> ( ( member1318342207407915856list_a @ X @ A4 )
=> ( member1318342207407915856list_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_742_subsetD,axiom,
! [A4: set_Pr4048851178543822343list_a,B4: set_Pr4048851178543822343list_a,C: produc9164743771328383783list_a] :
( ( ord_le7857023143581076903list_a @ A4 @ B4 )
=> ( ( member8191768239178080336list_a @ C @ A4 )
=> ( member8191768239178080336list_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_743_subsetD,axiom,
! [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B4 )
=> ( ( member1426531477525435216od_a_a @ C @ A4 )
=> ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_744_subsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
=> ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_745_subsetD,axiom,
! [A4: set_Pr3451248702717554689st_nat,B4: set_Pr3451248702717554689st_nat,C: produc1828647624359046049st_nat] :
( ( ord_le8406513867147106209st_nat @ A4 @ B4 )
=> ( ( member7340969449405702474st_nat @ C @ A4 )
=> ( member7340969449405702474st_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_746_subsetD,axiom,
! [A4: set_Pr5382606609415531783list_a,B4: set_Pr5382606609415531783list_a,C: produc7709606177366032167list_a] :
( ( ord_le251320724540548263list_a @ A4 @ B4 )
=> ( ( member1318342207407915856list_a @ C @ A4 )
=> ( member1318342207407915856list_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_747_subset__eq,axiom,
( ord_le7857023143581076903list_a
= ( ^ [A6: set_Pr4048851178543822343list_a,B6: set_Pr4048851178543822343list_a] :
! [X6: produc9164743771328383783list_a] :
( ( member8191768239178080336list_a @ X6 @ A6 )
=> ( member8191768239178080336list_a @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_748_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ A6 )
=> ( member1426531477525435216od_a_a @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_749_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ A6 )
=> ( member8440522571783428010at_nat @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_750_subset__eq,axiom,
( ord_le8406513867147106209st_nat
= ( ^ [A6: set_Pr3451248702717554689st_nat,B6: set_Pr3451248702717554689st_nat] :
! [X6: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X6 @ A6 )
=> ( member7340969449405702474st_nat @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_751_subset__eq,axiom,
( ord_le251320724540548263list_a
= ( ^ [A6: set_Pr5382606609415531783list_a,B6: set_Pr5382606609415531783list_a] :
! [X6: produc7709606177366032167list_a] :
( ( member1318342207407915856list_a @ X6 @ A6 )
=> ( member1318342207407915856list_a @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_752_subset__iff,axiom,
( ord_le7857023143581076903list_a
= ( ^ [A6: set_Pr4048851178543822343list_a,B6: set_Pr4048851178543822343list_a] :
! [T3: produc9164743771328383783list_a] :
( ( member8191768239178080336list_a @ T3 @ A6 )
=> ( member8191768239178080336list_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_753_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T3 @ A6 )
=> ( member1426531477525435216od_a_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_754_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [T3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T3 @ A6 )
=> ( member8440522571783428010at_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_755_subset__iff,axiom,
( ord_le8406513867147106209st_nat
= ( ^ [A6: set_Pr3451248702717554689st_nat,B6: set_Pr3451248702717554689st_nat] :
! [T3: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ T3 @ A6 )
=> ( member7340969449405702474st_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_756_subset__iff,axiom,
( ord_le251320724540548263list_a
= ( ^ [A6: set_Pr5382606609415531783list_a,B6: set_Pr5382606609415531783list_a] :
! [T3: produc7709606177366032167list_a] :
( ( member1318342207407915856list_a @ T3 @ A6 )
=> ( member1318342207407915856list_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_757_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_758_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_759_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_760_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_761_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_762_subrelI,axiom,
! [R2: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
( ! [X3: list_a,Y3: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y3 ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le7857023143581076903list_a @ R2 @ S ) ) ).
% subrelI
thf(fact_763_subrelI,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X3: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R2 @ S ) ) ).
% subrelI
thf(fact_764_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_765_subrelI,axiom,
! [R2: set_Pr3451248702717554689st_nat,S: set_Pr3451248702717554689st_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le8406513867147106209st_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_766_subrelI,axiom,
! [R2: set_Pr5382606609415531783list_a,S: set_Pr5382606609415531783list_a] :
( ! [X3: list_list_a,Y3: list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ Y3 ) @ R2 )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le251320724540548263list_a @ R2 @ S ) ) ).
% subrelI
thf(fact_767_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_768_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_769_enumerate__simps_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( enumerate_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_P8443330267410185325_nat_a @ ( product_Pair_nat_a @ N @ X ) @ ( enumerate_a @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_770_enumerate__simps_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_771_length__enumerate,axiom,
! [N: nat,Xs: list_a] :
( ( size_s243904063682394823_nat_a @ ( enumerate_a @ N @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_enumerate
thf(fact_772_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_773_length__enumerate,axiom,
! [N: nat,Xs: list_list_a] :
( ( size_s1629943811222010829list_a @ ( enumerate_list_a @ N @ Xs ) )
= ( size_s349497388124573686list_a @ Xs ) ) ).
% length_enumerate
thf(fact_774_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_a,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_a @ Xs ) )
=> ( ( nth_Pr2476257081389315164_nat_a @ ( enumerate_a @ N @ Xs ) @ M )
= ( product_Pair_nat_a @ ( plus_plus_nat @ N @ M ) @ ( nth_a @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_775_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_776_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_list_a,N: nat] :
( ( ord_less_nat @ M @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_Pr6032962307111270498list_a @ ( enumerate_list_a @ N @ Xs ) @ M )
= ( produc7903367357317368283list_a @ ( plus_plus_nat @ N @ M ) @ ( nth_list_a @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_777_nth__drop__0,axiom,
! [Ss: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Ss ) )
=> ( ( cons_a @ ( nth_a @ Ss @ zero_zero_nat ) @ ( drop_a @ ( suc @ zero_zero_nat ) @ Ss ) )
= Ss ) ) ).
% nth_drop_0
thf(fact_778_nth__drop__0,axiom,
! [Ss: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Ss ) )
=> ( ( cons_nat @ ( nth_nat @ Ss @ zero_zero_nat ) @ ( drop_nat @ ( suc @ zero_zero_nat ) @ Ss ) )
= Ss ) ) ).
% nth_drop_0
thf(fact_779_nth__drop__0,axiom,
! [Ss: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Ss ) )
=> ( ( cons_list_a @ ( nth_list_a @ Ss @ zero_zero_nat ) @ ( drop_list_a @ ( suc @ zero_zero_nat ) @ Ss ) )
= Ss ) ) ).
% nth_drop_0
thf(fact_780_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_781_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_782_length__code,axiom,
( size_s349497388124573686list_a
= ( gen_length_list_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_783_gen__length__code_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_784_nth__append__take__is__nth__conv,axiom,
! [I4: nat,J: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ ( take_a @ J @ Xs ) @ Ys ) @ I4 )
= ( nth_a @ Xs @ I4 ) ) ) ) ).
% nth_append_take_is_nth_conv
thf(fact_785_nth__append__take__is__nth__conv,axiom,
! [I4: nat,J: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs ) @ Ys ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ) ).
% nth_append_take_is_nth_conv
thf(fact_786_nth__append__take__is__nth__conv,axiom,
! [I4: nat,J: nat,Xs: list_list_a,Ys: list_list_a] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ ( take_list_a @ J @ Xs ) @ Ys ) @ I4 )
= ( nth_list_a @ Xs @ I4 ) ) ) ) ).
% nth_append_take_is_nth_conv
thf(fact_787_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_788_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_789_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_790_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_791_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_792_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_793_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_794_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_795_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_796_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_797_append__eq__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a,Us: list_list_a,Vs: list_list_a] :
( ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
| ( ( size_s349497388124573686list_a @ Us )
= ( size_s349497388124573686list_a @ Vs ) ) )
=> ( ( ( append_list_a @ Xs @ Us )
= ( append_list_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_798_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_799_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_800_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_801_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_802_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_803_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_804_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_805_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_806_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_807_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_808_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_809_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_810_length__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( size_s349497388124573686list_a @ ( append_list_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) ) ) ).
% length_append
thf(fact_811_drop__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( drop_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_812_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_813_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_814_nth__append__length,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a] :
( ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) ) @ ( size_s349497388124573686list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_815_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
= ( nth_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_816_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_817_nth__append__length__plus,axiom,
! [Xs: list_list_a,Ys: list_list_a,N: nat] :
( ( nth_list_a @ ( append_list_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) )
= ( nth_list_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_818_list__update__length,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_819_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_820_list__update__length,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( list_update_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) ) @ ( size_s349497388124573686list_a @ Xs ) @ Y )
= ( append_list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_821_nth__drop,axiom,
! [N: nat,Xs: list_a,I4: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( drop_a @ N @ Xs ) @ I4 )
= ( nth_a @ Xs @ ( plus_plus_nat @ N @ I4 ) ) ) ) ).
% nth_drop
thf(fact_822_nth__drop,axiom,
! [N: nat,Xs: list_nat,I4: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I4 )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I4 ) ) ) ) ).
% nth_drop
thf(fact_823_nth__drop,axiom,
! [N: nat,Xs: list_list_a,I4: nat] :
( ( ord_less_eq_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( drop_list_a @ N @ Xs ) @ I4 )
= ( nth_list_a @ Xs @ ( plus_plus_nat @ N @ I4 ) ) ) ) ).
% nth_drop
thf(fact_824_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_825_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_826_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_827_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_828_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_829_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_830_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_831_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_832_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs2: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs2 ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_833_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_834_add__lessD1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
=> ( ord_less_nat @ I4 @ K ) ) ).
% add_lessD1
thf(fact_835_add__less__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_836_not__add__less1,axiom,
! [I4: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ I4 ) ).
% not_add_less1
thf(fact_837_not__add__less2,axiom,
! [J: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_838_add__less__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_839_trans__less__add1,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_840_trans__less__add2,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_841_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_842_enumerate__append__eq,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( enumerate_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append1694031006427026248_nat_a @ ( enumerate_a @ N @ Xs ) @ ( enumerate_a @ ( plus_plus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_843_enumerate__append__eq,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_844_enumerate__append__eq,axiom,
! [N: nat,Xs: list_list_a,Ys: list_list_a] :
( ( enumerate_list_a @ N @ ( append_list_a @ Xs @ Ys ) )
= ( append4802409771775881550list_a @ ( enumerate_list_a @ N @ Xs ) @ ( enumerate_list_a @ ( plus_plus_nat @ N @ ( size_s349497388124573686list_a @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_845_gen__length__def,axiom,
( gen_length_a
= ( ^ [N2: nat,Xs3: list_a] : ( plus_plus_nat @ N2 @ ( size_size_list_a @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_846_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_847_gen__length__def,axiom,
( gen_length_list_a
= ( ^ [N2: nat,Xs3: list_list_a] : ( plus_plus_nat @ N2 @ ( size_s349497388124573686list_a @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_848_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K5: nat] :
( N2
= ( plus_plus_nat @ M2 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_849_trans__le__add2,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_850_trans__le__add1,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_851_add__le__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_852_add__le__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_853_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_854_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_855_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_856_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_857_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_858_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_859_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_860_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_861_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_862_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_863_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_864_in__set__dropD,axiom,
! [X: produc9164743771328383783list_a,N: nat,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ ( drop_P8401437906483696749list_a @ N @ Xs ) ) )
=> ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) ) ) ).
% in_set_dropD
thf(fact_865_in__set__dropD,axiom,
! [X: product_prod_a_a,N: nat,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( drop_P8456769997282094189od_a_a @ N @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_866_in__set__dropD,axiom,
! [X: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% in_set_dropD
thf(fact_867_in__set__dropD,axiom,
! [X: produc1828647624359046049st_nat,N: nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ ( drop_P3617384654841569255st_nat @ N @ Xs ) ) )
=> ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) ) ) ).
% in_set_dropD
thf(fact_868_in__set__dropD,axiom,
! [X: produc7709606177366032167list_a,N: nat,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ ( drop_P4251048118207763053list_a @ N @ Xs ) ) )
=> ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) ) ) ).
% in_set_dropD
thf(fact_869_append__eq__conv__conj,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_a] :
( ( ( append_a @ Xs @ Ys )
= Zs2 )
= ( ( Xs
= ( take_a @ ( size_size_list_a @ Xs ) @ Zs2 ) )
& ( Ys
= ( drop_a @ ( size_size_list_a @ Xs ) @ Zs2 ) ) ) ) ).
% append_eq_conv_conj
thf(fact_870_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs2 )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs2 ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs2 ) ) ) ) ).
% append_eq_conv_conj
thf(fact_871_append__eq__conv__conj,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Zs2 )
= ( ( Xs
= ( take_list_a @ ( size_s349497388124573686list_a @ Xs ) @ Zs2 ) )
& ( Ys
= ( drop_list_a @ ( size_s349497388124573686list_a @ Xs ) @ Zs2 ) ) ) ) ).
% append_eq_conv_conj
thf(fact_872_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_873_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_874_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_875_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_876_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_877_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_878_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_879_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_880_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_881_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_882_append__eq__append__conv__if,axiom,
! [Xs_1: list_a,Xs_2: list_a,Ys_1: list_a,Ys_2: list_a] :
( ( ( append_a @ Xs_1 @ Xs_2 )
= ( append_a @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( Xs_1
= ( take_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_a @ ( drop_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( ( take_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_a @ ( drop_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_883_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_884_append__eq__append__conv__if,axiom,
! [Xs_1: list_list_a,Xs_2: list_list_a,Ys_1: list_list_a,Ys_2: list_list_a] :
( ( ( append_list_a @ Xs_1 @ Xs_2 )
= ( append_list_a @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs_1 ) @ ( size_s349497388124573686list_a @ Ys_1 ) )
=> ( ( Xs_1
= ( take_list_a @ ( size_s349497388124573686list_a @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_list_a @ ( drop_list_a @ ( size_s349497388124573686list_a @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs_1 ) @ ( size_s349497388124573686list_a @ Ys_1 ) )
=> ( ( ( take_list_a @ ( size_s349497388124573686list_a @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_list_a @ ( drop_list_a @ ( size_s349497388124573686list_a @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_885_take__drop__imp__nth,axiom,
! [I4: nat,Ss: list_nat,X: nat] :
( ( ( append_nat @ ( take_nat @ I4 @ Ss ) @ ( cons_nat @ X @ ( drop_nat @ ( suc @ I4 ) @ Ss ) ) )
= Ss )
=> ( X
= ( nth_nat @ Ss @ I4 ) ) ) ).
% take_drop_imp_nth
thf(fact_886_take__drop__imp__nth,axiom,
! [I4: nat,Ss: list_list_a,X: list_a] :
( ( ( append_list_a @ ( take_list_a @ I4 @ Ss ) @ ( cons_list_a @ X @ ( drop_list_a @ ( suc @ I4 ) @ Ss ) ) )
= Ss )
=> ( X
= ( nth_list_a @ Ss @ I4 ) ) ) ).
% take_drop_imp_nth
thf(fact_887_take__drop__imp__nth,axiom,
! [I4: nat,Ss: list_a,X: a] :
( ( ( append_a @ ( take_a @ I4 @ Ss ) @ ( cons_a @ X @ ( drop_a @ ( suc @ I4 ) @ Ss ) ) )
= Ss )
=> ( X
= ( nth_a @ Ss @ I4 ) ) ) ).
% take_drop_imp_nth
thf(fact_888_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_889_nth__via__drop,axiom,
! [N: nat,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( ( drop_list_a @ N @ Xs )
= ( cons_list_a @ Y @ Ys ) )
=> ( ( nth_list_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_890_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_891_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_892_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_893_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_894_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_895_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_896_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_897_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_898_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_899_split__list,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ? [Ys4: list_P321204300973800749list_a,Zs: list_P321204300973800749list_a] :
( Xs
= ( append622457529216362434list_a @ Ys4 @ ( cons_P5184657343811988189list_a @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_900_split__list,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys4: list_P1396940483166286381od_a_a,Zs: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_901_split__list,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [Ys4: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( Xs
= ( append985823374593552924at_nat @ Ys4 @ ( cons_P6512896166579812791at_nat @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_902_split__list,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ? [Ys4: list_P7940050157051400743st_nat,Zs: list_P7940050157051400743st_nat] :
( Xs
= ( append2623875052807961020st_nat @ Ys4 @ ( cons_P5007559046487125591st_nat @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_903_split__list,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ? [Ys4: list_P3352141576573427245list_a,Zs: list_P3352141576573427245list_a] :
( Xs
= ( append8923838579933563074list_a @ Ys4 @ ( cons_P4475838960125360349list_a @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_904_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_905_split__list__last,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ? [Ys4: list_P321204300973800749list_a,Zs: list_P321204300973800749list_a] :
( ( Xs
= ( append622457529216362434list_a @ Ys4 @ ( cons_P5184657343811988189list_a @ X @ Zs ) ) )
& ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Zs ) ) ) ) ).
% split_list_last
thf(fact_906_split__list__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys4: list_P1396940483166286381od_a_a,Zs: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X @ Zs ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_907_split__list__last,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [Ys4: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Ys4 @ ( cons_P6512896166579812791at_nat @ X @ Zs ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Zs ) ) ) ) ).
% split_list_last
thf(fact_908_split__list__last,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ? [Ys4: list_P7940050157051400743st_nat,Zs: list_P7940050157051400743st_nat] :
( ( Xs
= ( append2623875052807961020st_nat @ Ys4 @ ( cons_P5007559046487125591st_nat @ X @ Zs ) ) )
& ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Zs ) ) ) ) ).
% split_list_last
thf(fact_909_split__list__last,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ? [Ys4: list_P3352141576573427245list_a,Zs: list_P3352141576573427245list_a] :
( ( Xs
= ( append8923838579933563074list_a @ Ys4 @ ( cons_P4475838960125360349list_a @ X @ Zs ) ) )
& ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Zs ) ) ) ) ).
% split_list_last
thf(fact_910_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_911_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list_a,X3: a] :
( ? [Zs: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_912_split__list__first,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ? [Ys4: list_P321204300973800749list_a,Zs: list_P321204300973800749list_a] :
( ( Xs
= ( append622457529216362434list_a @ Ys4 @ ( cons_P5184657343811988189list_a @ X @ Zs ) ) )
& ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_913_split__list__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys4: list_P1396940483166286381od_a_a,Zs: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X @ Zs ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_914_split__list__first,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [Ys4: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Ys4 @ ( cons_P6512896166579812791at_nat @ X @ Zs ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_915_split__list__first,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ? [Ys4: list_P7940050157051400743st_nat,Zs: list_P7940050157051400743st_nat] :
( ( Xs
= ( append2623875052807961020st_nat @ Ys4 @ ( cons_P5007559046487125591st_nat @ X @ Zs ) ) )
& ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_916_split__list__first,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ? [Ys4: list_P3352141576573427245list_a,Zs: list_P3352141576573427245list_a] :
( ( Xs
= ( append8923838579933563074list_a @ Ys4 @ ( cons_P4475838960125360349list_a @ X @ Zs ) ) )
& ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_917_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_918_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list_a,X3: a] :
( ? [Zs: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_919_append__Cons__eq__iff,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a,Ys: list_P321204300973800749list_a,Xs4: list_P321204300973800749list_a,Ys5: list_P321204300973800749list_a] :
( ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
=> ( ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Ys ) )
=> ( ( ( append622457529216362434list_a @ Xs @ ( cons_P5184657343811988189list_a @ X @ Ys ) )
= ( append622457529216362434list_a @ Xs4 @ ( cons_P5184657343811988189list_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_920_append__Cons__eq__iff,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Xs4: list_P1396940483166286381od_a_a,Ys5: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( ( append5335208819046833346od_a_a @ Xs @ ( cons_P7316939126706565853od_a_a @ X @ Ys ) )
= ( append5335208819046833346od_a_a @ Xs4 @ ( cons_P7316939126706565853od_a_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_921_append__Cons__eq__iff,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xs4: list_P6011104703257516679at_nat,Ys5: list_P6011104703257516679at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys ) )
=> ( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ Ys ) )
= ( append985823374593552924at_nat @ Xs4 @ ( cons_P6512896166579812791at_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_922_append__Cons__eq__iff,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat,Ys: list_P7940050157051400743st_nat,Xs4: list_P7940050157051400743st_nat,Ys5: list_P7940050157051400743st_nat] :
( ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
=> ( ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Ys ) )
=> ( ( ( append2623875052807961020st_nat @ Xs @ ( cons_P5007559046487125591st_nat @ X @ Ys ) )
= ( append2623875052807961020st_nat @ Xs4 @ ( cons_P5007559046487125591st_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_923_append__Cons__eq__iff,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a,Ys: list_P3352141576573427245list_a,Xs4: list_P3352141576573427245list_a,Ys5: list_P3352141576573427245list_a] :
( ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
=> ( ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Ys ) )
=> ( ( ( append8923838579933563074list_a @ Xs @ ( cons_P4475838960125360349list_a @ X @ Ys ) )
= ( append8923838579933563074list_a @ Xs4 @ ( cons_P4475838960125360349list_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_924_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys5: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_925_in__set__conv__decomp,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
= ( ? [Ys3: list_P321204300973800749list_a,Zs3: list_P321204300973800749list_a] :
( Xs
= ( append622457529216362434list_a @ Ys3 @ ( cons_P5184657343811988189list_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_926_in__set__conv__decomp,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_927_in__set__conv__decomp,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [Ys3: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
( Xs
= ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_928_in__set__conv__decomp,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
= ( ? [Ys3: list_P7940050157051400743st_nat,Zs3: list_P7940050157051400743st_nat] :
( Xs
= ( append2623875052807961020st_nat @ Ys3 @ ( cons_P5007559046487125591st_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_929_in__set__conv__decomp,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
= ( ? [Ys3: list_P3352141576573427245list_a,Zs3: list_P3352141576573427245list_a] :
( Xs
= ( append8923838579933563074list_a @ Ys3 @ ( cons_P4475838960125360349list_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_930_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_931_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list_a,X3: a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs ) ) )
& ( P2 @ X3 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_932_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list_a,X3: a] :
( ? [Zs: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs ) ) )
& ( P2 @ X3 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_933_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list_a,X3: a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_934_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list_a,X3: a] :
( ? [Zs: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_935_in__set__conv__decomp__last,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
= ( ? [Ys3: list_P321204300973800749list_a,Zs3: list_P321204300973800749list_a] :
( ( Xs
= ( append622457529216362434list_a @ Ys3 @ ( cons_P5184657343811988189list_a @ X @ Zs3 ) ) )
& ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_936_in__set__conv__decomp__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_937_in__set__conv__decomp__last,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [Ys3: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_938_in__set__conv__decomp__last,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
= ( ? [Ys3: list_P7940050157051400743st_nat,Zs3: list_P7940050157051400743st_nat] :
( ( Xs
= ( append2623875052807961020st_nat @ Ys3 @ ( cons_P5007559046487125591st_nat @ X @ Zs3 ) ) )
& ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_939_in__set__conv__decomp__last,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
= ( ? [Ys3: list_P3352141576573427245list_a,Zs3: list_P3352141576573427245list_a] :
( ( Xs
= ( append8923838579933563074list_a @ Ys3 @ ( cons_P4475838960125360349list_a @ X @ Zs3 ) ) )
& ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_940_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_941_in__set__conv__decomp__first,axiom,
! [X: produc9164743771328383783list_a,Xs: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Xs ) )
= ( ? [Ys3: list_P321204300973800749list_a,Zs3: list_P321204300973800749list_a] :
( ( Xs
= ( append622457529216362434list_a @ Ys3 @ ( cons_P5184657343811988189list_a @ X @ Zs3 ) ) )
& ~ ( member8191768239178080336list_a @ X @ ( set_Pr2906193453920572092list_a @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_942_in__set__conv__decomp__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_943_in__set__conv__decomp__first,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [Ys3: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_944_in__set__conv__decomp__first,axiom,
! [X: produc1828647624359046049st_nat,Xs: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs ) )
= ( ? [Ys3: list_P7940050157051400743st_nat,Zs3: list_P7940050157051400743st_nat] :
( ( Xs
= ( append2623875052807961020st_nat @ Ys3 @ ( cons_P5007559046487125591st_nat @ X @ Zs3 ) ) )
& ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_945_in__set__conv__decomp__first,axiom,
! [X: produc7709606177366032167list_a,Xs: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Xs ) )
= ( ? [Ys3: list_P3352141576573427245list_a,Zs3: list_P3352141576573427245list_a] :
( ( Xs
= ( append8923838579933563074list_a @ Ys3 @ ( cons_P4475838960125360349list_a @ X @ Zs3 ) ) )
& ~ ( member1318342207407915856list_a @ X @ ( set_Pr1480943573297899964list_a @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_946_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_947_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) ) )
= ( ? [Ys3: list_a,X6: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs3 ) ) )
& ( P2 @ X6 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_948_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) ) )
= ( ? [Ys3: list_a,X6: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs3 ) ) )
& ( P2 @ X6 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_949_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_950_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_951_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_952_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_953_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_954_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_955_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_956_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_957_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_958_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_959_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_960_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K5: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_961_less__add__Suc2,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M @ I4 ) ) ) ).
% less_add_Suc2
thf(fact_962_less__add__Suc1,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M ) ) ) ).
% less_add_Suc1
thf(fact_963_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_964_less__imp__add__positive,axiom,
! [I4: nat,J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I4 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_965_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_966_append__listrel1I,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,Us: list_a,Vs: list_a] :
( ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Vs ) @ ( listrel1_a @ R2 ) ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( listrel1_a @ R2 ) ) ) ).
% append_listrel1I
thf(fact_967_append__listrel1I,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R2 ) ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R2 ) ) ) ).
% append_listrel1I
thf(fact_968_append__listrel1I,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a,Us: list_list_a,Vs: list_list_a] :
( ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Us @ Vs ) @ ( listrel1_list_a @ R2 ) ) ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Us ) @ ( append_list_a @ Ys @ Vs ) ) @ ( listrel1_list_a @ R2 ) ) ) ).
% append_listrel1I
thf(fact_969_distinct__take__drop,axiom,
! [Vs: list_a,I4: nat] :
( ( distinct_a @ Vs )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Vs ) )
=> ( distinct_a @ ( append_a @ ( take_a @ I4 @ Vs ) @ ( drop_a @ ( suc @ I4 ) @ Vs ) ) ) ) ) ).
% distinct_take_drop
thf(fact_970_distinct__take__drop,axiom,
! [Vs: list_nat,I4: nat] :
( ( distinct_nat @ Vs )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Vs ) )
=> ( distinct_nat @ ( append_nat @ ( take_nat @ I4 @ Vs ) @ ( drop_nat @ ( suc @ I4 ) @ Vs ) ) ) ) ) ).
% distinct_take_drop
thf(fact_971_distinct__take__drop,axiom,
! [Vs: list_list_a,I4: nat] :
( ( distinct_list_a @ Vs )
=> ( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Vs ) )
=> ( distinct_list_a @ ( append_list_a @ ( take_list_a @ I4 @ Vs ) @ ( drop_list_a @ ( suc @ I4 ) @ Vs ) ) ) ) ) ).
% distinct_take_drop
thf(fact_972_lex__append__leftI,axiom,
! [Ys: list_a,Zs2: list_a,R2: set_Product_prod_a_a,Xs: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs2 ) @ ( lex_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs2 ) ) @ ( lex_a @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_973_lex__append__leftI,axiom,
! [Ys: list_nat,Zs2: list_nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs2 ) @ ( lex_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs2 ) ) @ ( lex_nat @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_974_lex__append__leftI,axiom,
! [Ys: list_list_a,Zs2: list_list_a,R2: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs2 ) @ ( lex_list_a @ R2 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs2 ) ) @ ( lex_list_a @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_975_take__drop__update__second,axiom,
! [J: nat,Ds: list_a,Cs: list_a] :
( ( ord_less_nat @ J @ ( size_size_list_a @ Ds ) )
=> ( ( ( size_size_list_a @ Cs )
= ( size_size_list_a @ Ds ) )
=> ( ( list_update_a @ ( append_a @ ( take_a @ J @ Ds ) @ ( drop_a @ J @ Cs ) ) @ J @ ( nth_a @ Cs @ J ) )
= ( append_a @ ( take_a @ J @ Ds ) @ ( drop_a @ J @ Cs ) ) ) ) ) ).
% take_drop_update_second
thf(fact_976_take__drop__update__second,axiom,
! [J: nat,Ds: list_nat,Cs: list_nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Ds ) )
=> ( ( ( size_size_list_nat @ Cs )
= ( size_size_list_nat @ Ds ) )
=> ( ( list_update_nat @ ( append_nat @ ( take_nat @ J @ Ds ) @ ( drop_nat @ J @ Cs ) ) @ J @ ( nth_nat @ Cs @ J ) )
= ( append_nat @ ( take_nat @ J @ Ds ) @ ( drop_nat @ J @ Cs ) ) ) ) ) ).
% take_drop_update_second
thf(fact_977_take__drop__update__second,axiom,
! [J: nat,Ds: list_list_a,Cs: list_list_a] :
( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Ds ) )
=> ( ( ( size_s349497388124573686list_a @ Cs )
= ( size_s349497388124573686list_a @ Ds ) )
=> ( ( list_update_list_a @ ( append_list_a @ ( take_list_a @ J @ Ds ) @ ( drop_list_a @ J @ Cs ) ) @ J @ ( nth_list_a @ Cs @ J ) )
= ( append_list_a @ ( take_list_a @ J @ Ds ) @ ( drop_list_a @ J @ Cs ) ) ) ) ) ).
% take_drop_update_second
thf(fact_978_id__take__nth__drop,axiom,
! [I4: nat,Xs: list_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I4 @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I4 ) @ ( drop_a @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_979_id__take__nth__drop,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I4 ) @ ( drop_nat @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_980_id__take__nth__drop,axiom,
! [I4: nat,Xs: list_list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( Xs
= ( append_list_a @ ( take_list_a @ I4 @ Xs ) @ ( cons_list_a @ ( nth_list_a @ Xs @ I4 ) @ ( drop_list_a @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_981_nth__append__take__drop__is__nth__conv,axiom,
! [I4: nat,Xs: list_a,J: nat,Y: a] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( I4 != J )
=> ( ( nth_a @ ( append_a @ ( take_a @ J @ Xs ) @ ( cons_a @ Y @ ( drop_a @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_a @ Xs @ I4 ) ) ) ) ) ).
% nth_append_take_drop_is_nth_conv
thf(fact_982_nth__append__take__drop__is__nth__conv,axiom,
! [I4: nat,Xs: list_nat,J: nat,Y: nat] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( I4 != J )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs ) @ ( cons_nat @ Y @ ( drop_nat @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% nth_append_take_drop_is_nth_conv
thf(fact_983_nth__append__take__drop__is__nth__conv,axiom,
! [I4: nat,Xs: list_list_a,J: nat,Y: list_a] :
( ( ord_less_eq_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( I4 != J )
=> ( ( nth_list_a @ ( append_list_a @ ( take_list_a @ J @ Xs ) @ ( cons_list_a @ Y @ ( drop_list_a @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_list_a @ Xs @ I4 ) ) ) ) ) ).
% nth_append_take_drop_is_nth_conv
thf(fact_984_upd__conv__take__nth__drop,axiom,
! [I4: nat,Xs: list_a,A: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I4 @ A )
= ( append_a @ ( take_a @ I4 @ Xs ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_985_upd__conv__take__nth__drop,axiom,
! [I4: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I4 @ A )
= ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_986_upd__conv__take__nth__drop,axiom,
! [I4: nat,Xs: list_list_a,A: list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( list_update_list_a @ Xs @ I4 @ A )
= ( append_list_a @ ( take_list_a @ I4 @ Xs ) @ ( cons_list_a @ A @ ( drop_list_a @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_987_take__drop__update__first,axiom,
! [J: nat,Ds: list_a,Cs: list_a] :
( ( ord_less_nat @ J @ ( size_size_list_a @ Ds ) )
=> ( ( ( size_size_list_a @ Cs )
= ( size_size_list_a @ Ds ) )
=> ( ( list_update_a @ ( append_a @ ( take_a @ J @ Ds ) @ ( drop_a @ J @ Cs ) ) @ J @ ( nth_a @ Ds @ J ) )
= ( append_a @ ( take_a @ ( suc @ J ) @ Ds ) @ ( drop_a @ ( suc @ J ) @ Cs ) ) ) ) ) ).
% take_drop_update_first
thf(fact_988_take__drop__update__first,axiom,
! [J: nat,Ds: list_nat,Cs: list_nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Ds ) )
=> ( ( ( size_size_list_nat @ Cs )
= ( size_size_list_nat @ Ds ) )
=> ( ( list_update_nat @ ( append_nat @ ( take_nat @ J @ Ds ) @ ( drop_nat @ J @ Cs ) ) @ J @ ( nth_nat @ Ds @ J ) )
= ( append_nat @ ( take_nat @ ( suc @ J ) @ Ds ) @ ( drop_nat @ ( suc @ J ) @ Cs ) ) ) ) ) ).
% take_drop_update_first
thf(fact_989_take__drop__update__first,axiom,
! [J: nat,Ds: list_list_a,Cs: list_list_a] :
( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Ds ) )
=> ( ( ( size_s349497388124573686list_a @ Cs )
= ( size_s349497388124573686list_a @ Ds ) )
=> ( ( list_update_list_a @ ( append_list_a @ ( take_list_a @ J @ Ds ) @ ( drop_list_a @ J @ Cs ) ) @ J @ ( nth_list_a @ Ds @ J ) )
= ( append_list_a @ ( take_list_a @ ( suc @ J ) @ Ds ) @ ( drop_list_a @ ( suc @ J ) @ Cs ) ) ) ) ) ).
% take_drop_update_first
thf(fact_990_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_991_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_992_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_993_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_994_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_995_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_996_append__Cons__nth__middle,axiom,
! [I4: nat,Xs: list_a,Y: a,Zs2: list_a] :
( ( I4
= ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ ( cons_a @ Y @ Zs2 ) ) @ I4 )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_997_append__Cons__nth__middle,axiom,
! [I4: nat,Xs: list_nat,Y: nat,Zs2: list_nat] :
( ( I4
= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs2 ) ) @ I4 )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_998_append__Cons__nth__middle,axiom,
! [I4: nat,Xs: list_list_a,Y: list_a,Zs2: list_list_a] :
( ( I4
= ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ Y @ Zs2 ) ) @ I4 )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_999_append__Cons__nth__not__middle,axiom,
! [I4: nat,Xs: list_a,U: a,Ys: list_a,Z2: a] :
( ( I4
!= ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ ( cons_a @ U @ Ys ) ) @ I4 )
= ( nth_a @ ( append_a @ Xs @ ( cons_a @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_1000_append__Cons__nth__not__middle,axiom,
! [I4: nat,Xs: list_nat,U: nat,Ys: list_nat,Z2: nat] :
( ( I4
!= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I4 )
= ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_1001_append__Cons__nth__not__middle,axiom,
! [I4: nat,Xs: list_list_a,U: list_a,Ys: list_list_a,Z2: list_a] :
( ( I4
!= ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ U @ Ys ) ) @ I4 )
= ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_1002_P__as__bs__extend,axiom,
! [As: list_a,Bs: list_a,Cs: list_a,Ds: list_a,P2: a > a > $o] :
( ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ( ( size_size_list_a @ Cs )
= ( size_size_list_a @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Bs ) )
=> ( P2 @ ( nth_a @ As @ I2 ) @ ( nth_a @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ds ) )
=> ( P2 @ ( nth_a @ Cs @ I2 ) @ ( nth_a @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ( append_a @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_a @ ( append_a @ As @ Cs ) @ I ) @ ( nth_a @ ( append_a @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1003_P__as__bs__extend,axiom,
! [As: list_a,Bs: list_nat,Cs: list_a,Ds: list_nat,P2: a > nat > $o] :
( ( ( size_size_list_a @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ( ( size_size_list_a @ Cs )
= ( size_size_list_nat @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
=> ( P2 @ ( nth_a @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
=> ( P2 @ ( nth_a @ Cs @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_a @ ( append_a @ As @ Cs ) @ I ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1004_P__as__bs__extend,axiom,
! [As: list_a,Bs: list_list_a,Cs: list_a,Ds: list_list_a,P2: a > list_a > $o] :
( ( ( size_size_list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) )
=> ( ( ( size_size_list_a @ Cs )
= ( size_s349497388124573686list_a @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Bs ) )
=> ( P2 @ ( nth_a @ As @ I2 ) @ ( nth_list_a @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ds ) )
=> ( P2 @ ( nth_a @ Cs @ I2 ) @ ( nth_list_a @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ ( append_list_a @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_a @ ( append_a @ As @ Cs ) @ I ) @ ( nth_list_a @ ( append_list_a @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1005_P__as__bs__extend,axiom,
! [As: list_nat,Bs: list_a,Cs: list_nat,Ds: list_a,P2: nat > a > $o] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_a @ Bs ) )
=> ( ( ( size_size_list_nat @ Cs )
= ( size_size_list_a @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Bs ) )
=> ( P2 @ ( nth_nat @ As @ I2 ) @ ( nth_a @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ds ) )
=> ( P2 @ ( nth_nat @ Cs @ I2 ) @ ( nth_a @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ( append_a @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_nat @ ( append_nat @ As @ Cs ) @ I ) @ ( nth_a @ ( append_a @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1006_P__as__bs__extend,axiom,
! [As: list_nat,Bs: list_nat,Cs: list_nat,Ds: list_nat,P2: nat > nat > $o] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ( ( size_size_list_nat @ Cs )
= ( size_size_list_nat @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
=> ( P2 @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
=> ( P2 @ ( nth_nat @ Cs @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_nat @ ( append_nat @ As @ Cs ) @ I ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1007_P__as__bs__extend,axiom,
! [As: list_nat,Bs: list_list_a,Cs: list_nat,Ds: list_list_a,P2: nat > list_a > $o] :
( ( ( size_size_list_nat @ As )
= ( size_s349497388124573686list_a @ Bs ) )
=> ( ( ( size_size_list_nat @ Cs )
= ( size_s349497388124573686list_a @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Bs ) )
=> ( P2 @ ( nth_nat @ As @ I2 ) @ ( nth_list_a @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ds ) )
=> ( P2 @ ( nth_nat @ Cs @ I2 ) @ ( nth_list_a @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ ( append_list_a @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_nat @ ( append_nat @ As @ Cs ) @ I ) @ ( nth_list_a @ ( append_list_a @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1008_P__as__bs__extend,axiom,
! [As: list_list_a,Bs: list_a,Cs: list_list_a,Ds: list_a,P2: list_a > a > $o] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ( ( size_s349497388124573686list_a @ Cs )
= ( size_size_list_a @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Bs ) )
=> ( P2 @ ( nth_list_a @ As @ I2 ) @ ( nth_a @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ds ) )
=> ( P2 @ ( nth_list_a @ Cs @ I2 ) @ ( nth_a @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ( append_a @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_list_a @ ( append_list_a @ As @ Cs ) @ I ) @ ( nth_a @ ( append_a @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1009_P__as__bs__extend,axiom,
! [As: list_list_a,Bs: list_nat,Cs: list_list_a,Ds: list_nat,P2: list_a > nat > $o] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ( ( size_s349497388124573686list_a @ Cs )
= ( size_size_list_nat @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
=> ( P2 @ ( nth_list_a @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
=> ( P2 @ ( nth_list_a @ Cs @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_list_a @ ( append_list_a @ As @ Cs ) @ I ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1010_P__as__bs__extend,axiom,
! [As: list_list_a,Bs: list_list_a,Cs: list_list_a,Ds: list_list_a,P2: list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) )
=> ( ( ( size_s349497388124573686list_a @ Cs )
= ( size_s349497388124573686list_a @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Bs ) )
=> ( P2 @ ( nth_list_a @ As @ I2 ) @ ( nth_list_a @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ds ) )
=> ( P2 @ ( nth_list_a @ Cs @ I2 ) @ ( nth_list_a @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ ( append_list_a @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_list_a @ ( append_list_a @ As @ Cs ) @ I ) @ ( nth_list_a @ ( append_list_a @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1011_not__distinct__conv__prefix,axiom,
! [As: list_P321204300973800749list_a] :
( ( ~ ( distin9160064362609808800list_a @ As ) )
= ( ? [Xs3: list_P321204300973800749list_a,Y5: produc9164743771328383783list_a,Ys3: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ Y5 @ ( set_Pr2906193453920572092list_a @ Xs3 ) )
& ( distin9160064362609808800list_a @ Xs3 )
& ( As
= ( append622457529216362434list_a @ Xs3 @ ( cons_P5184657343811988189list_a @ Y5 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1012_not__distinct__conv__prefix,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( ~ ( distin132333870042060960od_a_a @ As ) )
= ( ? [Xs3: list_P1396940483166286381od_a_a,Y5: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_Product_prod_a_a2 @ Xs3 ) )
& ( distin132333870042060960od_a_a @ Xs3 )
& ( As
= ( append5335208819046833346od_a_a @ Xs3 @ ( cons_P7316939126706565853od_a_a @ Y5 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1013_not__distinct__conv__prefix,axiom,
! [As: list_P6011104703257516679at_nat] :
( ( ~ ( distin6923225563576452346at_nat @ As ) )
= ( ? [Xs3: list_P6011104703257516679at_nat,Y5: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Xs3 ) )
& ( distin6923225563576452346at_nat @ Xs3 )
& ( As
= ( append985823374593552924at_nat @ Xs3 @ ( cons_P6512896166579812791at_nat @ Y5 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1014_not__distinct__conv__prefix,axiom,
! [As: list_P7940050157051400743st_nat] :
( ( ~ ( distin347524420504911514st_nat @ As ) )
= ( ? [Xs3: list_P7940050157051400743st_nat,Y5: produc1828647624359046049st_nat,Ys3: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ Y5 @ ( set_Pr3842133991353686454st_nat @ Xs3 ) )
& ( distin347524420504911514st_nat @ Xs3 )
& ( As
= ( append2623875052807961020st_nat @ Xs3 @ ( cons_P5007559046487125591st_nat @ Y5 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1015_not__distinct__conv__prefix,axiom,
! [As: list_P3352141576573427245list_a] :
( ( ~ ( distin5851120023645153440list_a @ As ) )
= ( ? [Xs3: list_P3352141576573427245list_a,Y5: produc7709606177366032167list_a,Ys3: list_P3352141576573427245list_a] :
( ( member1318342207407915856list_a @ Y5 @ ( set_Pr1480943573297899964list_a @ Xs3 ) )
& ( distin5851120023645153440list_a @ Xs3 )
& ( As
= ( append8923838579933563074list_a @ Xs3 @ ( cons_P4475838960125360349list_a @ Y5 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1016_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs3: list_a,Y5: a,Ys3: list_a] :
( ( member_a @ Y5 @ ( set_a2 @ Xs3 ) )
& ( distinct_a @ Xs3 )
& ( As
= ( append_a @ Xs3 @ ( cons_a @ Y5 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1017_list__update__append1,axiom,
! [I4: nat,Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I4 @ X )
= ( append_a @ ( list_update_a @ Xs @ I4 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1018_list__update__append1,axiom,
! [I4: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I4 @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I4 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1019_list__update__append1,axiom,
! [I4: nat,Xs: list_list_a,Ys: list_list_a,X: list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( list_update_list_a @ ( append_list_a @ Xs @ Ys ) @ I4 @ X )
= ( append_list_a @ ( list_update_list_a @ Xs @ I4 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1020_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I4: nat,Xs: list_a,Y: a] :
( ( ord_less_nat @ J @ I4 )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ ( take_a @ J @ Xs ) @ ( cons_a @ Y @ ( drop_a @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_a @ Xs @ I4 ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_1021_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I4: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ J @ I4 )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs ) @ ( cons_nat @ Y @ ( drop_nat @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_1022_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I4: nat,Xs: list_list_a,Y: list_a] :
( ( ord_less_nat @ J @ I4 )
=> ( ( ord_less_eq_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ ( take_list_a @ J @ Xs ) @ ( cons_list_a @ Y @ ( drop_list_a @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_list_a @ Xs @ I4 ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_1023_lex__append__left__iff,axiom,
! [R2: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ! [X3: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ X3 ) @ R2 )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs2 ) ) @ ( lex_list_a @ R2 ) )
= ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs2 ) @ ( lex_list_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1024_lex__append__left__iff,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs2: list_a] :
( ! [X3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs2 ) ) @ ( lex_a @ R2 ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs2 ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1025_lex__append__left__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs2: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs2 ) ) @ ( lex_nat @ R2 ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs2 ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1026_lex__append__left__iff,axiom,
! [R2: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ! [X3: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ R2 )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs @ Ys ) @ ( append_list_nat @ Xs @ Zs2 ) ) @ ( lex_list_nat @ R2 ) )
= ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Zs2 ) @ ( lex_list_nat @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1027_lex__append__left__iff,axiom,
! [R2: set_Pr5382606609415531783list_a,Xs: list_list_list_a,Ys: list_list_list_a,Zs2: list_list_list_a] :
( ! [X3: list_list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ X3 ) @ R2 )
=> ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( append_list_list_a @ Xs @ Ys ) @ ( append_list_list_a @ Xs @ Zs2 ) ) @ ( lex_list_list_a @ R2 ) )
= ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Ys @ Zs2 ) @ ( lex_list_list_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1028_lex__append__leftD,axiom,
! [R2: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ! [X3: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ X3 ) @ R2 )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs2 ) ) @ ( lex_list_a @ R2 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs2 ) @ ( lex_list_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1029_lex__append__leftD,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs2: list_a] :
( ! [X3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs2 ) ) @ ( lex_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs2 ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1030_lex__append__leftD,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs2: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs2 ) ) @ ( lex_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs2 ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1031_lex__append__leftD,axiom,
! [R2: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ! [X3: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ R2 )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs @ Ys ) @ ( append_list_nat @ Xs @ Zs2 ) ) @ ( lex_list_nat @ R2 ) )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Zs2 ) @ ( lex_list_nat @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1032_lex__append__leftD,axiom,
! [R2: set_Pr5382606609415531783list_a,Xs: list_list_list_a,Ys: list_list_list_a,Zs2: list_list_list_a] :
( ! [X3: list_list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ X3 ) @ R2 )
=> ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( append_list_list_a @ Xs @ Ys ) @ ( append_list_list_a @ Xs @ Zs2 ) ) @ ( lex_list_list_a @ R2 ) )
=> ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Ys @ Zs2 ) @ ( lex_list_list_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1033_lex__append__rightI,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_1034_lex__append__rightI,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_1035_lex__append__rightI,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a,Vs: list_list_a,Us: list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R2 ) )
=> ( ( ( size_s349497388124573686list_a @ Vs )
= ( size_s349497388124573686list_a @ Us ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Us ) @ ( append_list_a @ Ys @ Vs ) ) @ ( lex_list_a @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_1036_lenlex__append1,axiom,
! [Us: list_a,Xs: list_a,R4: set_Product_prod_a_a,Vs: list_a,Ys: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs ) @ ( lenlex_a @ R4 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Ys ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs @ Ys ) ) @ ( lenlex_a @ R4 ) ) ) ) ).
% lenlex_append1
thf(fact_1037_lenlex__append1,axiom,
! [Us: list_nat,Xs: list_nat,R4: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R4 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R4 ) ) ) ) ).
% lenlex_append1
thf(fact_1038_lenlex__append1,axiom,
! [Us: list_list_a,Xs: list_list_a,R4: set_Pr4048851178543822343list_a,Vs: list_list_a,Ys: list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Us @ Xs ) @ ( lenlex_list_a @ R4 ) )
=> ( ( ( size_s349497388124573686list_a @ Vs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Us @ Vs ) @ ( append_list_a @ Xs @ Ys ) ) @ ( lenlex_list_a @ R4 ) ) ) ) ).
% lenlex_append1
thf(fact_1039_append__Cons__nth__left,axiom,
! [I4: nat,Xs: list_a,U: a,Ys: list_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ ( cons_a @ U @ Ys ) ) @ I4 )
= ( nth_a @ Xs @ I4 ) ) ) ).
% append_Cons_nth_left
thf(fact_1040_append__Cons__nth__left,axiom,
! [I4: nat,Xs: list_nat,U: nat,Ys: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ).
% append_Cons_nth_left
thf(fact_1041_append__Cons__nth__left,axiom,
! [I4: nat,Xs: list_list_a,U: list_a,Ys: list_list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ U @ Ys ) ) @ I4 )
= ( nth_list_a @ Xs @ I4 ) ) ) ).
% append_Cons_nth_left
thf(fact_1042_append__Cons__nth__right,axiom,
! [Xs: list_a,I4: nat,U: a,Ys: list_a,Z2: a] :
( ( ord_less_nat @ ( size_size_list_a @ Xs ) @ I4 )
=> ( ( nth_a @ ( append_a @ Xs @ ( cons_a @ U @ Ys ) ) @ I4 )
= ( nth_a @ ( append_a @ Xs @ ( cons_a @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_right
thf(fact_1043_append__Cons__nth__right,axiom,
! [Xs: list_nat,I4: nat,U: nat,Ys: list_nat,Z2: nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ I4 )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I4 )
= ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_right
thf(fact_1044_append__Cons__nth__right,axiom,
! [Xs: list_list_a,I4: nat,U: list_a,Ys: list_list_a,Z2: list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Xs ) @ I4 )
=> ( ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ U @ Ys ) ) @ I4 )
= ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_right
thf(fact_1045_list_Osize_I4_J,axiom,
! [X21: a,X22: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1046_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1047_list_Osize_I4_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( size_s349497388124573686list_a @ ( cons_list_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s349497388124573686list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1048_listrel1I,axiom,
! [X: list_a,Y: list_a,R2: set_Pr4048851178543822343list_a,Xs: list_list_a,Us: list_list_a,Vs: list_list_a,Ys: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_list_a @ Us @ ( cons_list_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_list_a @ Us @ ( cons_list_a @ Y @ Vs ) ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1049_listrel1I,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1050_listrel1I,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
=> ( ( Ys
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1051_listrel1I,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Us: list_list_nat,Vs: list_list_nat,Ys: list_list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_list_nat @ Us @ ( cons_list_nat @ X @ Vs ) ) )
=> ( ( Ys
= ( append_list_nat @ Us @ ( cons_list_nat @ Y @ Vs ) ) )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1052_listrel1I,axiom,
! [X: list_list_a,Y: list_list_a,R2: set_Pr5382606609415531783list_a,Xs: list_list_list_a,Us: list_list_list_a,Vs: list_list_list_a,Ys: list_list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_list_list_a @ Us @ ( cons_list_list_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_list_list_a @ Us @ ( cons_list_list_a @ Y @ Vs ) ) )
=> ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1053_listrel1E,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R2 ) )
=> ~ ! [X3: list_nat,Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
=> ! [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ ( cons_list_nat @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_list_nat @ Us2 @ ( cons_list_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1054_listrel1E,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a,R2: set_Pr5382606609415531783list_a] :
( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R2 ) )
=> ~ ! [X3: list_list_a,Y3: list_list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ Y3 ) @ R2 )
=> ! [Us2: list_list_list_a,Vs2: list_list_list_a] :
( ( Xs
= ( append_list_list_a @ Us2 @ ( cons_list_list_a @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_list_list_a @ Us2 @ ( cons_list_list_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1055_listrel1E,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ~ ! [X3: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ! [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ ( cons_a @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_a @ Us2 @ ( cons_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1056_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ! [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ ( cons_nat @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_nat @ Us2 @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1057_listrel1E,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) )
=> ~ ! [X3: list_a,Y3: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y3 ) @ R2 )
=> ! [Us2: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us2 @ ( cons_list_a @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_list_a @ Us2 @ ( cons_list_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1058_Cons__nth__drop__Suc,axiom,
! [I4: nat,Xs: list_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I4 ) @ ( drop_a @ ( suc @ I4 ) @ Xs ) )
= ( drop_a @ I4 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1059_Cons__nth__drop__Suc,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I4 ) @ ( drop_nat @ ( suc @ I4 ) @ Xs ) )
= ( drop_nat @ I4 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1060_Cons__nth__drop__Suc,axiom,
! [I4: nat,Xs: list_list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( cons_list_a @ ( nth_list_a @ Xs @ I4 ) @ ( drop_list_a @ ( suc @ I4 ) @ Xs ) )
= ( drop_list_a @ I4 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1061_nth__append__take,axiom,
! [I4: nat,Xs: list_a,Y: a,Ys: list_a] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ ( take_a @ I4 @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ I4 )
= Y ) ) ).
% nth_append_take
thf(fact_1062_nth__append__take,axiom,
! [I4: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ I4 )
= Y ) ) ).
% nth_append_take
thf(fact_1063_nth__append__take,axiom,
! [I4: nat,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( ord_less_eq_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( append_list_a @ ( take_list_a @ I4 @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ I4 )
= Y ) ) ).
% nth_append_take
thf(fact_1064_add__elem__list__listsE,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( basic_189371981877014951ists_a @ X @ Xs ) ) )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ N3 @ ( size_size_list_a @ Xs ) )
& ( Ys
= ( append_a @ ( take_a @ N3 @ Xs ) @ ( cons_a @ X @ ( drop_a @ N3 @ Xs ) ) ) ) ) ) ).
% add_elem_list_listsE
thf(fact_1065_add__elem__list__listsE,axiom,
! [Ys: list_nat,X: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs ) ) )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ N3 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( append_nat @ ( take_nat @ N3 @ Xs ) @ ( cons_nat @ X @ ( drop_nat @ N3 @ Xs ) ) ) ) ) ) ).
% add_elem_list_listsE
thf(fact_1066_add__elem__list__listsE,axiom,
! [Ys: list_list_a,X: list_a,Xs: list_list_a] :
( ( member_list_list_a @ Ys @ ( set_list_list_a2 @ ( basic_8052073458081932717list_a @ X @ Xs ) ) )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ N3 @ ( size_s349497388124573686list_a @ Xs ) )
& ( Ys
= ( append_list_a @ ( take_list_a @ N3 @ Xs ) @ ( cons_list_a @ X @ ( drop_list_a @ N3 @ Xs ) ) ) ) ) ) ).
% add_elem_list_listsE
thf(fact_1067_add__elem__list__listsI,axiom,
! [N: nat,Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( Ys
= ( append_a @ ( take_a @ N @ Xs ) @ ( cons_a @ X @ ( drop_a @ N @ Xs ) ) ) )
=> ( member_list_a @ Ys @ ( set_list_a2 @ ( basic_189371981877014951ists_a @ X @ Xs ) ) ) ) ) ).
% add_elem_list_listsI
thf(fact_1068_add__elem__list__listsI,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( Ys
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ X @ ( drop_nat @ N @ Xs ) ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs ) ) ) ) ) ).
% add_elem_list_listsI
thf(fact_1069_add__elem__list__listsI,axiom,
! [N: nat,Xs: list_list_a,Ys: list_list_a,X: list_a] :
( ( ord_less_eq_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( Ys
= ( append_list_a @ ( take_list_a @ N @ Xs ) @ ( cons_list_a @ X @ ( drop_list_a @ N @ Xs ) ) ) )
=> ( member_list_list_a @ Ys @ ( set_list_list_a2 @ ( basic_8052073458081932717list_a @ X @ Xs ) ) ) ) ) ).
% add_elem_list_listsI
thf(fact_1070_length__add__elem__list__lists,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( basic_189371981877014951ists_a @ X @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_1071_length__add__elem__list__lists,axiom,
! [Ys: list_nat,X: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_1072_length__add__elem__list__lists,axiom,
! [Ys: list_list_a,X: list_a,Xs: list_list_a] :
( ( member_list_list_a @ Ys @ ( set_list_list_a2 @ ( basic_8052073458081932717list_a @ X @ Xs ) ) )
=> ( ( size_s349497388124573686list_a @ Ys )
= ( suc @ ( size_s349497388124573686list_a @ Xs ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_1073_take__Suc__conv__app__nth,axiom,
! [I4: nat,Xs: list_a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I4 ) @ Xs )
= ( append_a @ ( take_a @ I4 @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I4 ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1074_take__Suc__conv__app__nth,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I4 ) @ Xs )
= ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I4 ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1075_take__Suc__conv__app__nth,axiom,
! [I4: nat,Xs: list_list_a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( take_list_a @ ( suc @ I4 ) @ Xs )
= ( append_list_a @ ( take_list_a @ I4 @ Xs ) @ ( cons_list_a @ ( nth_list_a @ Xs @ I4 ) @ nil_list_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1076_take__nth__drop__concat,axiom,
! [I4: nat,Xss: list_list_nat,Ys: list_nat,J: nat,Z2: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xss ) )
=> ( ( ( nth_list_nat @ Xss @ I4 )
= Ys )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ys ) )
=> ( ( ( nth_nat @ Ys @ J )
= Z2 )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xss ) ) )
& ( ( take_nat @ K2 @ ( concat_nat @ Xss ) )
= ( append_nat @ ( concat_nat @ ( take_list_nat @ I4 @ Xss ) ) @ ( take_nat @ J @ Ys ) ) )
& ( ( nth_nat @ ( concat_nat @ Xss ) @ K2 )
= ( nth_nat @ ( nth_list_nat @ Xss @ I4 ) @ J ) )
& ( ( drop_nat @ ( suc @ K2 ) @ ( concat_nat @ Xss ) )
= ( append_nat @ ( drop_nat @ ( suc @ J ) @ Ys ) @ ( concat_nat @ ( drop_list_nat @ ( suc @ I4 ) @ Xss ) ) ) ) ) ) ) ) ) ).
% take_nth_drop_concat
thf(fact_1077_take__nth__drop__concat,axiom,
! [I4: nat,Xss: list_list_list_a,Ys: list_list_a,J: nat,Z2: list_a] :
( ( ord_less_nat @ I4 @ ( size_s2403821588304063868list_a @ Xss ) )
=> ( ( ( nth_list_list_a @ Xss @ I4 )
= Ys )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( nth_list_a @ Ys @ J )
= Z2 )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_s349497388124573686list_a @ ( concat_list_a @ Xss ) ) )
& ( ( take_list_a @ K2 @ ( concat_list_a @ Xss ) )
= ( append_list_a @ ( concat_list_a @ ( take_list_list_a @ I4 @ Xss ) ) @ ( take_list_a @ J @ Ys ) ) )
& ( ( nth_list_a @ ( concat_list_a @ Xss ) @ K2 )
= ( nth_list_a @ ( nth_list_list_a @ Xss @ I4 ) @ J ) )
& ( ( drop_list_a @ ( suc @ K2 ) @ ( concat_list_a @ Xss ) )
= ( append_list_a @ ( drop_list_a @ ( suc @ J ) @ Ys ) @ ( concat_list_a @ ( drop_list_list_a @ ( suc @ I4 ) @ Xss ) ) ) ) ) ) ) ) ) ).
% take_nth_drop_concat
thf(fact_1078_take__nth__drop__concat,axiom,
! [I4: nat,Xss: list_list_a,Ys: list_a,J: nat,Z2: a] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xss ) )
=> ( ( ( nth_list_a @ Xss @ I4 )
= Ys )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Ys ) )
=> ( ( ( nth_a @ Ys @ J )
= Z2 )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_size_list_a @ ( concat_a @ Xss ) ) )
& ( ( take_a @ K2 @ ( concat_a @ Xss ) )
= ( append_a @ ( concat_a @ ( take_list_a @ I4 @ Xss ) ) @ ( take_a @ J @ Ys ) ) )
& ( ( nth_a @ ( concat_a @ Xss ) @ K2 )
= ( nth_a @ ( nth_list_a @ Xss @ I4 ) @ J ) )
& ( ( drop_a @ ( suc @ K2 ) @ ( concat_a @ Xss ) )
= ( append_a @ ( drop_a @ ( suc @ J ) @ Ys ) @ ( concat_a @ ( drop_list_a @ ( suc @ I4 ) @ Xss ) ) ) ) ) ) ) ) ) ).
% take_nth_drop_concat
thf(fact_1079_in__measures_I1_J,axiom,
! [X: list_a,Y: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ nil_list_a_nat ) ) ).
% in_measures(1)
thf(fact_1080_in__measures_I1_J,axiom,
! [X: a,Y: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ nil_a_nat ) ) ).
% in_measures(1)
thf(fact_1081_in__measures_I1_J,axiom,
! [X: nat,Y: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).
% in_measures(1)
thf(fact_1082_in__measures_I1_J,axiom,
! [X: list_nat,Y: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ nil_list_nat_nat ) ) ).
% in_measures(1)
thf(fact_1083_in__measures_I1_J,axiom,
! [X: list_list_a,Y: list_list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ ( measures_list_list_a @ nil_list_list_a_nat ) ) ).
% in_measures(1)
thf(fact_1084_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_1085_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1086_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_1087_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_1088_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_1089_Nil__lenlex__iff1,axiom,
! [Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ns ) @ ( lenlex_a @ R2 ) )
= ( Ns != nil_a ) ) ).
% Nil_lenlex_iff1
thf(fact_1090_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_1091_Nil__lenlex__iff1,axiom,
! [Ns: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Ns ) @ ( lenlex_list_a @ R2 ) )
= ( Ns != nil_list_a ) ) ).
% Nil_lenlex_iff1
thf(fact_1092_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_1093_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_1094_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_1095_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_1096_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_1097_drop__all,axiom,
! [Xs: list_list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N )
=> ( ( drop_list_a @ N @ Xs )
= nil_list_a ) ) ).
% drop_all
thf(fact_1098_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1099_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_1100_drop__eq__Nil,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( drop_list_a @ N @ Xs )
= nil_list_a )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1101_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1102_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_1103_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_list_a] :
( ( nil_list_a
= ( drop_list_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1104_add__elem__list__lists_Osimps_I1_J,axiom,
! [X: a] :
( ( basic_189371981877014951ists_a @ X @ nil_a )
= ( cons_list_a @ ( cons_a @ X @ nil_a ) @ nil_list_a ) ) ).
% add_elem_list_lists.simps(1)
thf(fact_1105_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss2: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss2 ) )
=> ~ ! [X3: a,Xs2: list_a,Xss2: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_1106_list__3__cases,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X3: a] :
( Xs
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Ys4: list_a] :
( Xs
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ Ys4 ) ) ) ) ) ).
% list_3_cases
thf(fact_1107_list__4__cases,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X3: a] :
( Xs
!= ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Y3: a] :
( Xs
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ nil_a ) ) )
=> ~ ! [X3: a,Y3: a,Z4: a,Zs: list_a] :
( Xs
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ ( cons_a @ Z4 @ Zs ) ) ) ) ) ) ) ).
% list_4_cases
thf(fact_1108_distinct__eq_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [Uu: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ Uu @ nil_a ) )
=> ~ ! [Eq: a > a > $o,X3: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ Eq @ ( cons_a @ X3 @ Xs2 ) ) ) ) ).
% distinct_eq.cases
thf(fact_1109_span_Ocases,axiom,
! [X: produc7567695631791103674list_a] :
( ! [P6: a > $o,X3: a,Xs2: list_a] :
( X
!= ( produc5033538197335023340list_a @ P6 @ ( cons_a @ X3 @ Xs2 ) ) )
=> ~ ! [Uu: a > $o] :
( X
!= ( produc5033538197335023340list_a @ Uu @ nil_a ) ) ) ).
% span.cases
thf(fact_1110_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_1111_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_1112_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_1113_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1114_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y5: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1115_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P2 @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys4: list_a] : ( P2 @ nil_a @ ( cons_a @ Y3 @ Ys4 ) )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1116_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1117_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P6: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P6 @ nil_a ) )
=> ( ! [P6: a > a > $o,X3: a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [P6: a > a > $o,X3: a,Y3: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1118_eq__length__concat__nth,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
=> ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
= ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1119_eq__length__concat__nth,axiom,
! [Xs: list_list_nat,Ys: list_list_list_a] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s2403821588304063868list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
= ( size_s349497388124573686list_a @ ( nth_list_list_a @ Ys @ I2 ) ) ) )
=> ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
= ( size_s349497388124573686list_a @ ( concat_list_a @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1120_eq__length__concat__nth,axiom,
! [Xs: list_list_list_a,Ys: list_list_nat] :
( ( ( size_s2403821588304063868list_a @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2403821588304063868list_a @ Xs ) )
=> ( ( size_s349497388124573686list_a @ ( nth_list_list_a @ Xs @ I2 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
=> ( ( size_s349497388124573686list_a @ ( concat_list_a @ Xs ) )
= ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1121_eq__length__concat__nth,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a] :
( ( ( size_s2403821588304063868list_a @ Xs )
= ( size_s2403821588304063868list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2403821588304063868list_a @ Xs ) )
=> ( ( size_s349497388124573686list_a @ ( nth_list_list_a @ Xs @ I2 ) )
= ( size_s349497388124573686list_a @ ( nth_list_list_a @ Ys @ I2 ) ) ) )
=> ( ( size_s349497388124573686list_a @ ( concat_list_a @ Xs ) )
= ( size_s349497388124573686list_a @ ( concat_list_a @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1122_eq__length__concat__nth,axiom,
! [Xs: list_list_nat,Ys: list_list_a] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
= ( size_size_list_a @ ( nth_list_a @ Ys @ I2 ) ) ) )
=> ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
= ( size_size_list_a @ ( concat_a @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1123_eq__length__concat__nth,axiom,
! [Xs: list_list_list_a,Ys: list_list_a] :
( ( ( size_s2403821588304063868list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2403821588304063868list_a @ Xs ) )
=> ( ( size_s349497388124573686list_a @ ( nth_list_list_a @ Xs @ I2 ) )
= ( size_size_list_a @ ( nth_list_a @ Ys @ I2 ) ) ) )
=> ( ( size_s349497388124573686list_a @ ( concat_list_a @ Xs ) )
= ( size_size_list_a @ ( concat_a @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1124_eq__length__concat__nth,axiom,
! [Xs: list_list_a,Ys: list_list_nat] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( size_size_list_a @ ( nth_list_a @ Xs @ I2 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
=> ( ( size_size_list_a @ ( concat_a @ Xs ) )
= ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1125_eq__length__concat__nth,axiom,
! [Xs: list_list_a,Ys: list_list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s2403821588304063868list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( size_size_list_a @ ( nth_list_a @ Xs @ I2 ) )
= ( size_s349497388124573686list_a @ ( nth_list_list_a @ Ys @ I2 ) ) ) )
=> ( ( size_size_list_a @ ( concat_a @ Xs ) )
= ( size_s349497388124573686list_a @ ( concat_list_a @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1126_eq__length__concat__nth,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( size_size_list_a @ ( nth_list_a @ Xs @ I2 ) )
= ( size_size_list_a @ ( nth_list_a @ Ys @ I2 ) ) ) )
=> ( ( size_size_list_a @ ( concat_a @ Xs ) )
= ( size_size_list_a @ ( concat_a @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1127_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a,Z4: a,Zs: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1128_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_a,Ws: list_nat,P2: list_a > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a,Z4: a,Zs: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1129_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_nat,Ws: list_a,P2: list_a > list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a,Z4: nat,Zs: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1130_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_nat,Ws: list_nat,P2: list_a > list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a,Z4: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1131_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs2: list_a,Ws: list_a,P2: list_a > list_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_nat @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z4: a,Zs: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1132_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs2: list_a,Ws: list_nat,P2: list_a > list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z4: a,Zs: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1133_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs2: list_nat,Ws: list_a,P2: list_a > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z4: nat,Zs: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_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_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1134_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs2: list_nat,Ws: list_nat,P2: list_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z4: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_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 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1135_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs2: list_a,Ws: list_a,P2: list_nat > list_a > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z4: a,Zs: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1136_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs2: list_a,Ws: list_nat,P2: list_nat > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z4: a,Zs: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1137_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a,Z4: a,Zs: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1138_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_nat,P2: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a,Z4: nat,Zs: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1139_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs2: list_a,P2: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ nil_a @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z4: a,Zs: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1140_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs2: list_nat,P2: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z4: nat,Zs: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1141_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs2: list_a,P2: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ nil_nat @ nil_a @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z4: a,Zs: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1142_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs2: list_nat,P2: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z4: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1143_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs2: list_a,P2: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z4: a,Zs: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1144_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,P2: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z4: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1145_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_list_a,P2: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a,Z4: list_a,Zs: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_list_a @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1146_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs2: list_list_a,P2: list_a > list_nat > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ nil_a @ nil_nat @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z4: list_a,Zs: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_list_a @ Z4 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1147_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1148_list__induct2,axiom,
! [Xs: list_a,Ys: list_nat,P2: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P2 @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1149_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P2: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys4: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1150_list__induct2,axiom,
! [Xs: list_nat,Ys: list_a,P2: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys4: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1151_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P2: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1152_list__induct2,axiom,
! [Xs: list_nat,Ys: list_list_a,P2: list_nat > list_list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_nat @ nil_list_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_a,Ys4: list_list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1153_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P2: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys4: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1154_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_nat,P2: list_list_a > list_nat > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_nat )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: nat,Ys4: list_nat] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1155_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P2: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys4: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1156_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1157_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1158_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1159_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X3: a,Xs2: list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_1160_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys4: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys4 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_1161_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs2: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs2 ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_a @ Ys6 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1162_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs2: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs2 )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs2
= ( cons_a @ X @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1163_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1164_mem__idx_Ocases,axiom,
! [X: produc8685980395799941037list_a] :
( ! [Uu: a] :
( X
!= ( produc6670463072477821725list_a @ Uu @ nil_a ) )
=> ~ ! [X3: a,A3: a,As2: list_a] :
( X
!= ( produc6670463072477821725list_a @ X3 @ ( cons_a @ A3 @ As2 ) ) ) ) ).
% mem_idx.cases
thf(fact_1165_map__tailrec__rev_Ocases,axiom,
! [X: produc1473018763691903991list_a] :
( ! [F2: a > a,Bs2: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ nil_a @ Bs2 ) ) )
=> ~ ! [F2: a > a,A3: a,As2: list_a,Bs2: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ ( cons_a @ A3 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_1166_map__tailrec__rev_Ocases,axiom,
! [X: produc1616951275169580055st_nat] :
( ! [F2: nat > nat,Bs2: list_nat] :
( X
!= ( produc4626581765195395529st_nat @ F2 @ ( produc2694037385005941721st_nat @ nil_nat @ Bs2 ) ) )
=> ~ ! [F2: nat > nat,A3: nat,As2: list_nat,Bs2: list_nat] :
( X
!= ( produc4626581765195395529st_nat @ F2 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_1167_map__tailrec__rev_Ocases,axiom,
! [X: produc4304492727526933495list_a] :
( ! [F2: list_a > list_a,Bs2: list_list_a] :
( X
!= ( produc7935328968026918825list_a @ F2 @ ( produc8696003437204565271list_a @ nil_list_a @ Bs2 ) ) )
=> ~ ! [F2: list_a > list_a,A3: list_a,As2: list_list_a,Bs2: list_list_a] :
( X
!= ( produc7935328968026918825list_a @ F2 @ ( produc8696003437204565271list_a @ ( cons_list_a @ A3 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_1168_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys4: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys4 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X3: a,Xs2: list_a,Y3: a,Ys4: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1169_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ( ! [Xs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ~ ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1170_shuffles_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Ys4: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys4 ) )
=> ( ! [Xs2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Xs2 @ nil_list_a ) )
=> ~ ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys4: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1171_filter2_Ocases,axiom,
! [X: produc1392733710753813600list_a] :
( ! [P6: a > a > $o,Uu: list_a] :
( X
!= ( produc8684083420967988176list_a @ P6 @ ( produc6837034575241423639list_a @ nil_a @ Uu ) ) )
=> ( ! [P6: a > a > $o,V: a,Va: list_a] :
( X
!= ( produc8684083420967988176list_a @ P6 @ ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ nil_a ) ) )
=> ~ ! [P6: a > a > $o,A3: a,As2: list_a,B3: a,Bs2: list_a] :
( X
!= ( produc8684083420967988176list_a @ P6 @ ( produc6837034575241423639list_a @ ( cons_a @ A3 @ As2 ) @ ( cons_a @ B3 @ Bs2 ) ) ) ) ) ) ).
% filter2.cases
thf(fact_1172_filter2_Ocases,axiom,
! [X: produc4787317212837456354st_nat] :
( ! [P6: nat > nat > $o,Uu: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ nil_nat @ Uu ) ) )
=> ( ! [P6: nat > nat > $o,V: nat,Va: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) )
=> ~ ! [P6: nat > nat > $o,A3: nat,As2: list_nat,B3: nat,Bs2: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As2 ) @ ( cons_nat @ B3 @ Bs2 ) ) ) ) ) ) ).
% filter2.cases
thf(fact_1173_filter2_Ocases,axiom,
! [X: produc2719896074235046368list_a] :
( ! [P6: list_a > list_a > $o,Uu: list_list_a] :
( X
!= ( produc3685947910518003536list_a @ P6 @ ( produc8696003437204565271list_a @ nil_list_a @ Uu ) ) )
=> ( ! [P6: list_a > list_a > $o,V: list_a,Va: list_list_a] :
( X
!= ( produc3685947910518003536list_a @ P6 @ ( produc8696003437204565271list_a @ ( cons_list_a @ V @ Va ) @ nil_list_a ) ) )
=> ~ ! [P6: list_a > list_a > $o,A3: list_a,As2: list_list_a,B3: list_a,Bs2: list_list_a] :
( X
!= ( produc3685947910518003536list_a @ P6 @ ( produc8696003437204565271list_a @ ( cons_list_a @ A3 @ As2 ) @ ( cons_list_a @ B3 @ Bs2 ) ) ) ) ) ) ).
% filter2.cases
thf(fact_1174_subtract__list__sorted_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ~ ! [V: nat,Va: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).
% subtract_list_sorted.cases
thf(fact_1175_union__list__sorted_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ~ ! [V: nat,Va: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).
% union_list_sorted.cases
thf(fact_1176_list__inter_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Bs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Bs2 ) )
=> ~ ! [A3: a,As2: list_a,Bs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ A3 @ As2 ) @ Bs2 ) ) ) ).
% list_inter.cases
thf(fact_1177_list__inter_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Bs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Bs2 ) )
=> ~ ! [A3: nat,As2: list_nat,Bs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As2 ) @ Bs2 ) ) ) ).
% list_inter.cases
thf(fact_1178_list__inter_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Bs2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Bs2 ) )
=> ~ ! [A3: list_a,As2: list_list_a,Bs2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ A3 @ As2 ) @ Bs2 ) ) ) ).
% list_inter.cases
thf(fact_1179_distinct__singleton,axiom,
! [X: a] : ( distinct_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_singleton
thf(fact_1180_not__Nil__listrel1,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_1181_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_1182_not__Nil__listrel1,axiom,
! [Xs: list_list_a,R2: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Xs ) @ ( listrel1_list_a @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_1183_not__listrel1__Nil,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_1184_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_1185_not__listrel1__Nil,axiom,
! [Xs: list_list_a,R2: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ nil_list_a ) @ ( listrel1_list_a @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_1186_listrel_ONil,axiom,
! [R2: set_Pr4048851178543822343list_a] : ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ nil_list_a ) @ ( listre6772471554020304241list_a @ R2 ) ) ).
% listrel.Nil
thf(fact_1187_sum__list__1__E,axiom,
! [Xs: list_nat] :
( ( ( groups4561878855575611511st_nat @ Xs )
= ( suc @ zero_zero_nat ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I2 )
= ( suc @ zero_zero_nat ) )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_size_list_nat @ Xs ) )
=> ( ( J4 != I2 )
=> ( ( nth_nat @ Xs @ J4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_list_1_E
thf(fact_1188_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_1189_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_1190_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1191_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1192_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_1193_diff__diff__left,axiom,
! [I4: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1194_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_1195_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_1196_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_1197_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1198_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1199_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1200_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_1201_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I4 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1202_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1203_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_1204_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1205_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_1206_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_1207_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1208_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1209_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_1210_less__diff__conv,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1211_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1212_le__diff__conv,axiom,
! [J: nat,K: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I4 @ K ) ) ) ).
% le_diff_conv
thf(fact_1213_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1214_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1215_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1216_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ( minus_minus_nat @ J @ I4 )
= K )
= ( J
= ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1217_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1218_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1219_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1220_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1221_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_1222_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_1223_diff__commute,axiom,
! [I4: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1224_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_1225_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1226_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1227_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_1228_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_1229_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_1230_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_1231_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1232_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_1233_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I4: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1234_diff__Suc__less,axiom,
! [N: nat,I4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1235_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1236_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1237_less__diff__conv2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1238_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I3: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_1239_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_1240_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K5: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K5 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K5 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K5 ) @ ( minus_minus_nat @ M2 @ ( suc @ K5 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_1241_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1242_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1243_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1244_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1245_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1246_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1247_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1248_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1249_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1250_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1251_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1252_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1253_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M7: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X5: nat] :
( ( P2 @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
% Helper facts (17)
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_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_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_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X: list_P1396940483166286381od_a_a,Y: list_P1396940483166286381od_a_a] :
( ( if_lis931442767461590515od_a_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X: list_P1396940483166286381od_a_a,Y: list_P1396940483166286381od_a_a] :
( ( if_lis931442767461590515od_a_a @ $true @ X @ Y )
= X ) ).
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 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_T,axiom,
! [X: list_P321204300973800749list_a,Y: list_P321204300973800749list_a] :
( ( if_lis2255084897382415091list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_T,axiom,
! [X: list_P321204300973800749list_a,Y: list_P321204300973800749list_a] :
( ( if_lis2255084897382415091list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
! [X: list_P7940050157051400743st_nat,Y: list_P7940050157051400743st_nat] :
( ( if_lis5496265267951696621st_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
! [X: list_P7940050157051400743st_nat,Y: list_P7940050157051400743st_nat] :
( ( if_lis5496265267951696621st_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
! [X: list_P3352141576573427245list_a,Y: list_P3352141576573427245list_a] :
( ( if_lis8043406425296152051list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
! [X: list_P3352141576573427245list_a,Y: list_P3352141576573427245list_a] :
( ( if_lis8043406425296152051list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ x @ xsa ) @ ( cons_a @ y @ ysa ) ) @ ( trancl_list_a @ r ) ).
%------------------------------------------------------------------------------