TPTP Problem File: ITP092^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP092^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Kuratowski problem prob_158__5524402_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Kuratowski/prob_158__5524402_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 566 ( 174 unt; 207 typ; 0 def)
% Number of atoms : 1080 ( 694 equ; 0 cnn)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 4874 ( 229 ~; 59 |; 141 &;3982 @)
% ( 0 <=>; 463 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 56 ( 55 usr)
% Number of type conns : 475 ( 475 >; 0 *; 0 +; 0 <<)
% Number of symbols : 155 ( 152 usr; 9 con; 0-3 aty)
% Number of variables : 1599 ( 12 ^;1498 !; 89 ?;1599 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:44:40.460
%------------------------------------------------------------------------------
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
collec1811041560od_a_a: ( list_P1467822637od_a_a > $o ) > set_li562223245od_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
collec1566074578od_a_a: ( produc651658087od_a_a > $o ) > set_Pr800560199od_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
collec656258194list_a: ( produc864062247list_a > $o ) > set_Pr1971762695list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
the_el1640649244od_a_a: set_Product_prod_a_a > product_prod_a_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mtf__a_J,type,
accp_P941981633_a_a_a: ( produc657167928_a_a_a > produc657167928_a_a_a > $o ) > produc657167928_a_a_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
accp_P1029463583od_a_a: ( produc821581462od_a_a > produc821581462od_a_a > $o ) > produc821581462od_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1578987478od_a_a: list_P1467822637od_a_a > set_li562223245od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member372644687od_a_a: produc501911462od_a_a > set_Pr727105414od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__a_Mt__Nat__Onat_J_M_062_Itf__a_Mt__Nat__Onat_J_J,type,
member125757712_a_nat: produc1178641895_a_nat > set_Pr1940702919_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_062_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_I_062_Itf__a_Mt__Nat__Onat_J_J_J,type,
member688761744_a_nat: produc1334933863_a_nat > set_Pr753687879_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member8108048od_a_a: produc657724903od_a_a > set_Pr1239640519od_a_a > $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,
member888406864list_a: produc403652391list_a > set_Pr1742859527list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member1254757008od_a_a: produc205693799od_a_a > set_Pr779272263od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1691432592od_a_a: produc651658087od_a_a > set_Pr800560199od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mtf__a_J,type,
member430654177_a_a_a: produc657167928_a_a_a > set_Pr1525541400_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member1250517072list_a: produc864062247list_a > set_Pr1971762695list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1474739327od_a_a: produc1701253078od_a_a > set_Pr248149174od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mtf__a_J_J,type,
member1904151562_a_a_a: produc1197928545_a_a_a > set_Pr110478017_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_e,type,
e: product_prod_a_a ).
thf(sy_v_es,type,
es: list_P1467822637od_a_a ).
% Relevant facts (355)
thf(fact_0_assms,axiom,
kurato1072996284sing_a @ ( cons_P1921020637od_a_a @ e @ es ) ).
% assms
thf(fact_1_progressing__def,axiom,
( kurato1404746725od_a_a
= ( ^ [P: list_P1891567085od_a_a] :
! [Xs: list_P1891567085od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Ys: list_P1891567085od_a_a] :
( P
!= ( append864718978od_a_a @ Xs @ ( cons_P1070399645od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( cons_P1070399645od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Ys ) ) ) ) ) ) ).
% progressing_def
thf(fact_2_progressing__def,axiom,
( kurato292963650list_a
= ( ^ [P: list_P601816365list_a] :
! [Xs: list_P601816365list_a,X: list_a,Y: list_a,Ys: list_P601816365list_a] :
( P
!= ( append750086082list_a @ Xs @ ( cons_P2080942813list_a @ ( produc1448356631list_a @ X @ Y ) @ ( cons_P2080942813list_a @ ( produc1448356631list_a @ Y @ X ) @ Ys ) ) ) ) ) ) ).
% progressing_def
thf(fact_3_progressing__def,axiom,
( kurato1072996284sing_a
= ( ^ [P: list_P1467822637od_a_a] :
! [Xs: list_P1467822637od_a_a,X: a,Y: a,Ys: list_P1467822637od_a_a] :
( P
!= ( append1574001858od_a_a @ Xs @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Ys ) ) ) ) ) ) ).
% progressing_def
thf(fact_4_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_5_append_Oassoc,axiom,
! [A: list_P1891567085od_a_a,B: list_P1891567085od_a_a,C: list_P1891567085od_a_a] :
( ( append864718978od_a_a @ ( append864718978od_a_a @ A @ B ) @ C )
= ( append864718978od_a_a @ A @ ( append864718978od_a_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_6_append_Oassoc,axiom,
! [A: list_l1108175539od_a_a,B: list_l1108175539od_a_a,C: list_l1108175539od_a_a] :
( ( append2141280328od_a_a @ ( append2141280328od_a_a @ A @ B ) @ C )
= ( append2141280328od_a_a @ A @ ( append2141280328od_a_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_7_append_Oassoc,axiom,
! [A: list_P1467822637od_a_a,B: list_P1467822637od_a_a,C: list_P1467822637od_a_a] :
( ( append1574001858od_a_a @ ( append1574001858od_a_a @ A @ B ) @ C )
= ( append1574001858od_a_a @ A @ ( append1574001858od_a_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_8_append__assoc,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs2 @ Ys2 ) @ Zs )
= ( append_a @ Xs2 @ ( append_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_9_append__assoc,axiom,
! [Xs2: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,Zs: list_P1891567085od_a_a] :
( ( append864718978od_a_a @ ( append864718978od_a_a @ Xs2 @ Ys2 ) @ Zs )
= ( append864718978od_a_a @ Xs2 @ ( append864718978od_a_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_10_append__assoc,axiom,
! [Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,Zs: list_l1108175539od_a_a] :
( ( append2141280328od_a_a @ ( append2141280328od_a_a @ Xs2 @ Ys2 ) @ Zs )
= ( append2141280328od_a_a @ Xs2 @ ( append2141280328od_a_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_11_append__assoc,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( append1574001858od_a_a @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) @ Zs )
= ( append1574001858od_a_a @ Xs2 @ ( append1574001858od_a_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_12_append__same__eq,axiom,
! [Ys2: list_a,Xs2: list_a,Zs: list_a] :
( ( ( append_a @ Ys2 @ Xs2 )
= ( append_a @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_13_append__same__eq,axiom,
! [Ys2: list_P1891567085od_a_a,Xs2: list_P1891567085od_a_a,Zs: list_P1891567085od_a_a] :
( ( ( append864718978od_a_a @ Ys2 @ Xs2 )
= ( append864718978od_a_a @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_14_append__same__eq,axiom,
! [Ys2: list_l1108175539od_a_a,Xs2: list_l1108175539od_a_a,Zs: list_l1108175539od_a_a] :
( ( ( append2141280328od_a_a @ Ys2 @ Xs2 )
= ( append2141280328od_a_a @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_15_append__same__eq,axiom,
! [Ys2: list_P1467822637od_a_a,Xs2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Ys2 @ Xs2 )
= ( append1574001858od_a_a @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_16_same__append__eq,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= ( append_a @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_17_same__append__eq,axiom,
! [Xs2: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,Zs: list_P1891567085od_a_a] :
( ( ( append864718978od_a_a @ Xs2 @ Ys2 )
= ( append864718978od_a_a @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_18_same__append__eq,axiom,
! [Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,Zs: list_l1108175539od_a_a] :
( ( ( append2141280328od_a_a @ Xs2 @ Ys2 )
= ( append2141280328od_a_a @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_19_same__append__eq,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Xs2 @ Ys2 )
= ( append1574001858od_a_a @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_20_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_21_list_Oinject,axiom,
! [X21: produc1572603623od_a_a,X22: list_P1891567085od_a_a,Y21: produc1572603623od_a_a,Y22: list_P1891567085od_a_a] :
( ( ( cons_P1070399645od_a_a @ X21 @ X22 )
= ( cons_P1070399645od_a_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_22_list_Oinject,axiom,
! [X21: list_P1467822637od_a_a,X22: list_l1108175539od_a_a,Y21: list_P1467822637od_a_a,Y22: list_l1108175539od_a_a] :
( ( ( cons_l724116707od_a_a @ X21 @ X22 )
= ( cons_l724116707od_a_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_23_list_Oinject,axiom,
! [X21: a > nat,X22: list_a_nat,Y21: a > nat,Y22: list_a_nat] :
( ( ( cons_a_nat @ X21 @ X22 )
= ( cons_a_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_24_list_Oinject,axiom,
! [X21: product_prod_a_a,X22: list_P1467822637od_a_a,Y21: product_prod_a_a,Y22: list_P1467822637od_a_a] :
( ( ( cons_P1921020637od_a_a @ X21 @ X22 )
= ( cons_P1921020637od_a_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_25_prod_Oinject,axiom,
! [X1: list_P1467822637od_a_a,X2: a,Y1: list_P1467822637od_a_a,Y2: a] :
( ( ( produc998622632_a_a_a @ X1 @ X2 )
= ( produc998622632_a_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_26_prod_Oinject,axiom,
! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
( ( ( produc1448356631list_a @ X1 @ X2 )
= ( produc1448356631list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_27_prod_Oinject,axiom,
! [X1: product_prod_a_a > product_prod_a_a > $o,X2: list_P1467822637od_a_a,Y1: product_prod_a_a > product_prod_a_a > $o,Y2: list_P1467822637od_a_a] :
( ( ( produc214132118od_a_a @ X1 @ X2 )
= ( produc214132118od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_28_prod_Oinject,axiom,
! [X1: a,X2: produc657167928_a_a_a,Y1: a,Y2: produc657167928_a_a_a] :
( ( ( produc1440321489_a_a_a @ X1 @ X2 )
= ( produc1440321489_a_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_29_prod_Oinject,axiom,
! [X1: a,X2: list_P1467822637od_a_a,Y1: a,Y2: list_P1467822637od_a_a] :
( ( ( produc575024326od_a_a @ X1 @ X2 )
= ( produc575024326od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_30_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_31_old_Oprod_Oinject,axiom,
! [A: list_P1467822637od_a_a,B: a,A2: list_P1467822637od_a_a,B2: a] :
( ( ( produc998622632_a_a_a @ A @ B )
= ( produc998622632_a_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_32_old_Oprod_Oinject,axiom,
! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
( ( ( produc1448356631list_a @ A @ B )
= ( produc1448356631list_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_33_old_Oprod_Oinject,axiom,
! [A: product_prod_a_a > product_prod_a_a > $o,B: list_P1467822637od_a_a,A2: product_prod_a_a > product_prod_a_a > $o,B2: list_P1467822637od_a_a] :
( ( ( produc214132118od_a_a @ A @ B )
= ( produc214132118od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_34_old_Oprod_Oinject,axiom,
! [A: a,B: produc657167928_a_a_a,A2: a,B2: produc657167928_a_a_a] :
( ( ( produc1440321489_a_a_a @ A @ B )
= ( produc1440321489_a_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_35_old_Oprod_Oinject,axiom,
! [A: a,B: list_P1467822637od_a_a,A2: a,B2: list_P1467822637od_a_a] :
( ( ( produc575024326od_a_a @ A @ B )
= ( produc575024326od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_36_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_37_Cons__eq__appendI,axiom,
! [X3: a,Xs1: list_a,Ys2: list_a,Xs2: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X3 @ Xs2 )
= ( append_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_38_Cons__eq__appendI,axiom,
! [X3: produc1572603623od_a_a,Xs1: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,Xs2: list_P1891567085od_a_a,Zs: list_P1891567085od_a_a] :
( ( ( cons_P1070399645od_a_a @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append864718978od_a_a @ Xs1 @ Zs ) )
=> ( ( cons_P1070399645od_a_a @ X3 @ Xs2 )
= ( append864718978od_a_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_39_Cons__eq__appendI,axiom,
! [X3: list_P1467822637od_a_a,Xs1: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,Xs2: list_l1108175539od_a_a,Zs: list_l1108175539od_a_a] :
( ( ( cons_l724116707od_a_a @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append2141280328od_a_a @ Xs1 @ Zs ) )
=> ( ( cons_l724116707od_a_a @ X3 @ Xs2 )
= ( append2141280328od_a_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_40_Cons__eq__appendI,axiom,
! [X3: a > nat,Xs1: list_a_nat,Ys2: list_a_nat,Xs2: list_a_nat,Zs: list_a_nat] :
( ( ( cons_a_nat @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_a_nat @ Xs1 @ Zs ) )
=> ( ( cons_a_nat @ X3 @ Xs2 )
= ( append_a_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_41_Cons__eq__appendI,axiom,
! [X3: product_prod_a_a,Xs1: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Xs2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( ( cons_P1921020637od_a_a @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append1574001858od_a_a @ Xs1 @ Zs ) )
=> ( ( cons_P1921020637od_a_a @ X3 @ Xs2 )
= ( append1574001858od_a_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_42_append__Cons,axiom,
! [X3: a,Xs2: list_a,Ys2: list_a] :
( ( append_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 )
= ( cons_a @ X3 @ ( append_a @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_43_append__Cons,axiom,
! [X3: produc1572603623od_a_a,Xs2: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a] :
( ( append864718978od_a_a @ ( cons_P1070399645od_a_a @ X3 @ Xs2 ) @ Ys2 )
= ( cons_P1070399645od_a_a @ X3 @ ( append864718978od_a_a @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_44_append__Cons,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a] :
( ( append2141280328od_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) @ Ys2 )
= ( cons_l724116707od_a_a @ X3 @ ( append2141280328od_a_a @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_45_append__Cons,axiom,
! [X3: a > nat,Xs2: list_a_nat,Ys2: list_a_nat] :
( ( append_a_nat @ ( cons_a_nat @ X3 @ Xs2 ) @ Ys2 )
= ( cons_a_nat @ X3 @ ( append_a_nat @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_46_append__Cons,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( append1574001858od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ Ys2 )
= ( cons_P1921020637od_a_a @ X3 @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_47_append__eq__appendI,axiom,
! [Xs2: list_a,Xs1: list_a,Zs: list_a,Ys2: list_a,Us: list_a] :
( ( ( append_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs2 @ Ys2 )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_48_append__eq__appendI,axiom,
! [Xs2: list_P1891567085od_a_a,Xs1: list_P1891567085od_a_a,Zs: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,Us: list_P1891567085od_a_a] :
( ( ( append864718978od_a_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append864718978od_a_a @ Xs1 @ Us ) )
=> ( ( append864718978od_a_a @ Xs2 @ Ys2 )
= ( append864718978od_a_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_49_append__eq__appendI,axiom,
! [Xs2: list_l1108175539od_a_a,Xs1: list_l1108175539od_a_a,Zs: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,Us: list_l1108175539od_a_a] :
( ( ( append2141280328od_a_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append2141280328od_a_a @ Xs1 @ Us ) )
=> ( ( append2141280328od_a_a @ Xs2 @ Ys2 )
= ( append2141280328od_a_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_50_append__eq__appendI,axiom,
! [Xs2: list_P1467822637od_a_a,Xs1: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Us: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append1574001858od_a_a @ Xs1 @ Us ) )
=> ( ( append1574001858od_a_a @ Xs2 @ Ys2 )
= ( append1574001858od_a_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_51_append__eq__append__conv2,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs2
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_a @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_52_append__eq__append__conv2,axiom,
! [Xs2: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,Zs: list_P1891567085od_a_a,Ts: list_P1891567085od_a_a] :
( ( ( append864718978od_a_a @ Xs2 @ Ys2 )
= ( append864718978od_a_a @ Zs @ Ts ) )
= ( ? [Us2: list_P1891567085od_a_a] :
( ( ( Xs2
= ( append864718978od_a_a @ Zs @ Us2 ) )
& ( ( append864718978od_a_a @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append864718978od_a_a @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append864718978od_a_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_53_append__eq__append__conv2,axiom,
! [Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,Zs: list_l1108175539od_a_a,Ts: list_l1108175539od_a_a] :
( ( ( append2141280328od_a_a @ Xs2 @ Ys2 )
= ( append2141280328od_a_a @ Zs @ Ts ) )
= ( ? [Us2: list_l1108175539od_a_a] :
( ( ( Xs2
= ( append2141280328od_a_a @ Zs @ Us2 ) )
& ( ( append2141280328od_a_a @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append2141280328od_a_a @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append2141280328od_a_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_54_append__eq__append__conv2,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,Ts: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Xs2 @ Ys2 )
= ( append1574001858od_a_a @ Zs @ Ts ) )
= ( ? [Us2: list_P1467822637od_a_a] :
( ( ( Xs2
= ( append1574001858od_a_a @ Zs @ Us2 ) )
& ( ( append1574001858od_a_a @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append1574001858od_a_a @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append1574001858od_a_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_55_old_Oprod_Oinducts,axiom,
! [P2: produc657167928_a_a_a > $o,Prod: produc657167928_a_a_a] :
( ! [A3: list_P1467822637od_a_a,B3: a] : ( P2 @ ( produc998622632_a_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_56_old_Oprod_Oinducts,axiom,
! [P2: produc864062247list_a > $o,Prod: produc864062247list_a] :
( ! [A3: list_a,B3: list_a] : ( P2 @ ( produc1448356631list_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_57_old_Oprod_Oinducts,axiom,
! [P2: produc501911462od_a_a > $o,Prod: produc501911462od_a_a] :
( ! [A3: product_prod_a_a > product_prod_a_a > $o,B3: list_P1467822637od_a_a] : ( P2 @ ( produc214132118od_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_58_old_Oprod_Oinducts,axiom,
! [P2: produc1197928545_a_a_a > $o,Prod: produc1197928545_a_a_a] :
( ! [A3: a,B3: produc657167928_a_a_a] : ( P2 @ ( produc1440321489_a_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_59_old_Oprod_Oinducts,axiom,
! [P2: produc1701253078od_a_a > $o,Prod: produc1701253078od_a_a] :
( ! [A3: a,B3: list_P1467822637od_a_a] : ( P2 @ ( produc575024326od_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_60_old_Oprod_Oinducts,axiom,
! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_61_old_Oprod_Oexhaust,axiom,
! [Y3: produc657167928_a_a_a] :
~ ! [A3: list_P1467822637od_a_a,B3: a] :
( Y3
!= ( produc998622632_a_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_62_old_Oprod_Oexhaust,axiom,
! [Y3: produc864062247list_a] :
~ ! [A3: list_a,B3: list_a] :
( Y3
!= ( produc1448356631list_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_63_old_Oprod_Oexhaust,axiom,
! [Y3: produc501911462od_a_a] :
~ ! [A3: product_prod_a_a > product_prod_a_a > $o,B3: list_P1467822637od_a_a] :
( Y3
!= ( produc214132118od_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_64_old_Oprod_Oexhaust,axiom,
! [Y3: produc1197928545_a_a_a] :
~ ! [A3: a,B3: produc657167928_a_a_a] :
( Y3
!= ( produc1440321489_a_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_65_old_Oprod_Oexhaust,axiom,
! [Y3: produc1701253078od_a_a] :
~ ! [A3: a,B3: list_P1467822637od_a_a] :
( Y3
!= ( produc575024326od_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_66_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_a_a] :
~ ! [A3: a,B3: a] :
( Y3
!= ( product_Pair_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_67_prod__induct3,axiom,
! [P2: produc1197928545_a_a_a > $o,X3: produc1197928545_a_a_a] :
( ! [A3: a,B3: list_P1467822637od_a_a,C2: a] : ( P2 @ ( produc1440321489_a_a_a @ A3 @ ( produc998622632_a_a_a @ B3 @ C2 ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_68_prod__cases3,axiom,
! [Y3: produc1197928545_a_a_a] :
~ ! [A3: a,B3: list_P1467822637od_a_a,C2: a] :
( Y3
!= ( produc1440321489_a_a_a @ A3 @ ( produc998622632_a_a_a @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_69_Pair__inject,axiom,
! [A: list_P1467822637od_a_a,B: a,A2: list_P1467822637od_a_a,B2: a] :
( ( ( produc998622632_a_a_a @ A @ B )
= ( produc998622632_a_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_70_Pair__inject,axiom,
! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
( ( ( produc1448356631list_a @ A @ B )
= ( produc1448356631list_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_71_Pair__inject,axiom,
! [A: product_prod_a_a > product_prod_a_a > $o,B: list_P1467822637od_a_a,A2: product_prod_a_a > product_prod_a_a > $o,B2: list_P1467822637od_a_a] :
( ( ( produc214132118od_a_a @ A @ B )
= ( produc214132118od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_72_Pair__inject,axiom,
! [A: a,B: produc657167928_a_a_a,A2: a,B2: produc657167928_a_a_a] :
( ( ( produc1440321489_a_a_a @ A @ B )
= ( produc1440321489_a_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_73_Pair__inject,axiom,
! [A: a,B: list_P1467822637od_a_a,A2: a,B2: list_P1467822637od_a_a] :
( ( ( produc575024326od_a_a @ A @ B )
= ( produc575024326od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_74_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_75_prod__cases,axiom,
! [P2: produc657167928_a_a_a > $o,P3: produc657167928_a_a_a] :
( ! [A3: list_P1467822637od_a_a,B3: a] : ( P2 @ ( produc998622632_a_a_a @ A3 @ B3 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_76_prod__cases,axiom,
! [P2: produc864062247list_a > $o,P3: produc864062247list_a] :
( ! [A3: list_a,B3: list_a] : ( P2 @ ( produc1448356631list_a @ A3 @ B3 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_77_prod__cases,axiom,
! [P2: produc501911462od_a_a > $o,P3: produc501911462od_a_a] :
( ! [A3: product_prod_a_a > product_prod_a_a > $o,B3: list_P1467822637od_a_a] : ( P2 @ ( produc214132118od_a_a @ A3 @ B3 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_78_prod__cases,axiom,
! [P2: produc1197928545_a_a_a > $o,P3: produc1197928545_a_a_a] :
( ! [A3: a,B3: produc657167928_a_a_a] : ( P2 @ ( produc1440321489_a_a_a @ A3 @ B3 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_79_prod__cases,axiom,
! [P2: produc1701253078od_a_a > $o,P3: produc1701253078od_a_a] :
( ! [A3: a,B3: list_P1467822637od_a_a] : ( P2 @ ( produc575024326od_a_a @ A3 @ B3 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_80_prod__cases,axiom,
! [P2: product_prod_a_a > $o,P3: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_81_surj__pair,axiom,
! [P3: produc657167928_a_a_a] :
? [X4: list_P1467822637od_a_a,Y4: a] :
( P3
= ( produc998622632_a_a_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_82_surj__pair,axiom,
! [P3: produc864062247list_a] :
? [X4: list_a,Y4: list_a] :
( P3
= ( produc1448356631list_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_83_surj__pair,axiom,
! [P3: produc501911462od_a_a] :
? [X4: product_prod_a_a > product_prod_a_a > $o,Y4: list_P1467822637od_a_a] :
( P3
= ( produc214132118od_a_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_84_surj__pair,axiom,
! [P3: produc1197928545_a_a_a] :
? [X4: a,Y4: produc657167928_a_a_a] :
( P3
= ( produc1440321489_a_a_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_85_surj__pair,axiom,
! [P3: produc1701253078od_a_a] :
? [X4: a,Y4: list_P1467822637od_a_a] :
( P3
= ( produc575024326od_a_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_86_surj__pair,axiom,
! [P3: product_prod_a_a] :
? [X4: a,Y4: a] :
( P3
= ( product_Pair_a_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_87_not__Cons__self2,axiom,
! [X3: a,Xs2: list_a] :
( ( cons_a @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_88_not__Cons__self2,axiom,
! [X3: produc1572603623od_a_a,Xs2: list_P1891567085od_a_a] :
( ( cons_P1070399645od_a_a @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_89_not__Cons__self2,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a] :
( ( cons_l724116707od_a_a @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_90_not__Cons__self2,axiom,
! [X3: a > nat,Xs2: list_a_nat] :
( ( cons_a_nat @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_91_not__Cons__self2,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( cons_P1921020637od_a_a @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_92_bind__simps_I2_J,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,F: product_prod_a_a > list_P1467822637od_a_a] :
( ( bind_P1783625031od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ F )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( bind_P1783625031od_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_93_bind__simps_I2_J,axiom,
! [X3: a,Xs2: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X3 @ Xs2 ) @ F )
= ( append_a @ ( F @ X3 ) @ ( bind_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_94_bind__simps_I2_J,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,F: product_prod_a_a > list_a] :
( ( bind_P293964574_a_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ F )
= ( append_a @ ( F @ X3 ) @ ( bind_P293964574_a_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_95_bind__simps_I2_J,axiom,
! [X3: a,Xs2: list_a,F: a > list_P1467822637od_a_a] :
( ( bind_a1679112752od_a_a @ ( cons_a @ X3 @ Xs2 ) @ F )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( bind_a1679112752od_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_96_bind__simps_I2_J,axiom,
! [X3: a > nat,Xs2: list_a_nat,F: ( a > nat ) > list_a] :
( ( bind_a_nat_a @ ( cons_a_nat @ X3 @ Xs2 ) @ F )
= ( append_a @ ( F @ X3 ) @ ( bind_a_nat_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_97_bind__simps_I2_J,axiom,
! [X3: a,Xs2: list_a,F: a > list_l1108175539od_a_a] :
( ( bind_a883992246od_a_a @ ( cons_a @ X3 @ Xs2 ) @ F )
= ( append2141280328od_a_a @ ( F @ X3 ) @ ( bind_a883992246od_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_98_bind__simps_I2_J,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a,F: list_P1467822637od_a_a > list_a] :
( ( bind_l1307590552_a_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) @ F )
= ( append_a @ ( F @ X3 ) @ ( bind_l1307590552_a_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_99_bind__simps_I2_J,axiom,
! [X3: a > nat,Xs2: list_a_nat,F: ( a > nat ) > list_P1467822637od_a_a] :
( ( bind_a1190333103od_a_a @ ( cons_a_nat @ X3 @ Xs2 ) @ F )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( bind_a1190333103od_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_100_bind__simps_I2_J,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,F: product_prod_a_a > list_l1108175539od_a_a] :
( ( bind_P240085581od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ F )
= ( append2141280328od_a_a @ ( F @ X3 ) @ ( bind_P240085581od_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_101_bind__simps_I2_J,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a,F: list_P1467822637od_a_a > list_P1467822637od_a_a] :
( ( bind_l618536897od_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) @ F )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( bind_l618536897od_a_a @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_102_maps__simps_I1_J,axiom,
! [F: product_prod_a_a > list_P1467822637od_a_a,X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( maps_P1529886977od_a_a @ F @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( maps_P1529886977od_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_103_maps__simps_I1_J,axiom,
! [F: a > list_a,X3: a,Xs2: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X3 @ Xs2 ) )
= ( append_a @ ( F @ X3 ) @ ( maps_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_104_maps__simps_I1_J,axiom,
! [F: product_prod_a_a > list_a,X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( maps_P321755352_a_a_a @ F @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) )
= ( append_a @ ( F @ X3 ) @ ( maps_P321755352_a_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_105_maps__simps_I1_J,axiom,
! [F: a > list_P1467822637od_a_a,X3: a,Xs2: list_a] :
( ( maps_a1706903530od_a_a @ F @ ( cons_a @ X3 @ Xs2 ) )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( maps_a1706903530od_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_106_maps__simps_I1_J,axiom,
! [F: ( a > nat ) > list_a,X3: a > nat,Xs2: list_a_nat] :
( ( maps_a_nat_a @ F @ ( cons_a_nat @ X3 @ Xs2 ) )
= ( append_a @ ( F @ X3 ) @ ( maps_a_nat_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_107_maps__simps_I1_J,axiom,
! [F: a > list_l1108175539od_a_a,X3: a,Xs2: list_a] :
( ( maps_a667520880od_a_a @ F @ ( cons_a @ X3 @ Xs2 ) )
= ( append2141280328od_a_a @ ( F @ X3 ) @ ( maps_a667520880od_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_108_maps__simps_I1_J,axiom,
! [F: list_P1467822637od_a_a > list_a,X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a] :
( ( maps_l1091119186_a_a_a @ F @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) )
= ( append_a @ ( F @ X3 ) @ ( maps_l1091119186_a_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_109_maps__simps_I1_J,axiom,
! [F: ( a > nat ) > list_P1467822637od_a_a,X3: a > nat,Xs2: list_a_nat] :
( ( maps_a836827497od_a_a @ F @ ( cons_a_nat @ X3 @ Xs2 ) )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( maps_a836827497od_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_110_maps__simps_I1_J,axiom,
! [F: product_prod_a_a > list_l1108175539od_a_a,X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( maps_P739084551od_a_a @ F @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) )
= ( append2141280328od_a_a @ ( F @ X3 ) @ ( maps_P739084551od_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_111_maps__simps_I1_J,axiom,
! [F: list_P1467822637od_a_a > list_P1467822637od_a_a,X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a] :
( ( maps_l1117535867od_a_a @ F @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) )
= ( append1574001858od_a_a @ ( F @ X3 ) @ ( maps_l1117535867od_a_a @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_112_lexord__append__left__rightI,axiom,
! [A: produc1572603623od_a_a,B: produc1572603623od_a_a,R: set_Pr1295299783od_a_a,U: list_P1891567085od_a_a,X3: list_P1891567085od_a_a,Y3: list_P1891567085od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R )
=> ( member1254757008od_a_a @ ( produc461662679od_a_a @ ( append864718978od_a_a @ U @ ( cons_P1070399645od_a_a @ A @ X3 ) ) @ ( append864718978od_a_a @ U @ ( cons_P1070399645od_a_a @ B @ Y3 ) ) ) @ ( lexord931792858od_a_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_113_lexord__append__left__rightI,axiom,
! [A: list_P1467822637od_a_a,B: list_P1467822637od_a_a,R: set_Pr800560199od_a_a,U: list_l1108175539od_a_a,X3: list_l1108175539od_a_a,Y3: list_l1108175539od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ A @ B ) @ R )
=> ( member8108048od_a_a @ ( produc915218007od_a_a @ ( append2141280328od_a_a @ U @ ( cons_l724116707od_a_a @ A @ X3 ) ) @ ( append2141280328od_a_a @ U @ ( cons_l724116707od_a_a @ B @ Y3 ) ) ) @ ( lexord1284731808od_a_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_114_lexord__append__left__rightI,axiom,
! [A: a > nat,B: a > nat,R: set_Pr1940702919_a_nat,U: list_a_nat,X3: list_a_nat,Y3: list_a_nat] :
( ( member125757712_a_nat @ ( produc1569872215_a_nat @ A @ B ) @ R )
=> ( member688761744_a_nat @ ( produc1430747863_a_nat @ ( append_a_nat @ U @ ( cons_a_nat @ A @ X3 ) ) @ ( append_a_nat @ U @ ( cons_a_nat @ B @ Y3 ) ) ) @ ( lexord_a_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_115_lexord__append__left__rightI,axiom,
! [A: list_a,B: list_a,R: set_Pr1971762695list_a,U: list_list_a,X3: list_list_a,Y3: list_list_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ A @ B ) @ R )
=> ( member888406864list_a @ ( produc2583831list_a @ ( append_list_a @ U @ ( cons_list_a @ A @ X3 ) ) @ ( append_list_a @ U @ ( cons_list_a @ B @ Y3 ) ) ) @ ( lexord_list_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_116_lexord__append__left__rightI,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,R: set_Pr1948701895od_a_a,U: list_P1467822637od_a_a,X3: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ U @ ( cons_P1921020637od_a_a @ A @ X3 ) ) @ ( append1574001858od_a_a @ U @ ( cons_P1921020637od_a_a @ B @ Y3 ) ) ) @ ( lexord673907738od_a_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_117_lexord__append__left__rightI,axiom,
! [A: a,B: a,R: set_Product_prod_a_a,U: list_a,X3: list_a,Y3: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( member1250517072list_a @ ( produc1448356631list_a @ ( append_a @ U @ ( cons_a @ A @ X3 ) ) @ ( append_a @ U @ ( cons_a @ B @ Y3 ) ) ) @ ( lexord_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_118_zip__Cons__Cons,axiom,
! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( zip_a_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) )
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( zip_a_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_119_zip__Cons__Cons,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a] :
( ( zip_Pr792472183od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) )
= ( cons_P1070399645od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ ( zip_Pr792472183od_a_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_120_zip__Cons__Cons,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Y3: a,Ys2: list_a] :
( ( zip_Pr201224270_a_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) )
= ( cons_P86081012_a_a_a @ ( produc2061588782_a_a_a @ X3 @ Y3 ) @ ( zip_Pr201224270_a_a_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_121_zip__Cons__Cons,axiom,
! [X3: a,Xs2: list_a,Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a] :
( ( zip_a_1586372448od_a_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) )
= ( cons_P991044742od_a_a @ ( produc1299253312od_a_a @ X3 @ Y3 ) @ ( zip_a_1586372448od_a_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_122_zip__Cons__Cons,axiom,
! [X3: a,Xs2: list_a,Y3: a > nat,Ys2: list_a_nat] :
( ( zip_a_a_nat @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a_nat @ Y3 @ Ys2 ) )
= ( cons_P1915851550_a_nat @ ( product_Pair_a_a_nat @ X3 @ Y3 ) @ ( zip_a_a_nat @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_123_zip__Cons__Cons,axiom,
! [X3: a > nat,Xs2: list_a_nat,Y3: a,Ys2: list_a] :
( ( zip_a_nat_a @ ( cons_a_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) )
= ( cons_P896312284_nat_a @ ( product_Pair_a_nat_a @ X3 @ Y3 ) @ ( zip_a_nat_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_124_zip__Cons__Cons,axiom,
! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a] :
( ( zip_list_a_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) )
= ( cons_P2080942813list_a @ ( produc1448356631list_a @ X3 @ Y3 ) @ ( zip_list_a_list_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_125_zip__Cons__Cons,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a,Y3: a,Ys2: list_a] :
( ( zip_li245157064_a_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) )
= ( cons_P169173742_a_a_a @ ( produc998622632_a_a_a @ X3 @ Y3 ) @ ( zip_li245157064_a_a_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_126_zip__Cons__Cons,axiom,
! [X3: a,Xs2: list_a,Y3: list_P1467822637od_a_a,Ys2: list_l1108175539od_a_a] :
( ( zip_a_1969042406od_a_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_l724116707od_a_a @ Y3 @ Ys2 ) )
= ( cons_P1213258892od_a_a @ ( produc575024326od_a_a @ X3 @ Y3 ) @ ( zip_a_1969042406od_a_a @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_127_zip__Cons__Cons,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Y3: a > nat,Ys2: list_a_nat] :
( ( zip_Pr24086287_a_nat @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_a_nat @ Y3 @ Ys2 ) )
= ( cons_P659009461_a_nat @ ( produc825466863_a_nat @ X3 @ Y3 ) @ ( zip_Pr24086287_a_nat @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_128_listrel1I,axiom,
! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a,R: set_Pr1295299783od_a_a,Xs2: list_P1891567085od_a_a,Us: list_P1891567085od_a_a,Vs: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ R )
=> ( ( Xs2
= ( append864718978od_a_a @ Us @ ( cons_P1070399645od_a_a @ X3 @ Vs ) ) )
=> ( ( Ys2
= ( append864718978od_a_a @ Us @ ( cons_P1070399645od_a_a @ Y3 @ Vs ) ) )
=> ( member1254757008od_a_a @ ( produc461662679od_a_a @ Xs2 @ Ys2 ) @ ( listre1832537542od_a_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_129_listrel1I,axiom,
! [X3: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a,R: set_Pr800560199od_a_a,Xs2: list_l1108175539od_a_a,Us: list_l1108175539od_a_a,Vs: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ X3 @ Y3 ) @ R )
=> ( ( Xs2
= ( append2141280328od_a_a @ Us @ ( cons_l724116707od_a_a @ X3 @ Vs ) ) )
=> ( ( Ys2
= ( append2141280328od_a_a @ Us @ ( cons_l724116707od_a_a @ Y3 @ Vs ) ) )
=> ( member8108048od_a_a @ ( produc915218007od_a_a @ Xs2 @ Ys2 ) @ ( listre651914124od_a_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_130_listrel1I,axiom,
! [X3: a > nat,Y3: a > nat,R: set_Pr1940702919_a_nat,Xs2: list_a_nat,Us: list_a_nat,Vs: list_a_nat,Ys2: list_a_nat] :
( ( member125757712_a_nat @ ( produc1569872215_a_nat @ X3 @ Y3 ) @ R )
=> ( ( Xs2
= ( append_a_nat @ Us @ ( cons_a_nat @ X3 @ Vs ) ) )
=> ( ( Ys2
= ( append_a_nat @ Us @ ( cons_a_nat @ Y3 @ Vs ) ) )
=> ( member688761744_a_nat @ ( produc1430747863_a_nat @ Xs2 @ Ys2 ) @ ( listrel1_a_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_131_listrel1I,axiom,
! [X3: list_a,Y3: list_a,R: set_Pr1971762695list_a,Xs2: list_list_a,Us: list_list_a,Vs: list_list_a,Ys2: list_list_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ X3 @ Y3 ) @ R )
=> ( ( Xs2
= ( append_list_a @ Us @ ( cons_list_a @ X3 @ Vs ) ) )
=> ( ( Ys2
= ( append_list_a @ Us @ ( cons_list_a @ Y3 @ Vs ) ) )
=> ( member888406864list_a @ ( produc2583831list_a @ Xs2 @ Ys2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_132_listrel1I,axiom,
! [X3: product_prod_a_a,Y3: product_prod_a_a,R: set_Pr1948701895od_a_a,Xs2: list_P1467822637od_a_a,Us: list_P1467822637od_a_a,Vs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
=> ( ( Xs2
= ( append1574001858od_a_a @ Us @ ( cons_P1921020637od_a_a @ X3 @ Vs ) ) )
=> ( ( Ys2
= ( append1574001858od_a_a @ Us @ ( cons_P1921020637od_a_a @ Y3 @ Vs ) ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( listre944824838od_a_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_133_listrel1I,axiom,
! [X3: a,Y3: a,R: set_Product_prod_a_a,Xs2: list_a,Us: list_a,Vs: list_a,Ys2: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
=> ( ( Xs2
= ( append_a @ Us @ ( cons_a @ X3 @ Vs ) ) )
=> ( ( Ys2
= ( append_a @ Us @ ( cons_a @ Y3 @ Vs ) ) )
=> ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_134_listrel1E,axiom,
! [Xs2: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,R: set_Pr1295299783od_a_a] :
( ( member1254757008od_a_a @ ( produc461662679od_a_a @ Xs2 @ Ys2 ) @ ( listre1832537542od_a_a @ R ) )
=> ~ ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ R )
=> ! [Us3: list_P1891567085od_a_a,Vs2: list_P1891567085od_a_a] :
( ( Xs2
= ( append864718978od_a_a @ Us3 @ ( cons_P1070399645od_a_a @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append864718978od_a_a @ Us3 @ ( cons_P1070399645od_a_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_135_listrel1E,axiom,
! [Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,R: set_Pr800560199od_a_a] :
( ( member8108048od_a_a @ ( produc915218007od_a_a @ Xs2 @ Ys2 ) @ ( listre651914124od_a_a @ R ) )
=> ~ ! [X4: list_P1467822637od_a_a,Y4: list_P1467822637od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ X4 @ Y4 ) @ R )
=> ! [Us3: list_l1108175539od_a_a,Vs2: list_l1108175539od_a_a] :
( ( Xs2
= ( append2141280328od_a_a @ Us3 @ ( cons_l724116707od_a_a @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append2141280328od_a_a @ Us3 @ ( cons_l724116707od_a_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_136_listrel1E,axiom,
! [Xs2: list_a_nat,Ys2: list_a_nat,R: set_Pr1940702919_a_nat] :
( ( member688761744_a_nat @ ( produc1430747863_a_nat @ Xs2 @ Ys2 ) @ ( listrel1_a_nat @ R ) )
=> ~ ! [X4: a > nat,Y4: a > nat] :
( ( member125757712_a_nat @ ( produc1569872215_a_nat @ X4 @ Y4 ) @ R )
=> ! [Us3: list_a_nat,Vs2: list_a_nat] :
( ( Xs2
= ( append_a_nat @ Us3 @ ( cons_a_nat @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append_a_nat @ Us3 @ ( cons_a_nat @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_137_listrel1E,axiom,
! [Xs2: list_list_a,Ys2: list_list_a,R: set_Pr1971762695list_a] :
( ( member888406864list_a @ ( produc2583831list_a @ Xs2 @ Ys2 ) @ ( listrel1_list_a @ R ) )
=> ~ ! [X4: list_a,Y4: list_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ X4 @ Y4 ) @ R )
=> ! [Us3: list_list_a,Vs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Us3 @ ( cons_list_a @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append_list_a @ Us3 @ ( cons_list_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_138_listrel1E,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( listre944824838od_a_a @ R ) )
=> ~ ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
=> ! [Us3: list_P1467822637od_a_a,Vs2: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Us3 @ ( cons_P1921020637od_a_a @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append1574001858od_a_a @ Us3 @ ( cons_P1921020637od_a_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_139_listrel1E,axiom,
! [Xs2: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) )
=> ~ ! [X4: a,Y4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
=> ! [Us3: list_a,Vs2: list_a] :
( ( Xs2
= ( append_a @ Us3 @ ( cons_a @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append_a @ Us3 @ ( cons_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_140_member__rec_I1_J,axiom,
! [X3: a,Xs2: list_a,Y3: a] :
( ( member_a @ ( cons_a @ X3 @ Xs2 ) @ Y3 )
= ( ( X3 = Y3 )
| ( member_a @ Xs2 @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_141_member__rec_I1_J,axiom,
! [X3: produc1572603623od_a_a,Xs2: list_P1891567085od_a_a,Y3: produc1572603623od_a_a] :
( ( member2070313346od_a_a @ ( cons_P1070399645od_a_a @ X3 @ Xs2 ) @ Y3 )
= ( ( X3 = Y3 )
| ( member2070313346od_a_a @ Xs2 @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_142_member__rec_I1_J,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a,Y3: list_P1467822637od_a_a] :
( ( member261538120od_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) @ Y3 )
= ( ( X3 = Y3 )
| ( member261538120od_a_a @ Xs2 @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_143_member__rec_I1_J,axiom,
! [X3: a > nat,Xs2: list_a_nat,Y3: a > nat] :
( ( member_a_nat @ ( cons_a_nat @ X3 @ Xs2 ) @ Y3 )
= ( ( X3 = Y3 )
| ( member_a_nat @ Xs2 @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_144_member__rec_I1_J,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Y3: product_prod_a_a] :
( ( member1112491458od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ Y3 )
= ( ( X3 = Y3 )
| ( member1112491458od_a_a @ Xs2 @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_145_ssubst__Pair__rhs,axiom,
! [R: list_P1467822637od_a_a,S: list_P1467822637od_a_a,R2: set_Pr800560199od_a_a,S2: list_P1467822637od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_146_ssubst__Pair__rhs,axiom,
! [R: list_P1467822637od_a_a,S: a,R2: set_Pr1525541400_a_a_a,S2: a] :
( ( member430654177_a_a_a @ ( produc998622632_a_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member430654177_a_a_a @ ( produc998622632_a_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_147_ssubst__Pair__rhs,axiom,
! [R: list_a,S: list_a,R2: set_Pr1971762695list_a,S2: list_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member1250517072list_a @ ( produc1448356631list_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_148_ssubst__Pair__rhs,axiom,
! [R: product_prod_a_a > product_prod_a_a > $o,S: list_P1467822637od_a_a,R2: set_Pr727105414od_a_a,S2: list_P1467822637od_a_a] :
( ( member372644687od_a_a @ ( produc214132118od_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member372644687od_a_a @ ( produc214132118od_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_149_ssubst__Pair__rhs,axiom,
! [R: a,S: produc657167928_a_a_a,R2: set_Pr110478017_a_a_a,S2: produc657167928_a_a_a] :
( ( member1904151562_a_a_a @ ( produc1440321489_a_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member1904151562_a_a_a @ ( produc1440321489_a_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_150_ssubst__Pair__rhs,axiom,
! [R: a,S: list_P1467822637od_a_a,R2: set_Pr248149174od_a_a,S2: list_P1467822637od_a_a] :
( ( member1474739327od_a_a @ ( produc575024326od_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member1474739327od_a_a @ ( produc575024326od_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_151_ssubst__Pair__rhs,axiom,
! [R: a,S: a,R2: set_Product_prod_a_a,S2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_152_subdivide_Oinduct,axiom,
! [P2: pair_p1891813445t_unit > produc864062247list_a > list_a > $o,A0: pair_p1891813445t_unit,A1: produc864062247list_a,A22: list_a] :
( ! [G: pair_p1891813445t_unit,U2: list_a,V: list_a,X_1: list_a] : ( P2 @ G @ ( produc1448356631list_a @ U2 @ V ) @ X_1 )
=> ( P2 @ A0 @ A1 @ A22 ) ) ).
% subdivide.induct
thf(fact_153_subdivide_Oinduct,axiom,
! [P2: pair_p125712459t_unit > product_prod_a_a > a > $o,A0: pair_p125712459t_unit,A1: product_prod_a_a,A22: a] :
( ! [G: pair_p125712459t_unit,U2: a,V: a,X_1: a] : ( P2 @ G @ ( product_Pair_a_a @ U2 @ V ) @ X_1 )
=> ( P2 @ A0 @ A1 @ A22 ) ) ).
% subdivide.induct
thf(fact_154_Cons__listrel1__Cons,axiom,
! [X3: produc1572603623od_a_a,Xs2: list_P1891567085od_a_a,Y3: produc1572603623od_a_a,Ys2: list_P1891567085od_a_a,R: set_Pr1295299783od_a_a] :
( ( member1254757008od_a_a @ ( produc461662679od_a_a @ ( cons_P1070399645od_a_a @ X3 @ Xs2 ) @ ( cons_P1070399645od_a_a @ Y3 @ Ys2 ) ) @ ( listre1832537542od_a_a @ R ) )
= ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X3 = Y3 )
& ( member1254757008od_a_a @ ( produc461662679od_a_a @ Xs2 @ Ys2 ) @ ( listre1832537542od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_155_Cons__listrel1__Cons,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a,Y3: list_P1467822637od_a_a,Ys2: list_l1108175539od_a_a,R: set_Pr800560199od_a_a] :
( ( member8108048od_a_a @ ( produc915218007od_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) @ ( cons_l724116707od_a_a @ Y3 @ Ys2 ) ) @ ( listre651914124od_a_a @ R ) )
= ( ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ X3 @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X3 = Y3 )
& ( member8108048od_a_a @ ( produc915218007od_a_a @ Xs2 @ Ys2 ) @ ( listre651914124od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_156_Cons__listrel1__Cons,axiom,
! [X3: a > nat,Xs2: list_a_nat,Y3: a > nat,Ys2: list_a_nat,R: set_Pr1940702919_a_nat] :
( ( member688761744_a_nat @ ( produc1430747863_a_nat @ ( cons_a_nat @ X3 @ Xs2 ) @ ( cons_a_nat @ Y3 @ Ys2 ) ) @ ( listrel1_a_nat @ R ) )
= ( ( ( member125757712_a_nat @ ( produc1569872215_a_nat @ X3 @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X3 = Y3 )
& ( member688761744_a_nat @ ( produc1430747863_a_nat @ Xs2 @ Ys2 ) @ ( listrel1_a_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_157_Cons__listrel1__Cons,axiom,
! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a,R: set_Pr1971762695list_a] :
( ( member888406864list_a @ ( produc2583831list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) @ ( listrel1_list_a @ R ) )
= ( ( ( member1250517072list_a @ ( produc1448356631list_a @ X3 @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X3 = Y3 )
& ( member888406864list_a @ ( produc2583831list_a @ Xs2 @ Ys2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_158_Cons__listrel1__Cons,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) ) @ ( listre944824838od_a_a @ R ) )
= ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X3 = Y3 )
& ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( listre944824838od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_159_Cons__listrel1__Cons,axiom,
! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) @ ( listrel1_a @ R ) )
= ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X3 = Y3 )
& ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_160_lexord__cons__cons,axiom,
! [A: produc1572603623od_a_a,X3: list_P1891567085od_a_a,B: produc1572603623od_a_a,Y3: list_P1891567085od_a_a,R: set_Pr1295299783od_a_a] :
( ( member1254757008od_a_a @ ( produc461662679od_a_a @ ( cons_P1070399645od_a_a @ A @ X3 ) @ ( cons_P1070399645od_a_a @ B @ Y3 ) ) @ ( lexord931792858od_a_a @ R ) )
= ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member1254757008od_a_a @ ( produc461662679od_a_a @ X3 @ Y3 ) @ ( lexord931792858od_a_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_161_lexord__cons__cons,axiom,
! [A: list_P1467822637od_a_a,X3: list_l1108175539od_a_a,B: list_P1467822637od_a_a,Y3: list_l1108175539od_a_a,R: set_Pr800560199od_a_a] :
( ( member8108048od_a_a @ ( produc915218007od_a_a @ ( cons_l724116707od_a_a @ A @ X3 ) @ ( cons_l724116707od_a_a @ B @ Y3 ) ) @ ( lexord1284731808od_a_a @ R ) )
= ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member8108048od_a_a @ ( produc915218007od_a_a @ X3 @ Y3 ) @ ( lexord1284731808od_a_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_162_lexord__cons__cons,axiom,
! [A: a > nat,X3: list_a_nat,B: a > nat,Y3: list_a_nat,R: set_Pr1940702919_a_nat] :
( ( member688761744_a_nat @ ( produc1430747863_a_nat @ ( cons_a_nat @ A @ X3 ) @ ( cons_a_nat @ B @ Y3 ) ) @ ( lexord_a_nat @ R ) )
= ( ( member125757712_a_nat @ ( produc1569872215_a_nat @ A @ B ) @ R )
| ( ( A = B )
& ( member688761744_a_nat @ ( produc1430747863_a_nat @ X3 @ Y3 ) @ ( lexord_a_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_163_lexord__cons__cons,axiom,
! [A: list_a,X3: list_list_a,B: list_a,Y3: list_list_a,R: set_Pr1971762695list_a] :
( ( member888406864list_a @ ( produc2583831list_a @ ( cons_list_a @ A @ X3 ) @ ( cons_list_a @ B @ Y3 ) ) @ ( lexord_list_a @ R ) )
= ( ( member1250517072list_a @ ( produc1448356631list_a @ A @ B ) @ R )
| ( ( A = B )
& ( member888406864list_a @ ( produc2583831list_a @ X3 @ Y3 ) @ ( lexord_list_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_164_lexord__cons__cons,axiom,
! [A: product_prod_a_a,X3: list_P1467822637od_a_a,B: product_prod_a_a,Y3: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ A @ X3 ) @ ( cons_P1921020637od_a_a @ B @ Y3 ) ) @ ( lexord673907738od_a_a @ R ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member1691432592od_a_a @ ( produc85179863od_a_a @ X3 @ Y3 ) @ ( lexord673907738od_a_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_165_lexord__cons__cons,axiom,
! [A: a,X3: list_a,B: a,Y3: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ A @ X3 ) @ ( cons_a @ B @ Y3 ) ) @ ( lexord_a @ R ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member1250517072list_a @ ( produc1448356631list_a @ X3 @ Y3 ) @ ( lexord_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_166_listrel1I2,axiom,
! [Xs2: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,R: set_Pr1295299783od_a_a,X3: produc1572603623od_a_a] :
( ( member1254757008od_a_a @ ( produc461662679od_a_a @ Xs2 @ Ys2 ) @ ( listre1832537542od_a_a @ R ) )
=> ( member1254757008od_a_a @ ( produc461662679od_a_a @ ( cons_P1070399645od_a_a @ X3 @ Xs2 ) @ ( cons_P1070399645od_a_a @ X3 @ Ys2 ) ) @ ( listre1832537542od_a_a @ R ) ) ) ).
% listrel1I2
thf(fact_167_listrel1I2,axiom,
! [Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,R: set_Pr800560199od_a_a,X3: list_P1467822637od_a_a] :
( ( member8108048od_a_a @ ( produc915218007od_a_a @ Xs2 @ Ys2 ) @ ( listre651914124od_a_a @ R ) )
=> ( member8108048od_a_a @ ( produc915218007od_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) @ ( cons_l724116707od_a_a @ X3 @ Ys2 ) ) @ ( listre651914124od_a_a @ R ) ) ) ).
% listrel1I2
thf(fact_168_listrel1I2,axiom,
! [Xs2: list_a_nat,Ys2: list_a_nat,R: set_Pr1940702919_a_nat,X3: a > nat] :
( ( member688761744_a_nat @ ( produc1430747863_a_nat @ Xs2 @ Ys2 ) @ ( listrel1_a_nat @ R ) )
=> ( member688761744_a_nat @ ( produc1430747863_a_nat @ ( cons_a_nat @ X3 @ Xs2 ) @ ( cons_a_nat @ X3 @ Ys2 ) ) @ ( listrel1_a_nat @ R ) ) ) ).
% listrel1I2
thf(fact_169_listrel1I2,axiom,
! [Xs2: list_a,Ys2: list_a,R: set_Product_prod_a_a,X3: a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) )
=> ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ X3 @ Ys2 ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I2
thf(fact_170_listrel1I2,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a,X3: product_prod_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( listre944824838od_a_a @ R ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_P1921020637od_a_a @ X3 @ Ys2 ) ) @ ( listre944824838od_a_a @ R ) ) ) ).
% listrel1I2
thf(fact_171_append__listrel1I,axiom,
! [Xs2: list_P1891567085od_a_a,Ys2: list_P1891567085od_a_a,R: set_Pr1295299783od_a_a,Us: list_P1891567085od_a_a,Vs: list_P1891567085od_a_a] :
( ( ( ( member1254757008od_a_a @ ( produc461662679od_a_a @ Xs2 @ Ys2 ) @ ( listre1832537542od_a_a @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member1254757008od_a_a @ ( produc461662679od_a_a @ Us @ Vs ) @ ( listre1832537542od_a_a @ R ) ) ) )
=> ( member1254757008od_a_a @ ( produc461662679od_a_a @ ( append864718978od_a_a @ Xs2 @ Us ) @ ( append864718978od_a_a @ Ys2 @ Vs ) ) @ ( listre1832537542od_a_a @ R ) ) ) ).
% append_listrel1I
thf(fact_172_append__listrel1I,axiom,
! [Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a,R: set_Pr800560199od_a_a,Us: list_l1108175539od_a_a,Vs: list_l1108175539od_a_a] :
( ( ( ( member8108048od_a_a @ ( produc915218007od_a_a @ Xs2 @ Ys2 ) @ ( listre651914124od_a_a @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member8108048od_a_a @ ( produc915218007od_a_a @ Us @ Vs ) @ ( listre651914124od_a_a @ R ) ) ) )
=> ( member8108048od_a_a @ ( produc915218007od_a_a @ ( append2141280328od_a_a @ Xs2 @ Us ) @ ( append2141280328od_a_a @ Ys2 @ Vs ) ) @ ( listre651914124od_a_a @ R ) ) ) ).
% append_listrel1I
thf(fact_173_append__listrel1I,axiom,
! [Xs2: list_a,Ys2: list_a,R: set_Product_prod_a_a,Us: list_a,Vs: list_a] :
( ( ( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member1250517072list_a @ ( produc1448356631list_a @ Us @ Vs ) @ ( listrel1_a @ R ) ) ) )
=> ( member1250517072list_a @ ( produc1448356631list_a @ ( append_a @ Xs2 @ Us ) @ ( append_a @ Ys2 @ Vs ) ) @ ( listrel1_a @ R ) ) ) ).
% append_listrel1I
thf(fact_174_append__listrel1I,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a,Us: list_P1467822637od_a_a,Vs: list_P1467822637od_a_a] :
( ( ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( listre944824838od_a_a @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member1691432592od_a_a @ ( produc85179863od_a_a @ Us @ Vs ) @ ( listre944824838od_a_a @ R ) ) ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Us ) @ ( append1574001858od_a_a @ Ys2 @ Vs ) ) @ ( listre944824838od_a_a @ R ) ) ) ).
% append_listrel1I
thf(fact_175_lexord__linear,axiom,
! [R: set_Pr800560199od_a_a,X3: list_l1108175539od_a_a,Y3: list_l1108175539od_a_a] :
( ! [A3: list_P1467822637od_a_a,B3: list_P1467822637od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member1691432592od_a_a @ ( produc85179863od_a_a @ B3 @ A3 ) @ R ) )
=> ( ( member8108048od_a_a @ ( produc915218007od_a_a @ X3 @ Y3 ) @ ( lexord1284731808od_a_a @ R ) )
| ( X3 = Y3 )
| ( member8108048od_a_a @ ( produc915218007od_a_a @ Y3 @ X3 ) @ ( lexord1284731808od_a_a @ R ) ) ) ) ).
% lexord_linear
thf(fact_176_lexord__linear,axiom,
! [R: set_Pr1948701895od_a_a,X3: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ! [A3: product_prod_a_a,B3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B3 @ A3 ) @ R ) )
=> ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ X3 @ Y3 ) @ ( lexord673907738od_a_a @ R ) )
| ( X3 = Y3 )
| ( member1691432592od_a_a @ ( produc85179863od_a_a @ Y3 @ X3 ) @ ( lexord673907738od_a_a @ R ) ) ) ) ).
% lexord_linear
thf(fact_177_lexord__linear,axiom,
! [R: set_Pr1971762695list_a,X3: list_list_a,Y3: list_list_a] :
( ! [A3: list_a,B3: list_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member1250517072list_a @ ( produc1448356631list_a @ B3 @ A3 ) @ R ) )
=> ( ( member888406864list_a @ ( produc2583831list_a @ X3 @ Y3 ) @ ( lexord_list_a @ R ) )
| ( X3 = Y3 )
| ( member888406864list_a @ ( produc2583831list_a @ Y3 @ X3 ) @ ( lexord_list_a @ R ) ) ) ) ).
% lexord_linear
thf(fact_178_lexord__linear,axiom,
! [R: set_Product_prod_a_a,X3: list_a,Y3: list_a] :
( ! [A3: a,B3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ R ) )
=> ( ( member1250517072list_a @ ( produc1448356631list_a @ X3 @ Y3 ) @ ( lexord_a @ R ) )
| ( X3 = Y3 )
| ( member1250517072list_a @ ( produc1448356631list_a @ Y3 @ X3 ) @ ( lexord_a @ R ) ) ) ) ).
% lexord_linear
thf(fact_179_mem__Collect__eq,axiom,
! [A: produc651658087od_a_a,P2: produc651658087od_a_a > $o] :
( ( member1691432592od_a_a @ A @ ( collec1566074578od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_180_mem__Collect__eq,axiom,
! [A: produc864062247list_a,P2: produc864062247list_a > $o] :
( ( member1250517072list_a @ A @ ( collec656258194list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_181_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_182_mem__Collect__eq,axiom,
! [A: list_P1467822637od_a_a,P2: list_P1467822637od_a_a > $o] :
( ( member1578987478od_a_a @ A @ ( collec1811041560od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_183_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a2 @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_184_Collect__mem__eq,axiom,
! [A4: set_Pr1971762695list_a] :
( ( collec656258194list_a
@ ^ [X: produc864062247list_a] : ( member1250517072list_a @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_185_Collect__mem__eq,axiom,
! [A4: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X: product_prod_a_a] : ( member449909584od_a_a @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_186_Collect__mem__eq,axiom,
! [A4: set_li562223245od_a_a] :
( ( collec1811041560od_a_a
@ ^ [X: list_P1467822637od_a_a] : ( member1578987478od_a_a @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_187_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a2 @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_188_lexord__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a] :
( ! [X4: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
=> ~ ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Xs2 ) @ ( lexord_a @ R ) ) ) ).
% lexord_irreflexive
thf(fact_189_lexord__append__leftI,axiom,
! [U: list_P1467822637od_a_a,V2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a,X3: list_P1467822637od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ U @ V2 ) @ ( lexord673907738od_a_a @ R ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ X3 @ U ) @ ( append1574001858od_a_a @ X3 @ V2 ) ) @ ( lexord673907738od_a_a @ R ) ) ) ).
% lexord_append_leftI
thf(fact_190_listrel1I1,axiom,
! [X3: product_prod_a_a,Y3: product_prod_a_a,R: set_Pr1948701895od_a_a,Xs2: list_P1467822637od_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_P1921020637od_a_a @ Y3 @ Xs2 ) ) @ ( listre944824838od_a_a @ R ) ) ) ).
% listrel1I1
thf(fact_191_listrel1I1,axiom,
! [X3: a,Y3: a,R: set_Product_prod_a_a,Xs2: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
=> ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Xs2 ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I1
thf(fact_192_Cons__listrel1E1,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ Ys2 ) @ ( listre944824838od_a_a @ R ) )
=> ( ! [Y4: product_prod_a_a] :
( ( Ys2
= ( cons_P1921020637od_a_a @ Y4 @ Xs2 ) )
=> ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y4 ) @ R ) )
=> ~ ! [Zs2: list_P1467822637od_a_a] :
( ( Ys2
= ( cons_P1921020637od_a_a @ X3 @ Zs2 ) )
=> ~ ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Zs2 ) @ ( listre944824838od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_193_Cons__listrel1E1,axiom,
! [X3: a,Xs2: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) @ ( listrel1_a @ R ) )
=> ( ! [Y4: a] :
( ( Ys2
= ( cons_a @ Y4 @ Xs2 ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Ys2
= ( cons_a @ X3 @ Zs2 ) )
=> ~ ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Zs2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_194_Cons__listrel1E2,axiom,
! [Xs2: list_P1467822637od_a_a,Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) ) @ ( listre944824838od_a_a @ R ) )
=> ( ! [X4: product_prod_a_a] :
( ( Xs2
= ( cons_P1921020637od_a_a @ X4 @ Ys2 ) )
=> ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_P1467822637od_a_a] :
( ( Xs2
= ( cons_P1921020637od_a_a @ Y3 @ Zs2 ) )
=> ~ ( member1691432592od_a_a @ ( produc85179863od_a_a @ Zs2 @ Ys2 ) @ ( listre944824838od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_195_Cons__listrel1E2,axiom,
! [Xs2: list_a,Y3: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ ( cons_a @ Y3 @ Ys2 ) ) @ ( listrel1_a @ R ) )
=> ( ! [X4: a] :
( ( Xs2
= ( cons_a @ X4 @ Ys2 ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Xs2
= ( cons_a @ Y3 @ Zs2 ) )
=> ~ ( member1250517072list_a @ ( produc1448356631list_a @ Zs2 @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_196_lexord__append__leftD,axiom,
! [X3: list_P1467822637od_a_a,U: list_P1467822637od_a_a,V2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ X3 @ U ) @ ( append1574001858od_a_a @ X3 @ V2 ) ) @ ( lexord673907738od_a_a @ R ) )
=> ( ! [A3: product_prod_a_a] :
~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A3 @ A3 ) @ R )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ U @ V2 ) @ ( lexord673907738od_a_a @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_197_lexord__append__leftD,axiom,
! [X3: list_a,U: list_a,V2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( append_a @ X3 @ U ) @ ( append_a @ X3 @ V2 ) ) @ ( lexord_a @ R ) )
=> ( ! [A3: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R )
=> ( member1250517072list_a @ ( produc1448356631list_a @ U @ V2 ) @ ( lexord_a @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_198_lexord__append__rightI,axiom,
! [Y3: list_P1467822637od_a_a,X3: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ? [B4: product_prod_a_a,Z: list_P1467822637od_a_a] :
( Y3
= ( cons_P1921020637od_a_a @ B4 @ Z ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ X3 @ ( append1574001858od_a_a @ X3 @ Y3 ) ) @ ( lexord673907738od_a_a @ R ) ) ) ).
% lexord_append_rightI
thf(fact_199_zip__eq__ConsE,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Xy: produc1572603623od_a_a,Xys: list_P1891567085od_a_a] :
( ( ( zip_Pr792472183od_a_a @ Xs2 @ Ys2 )
= ( cons_P1070399645od_a_a @ Xy @ Xys ) )
=> ~ ! [X4: product_prod_a_a,Xs3: list_P1467822637od_a_a] :
( ( Xs2
= ( cons_P1921020637od_a_a @ X4 @ Xs3 ) )
=> ! [Y4: product_prod_a_a,Ys3: list_P1467822637od_a_a] :
( ( Ys2
= ( cons_P1921020637od_a_a @ Y4 @ Ys3 ) )
=> ( ( Xy
= ( produc1474507607od_a_a @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_Pr792472183od_a_a @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_200_zip__eq__ConsE,axiom,
! [Xs2: list_a,Ys2: list_a,Xy: product_prod_a_a,Xys: list_P1467822637od_a_a] :
( ( ( zip_a_a @ Xs2 @ Ys2 )
= ( cons_P1921020637od_a_a @ Xy @ Xys ) )
=> ~ ! [X4: a,Xs3: list_a] :
( ( Xs2
= ( cons_a @ X4 @ Xs3 ) )
=> ! [Y4: a,Ys3: list_a] :
( ( Ys2
= ( cons_a @ Y4 @ Ys3 ) )
=> ( ( Xy
= ( product_Pair_a_a @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_a_a @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_201_subdivide_Ocases,axiom,
! [X3: produc1912141786_a_a_a] :
~ ! [G: pair_p125712459t_unit,U2: a,V: a,W: a] :
( X3
!= ( produc74075722_a_a_a @ G @ ( produc2061588782_a_a_a @ ( product_Pair_a_a @ U2 @ V ) @ W ) ) ) ).
% subdivide.cases
thf(fact_202_lexord__same__pref__if__irrefl,axiom,
! [R: set_Pr1948701895od_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( irrefl76447196od_a_a @ R )
=> ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) @ ( append1574001858od_a_a @ Xs2 @ Zs ) ) @ ( lexord673907738od_a_a @ R ) )
= ( member1691432592od_a_a @ ( produc85179863od_a_a @ Ys2 @ Zs ) @ ( lexord673907738od_a_a @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_203_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_P1467822637od_a_a,X3: product_prod_a_a,Ys2: list_P1467822637od_a_a,Y3: product_prod_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ ( cons_P1921020637od_a_a @ X3 @ nil_Product_prod_a_a ) ) @ ( append1574001858od_a_a @ Ys2 @ ( cons_P1921020637od_a_a @ Y3 @ nil_Product_prod_a_a ) ) ) @ ( listre944824838od_a_a @ R ) )
= ( ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( listre944824838od_a_a @ R ) )
& ( X3 = Y3 ) )
| ( ( Xs2 = Ys2 )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_204_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_a,X3: a,Ys2: list_a,Y3: a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) )
& ( X3 = Y3 ) )
| ( ( Xs2 = Ys2 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_205_lexord__Nil__left,axiom,
! [Y3: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ nil_Product_prod_a_a @ Y3 ) @ ( lexord673907738od_a_a @ R ) )
= ( ? [A5: product_prod_a_a,X: list_P1467822637od_a_a] :
( Y3
= ( cons_P1921020637od_a_a @ A5 @ X ) ) ) ) ).
% lexord_Nil_left
thf(fact_206_lexord__same__pref__iff,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) @ ( append1574001858od_a_a @ Xs2 @ Zs ) ) @ ( lexord673907738od_a_a @ R ) )
= ( ? [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ X ) @ R ) )
| ( member1691432592od_a_a @ ( produc85179863od_a_a @ Ys2 @ Zs ) @ ( lexord673907738od_a_a @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_207_lexord__same__pref__iff,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( append_a @ Xs2 @ Ys2 ) @ ( append_a @ Xs2 @ Zs ) ) @ ( lexord_a @ R ) )
= ( ? [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X @ X ) @ R ) )
| ( member1250517072list_a @ ( produc1448356631list_a @ Ys2 @ Zs ) @ ( lexord_a @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_208_zip__append,axiom,
! [Xs2: list_a,Us: list_a,Ys2: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Us ) )
=> ( ( zip_a_a @ ( append_a @ Xs2 @ Ys2 ) @ ( append_a @ Us @ Vs ) )
= ( append1574001858od_a_a @ ( zip_a_a @ Xs2 @ Us ) @ ( zip_a_a @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_209_zip__append,axiom,
! [Xs2: list_P1467822637od_a_a,Us: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Vs: list_P1467822637od_a_a] :
( ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Us ) )
=> ( ( zip_Pr792472183od_a_a @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) @ ( append1574001858od_a_a @ Us @ Vs ) )
= ( append864718978od_a_a @ ( zip_Pr792472183od_a_a @ Xs2 @ Us ) @ ( zip_Pr792472183od_a_a @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_210_lex__append__leftD,axiom,
! [R: set_Pr1948701895od_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ! [X4: product_prod_a_a] :
~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ X4 ) @ R )
=> ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) @ ( append1574001858od_a_a @ Xs2 @ Zs ) ) @ ( lex_Product_prod_a_a @ R ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ Ys2 @ Zs ) @ ( lex_Product_prod_a_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_211_lex__append__leftD,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a,Ys2: list_a,Zs: list_a] :
( ! [X4: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
=> ( ( member1250517072list_a @ ( produc1448356631list_a @ ( append_a @ Xs2 @ Ys2 ) @ ( append_a @ Xs2 @ Zs ) ) @ ( lex_a @ R ) )
=> ( member1250517072list_a @ ( produc1448356631list_a @ Ys2 @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_212_lex__append__left__iff,axiom,
! [R: set_Pr1948701895od_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ! [X4: product_prod_a_a] :
~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ X4 ) @ R )
=> ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) @ ( append1574001858od_a_a @ Xs2 @ Zs ) ) @ ( lex_Product_prod_a_a @ R ) )
= ( member1691432592od_a_a @ ( produc85179863od_a_a @ Ys2 @ Zs ) @ ( lex_Product_prod_a_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_213_lex__append__left__iff,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a,Ys2: list_a,Zs: list_a] :
( ! [X4: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
=> ( ( member1250517072list_a @ ( produc1448356631list_a @ ( append_a @ Xs2 @ Ys2 ) @ ( append_a @ Xs2 @ Zs ) ) @ ( lex_a @ R ) )
= ( member1250517072list_a @ ( produc1448356631list_a @ Ys2 @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_214_append_Oright__neutral,axiom,
! [A: list_P1467822637od_a_a] :
( ( append1574001858od_a_a @ A @ nil_Product_prod_a_a )
= A ) ).
% append.right_neutral
thf(fact_215_append__is__Nil__conv,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Xs2 @ Ys2 )
= nil_Product_prod_a_a )
= ( ( Xs2 = nil_Product_prod_a_a )
& ( Ys2 = nil_Product_prod_a_a ) ) ) ).
% append_is_Nil_conv
thf(fact_216_Nil__is__append__conv,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( nil_Product_prod_a_a
= ( append1574001858od_a_a @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_Product_prod_a_a )
& ( Ys2 = nil_Product_prod_a_a ) ) ) ).
% Nil_is_append_conv
thf(fact_217_self__append__conv2,axiom,
! [Ys2: list_P1467822637od_a_a,Xs2: list_P1467822637od_a_a] :
( ( Ys2
= ( append1574001858od_a_a @ Xs2 @ Ys2 ) )
= ( Xs2 = nil_Product_prod_a_a ) ) ).
% self_append_conv2
thf(fact_218_append__self__conv2,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_Product_prod_a_a ) ) ).
% append_self_conv2
thf(fact_219_self__append__conv,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Xs2 @ Ys2 ) )
= ( Ys2 = nil_Product_prod_a_a ) ) ).
% self_append_conv
thf(fact_220_append__self__conv,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_Product_prod_a_a ) ) ).
% append_self_conv
thf(fact_221_append__Nil2,axiom,
! [Xs2: list_P1467822637od_a_a] :
( ( append1574001858od_a_a @ Xs2 @ nil_Product_prod_a_a )
= Xs2 ) ).
% append_Nil2
thf(fact_222_append__eq__append__conv,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Us: list_P1467822637od_a_a,Vs: list_P1467822637od_a_a] :
( ( ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Ys2 ) )
| ( ( size_s642698393od_a_a @ Us )
= ( size_s642698393od_a_a @ Vs ) ) )
=> ( ( ( append1574001858od_a_a @ Xs2 @ Us )
= ( append1574001858od_a_a @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_223_append1__eq__conv,axiom,
! [Xs2: list_P1467822637od_a_a,X3: product_prod_a_a,Ys2: list_P1467822637od_a_a,Y3: product_prod_a_a] :
( ( ( append1574001858od_a_a @ Xs2 @ ( cons_P1921020637od_a_a @ X3 @ nil_Product_prod_a_a ) )
= ( append1574001858od_a_a @ Ys2 @ ( cons_P1921020637od_a_a @ Y3 @ nil_Product_prod_a_a ) ) )
= ( ( Xs2 = Ys2 )
& ( X3 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_224_Cons__in__lex,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) ) @ ( lex_Product_prod_a_a @ R ) )
= ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
& ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Ys2 ) ) )
| ( ( X3 = Y3 )
& ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( lex_Product_prod_a_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_225_Cons__in__lex,axiom,
! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) @ ( lex_a @ R ) )
= ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
& ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) ) )
| ( ( X3 = Y3 )
& ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_226_in__set__impl__in__set__zip1,axiom,
! [Xs2: list_a,Ys2: list_a,X3: a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
=> ~ ! [Y4: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_227_in__set__impl__in__set__zip2,axiom,
! [Xs2: list_a,Ys2: list_a,Y3: a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( member_a2 @ Y3 @ ( set_a2 @ Ys2 ) )
=> ~ ! [X4: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_228_list__induct4,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,Ws: list_P1467822637od_a_a,P2: list_P1467822637od_a_a > list_P1467822637od_a_a > list_P1467822637od_a_a > list_P1467822637od_a_a > $o] :
( ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Ys2 ) )
=> ( ( ( size_s642698393od_a_a @ Ys2 )
= ( size_s642698393od_a_a @ Zs ) )
=> ( ( ( size_s642698393od_a_a @ Zs )
= ( size_s642698393od_a_a @ Ws ) )
=> ( ( P2 @ nil_Product_prod_a_a @ nil_Product_prod_a_a @ nil_Product_prod_a_a @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a,Z2: product_prod_a_a,Zs2: list_P1467822637od_a_a,W: product_prod_a_a,Ws2: list_P1467822637od_a_a] :
( ( ( size_s642698393od_a_a @ Xs4 )
= ( size_s642698393od_a_a @ Ys4 ) )
=> ( ( ( size_s642698393od_a_a @ Ys4 )
= ( size_s642698393od_a_a @ Zs2 ) )
=> ( ( ( size_s642698393od_a_a @ Zs2 )
= ( size_s642698393od_a_a @ Ws2 ) )
=> ( ( P2 @ Xs4 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) @ ( cons_P1921020637od_a_a @ Z2 @ Zs2 ) @ ( cons_P1921020637od_a_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_229_list__induct3,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,P2: list_P1467822637od_a_a > list_P1467822637od_a_a > list_P1467822637od_a_a > $o] :
( ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Ys2 ) )
=> ( ( ( size_s642698393od_a_a @ Ys2 )
= ( size_s642698393od_a_a @ Zs ) )
=> ( ( P2 @ nil_Product_prod_a_a @ nil_Product_prod_a_a @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a,Z2: product_prod_a_a,Zs2: list_P1467822637od_a_a] :
( ( ( size_s642698393od_a_a @ Xs4 )
= ( size_s642698393od_a_a @ Ys4 ) )
=> ( ( ( size_s642698393od_a_a @ Ys4 )
= ( size_s642698393od_a_a @ Zs2 ) )
=> ( ( P2 @ Xs4 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) @ ( cons_P1921020637od_a_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_230_list__induct2,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,P2: list_P1467822637od_a_a > list_P1467822637od_a_a > $o] :
( ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Ys2 ) )
=> ( ( P2 @ nil_Product_prod_a_a @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( ( ( size_s642698393od_a_a @ Xs4 )
= ( size_s642698393od_a_a @ Ys4 ) )
=> ( ( P2 @ Xs4 @ Ys4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) ) ) )
=> ( P2 @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_231_transpose_Ocases,axiom,
! [X3: list_l1108175539od_a_a] :
( ( X3 != nil_li1885598003od_a_a )
=> ( ! [Xss: list_l1108175539od_a_a] :
( X3
!= ( cons_l724116707od_a_a @ nil_Product_prod_a_a @ Xss ) )
=> ~ ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Xss: list_l1108175539od_a_a] :
( X3
!= ( cons_l724116707od_a_a @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_232_set__zip__rightD,axiom,
! [X3: a,Y3: a,Xs2: list_a,Ys2: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Ys2 ) ) )
=> ( member_a2 @ Y3 @ ( set_a2 @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_233_set__zip__leftD,axiom,
! [X3: a,Y3: a,Xs2: list_a,Ys2: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Ys2 ) ) )
=> ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) ) ) ).
% set_zip_leftD
thf(fact_234_in__set__zipE,axiom,
! [X3: a,Y3: a,Xs2: list_a,Ys2: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
=> ~ ( member_a2 @ Y3 @ ( set_a2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_235_zip__same,axiom,
! [A: a,B: a,Xs2: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Xs2 ) ) )
= ( ( member_a2 @ A @ ( set_a2 @ Xs2 ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_236_same__length__different,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( Xs2 != Ys2 )
=> ( ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Ys2 ) )
=> ? [Pre: list_P1467822637od_a_a,X4: product_prod_a_a,Xs3: list_P1467822637od_a_a,Y4: product_prod_a_a,Ys3: list_P1467822637od_a_a] :
( ( X4 != Y4 )
& ( Xs2
= ( append1574001858od_a_a @ Pre @ ( append1574001858od_a_a @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) @ Xs3 ) ) )
& ( Ys2
= ( append1574001858od_a_a @ Pre @ ( append1574001858od_a_a @ ( cons_P1921020637od_a_a @ Y4 @ nil_Product_prod_a_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_237_list_Oset__cases,axiom,
! [E: product_prod_a_a,A: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ E @ ( set_Product_prod_a_a2 @ A ) )
=> ( ! [Z22: list_P1467822637od_a_a] :
( A
!= ( cons_P1921020637od_a_a @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_a_a,Z22: list_P1467822637od_a_a] :
( ( A
= ( cons_P1921020637od_a_a @ Z1 @ Z22 ) )
=> ~ ( member449909584od_a_a @ E @ ( set_Product_prod_a_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_238_set__ConsD,axiom,
! [Y3: product_prod_a_a,X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) ) )
=> ( ( Y3 = X3 )
| ( member449909584od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_239_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1467822637od_a_a] : ( member449909584od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P1921020637od_a_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_240_list_Oset__intros_I2_J,axiom,
! [Y3: product_prod_a_a,X22: list_P1467822637od_a_a,X21: product_prod_a_a] :
( ( member449909584od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ X22 ) )
=> ( member449909584od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ ( cons_P1921020637od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_241_lex__append__rightI,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a,Vs: list_P1467822637od_a_a,Us: list_P1467822637od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( lex_Product_prod_a_a @ R ) )
=> ( ( ( size_s642698393od_a_a @ Vs )
= ( size_s642698393od_a_a @ Us ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Us ) @ ( append1574001858od_a_a @ Ys2 @ Vs ) ) @ ( lex_Product_prod_a_a @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_242_sorted__wrt_Ocases,axiom,
! [X3: produc501911462od_a_a] :
( ! [P4: product_prod_a_a > product_prod_a_a > $o] :
( X3
!= ( produc214132118od_a_a @ P4 @ nil_Product_prod_a_a ) )
=> ~ ! [P4: product_prod_a_a > product_prod_a_a > $o,X4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( X3
!= ( produc214132118od_a_a @ P4 @ ( cons_P1921020637od_a_a @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_243_successively_Ocases,axiom,
! [X3: produc501911462od_a_a] :
( ! [P4: product_prod_a_a > product_prod_a_a > $o] :
( X3
!= ( produc214132118od_a_a @ P4 @ nil_Product_prod_a_a ) )
=> ( ! [P4: product_prod_a_a > product_prod_a_a > $o,X4: product_prod_a_a] :
( X3
!= ( produc214132118od_a_a @ P4 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) ) )
=> ~ ! [P4: product_prod_a_a > product_prod_a_a > $o,X4: product_prod_a_a,Y4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( X3
!= ( produc214132118od_a_a @ P4 @ ( cons_P1921020637od_a_a @ X4 @ ( cons_P1921020637od_a_a @ Y4 @ Xs4 ) ) ) ) ) ) ).
% successively.cases
thf(fact_244_map__tailrec__rev_Oinduct,axiom,
! [P2: ( product_prod_a_a > product_prod_a_a ) > list_P1467822637od_a_a > list_P1467822637od_a_a > $o,A0: product_prod_a_a > product_prod_a_a,A1: list_P1467822637od_a_a,A22: list_P1467822637od_a_a] :
( ! [F2: product_prod_a_a > product_prod_a_a,X_1: list_P1467822637od_a_a] : ( P2 @ F2 @ nil_Product_prod_a_a @ X_1 )
=> ( ! [F2: product_prod_a_a > product_prod_a_a,A3: product_prod_a_a,As: list_P1467822637od_a_a,Bs: list_P1467822637od_a_a] :
( ( P2 @ F2 @ As @ ( cons_P1921020637od_a_a @ ( F2 @ A3 ) @ Bs ) )
=> ( P2 @ F2 @ ( cons_P1921020637od_a_a @ A3 @ As ) @ Bs ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_245_list__nonempty__induct,axiom,
! [Xs2: list_P1467822637od_a_a,P2: list_P1467822637od_a_a > $o] :
( ( Xs2 != nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a] : ( P2 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( ( Xs4 != nil_Product_prod_a_a )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_246_successively_Oinduct,axiom,
! [P2: ( product_prod_a_a > product_prod_a_a > $o ) > list_P1467822637od_a_a > $o,A0: product_prod_a_a > product_prod_a_a > $o,A1: list_P1467822637od_a_a] :
( ! [P4: product_prod_a_a > product_prod_a_a > $o] : ( P2 @ P4 @ nil_Product_prod_a_a )
=> ( ! [P4: product_prod_a_a > product_prod_a_a > $o,X4: product_prod_a_a] : ( P2 @ P4 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) )
=> ( ! [P4: product_prod_a_a > product_prod_a_a > $o,X4: product_prod_a_a,Y4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( ( P2 @ P4 @ ( cons_P1921020637od_a_a @ Y4 @ Xs4 ) )
=> ( P2 @ P4 @ ( cons_P1921020637od_a_a @ X4 @ ( cons_P1921020637od_a_a @ Y4 @ Xs4 ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_247_remdups__adj_Oinduct,axiom,
! [P2: list_P1467822637od_a_a > $o,A0: list_P1467822637od_a_a] :
( ( P2 @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a] : ( P2 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) )
=> ( ! [X4: product_prod_a_a,Y4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( ( ( X4 = Y4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) ) )
=> ( ( ( X4 != Y4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ Y4 @ Xs4 ) ) )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ ( cons_P1921020637od_a_a @ Y4 @ Xs4 ) ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_248_sorted__wrt_Oinduct,axiom,
! [P2: ( product_prod_a_a > product_prod_a_a > $o ) > list_P1467822637od_a_a > $o,A0: product_prod_a_a > product_prod_a_a > $o,A1: list_P1467822637od_a_a] :
( ! [P4: product_prod_a_a > product_prod_a_a > $o] : ( P2 @ P4 @ nil_Product_prod_a_a )
=> ( ! [P4: product_prod_a_a > product_prod_a_a > $o,X4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( ( P2 @ P4 @ Ys4 )
=> ( P2 @ P4 @ ( cons_P1921020637od_a_a @ X4 @ Ys4 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_249_remdups__adj_Ocases,axiom,
! [X3: list_P1467822637od_a_a] :
( ( X3 != nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a] :
( X3
!= ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) )
=> ~ ! [X4: product_prod_a_a,Y4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( X3
!= ( cons_P1921020637od_a_a @ X4 @ ( cons_P1921020637od_a_a @ Y4 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_250_shuffles_Oinduct,axiom,
! [P2: list_P1467822637od_a_a > list_P1467822637od_a_a > $o,A0: list_P1467822637od_a_a,A1: list_P1467822637od_a_a] :
( ! [X_1: list_P1467822637od_a_a] : ( P2 @ nil_Product_prod_a_a @ X_1 )
=> ( ! [Xs4: list_P1467822637od_a_a] : ( P2 @ Xs4 @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( ( P2 @ Xs4 @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) )
=> ( ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ Ys4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_251_induct__list012,axiom,
! [P2: list_P1467822637od_a_a > $o,Xs2: list_P1467822637od_a_a] :
( ( P2 @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a] : ( P2 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) )
=> ( ! [X4: product_prod_a_a,Y4: product_prod_a_a,Zs2: list_P1467822637od_a_a] :
( ( P2 @ Zs2 )
=> ( ( P2 @ ( cons_P1921020637od_a_a @ Y4 @ Zs2 ) )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ ( cons_P1921020637od_a_a @ Y4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% induct_list012
thf(fact_252_splice_Oinduct,axiom,
! [P2: list_P1467822637od_a_a > list_P1467822637od_a_a > $o,A0: list_P1467822637od_a_a,A1: list_P1467822637od_a_a] :
( ! [X_1: list_P1467822637od_a_a] : ( P2 @ nil_Product_prod_a_a @ X_1 )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Ys4: list_P1467822637od_a_a] :
( ( P2 @ Ys4 @ Xs4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ Ys4 ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_253_list__induct2_H,axiom,
! [P2: list_P1467822637od_a_a > list_P1467822637od_a_a > $o,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( P2 @ nil_Product_prod_a_a @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a] : ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ nil_Product_prod_a_a )
=> ( ! [Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a] : ( P2 @ nil_Product_prod_a_a @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( ( P2 @ Xs4 @ Ys4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) ) )
=> ( P2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_254_neq__Nil__conv,axiom,
! [Xs2: list_P1467822637od_a_a] :
( ( Xs2 != nil_Product_prod_a_a )
= ( ? [Y: product_prod_a_a,Ys: list_P1467822637od_a_a] :
( Xs2
= ( cons_P1921020637od_a_a @ Y @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_255_list_Oinducts,axiom,
! [P2: list_P1467822637od_a_a > $o,List: list_P1467822637od_a_a] :
( ( P2 @ nil_Product_prod_a_a )
=> ( ! [X12: product_prod_a_a,X23: list_P1467822637od_a_a] :
( ( P2 @ X23 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X12 @ X23 ) ) )
=> ( P2 @ List ) ) ) ).
% list.inducts
thf(fact_256_list_Oexhaust,axiom,
! [Y3: list_P1467822637od_a_a] :
( ( Y3 != nil_Product_prod_a_a )
=> ~ ! [X212: product_prod_a_a,X222: list_P1467822637od_a_a] :
( Y3
!= ( cons_P1921020637od_a_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_257_list_OdiscI,axiom,
! [List: list_P1467822637od_a_a,X21: product_prod_a_a,X22: list_P1467822637od_a_a] :
( ( List
= ( cons_P1921020637od_a_a @ X21 @ X22 ) )
=> ( List != nil_Product_prod_a_a ) ) ).
% list.discI
thf(fact_258_list_Odistinct_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1467822637od_a_a] :
( nil_Product_prod_a_a
!= ( cons_P1921020637od_a_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_259_append_Oleft__neutral,axiom,
! [A: list_P1467822637od_a_a] :
( ( append1574001858od_a_a @ nil_Product_prod_a_a @ A )
= A ) ).
% append.left_neutral
thf(fact_260_append__Nil,axiom,
! [Ys2: list_P1467822637od_a_a] :
( ( append1574001858od_a_a @ nil_Product_prod_a_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_261_eq__Nil__appendI,axiom,
! [Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append1574001858od_a_a @ nil_Product_prod_a_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_262_pcas_Ocases,axiom,
! [X3: produc1197928545_a_a_a] :
( ! [U2: a,V: a] :
( X3
!= ( produc1440321489_a_a_a @ U2 @ ( produc998622632_a_a_a @ nil_Product_prod_a_a @ V ) ) )
=> ~ ! [U2: a,E2: product_prod_a_a,Es: list_P1467822637od_a_a,V: a] :
( X3
!= ( produc1440321489_a_a_a @ U2 @ ( produc998622632_a_a_a @ ( cons_P1921020637od_a_a @ E2 @ Es ) @ V ) ) ) ) ).
% pcas.cases
thf(fact_263_progressing__Nil,axiom,
kurato1072996284sing_a @ nil_Product_prod_a_a ).
% progressing_Nil
thf(fact_264_split__list__first__prop__iff,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ( ? [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X ) ) )
= ( ? [Ys: list_P1467822637od_a_a,X: product_prod_a_a] :
( ? [Zs3: list_P1467822637od_a_a] :
( Xs2
= ( append1574001858od_a_a @ Ys @ ( cons_P1921020637od_a_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Y: product_prod_a_a] :
( ( member449909584od_a_a @ Y @ ( set_Product_prod_a_a2 @ Ys ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_265_split__list__last__prop__iff,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ( ? [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X ) ) )
= ( ? [Ys: list_P1467822637od_a_a,X: product_prod_a_a,Zs3: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Ys @ ( cons_P1921020637od_a_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Y: product_prod_a_a] :
( ( member449909584od_a_a @ Y @ ( set_Product_prod_a_a2 @ Zs3 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_266_in__set__conv__decomp__first,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
= ( ? [Ys: list_P1467822637od_a_a,Zs3: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Ys @ ( cons_P1921020637od_a_a @ X3 @ Zs3 ) ) )
& ~ ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_267_in__set__conv__decomp__last,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
= ( ? [Ys: list_P1467822637od_a_a,Zs3: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Ys @ ( cons_P1921020637od_a_a @ X3 @ Zs3 ) ) )
& ~ ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_268_split__list__first__propE,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ? [X5: product_prod_a_a] :
( ( member449909584od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list_P1467822637od_a_a,X4: product_prod_a_a] :
( ? [Zs2: list_P1467822637od_a_a] :
( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_269_split__list__last__propE,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ? [X5: product_prod_a_a] :
( ( member449909584od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list_P1467822637od_a_a,X4: product_prod_a_a,Zs2: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_270_split__list__first__prop,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ? [X5: product_prod_a_a] :
( ( member449909584od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list_P1467822637od_a_a,X4: product_prod_a_a] :
( ? [Zs2: list_P1467822637od_a_a] :
( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_271_split__list__last__prop,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ? [X5: product_prod_a_a] :
( ( member449909584od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list_P1467822637od_a_a,X4: product_prod_a_a,Zs2: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_272_in__set__conv__decomp,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
= ( ? [Ys: list_P1467822637od_a_a,Zs3: list_P1467822637od_a_a] :
( Xs2
= ( append1574001858od_a_a @ Ys @ ( cons_P1921020637od_a_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_273_append__Cons__eq__iff,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Xs5: list_P1467822637od_a_a,Ys5: list_P1467822637od_a_a] :
( ~ ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ( ~ ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys2 ) )
=> ( ( ( append1574001858od_a_a @ Xs2 @ ( cons_P1921020637od_a_a @ X3 @ Ys2 ) )
= ( append1574001858od_a_a @ Xs5 @ ( cons_P1921020637od_a_a @ X3 @ Ys5 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_274_split__list__propE,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ? [X5: product_prod_a_a] :
( ( member449909584od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list_P1467822637od_a_a,X4: product_prod_a_a] :
( ? [Zs2: list_P1467822637od_a_a] :
( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X4 @ Zs2 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_275_split__list__first,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ? [Ys4: list_P1467822637od_a_a,Zs2: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X3 @ Zs2 ) ) )
& ~ ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_276_split__list__prop,axiom,
! [Xs2: list_P1467822637od_a_a,P2: product_prod_a_a > $o] :
( ? [X5: product_prod_a_a] :
( ( member449909584od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list_P1467822637od_a_a,X4: product_prod_a_a] :
( ? [Zs2: list_P1467822637od_a_a] :
( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_277_split__list__last,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ? [Ys4: list_P1467822637od_a_a,Zs2: list_P1467822637od_a_a] :
( ( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X3 @ Zs2 ) ) )
& ~ ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_278_split__list,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ? [Ys4: list_P1467822637od_a_a,Zs2: list_P1467822637od_a_a] :
( Xs2
= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_279_rev__nonempty__induct,axiom,
! [Xs2: list_P1467822637od_a_a,P2: list_P1467822637od_a_a > $o] :
( ( Xs2 != nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a] : ( P2 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( ( Xs4 != nil_Product_prod_a_a )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( append1574001858od_a_a @ Xs4 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_280_append__eq__Cons__conv,axiom,
! [Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ( ( append1574001858od_a_a @ Ys2 @ Zs )
= ( cons_P1921020637od_a_a @ X3 @ Xs2 ) )
= ( ( ( Ys2 = nil_Product_prod_a_a )
& ( Zs
= ( cons_P1921020637od_a_a @ X3 @ Xs2 ) ) )
| ? [Ys6: list_P1467822637od_a_a] :
( ( Ys2
= ( cons_P1921020637od_a_a @ X3 @ Ys6 ) )
& ( ( append1574001858od_a_a @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_281_Cons__eq__append__conv,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( ( cons_P1921020637od_a_a @ X3 @ Xs2 )
= ( append1574001858od_a_a @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_Product_prod_a_a )
& ( ( cons_P1921020637od_a_a @ X3 @ Xs2 )
= Zs ) )
| ? [Ys6: list_P1467822637od_a_a] :
( ( ( cons_P1921020637od_a_a @ X3 @ Ys6 )
= Ys2 )
& ( Xs2
= ( append1574001858od_a_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_282_rev__exhaust,axiom,
! [Xs2: list_P1467822637od_a_a] :
( ( Xs2 != nil_Product_prod_a_a )
=> ~ ! [Ys4: list_P1467822637od_a_a,Y4: product_prod_a_a] :
( Xs2
!= ( append1574001858od_a_a @ Ys4 @ ( cons_P1921020637od_a_a @ Y4 @ nil_Product_prod_a_a ) ) ) ) ).
% rev_exhaust
thf(fact_283_rev__induct,axiom,
! [P2: list_P1467822637od_a_a > $o,Xs2: list_P1467822637od_a_a] :
( ( P2 @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( ( P2 @ Xs4 )
=> ( P2 @ ( append1574001858od_a_a @ Xs4 @ ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_284_shuffles_Ocases,axiom,
! [X3: produc651658087od_a_a] :
( ! [Ys4: list_P1467822637od_a_a] :
( X3
!= ( produc85179863od_a_a @ nil_Product_prod_a_a @ Ys4 ) )
=> ( ! [Xs4: list_P1467822637od_a_a] :
( X3
!= ( produc85179863od_a_a @ Xs4 @ nil_Product_prod_a_a ) )
=> ~ ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( X3
!= ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_285_sd__path_Oinduct,axiom,
! [P2: product_prod_a_a > a > list_P1467822637od_a_a > $o,A0: product_prod_a_a,A1: a,A22: list_P1467822637od_a_a] :
( ! [Uu: product_prod_a_a,Uv: a] : ( P2 @ Uu @ Uv @ nil_Product_prod_a_a )
=> ( ! [U2: a,V: a,W: a,E2: product_prod_a_a,Es: list_P1467822637od_a_a] :
( ( P2 @ ( product_Pair_a_a @ U2 @ V ) @ W @ Es )
=> ( P2 @ ( product_Pair_a_a @ U2 @ V ) @ W @ ( cons_P1921020637od_a_a @ E2 @ Es ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ).
% sd_path.induct
thf(fact_286_co__path_Oinduct,axiom,
! [P2: product_prod_a_a > a > list_P1467822637od_a_a > $o,A0: product_prod_a_a,A1: a,A22: list_P1467822637od_a_a] :
( ! [Uu: product_prod_a_a,Uv: a] : ( P2 @ Uu @ Uv @ nil_Product_prod_a_a )
=> ( ! [Uw: product_prod_a_a,Ux: a,E2: product_prod_a_a] : ( P2 @ Uw @ Ux @ ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) )
=> ( ! [U2: a,V: a,W: a,E1: product_prod_a_a,E22: product_prod_a_a,Es: list_P1467822637od_a_a] :
( ( ( ( E1
= ( product_Pair_a_a @ U2 @ W ) )
& ( E22
= ( product_Pair_a_a @ W @ V ) ) )
=> ( P2 @ ( product_Pair_a_a @ U2 @ V ) @ W @ Es ) )
=> ( ( ~ ( ( E1
= ( product_Pair_a_a @ U2 @ W ) )
& ( E22
= ( product_Pair_a_a @ W @ V ) ) )
=> ( ( ( E1
= ( product_Pair_a_a @ V @ W ) )
& ( E22
= ( product_Pair_a_a @ W @ U2 ) ) )
=> ( P2 @ ( product_Pair_a_a @ U2 @ V ) @ W @ Es ) ) )
=> ( ( ~ ( ( E1
= ( product_Pair_a_a @ U2 @ W ) )
& ( E22
= ( product_Pair_a_a @ W @ V ) ) )
=> ( ~ ( ( E1
= ( product_Pair_a_a @ V @ W ) )
& ( E22
= ( product_Pair_a_a @ W @ U2 ) ) )
=> ( P2 @ ( product_Pair_a_a @ U2 @ V ) @ W @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) ) )
=> ( P2 @ ( product_Pair_a_a @ U2 @ V ) @ W @ ( cons_P1921020637od_a_a @ E1 @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) ) ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ) ).
% co_path.induct
thf(fact_287_sd__path_Ocases,axiom,
! [X3: produc821581462od_a_a] :
( ! [Uu: product_prod_a_a,Uv: a] :
( X3
!= ( produc600870406od_a_a @ Uu @ ( produc575024326od_a_a @ Uv @ nil_Product_prod_a_a ) ) )
=> ~ ! [U2: a,V: a,W: a,E2: product_prod_a_a,Es: list_P1467822637od_a_a] :
( X3
!= ( produc600870406od_a_a @ ( product_Pair_a_a @ U2 @ V ) @ ( produc575024326od_a_a @ W @ ( cons_P1921020637od_a_a @ E2 @ Es ) ) ) ) ) ).
% sd_path.cases
thf(fact_288_co__path_Ocases,axiom,
! [X3: produc821581462od_a_a] :
( ! [Uu: product_prod_a_a,Uv: a] :
( X3
!= ( produc600870406od_a_a @ Uu @ ( produc575024326od_a_a @ Uv @ nil_Product_prod_a_a ) ) )
=> ( ! [Uw: product_prod_a_a,Ux: a,E2: product_prod_a_a] :
( X3
!= ( produc600870406od_a_a @ Uw @ ( produc575024326od_a_a @ Ux @ ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) ) ) )
=> ~ ! [U2: a,V: a,W: a,E1: product_prod_a_a,E22: product_prod_a_a,Es: list_P1467822637od_a_a] :
( X3
!= ( produc600870406od_a_a @ ( product_Pair_a_a @ U2 @ V ) @ ( produc575024326od_a_a @ W @ ( cons_P1921020637od_a_a @ E1 @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) ) ) ) ) ) ).
% co_path.cases
thf(fact_289_lex__append__leftI,axiom,
! [Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a,Xs2: list_P1467822637od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Ys2 @ Zs ) @ ( lex_Product_prod_a_a @ R ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) @ ( append1574001858od_a_a @ Xs2 @ Zs ) ) @ ( lex_Product_prod_a_a @ R ) ) ) ).
% lex_append_leftI
thf(fact_290_progressing__single,axiom,
! [E: product_prod_a_a] : ( kurato1072996284sing_a @ ( cons_P1921020637od_a_a @ E @ nil_Product_prod_a_a ) ) ).
% progressing_single
thf(fact_291_lexord__partial__trans,axiom,
! [Xs2: list_a,R: set_Product_prod_a_a,Ys2: list_a,Zs: list_a] :
( ! [X4: a,Y4: a,Z2: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs2 ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z2 ) @ R ) ) ) )
=> ( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( lexord_a @ R ) )
=> ( ( member1250517072list_a @ ( produc1448356631list_a @ Ys2 @ Zs ) @ ( lexord_a @ R ) )
=> ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Zs ) @ ( lexord_a @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_292_lexord__sufE,axiom,
! [Xs2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a,Qs: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Xs2 @ Zs ) @ ( append1574001858od_a_a @ Ys2 @ Qs ) ) @ ( lexord673907738od_a_a @ R ) )
=> ( ( Xs2 != Ys2 )
=> ( ( ( size_s642698393od_a_a @ Xs2 )
= ( size_s642698393od_a_a @ Ys2 ) )
=> ( ( ( size_s642698393od_a_a @ Zs )
= ( size_s642698393od_a_a @ Qs ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( lexord673907738od_a_a @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_293_subset__eq__mset__impl_Ocases,axiom,
! [X3: produc651658087od_a_a] :
( ! [Ys4: list_P1467822637od_a_a] :
( X3
!= ( produc85179863od_a_a @ nil_Product_prod_a_a @ Ys4 ) )
=> ~ ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Ys4: list_P1467822637od_a_a] :
( X3
!= ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ Ys4 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_294_asym_Oinducts,axiom,
! [X3: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( asym_a @ X3 )
=> ( ! [R3: set_Product_prod_a_a] :
( ! [A6: a,B4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ B4 ) @ R3 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B4 @ A6 ) @ R3 ) )
=> ( P2 @ R3 ) )
=> ( P2 @ X3 ) ) ) ).
% asym.inducts
thf(fact_295_asym_Ointros,axiom,
! [R2: set_Product_prod_a_a] :
( ! [A3: a,B3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R2 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ R2 ) )
=> ( asym_a @ R2 ) ) ).
% asym.intros
thf(fact_296_asym_Osimps,axiom,
( asym_a
= ( ^ [A5: set_Product_prod_a_a] :
? [R4: set_Product_prod_a_a] :
( ( A5 = R4 )
& ! [X: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R4 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R4 ) ) ) ) ) ).
% asym.simps
thf(fact_297_asym_Ocases,axiom,
! [A: set_Product_prod_a_a] :
( ( asym_a @ A )
=> ! [A6: a,B4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ B4 ) @ A )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B4 @ A6 ) @ A ) ) ) ).
% asym.cases
thf(fact_298_asym__iff,axiom,
( asym_a
= ( ^ [R4: set_Product_prod_a_a] :
! [X: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R4 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R4 ) ) ) ) ).
% asym_iff
thf(fact_299_irreflI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [A3: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R2 )
=> ( irrefl_a @ R2 ) ) ).
% irreflI
thf(fact_300_irrefl__def,axiom,
( irrefl_a
= ( ^ [R5: set_Product_prod_a_a] :
! [A5: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ A5 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_301_asymD,axiom,
! [R2: set_Product_prod_a_a,X3: a,Y3: a] :
( ( asym_a @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ R2 ) ) ) ).
% asymD
thf(fact_302_inverse__permutation__of__list_Ocases,axiom,
! [X3: produc657167928_a_a_a] :
( ! [X4: a] :
( X3
!= ( produc998622632_a_a_a @ nil_Product_prod_a_a @ X4 ) )
=> ~ ! [Y4: a,X6: a,Xs4: list_P1467822637od_a_a,X4: a] :
( X3
!= ( produc998622632_a_a_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y4 @ X6 ) @ Xs4 ) @ X4 ) ) ) ).
% inverse_permutation_of_list.cases
thf(fact_303_inverse__permutation__of__list_Oinduct,axiom,
! [P2: list_P1467822637od_a_a > a > $o,A0: list_P1467822637od_a_a,A1: a] :
( ! [X_1: a] : ( P2 @ nil_Product_prod_a_a @ X_1 )
=> ( ! [Y4: a,X6: a,Xs4: list_P1467822637od_a_a,X4: a] :
( ( ( X4 != X6 )
=> ( P2 @ Xs4 @ X4 ) )
=> ( P2 @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y4 @ X6 ) @ Xs4 ) @ X4 ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% inverse_permutation_of_list.induct
thf(fact_304_sd__path_Osimps_I2_J,axiom,
! [U: a,V2: a,W2: a,E: product_prod_a_a,Es2: list_P1467822637od_a_a] :
( ( pair_sd_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ ( cons_P1921020637od_a_a @ E @ Es2 ) )
= ( append1574001858od_a_a
@ ( if_lis321365491od_a_a
@ ( E
= ( product_Pair_a_a @ U @ V2 ) )
@ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ U @ W2 ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ W2 @ V2 ) @ nil_Product_prod_a_a ) )
@ ( if_lis321365491od_a_a
@ ( E
= ( product_Pair_a_a @ V2 @ U ) )
@ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ V2 @ W2 ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ W2 @ U ) @ nil_Product_prod_a_a ) )
@ ( cons_P1921020637od_a_a @ E @ nil_Product_prod_a_a ) ) )
@ ( pair_sd_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ Es2 ) ) ) ).
% sd_path.simps(2)
thf(fact_305_sd__path__id,axiom,
! [X3: a,Y3: a,P3: list_P1467822637od_a_a,W2: a] :
( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ P3 ) )
=> ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ ( set_Product_prod_a_a2 @ P3 ) )
=> ( ( pair_sd_path_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ W2 @ P3 )
= P3 ) ) ) ).
% sd_path_id
thf(fact_306_sd__path_Oelims,axiom,
! [X3: product_prod_a_a,Xa2: a,Xb: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ( ( pair_sd_path_a @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( ( Xb = nil_Product_prod_a_a )
=> ( Y3 != nil_Product_prod_a_a ) )
=> ~ ! [U2: a,V: a] :
( ( X3
= ( product_Pair_a_a @ U2 @ V ) )
=> ! [E2: product_prod_a_a,Es: list_P1467822637od_a_a] :
( ( Xb
= ( cons_P1921020637od_a_a @ E2 @ Es ) )
=> ( Y3
!= ( append1574001858od_a_a
@ ( if_lis321365491od_a_a
@ ( E2
= ( product_Pair_a_a @ U2 @ V ) )
@ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ U2 @ Xa2 ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Xa2 @ V ) @ nil_Product_prod_a_a ) )
@ ( if_lis321365491od_a_a
@ ( E2
= ( product_Pair_a_a @ V @ U2 ) )
@ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ V @ Xa2 ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Xa2 @ U2 ) @ nil_Product_prod_a_a ) )
@ ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) ) )
@ ( pair_sd_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ Es ) ) ) ) ) ) ) ).
% sd_path.elims
thf(fact_307_the__elem__set,axiom,
! [X3: product_prod_a_a] :
( ( the_el1640649244od_a_a @ ( set_Product_prod_a_a2 @ ( cons_P1921020637od_a_a @ X3 @ nil_Product_prod_a_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_308_sd__path_Opelims,axiom,
! [X3: product_prod_a_a,Xa2: a,Xb: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ( ( pair_sd_path_a @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( accp_P1029463583od_a_a @ pair_sd_path_rel_a @ ( produc600870406od_a_a @ X3 @ ( produc575024326od_a_a @ Xa2 @ Xb ) ) )
=> ( ( ( Xb = nil_Product_prod_a_a )
=> ( ( Y3 = nil_Product_prod_a_a )
=> ~ ( accp_P1029463583od_a_a @ pair_sd_path_rel_a @ ( produc600870406od_a_a @ X3 @ ( produc575024326od_a_a @ Xa2 @ nil_Product_prod_a_a ) ) ) ) )
=> ~ ! [U2: a,V: a] :
( ( X3
= ( product_Pair_a_a @ U2 @ V ) )
=> ! [E2: product_prod_a_a,Es: list_P1467822637od_a_a] :
( ( Xb
= ( cons_P1921020637od_a_a @ E2 @ Es ) )
=> ( ( Y3
= ( append1574001858od_a_a
@ ( if_lis321365491od_a_a
@ ( E2
= ( product_Pair_a_a @ U2 @ V ) )
@ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ U2 @ Xa2 ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Xa2 @ V ) @ nil_Product_prod_a_a ) )
@ ( if_lis321365491od_a_a
@ ( E2
= ( product_Pair_a_a @ V @ U2 ) )
@ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ V @ Xa2 ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Xa2 @ U2 ) @ nil_Product_prod_a_a ) )
@ ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) ) )
@ ( pair_sd_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ Es ) ) )
=> ~ ( accp_P1029463583od_a_a @ pair_sd_path_rel_a @ ( produc600870406od_a_a @ ( product_Pair_a_a @ U2 @ V ) @ ( produc575024326od_a_a @ Xa2 @ ( cons_P1921020637od_a_a @ E2 @ Es ) ) ) ) ) ) ) ) ) ) ).
% sd_path.pelims
thf(fact_309_list__exhaust2,axiom,
! [Y3: list_P1467822637od_a_a,Ya: list_P1467822637od_a_a] :
( ( ( Y3 = nil_Product_prod_a_a )
=> ( Ya != nil_Product_prod_a_a ) )
=> ( ( ( Y3 = nil_Product_prod_a_a )
=> ! [X212: product_prod_a_a,X222: list_P1467822637od_a_a] :
( Ya
!= ( cons_P1921020637od_a_a @ X212 @ X222 ) ) )
=> ( ( ? [X212: product_prod_a_a,X222: list_P1467822637od_a_a] :
( Y3
= ( cons_P1921020637od_a_a @ X212 @ X222 ) )
=> ( Ya != nil_Product_prod_a_a ) )
=> ~ ( ? [X212: product_prod_a_a,X222: list_P1467822637od_a_a] :
( Y3
= ( cons_P1921020637od_a_a @ X212 @ X222 ) )
=> ! [X21a: product_prod_a_a,X22a: list_P1467822637od_a_a] :
( Ya
!= ( cons_P1921020637od_a_a @ X21a @ X22a ) ) ) ) ) ) ).
% list_exhaust2
thf(fact_310_list__exhaust__NSC,axiom,
! [Xs2: list_P1467822637od_a_a] :
( ( Xs2 != nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a] :
( Xs2
!= ( cons_P1921020637od_a_a @ X4 @ nil_Product_prod_a_a ) )
=> ~ ! [X4: product_prod_a_a,Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( Xs2
!= ( cons_P1921020637od_a_a @ X4 @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) ) ) ) ) ).
% list_exhaust_NSC
thf(fact_311_lenlex__append2,axiom,
! [R2: set_Pr1948701895od_a_a,Us: list_P1467822637od_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( irrefl76447196od_a_a @ R2 )
=> ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Us @ Xs2 ) @ ( append1574001858od_a_a @ Us @ Ys2 ) ) @ ( lenlex2038501170od_a_a @ R2 ) )
= ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( lenlex2038501170od_a_a @ R2 ) ) ) ) ).
% lenlex_append2
thf(fact_312_lenlex__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a] :
( ! [X4: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
=> ~ ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Xs2 ) @ ( lenlex_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_313_lenlex__append1,axiom,
! [Us: list_P1467822637od_a_a,Xs2: list_P1467822637od_a_a,R2: set_Pr1948701895od_a_a,Vs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Us @ Xs2 ) @ ( lenlex2038501170od_a_a @ R2 ) )
=> ( ( ( size_s642698393od_a_a @ Vs )
= ( size_s642698393od_a_a @ Ys2 ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( append1574001858od_a_a @ Us @ Vs ) @ ( append1574001858od_a_a @ Xs2 @ Ys2 ) ) @ ( lenlex2038501170od_a_a @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_314_co__path_Opelims,axiom,
! [X3: product_prod_a_a,Xa2: a,Xb: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ( ( pair_co_path_a @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( accp_P1029463583od_a_a @ pair_co_path_rel_a @ ( produc600870406od_a_a @ X3 @ ( produc575024326od_a_a @ Xa2 @ Xb ) ) )
=> ( ( ( Xb = nil_Product_prod_a_a )
=> ( ( Y3 = nil_Product_prod_a_a )
=> ~ ( accp_P1029463583od_a_a @ pair_co_path_rel_a @ ( produc600870406od_a_a @ X3 @ ( produc575024326od_a_a @ Xa2 @ nil_Product_prod_a_a ) ) ) ) )
=> ( ! [E2: product_prod_a_a] :
( ( Xb
= ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) )
=> ( ( Y3
= ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) )
=> ~ ( accp_P1029463583od_a_a @ pair_co_path_rel_a @ ( produc600870406od_a_a @ X3 @ ( produc575024326od_a_a @ Xa2 @ ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) ) ) ) ) )
=> ~ ! [U2: a,V: a] :
( ( X3
= ( product_Pair_a_a @ U2 @ V ) )
=> ! [E1: product_prod_a_a,E22: product_prod_a_a,Es: list_P1467822637od_a_a] :
( ( Xb
= ( cons_P1921020637od_a_a @ E1 @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) )
=> ( ( ( ( ( E1
= ( product_Pair_a_a @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ V ) ) )
=> ( Y3
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ U2 @ V ) @ ( pair_co_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair_a_a @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ V ) ) )
=> ( ( ( ( E1
= ( product_Pair_a_a @ V @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ V @ U2 ) @ ( pair_co_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair_a_a @ V @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons_P1921020637od_a_a @ E1 @ ( pair_co_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) ) ) ) ) ) )
=> ~ ( accp_P1029463583od_a_a @ pair_co_path_rel_a @ ( produc600870406od_a_a @ ( product_Pair_a_a @ U2 @ V ) @ ( produc575024326od_a_a @ Xa2 @ ( cons_P1921020637od_a_a @ E1 @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) ) ) ) ) ) ) ) ) ) ) ).
% co_path.pelims
thf(fact_315_pawalk__verts_Osimps_I1_J,axiom,
! [U: product_prod_a_a] :
( ( pair_p2140537249od_a_a @ U @ nil_Pr1049924845od_a_a )
= ( cons_P1921020637od_a_a @ U @ nil_Product_prod_a_a ) ) ).
% pawalk_verts.simps(1)
thf(fact_316_co__path_Osimps_I3_J,axiom,
! [E12: product_prod_a_a,U: a,W2: a,E23: product_prod_a_a,V2: a,Es2: list_P1467822637od_a_a] :
( ( ( ( E12
= ( product_Pair_a_a @ U @ W2 ) )
& ( E23
= ( product_Pair_a_a @ W2 @ V2 ) ) )
=> ( ( pair_co_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ ( cons_P1921020637od_a_a @ E12 @ ( cons_P1921020637od_a_a @ E23 @ Es2 ) ) )
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ U @ V2 ) @ ( pair_co_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ Es2 ) ) ) )
& ( ~ ( ( E12
= ( product_Pair_a_a @ U @ W2 ) )
& ( E23
= ( product_Pair_a_a @ W2 @ V2 ) ) )
=> ( ( ( ( E12
= ( product_Pair_a_a @ V2 @ W2 ) )
& ( E23
= ( product_Pair_a_a @ W2 @ U ) ) )
=> ( ( pair_co_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ ( cons_P1921020637od_a_a @ E12 @ ( cons_P1921020637od_a_a @ E23 @ Es2 ) ) )
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ V2 @ U ) @ ( pair_co_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ Es2 ) ) ) )
& ( ~ ( ( E12
= ( product_Pair_a_a @ V2 @ W2 ) )
& ( E23
= ( product_Pair_a_a @ W2 @ U ) ) )
=> ( ( pair_co_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ ( cons_P1921020637od_a_a @ E12 @ ( cons_P1921020637od_a_a @ E23 @ Es2 ) ) )
= ( cons_P1921020637od_a_a @ E12 @ ( pair_co_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ ( cons_P1921020637od_a_a @ E23 @ Es2 ) ) ) ) ) ) ) ) ).
% co_path.simps(3)
thf(fact_317_co__path_Osimps_I2_J,axiom,
! [Uw2: product_prod_a_a,Ux2: a,E: product_prod_a_a] :
( ( pair_co_path_a @ Uw2 @ Ux2 @ ( cons_P1921020637od_a_a @ E @ nil_Product_prod_a_a ) )
= ( cons_P1921020637od_a_a @ E @ nil_Product_prod_a_a ) ) ).
% co_path.simps(2)
thf(fact_318_co__path_Oelims,axiom,
! [X3: product_prod_a_a,Xa2: a,Xb: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ( ( pair_co_path_a @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( ( Xb = nil_Product_prod_a_a )
=> ( Y3 != nil_Product_prod_a_a ) )
=> ( ! [E2: product_prod_a_a] :
( ( Xb
= ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) )
=> ( Y3
!= ( cons_P1921020637od_a_a @ E2 @ nil_Product_prod_a_a ) ) )
=> ~ ! [U2: a,V: a] :
( ( X3
= ( product_Pair_a_a @ U2 @ V ) )
=> ! [E1: product_prod_a_a,E22: product_prod_a_a,Es: list_P1467822637od_a_a] :
( ( Xb
= ( cons_P1921020637od_a_a @ E1 @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) )
=> ~ ( ( ( ( E1
= ( product_Pair_a_a @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ V ) ) )
=> ( Y3
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ U2 @ V ) @ ( pair_co_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair_a_a @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ V ) ) )
=> ( ( ( ( E1
= ( product_Pair_a_a @ V @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ V @ U2 ) @ ( pair_co_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair_a_a @ V @ Xa2 ) )
& ( E22
= ( product_Pair_a_a @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons_P1921020637od_a_a @ E1 @ ( pair_co_path_a @ ( product_Pair_a_a @ U2 @ V ) @ Xa2 @ ( cons_P1921020637od_a_a @ E22 @ Es ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% co_path.elims
thf(fact_319_co__sd__id,axiom,
! [U: a,W2: a,P3: list_P1467822637od_a_a,V2: a] :
( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ U @ W2 ) @ ( set_Product_prod_a_a2 @ P3 ) )
=> ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ V2 @ W2 ) @ ( set_Product_prod_a_a2 @ P3 ) )
=> ( ( pair_co_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ ( pair_sd_path_a @ ( product_Pair_a_a @ U @ V2 ) @ W2 @ P3 ) )
= P3 ) ) ) ).
% co_sd_id
thf(fact_320_Cons__lenlex__iff,axiom,
! [M: product_prod_a_a,Ms: list_P1467822637od_a_a,N: product_prod_a_a,Ns: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ M @ Ms ) @ ( cons_P1921020637od_a_a @ N @ Ns ) ) @ ( lenlex2038501170od_a_a @ R ) )
= ( ( ord_less_nat @ ( size_s642698393od_a_a @ Ms ) @ ( size_s642698393od_a_a @ Ns ) )
| ( ( ( size_s642698393od_a_a @ Ms )
= ( size_s642698393od_a_a @ Ns ) )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ M @ N ) @ R ) )
| ( ( M = N )
& ( member1691432592od_a_a @ ( produc85179863od_a_a @ Ms @ Ns ) @ ( lenlex2038501170od_a_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_321_Cons__lenlex__iff,axiom,
! [M: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ M @ N ) @ R ) )
| ( ( M = N )
& ( member1250517072list_a @ ( produc1448356631list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_322_Cons__in__subseqsD,axiom,
! [Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a,Xs2: list_P1467822637od_a_a] :
( ( member1578987478od_a_a @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) @ ( set_li1284606018od_a_a @ ( subseq2115548028od_a_a @ Xs2 ) ) )
=> ( member1578987478od_a_a @ Ys2 @ ( set_li1284606018od_a_a @ ( subseq2115548028od_a_a @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_323_in__measures_I2_J,axiom,
! [X3: a,Y3: a,F: a > nat,Fs: list_a_nat] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
| ( ( ( F @ X3 )
= ( F @ Y3 ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( measures_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_324_inverse__permutation__of__list_Oelims,axiom,
! [X3: list_P1467822637od_a_a,Xa2: a,Y3: a] :
( ( ( invers1007791577list_a @ X3 @ Xa2 )
= Y3 )
=> ( ( ( X3 = nil_Product_prod_a_a )
=> ( Y3 != Xa2 ) )
=> ~ ! [Y4: a,X6: a,Xs4: list_P1467822637od_a_a] :
( ( X3
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y4 @ X6 ) @ Xs4 ) )
=> ~ ( ( ( Xa2 = X6 )
=> ( Y3 = Y4 ) )
& ( ( Xa2 != X6 )
=> ( Y3
= ( invers1007791577list_a @ Xs4 @ Xa2 ) ) ) ) ) ) ) ).
% inverse_permutation_of_list.elims
thf(fact_325_in__measures_I1_J,axiom,
! [X3: a,Y3: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( measures_a @ nil_a_nat ) ) ).
% in_measures(1)
thf(fact_326_eval__inverse__permutation__of__list_I3_J,axiom,
! [X3: a,X7: a,Y5: a,Xs2: list_P1467822637od_a_a] :
( ( X3 != X7 )
=> ( ( invers1007791577list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y5 @ X7 ) @ Xs2 ) @ X3 )
= ( invers1007791577list_a @ Xs2 @ X3 ) ) ) ).
% eval_inverse_permutation_of_list(3)
thf(fact_327_eval__inverse__permutation__of__list_I2_J,axiom,
! [X3: a,X7: a,Y3: a,Xs2: list_P1467822637od_a_a] :
( ( X3 = X7 )
=> ( ( invers1007791577list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y3 @ X7 ) @ Xs2 ) @ X3 )
= Y3 ) ) ).
% eval_inverse_permutation_of_list(2)
thf(fact_328_inverse__permutation__of__list_Osimps_I2_J,axiom,
! [X3: a,X7: a,Y3: a,Xs2: list_P1467822637od_a_a] :
( ( ( X3 = X7 )
=> ( ( invers1007791577list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y3 @ X7 ) @ Xs2 ) @ X3 )
= Y3 ) )
& ( ( X3 != X7 )
=> ( ( invers1007791577list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y3 @ X7 ) @ Xs2 ) @ X3 )
= ( invers1007791577list_a @ Xs2 @ X3 ) ) ) ) ).
% inverse_permutation_of_list.simps(2)
thf(fact_329_measures__less,axiom,
! [F: a > nat,X3: a,Y3: a,Fs: list_a_nat] :
( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_330_inverse__permutation__of__list__unique,axiom,
! [Xs2: list_P1467822637od_a_a,A4: set_a,X3: a,Y3: a] :
( ( list_permutes_a @ Xs2 @ A4 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ( ( invers1007791577list_a @ Xs2 @ Y3 )
= X3 ) ) ) ).
% inverse_permutation_of_list_unique
thf(fact_331_eval__permutation__of__list_I3_J,axiom,
! [X3: a,X7: a,Y5: a,Xs2: list_P1467822637od_a_a] :
( ( X3 != X7 )
=> ( ( permut1422637723list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ X7 @ Y5 ) @ Xs2 ) @ X3 )
= ( permut1422637723list_a @ Xs2 @ X3 ) ) ) ).
% eval_permutation_of_list(3)
thf(fact_332_eval__permutation__of__list_I2_J,axiom,
! [X3: a,X7: a,Y3: a,Xs2: list_P1467822637od_a_a] :
( ( X3 = X7 )
=> ( ( permut1422637723list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ X7 @ Y3 ) @ Xs2 ) @ X3 )
= Y3 ) ) ).
% eval_permutation_of_list(2)
thf(fact_333_permutation__of__list__unique,axiom,
! [Xs2: list_P1467822637od_a_a,A4: set_a,X3: a,Y3: a] :
( ( list_permutes_a @ Xs2 @ A4 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ( ( permut1422637723list_a @ Xs2 @ X3 )
= Y3 ) ) ) ).
% permutation_of_list_unique
thf(fact_334_permutation__of__list__Cons,axiom,
! [X3: a,X7: a,Y3: a,Xs2: list_P1467822637od_a_a] :
( ( ( X3 = X7 )
=> ( ( permut1422637723list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Xs2 ) @ X7 )
= Y3 ) )
& ( ( X3 != X7 )
=> ( ( permut1422637723list_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Xs2 ) @ X7 )
= ( permut1422637723list_a @ Xs2 @ X7 ) ) ) ) ).
% permutation_of_list_Cons
thf(fact_335_not__in__set__insert,axiom,
! [X3: product_prod_a_a,Xs2: list_P1467822637od_a_a] :
( ~ ( member449909584od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ( ( insert2123825731od_a_a @ X3 @ Xs2 )
= ( cons_P1921020637od_a_a @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_336_insert__Nil,axiom,
! [X3: product_prod_a_a] :
( ( insert2123825731od_a_a @ X3 @ nil_Product_prod_a_a )
= ( cons_P1921020637od_a_a @ X3 @ nil_Product_prod_a_a ) ) ).
% insert_Nil
thf(fact_337_List_Oinsert__def,axiom,
( insert2123825731od_a_a
= ( ^ [X: product_prod_a_a,Xs: list_P1467822637od_a_a] : ( if_lis321365491od_a_a @ ( member449909584od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) @ Xs @ ( cons_P1921020637od_a_a @ X @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_338_concat__eq__append__conv,axiom,
! [Xss2: list_l1108175539od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( ( concat1878217064od_a_a @ Xss2 )
= ( append1574001858od_a_a @ Ys2 @ Zs ) )
= ( ( ( Xss2 = nil_li1885598003od_a_a )
=> ( ( Ys2 = nil_Product_prod_a_a )
& ( Zs = nil_Product_prod_a_a ) ) )
& ( ( Xss2 != nil_li1885598003od_a_a )
=> ? [Xss1: list_l1108175539od_a_a,Xs: list_P1467822637od_a_a,Xs6: list_P1467822637od_a_a,Xss22: list_l1108175539od_a_a] :
( ( Xss2
= ( append2141280328od_a_a @ Xss1 @ ( cons_l724116707od_a_a @ ( append1574001858od_a_a @ Xs @ Xs6 ) @ Xss22 ) ) )
& ( Ys2
= ( append1574001858od_a_a @ ( concat1878217064od_a_a @ Xss1 ) @ Xs ) )
& ( Zs
= ( append1574001858od_a_a @ Xs6 @ ( concat1878217064od_a_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_339_inverse__permutation__of__list_Opelims,axiom,
! [X3: list_P1467822637od_a_a,Xa2: a,Y3: a] :
( ( ( invers1007791577list_a @ X3 @ Xa2 )
= Y3 )
=> ( ( accp_P941981633_a_a_a @ invers214109608_rel_a @ ( produc998622632_a_a_a @ X3 @ Xa2 ) )
=> ( ( ( X3 = nil_Product_prod_a_a )
=> ( ( Y3 = Xa2 )
=> ~ ( accp_P941981633_a_a_a @ invers214109608_rel_a @ ( produc998622632_a_a_a @ nil_Product_prod_a_a @ Xa2 ) ) ) )
=> ~ ! [Y4: a,X6: a,Xs4: list_P1467822637od_a_a] :
( ( X3
= ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y4 @ X6 ) @ Xs4 ) )
=> ( ( ( ( Xa2 = X6 )
=> ( Y3 = Y4 ) )
& ( ( Xa2 != X6 )
=> ( Y3
= ( invers1007791577list_a @ Xs4 @ Xa2 ) ) ) )
=> ~ ( accp_P941981633_a_a_a @ invers214109608_rel_a @ ( produc998622632_a_a_a @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y4 @ X6 ) @ Xs4 ) @ Xa2 ) ) ) ) ) ) ) ).
% inverse_permutation_of_list.pelims
thf(fact_340_concat__append,axiom,
! [Xs2: list_l1108175539od_a_a,Ys2: list_l1108175539od_a_a] :
( ( concat1878217064od_a_a @ ( append2141280328od_a_a @ Xs2 @ Ys2 ) )
= ( append1574001858od_a_a @ ( concat1878217064od_a_a @ Xs2 ) @ ( concat1878217064od_a_a @ Ys2 ) ) ) ).
% concat_append
thf(fact_341_concat_Osimps_I2_J,axiom,
! [X3: list_P1467822637od_a_a,Xs2: list_l1108175539od_a_a] :
( ( concat1878217064od_a_a @ ( cons_l724116707od_a_a @ X3 @ Xs2 ) )
= ( append1574001858od_a_a @ X3 @ ( concat1878217064od_a_a @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_342_concat__eq__appendD,axiom,
! [Xss2: list_l1108175539od_a_a,Ys2: list_P1467822637od_a_a,Zs: list_P1467822637od_a_a] :
( ( ( concat1878217064od_a_a @ Xss2 )
= ( append1574001858od_a_a @ Ys2 @ Zs ) )
=> ( ( Xss2 != nil_li1885598003od_a_a )
=> ? [Xss12: list_l1108175539od_a_a,Xs4: list_P1467822637od_a_a,Xs3: list_P1467822637od_a_a,Xss23: list_l1108175539od_a_a] :
( ( Xss2
= ( append2141280328od_a_a @ Xss12 @ ( cons_l724116707od_a_a @ ( append1574001858od_a_a @ Xs4 @ Xs3 ) @ Xss23 ) ) )
& ( Ys2
= ( append1574001858od_a_a @ ( concat1878217064od_a_a @ Xss12 ) @ Xs4 ) )
& ( Zs
= ( append1574001858od_a_a @ Xs3 @ ( concat1878217064od_a_a @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_343_listrel_Oinducts,axiom,
! [X1: list_P1467822637od_a_a,X2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a,P2: list_P1467822637od_a_a > list_P1467822637od_a_a > $o] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ X1 @ X2 ) @ ( listre2074280881od_a_a @ R ) )
=> ( ( P2 @ nil_Product_prod_a_a @ nil_Product_prod_a_a )
=> ( ! [X4: product_prod_a_a,Y4: product_prod_a_a,Xs4: list_P1467822637od_a_a,Ys4: list_P1467822637od_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
=> ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs4 @ Ys4 ) @ ( listre2074280881od_a_a @ R ) )
=> ( ( P2 @ Xs4 @ Ys4 )
=> ( P2 @ ( cons_P1921020637od_a_a @ X4 @ Xs4 ) @ ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) ) ) ) )
=> ( P2 @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_344_listrel_Oinducts,axiom,
! [X1: list_a,X2: list_a,R: set_Product_prod_a_a,P2: list_a > list_a > $o] :
( ( member1250517072list_a @ ( produc1448356631list_a @ X1 @ X2 ) @ ( listrel_a_a @ R ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X4: a,Y4: a,Xs4: list_a,Ys4: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
=> ( ( member1250517072list_a @ ( produc1448356631list_a @ Xs4 @ Ys4 ) @ ( listrel_a_a @ R ) )
=> ( ( P2 @ Xs4 @ Ys4 )
=> ( P2 @ ( cons_a @ X4 @ Xs4 ) @ ( cons_a @ Y4 @ Ys4 ) ) ) ) )
=> ( P2 @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_345_listrel_OCons,axiom,
! [X3: product_prod_a_a,Y3: product_prod_a_a,R: set_Pr1948701895od_a_a,Xs2: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
=> ( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ Ys2 ) @ ( listre2074280881od_a_a @ R ) )
=> ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ X3 @ Xs2 ) @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) ) @ ( listre2074280881od_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_346_listrel_OCons,axiom,
! [X3: a,Y3: a,R: set_Product_prod_a_a,Xs2: list_a,Ys2: list_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
=> ( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R ) )
=> ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) @ ( listrel_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_347_listrel__Cons1,axiom,
! [Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a,Xs2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre2074280881od_a_a @ R ) )
=> ~ ! [Y4: product_prod_a_a,Ys4: list_P1467822637od_a_a] :
( ( Xs2
= ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ Y4 ) @ R )
=> ~ ( member1691432592od_a_a @ ( produc85179863od_a_a @ Ys2 @ Ys4 ) @ ( listre2074280881od_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_348_listrel__Cons1,axiom,
! [Y3: a,Ys2: list_a,Xs2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ ( cons_a @ Y3 @ Ys2 ) @ Xs2 ) @ ( listrel_a_a @ R ) )
=> ~ ! [Y4: a,Ys4: list_a] :
( ( Xs2
= ( cons_a @ Y4 @ Ys4 ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Y4 ) @ R )
=> ~ ( member1250517072list_a @ ( produc1448356631list_a @ Ys2 @ Ys4 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_349_listrel__Cons2,axiom,
! [Xs2: list_P1467822637od_a_a,Y3: product_prod_a_a,Ys2: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs2 @ ( cons_P1921020637od_a_a @ Y3 @ Ys2 ) ) @ ( listre2074280881od_a_a @ R ) )
=> ~ ! [X4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( ( Xs2
= ( cons_P1921020637od_a_a @ X4 @ Xs4 ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R )
=> ~ ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs4 @ Ys2 ) @ ( listre2074280881od_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_350_listrel__Cons2,axiom,
! [Xs2: list_a,Y3: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ Xs2 @ ( cons_a @ Y3 @ Ys2 ) ) @ ( listrel_a_a @ R ) )
=> ~ ! [X4: a,Xs4: list_a] :
( ( Xs2
= ( cons_a @ X4 @ Xs4 ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
=> ~ ( member1250517072list_a @ ( produc1448356631list_a @ Xs4 @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_351_listrel_Ocases,axiom,
! [A1: list_P1467822637od_a_a,A22: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ A1 @ A22 ) @ ( listre2074280881od_a_a @ R ) )
=> ( ( ( A1 = nil_Product_prod_a_a )
=> ( A22 != nil_Product_prod_a_a ) )
=> ~ ! [X4: product_prod_a_a,Y4: product_prod_a_a,Xs4: list_P1467822637od_a_a] :
( ( A1
= ( cons_P1921020637od_a_a @ X4 @ Xs4 ) )
=> ! [Ys4: list_P1467822637od_a_a] :
( ( A22
= ( cons_P1921020637od_a_a @ Y4 @ Ys4 ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
=> ~ ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs4 @ Ys4 ) @ ( listre2074280881od_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_352_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X4: a,Y4: a,Xs4: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs4 ) )
=> ! [Ys4: list_a] :
( ( A22
= ( cons_a @ Y4 @ Ys4 ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
=> ~ ( member1250517072list_a @ ( produc1448356631list_a @ Xs4 @ Ys4 ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_353_listrel_Osimps,axiom,
! [A1: list_P1467822637od_a_a,A22: list_P1467822637od_a_a,R: set_Pr1948701895od_a_a] :
( ( member1691432592od_a_a @ ( produc85179863od_a_a @ A1 @ A22 ) @ ( listre2074280881od_a_a @ R ) )
= ( ( ( A1 = nil_Product_prod_a_a )
& ( A22 = nil_Product_prod_a_a ) )
| ? [X: product_prod_a_a,Y: product_prod_a_a,Xs: list_P1467822637od_a_a,Ys: list_P1467822637od_a_a] :
( ( A1
= ( cons_P1921020637od_a_a @ X @ Xs ) )
& ( A22
= ( cons_P1921020637od_a_a @ Y @ Ys ) )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R )
& ( member1691432592od_a_a @ ( produc85179863od_a_a @ Xs @ Ys ) @ ( listre2074280881od_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_354_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member1250517072list_a @ ( produc1448356631list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( A1
= ( cons_a @ X @ Xs ) )
& ( A22
= ( cons_a @ Y @ Ys ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( member1250517072list_a @ ( produc1448356631list_a @ Xs @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X3: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ( if_lis321365491od_a_a @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X3: list_P1467822637od_a_a,Y3: list_P1467822637od_a_a] :
( ( if_lis321365491od_a_a @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [Xs4: list_P1467822637od_a_a,X4: a,Y4: a,Ys4: list_P1467822637od_a_a] :
( es
!= ( append1574001858od_a_a @ Xs4 @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ ( cons_P1921020637od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ Ys4 ) ) ) ) ).
%------------------------------------------------------------------------------