TPTP Problem File: SLH0247^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Multiset_Ordering_NPC/0002_Multiset_Ordering_in_NP/prob_00200_008709__13378370_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1239 ( 377 unt; 426 typ; 0 def)
% Number of atoms : 2441 (1468 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 10382 ( 580 ~; 87 |; 360 &;8185 @)
% ( 0 <=>;1170 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number of types : 96 ( 95 usr)
% Number of type conns : 1521 (1521 >; 0 *; 0 +; 0 <<)
% Number of symbols : 334 ( 331 usr; 17 con; 0-14 aty)
% Number of variables : 3340 ( 73 ^;2922 !; 345 ?;3340 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:29:06.342
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__f,type,
f: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (331)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bNF_Gr2674349488622646127od_a_a: set_li6330610657153406774od_a_a > produc4044097585999906000od_a_a > set_li6330610657153406774od_a_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__f,type,
bNF_Greatest_Shift_f: set_list_f > f > set_list_f ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bNF_Gr428374559055706347od_a_a: set_li6330610657153406774od_a_a > list_P1802437025701573206od_a_a > set_Pr5530083903271594800od_a_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__f,type,
bNF_Greatest_Succ_f: set_list_f > list_f > set_f ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
if_lis3736317622110187548od_a_a: $o > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__List__Olist_Itf__f_J,type,
if_list_f: $o > list_f > list_f > list_f ).
thf(sy_c_List_Oappend_001_Eo,type,
append_o: list_o > list_o > list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
append2032168774138627441od_a_a: list_l4888837856047780700od_a_a > list_l4888837856047780700od_a_a > list_l4888837856047780700od_a_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__f_J,type,
append_list_f: list_list_f > list_list_f > list_list_f ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
append4725183380742660459od_a_a: list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Oappend_001tf__f,type,
append_f: list_f > list_f > list_f ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bind_n7657806052693191351od_a_a: list_nat > ( nat > list_P1802437025701573206od_a_a ) > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Obind_001t__Nat__Onat_001tf__f,type,
bind_nat_f: list_nat > ( nat > list_f ) > list_f ).
thf(sy_c_List_Obind_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Nat__Onat,type,
bind_P5255732232670723929_a_nat: list_P1802437025701573206od_a_a > ( produc4044097585999906000od_a_a > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bind_P6033372204491099015od_a_a: list_P1802437025701573206od_a_a > ( produc4044097585999906000od_a_a > list_P1802437025701573206od_a_a ) > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Obind_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001tf__f,type,
bind_P5070077749997897658_a_a_f: list_P1802437025701573206od_a_a > ( produc4044097585999906000od_a_a > list_f ) > list_f ).
thf(sy_c_List_Obind_001tf__f_001t__Nat__Onat,type,
bind_f_nat: list_f > ( f > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001tf__f_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bind_f6317164213574698516od_a_a: list_f > ( f > list_P1802437025701573206od_a_a ) > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Obind_001tf__f_001tf__f,type,
bind_f_f: list_f > ( f > list_f ) > list_f ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
butlas456703966728320546od_a_a: list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Obutlast_001tf__f,type,
butlast_f: list_f > list_f ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
can_se4212505445558963533od_a_a: ( produc4044097585999906000od_a_a > $o ) > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_List_Ocan__select_001tf__a,type,
can_select_a: ( a > $o ) > set_a > $o ).
thf(sy_c_List_Ocan__select_001tf__f,type,
can_select_f: ( f > $o ) > set_f > $o ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
concat5889154033839980561od_a_a: list_l4888837856047780700od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Oconcat_001tf__f,type,
concat_f: list_list_f > list_f ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
drop_P3280791721155218966od_a_a: nat > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldr_6871341030409798377st_nat: ( list_nat > list_nat > list_nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
foldr_1618125860701030823od_a_a: ( list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ) > list_l4888837856047780700od_a_a > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Ofoldr_001t__List__Olist_Itf__f_J_001t__List__Olist_Itf__f_J,type,
foldr_list_f_list_f: ( list_f > list_f > list_f ) > list_list_f > list_f > list_f ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
insert5320946937069198828od_a_a: produc4044097585999906000od_a_a > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Oinsert_001tf__f,type,
insert_f: f > list_f > list_f ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
last_P6719674514767459439od_a_a: list_P1802437025701573206od_a_a > produc4044097585999906000od_a_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olast_001tf__f,type,
last_f: list_f > f ).
thf(sy_c_List_Olenlex_001_Eo,type,
lenlex_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
lenlex2249140905286214619od_a_a: set_Pr9202094101234783879od_a_a > set_Pr1239236860231382151od_a_a ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olenlex_001tf__f,type,
lenlex_f: set_Product_prod_f_f > set_Pr3684263487712998727list_f ).
thf(sy_c_List_Olex_001_Eo,type,
lex_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
lex_Pr4501241201491742522od_a_a: set_Pr9202094101234783879od_a_a > set_Pr1239236860231382151od_a_a ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001tf__f,type,
lex_f: set_Product_prod_f_f > set_Pr3684263487712998727list_f ).
thf(sy_c_List_Olexord_001_Eo,type,
lexord_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
thf(sy_c_List_Olexord_001t__Nat__Onat,type,
lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
lexord2630910672304566467od_a_a: set_Pr9202094101234783879od_a_a > set_Pr1239236860231382151od_a_a ).
thf(sy_c_List_Olexord_001tf__a,type,
lexord_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olexord_001tf__f,type,
lexord_f: set_Product_prod_f_f > set_Pr3684263487712998727list_f ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001_Eo,type,
cons_o: $o > list_o > list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
cons_l7647087712702168844od_a_a: list_P1802437025701573206od_a_a > list_l4888837856047780700od_a_a > list_l4888837856047780700od_a_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__f_J,type,
cons_list_f: list_f > list_list_f > list_list_f ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
cons_P8766293264717362397od_o_o: product_prod_o_o > list_P4002435161011370285od_o_o > list_P4002435161011370285od_o_o ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
cons_P7930012998920551916od_o_o: produc2310465149704154422od_o_o > list_P7815964300992299068od_o_o > list_P7815964300992299068od_o_o ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
cons_P64011158483510406od_a_a: produc4044097585999906000od_a_a > list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
cons_P7316939126706565853od_a_a: product_prod_a_a > list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_OCons_001tf__f,type,
cons_f: f > list_f > list_f ).
thf(sy_c_List_Olist_ONil_001_Eo,type,
nil_o: list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
nil_li6406708984265833308od_a_a: list_l4888837856047780700od_a_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__f_J,type,
nil_list_f: list_list_f ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
nil_Product_prod_o_o: list_P4002435161011370285od_o_o ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
nil_Pr1406083253504547388od_o_o: list_P7815964300992299068od_o_o ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
nil_Pr7291147438282410198od_a_a: list_P1802437025701573206od_a_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
nil_Product_prod_a_a: list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_ONil_001tf__f,type,
nil_f: list_f ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001tf__a,type,
map_Pr4634343309551623282_a_a_a: ( produc4044097585999906000od_a_a > a ) > list_P1802437025701573206od_a_a > list_a ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
set_li8573657874255358827od_a_a: list_l4888837856047780700od_a_a > set_li6330610657153406774od_a_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__f_J,type,
set_list_f2: list_list_f > set_list_f ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Pr7008919305446870117od_a_a: list_P1802437025701573206od_a_a > set_Pr5530083903271594800od_a_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Oset_001tf__f,type,
set_f2: list_f > set_f ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_e867735100643366937od_a_a: ( produc4044097585999906000od_a_a > $o ) > list_P1802437025701573206od_a_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__f,type,
list_ex1_f: ( f > $o ) > list_f > $o ).
thf(sy_c_List_Olistrel1_001_Eo,type,
listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
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produc3537265031573258201od_a_a: ( a > a > produc1836594585875604997od_a_a ) > product_prod_a_a > produc1836594585875604997od_a_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc6516300539023753772od_a_a: produc4044097585999906000od_a_a > a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001tf__a,type,
image_1366754389297715763_a_a_a: ( produc4044097585999906000od_a_a > a ) > set_Pr5530083903271594800od_a_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
the_el7097569866588581061od_a_a: set_Pr5530083903271594800od_a_a > produc4044097585999906000od_a_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Othe__elem_001tf__f,type,
the_elem_f: set_f > f ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
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accp_P3924000266242761678st_nat: ( produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o ) > produc1616951275169580055st_nat > $o ).
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accp_P4649787823440830728od_a_a: ( produc1476193604856630481od_a_a > produc1476193604856630481od_a_a > $o ) > produc1476193604856630481od_a_a > $o ).
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accp_P7918123743290328942list_f: ( produc944695879195883319list_f > produc944695879195883319list_f > $o ) > produc944695879195883319list_f > $o ).
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accp_P4316615423486620892st_nat: ( produc1143021204902420645st_nat > produc1143021204902420645st_nat > $o ) > produc1143021204902420645st_nat > $o ).
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accp_P1059454084221188526od_a_a: ( produc2275148110972645623od_a_a > produc2275148110972645623od_a_a > $o ) > produc2275148110972645623od_a_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mtf__f_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__List__Olist_Itf__f_J_J_J,type,
accp_P2150597548507289454list_f: ( produc969922564335260343list_f > produc969922564335260343list_f > $o ) > produc969922564335260343list_f > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_Itf__f_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
accp_P180089516854618306st_nat: ( produc2430033689614948491st_nat > produc2430033689614948491st_nat > $o ) > produc2430033689614948491st_nat > $o ).
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accp_P953168501168768366od_a_a: ( produc8995865553851515063od_a_a > produc8995865553851515063od_a_a > $o ) > produc8995865553851515063od_a_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_Itf__f_Mtf__f_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__f_J_Mt__List__Olist_Itf__f_J_J_J,type,
accp_P5019072637223936814list_f: ( produc6083374541932913271list_f > produc6083374541932913271list_f > $o ) > produc6083374541932913271list_f > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member7908416913644183295od_a_a: list_P1802437025701573206od_a_a > set_li6330610657153406774od_a_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__f_J,type,
member_list_f: list_f > set_list_f > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member3102152822625457154od_a_a: produc8345577779301699435od_a_a > set_Pr1849233907213750689od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__f_J_J,type,
member6287320593503803189list_f: produc1868580951212782750list_f > set_Pr4040610612349571924list_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member1707075787549946620st_nat: produc6950500744226188901st_nat > set_Pr8204503298718716827st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member1628670812943486672od_a_a: produc3323165008400776615od_a_a > set_Pr1239236860231382151od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__List__Olist_Itf__f_J_J,type,
member4906821000576044547list_f: produc28465420239985370list_f > set_Pr6066905245664862778list_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
member5940269043467357561od_o_o: produc7168711000267980880od_o_o > set_Pr5863406110664949424od_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member4220520676121331065od_a_a: produc1388412609224671568od_a_a > set_Pr6371270201107065520od_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,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member8185636152533575727st_nat: produc3766896510242555288st_nat > set_Pr4719305120195102542st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__f_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member2314694291444880989od_a_a: produc6659710747963597620od_a_a > set_Pr7408352053642894740od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__f_J_Mt__List__Olist_Itf__f_J_J,type,
member631280506938425232list_f: produc1604256039088728679list_f > set_Pr3684263487712998727list_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member4271028325217606668od_a_a: produc7186635310996023413od_a_a > set_Pr7697616133272258731od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J,type,
member8962352073626239679_nat_f: product_prod_nat_f > set_Pr4548512049067879902_nat_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Nat__Onat_J,type,
member1741560536349867494_a_nat: produc4657167522128284239_a_nat > set_Pr8612406289294286213_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1309718761806480592od_a_a: produc7229718524382159527od_a_a > set_Pr9202094101234783879od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mtf__f_J,type,
member6782030652459103107_a_a_f: produc3949922585562443610_a_a_f > set_Pr4596002078154142650_a_a_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
member1089989612134903033od_o_o: produc4033586494401613520od_o_o > set_Pr6507870997995571760od_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member3071122053849602553od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
member2678038196313615385_f_nat: product_prod_f_nat > set_Pr5245412377734449720_f_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member8950279100819641181od_a_a: produc6118171033922981684od_a_a > set_Pr160775897027298964od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
member5935997662887750672od_f_f: product_prod_f_f > set_Product_prod_f_f > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__f,type,
member_f: f > set_f > $o ).
thf(sy_v__092_060alpha_062,type,
alpha: a > $o ).
thf(sy_v_enc__conj,type,
enc_conj: list_f > f ).
thf(sy_v_enc__different,type,
enc_different: a > a > f ).
thf(sy_v_enc__equiv__and__not,type,
enc_equiv_and_not: a > a > a > f ).
thf(sy_v_enc__equiv__ite,type,
enc_equiv_ite: a > a > a > a > f ).
thf(sy_v_enc__impl,type,
enc_impl: a > f > f ).
thf(sy_v_enc__not__all,type,
enc_not_all: list_a > f ).
thf(sy_v_eval,type,
eval: ( a > $o ) > f > $o ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_vars,type,
vars: list_P1802437025701573206od_a_a ).
% Relevant facts (800)
thf(fact_0_local_Oempty,axiom,
( ( vars = nil_Pr7291147438282410198od_a_a )
=> thesis ) ).
% local.empty
thf(fact_1_encode__sum__0__1__main_Ocases,axiom,
! [X: list_P1802437025701573206od_a_a] :
( ! [X2: a,Zero: a,One: a] :
( X
!= ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Zero @ One ) ) @ nil_Pr7291147438282410198od_a_a ) )
=> ( ! [X2: a,Zero: a,One: a,V: produc4044097585999906000od_a_a,Va: list_P1802437025701573206od_a_a] :
( X
!= ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Zero @ One ) ) @ ( cons_P64011158483510406od_a_a @ V @ Va ) ) )
=> ( X = nil_Pr7291147438282410198od_a_a ) ) ) ).
% encode_sum_0_1_main.cases
thf(fact_2_other,axiom,
! [X: a,Ze: a,On: a,V2: produc4044097585999906000od_a_a,Vs: list_P1802437025701573206od_a_a] :
( ( vars
= ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Ze @ On ) ) @ ( cons_P64011158483510406od_a_a @ V2 @ Vs ) ) )
=> thesis ) ).
% other
thf(fact_3_single,axiom,
! [X: a,Ze: a,On: a] :
( ( vars
= ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Ze @ On ) ) @ nil_Pr7291147438282410198od_a_a ) )
=> thesis ) ).
% single
thf(fact_4_list_Oinject,axiom,
! [X21: produc4044097585999906000od_a_a,X22: list_P1802437025701573206od_a_a,Y21: produc4044097585999906000od_a_a,Y22: list_P1802437025701573206od_a_a] :
( ( ( cons_P64011158483510406od_a_a @ X21 @ X22 )
= ( cons_P64011158483510406od_a_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_5_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_6_list_Oinject,axiom,
! [X21: f,X22: list_f,Y21: f,Y22: list_f] :
( ( ( cons_f @ X21 @ X22 )
= ( cons_f @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_7_prod_Oinject,axiom,
! [X1: $o,X23: $o,Y1: $o,Y2: $o] :
( ( ( product_Pair_o_o @ X1 @ X23 )
= ( product_Pair_o_o @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_prod_Oinject,axiom,
! [X1: a,X23: product_prod_o_o,Y1: a,Y2: product_prod_o_o] :
( ( ( produc1797438980157902144od_o_o @ X1 @ X23 )
= ( produc1797438980157902144od_o_o @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_prod_Oinject,axiom,
! [X1: a,X23: produc4033586494401613520od_o_o,Y1: a,Y2: produc4033586494401613520od_o_o] :
( ( ( produc580260797871235561od_o_o @ X1 @ X23 )
= ( produc580260797871235561od_o_o @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: a,X23: product_prod_a_a,Y1: a,Y2: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ X1 @ X23 )
= ( produc431845341423274048od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: a,X23: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X23 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: $o,B: $o,A2: $o,B2: $o] :
( ( ( product_Pair_o_o @ A @ B )
= ( product_Pair_o_o @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
! [A: a,B: product_prod_o_o,A2: a,B2: product_prod_o_o] :
( ( ( produc1797438980157902144od_o_o @ A @ B )
= ( produc1797438980157902144od_o_o @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
! [A: a,B: produc4033586494401613520od_o_o,A2: a,B2: produc4033586494401613520od_o_o] :
( ( ( produc580260797871235561od_o_o @ A @ B )
= ( produc580260797871235561od_o_o @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A: a,B: product_prod_a_a,A2: a,B2: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ A @ B )
= ( produc431845341423274048od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_16_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_17_prod__cases3,axiom,
! [Y: produc4033586494401613520od_o_o] :
~ ! [A3: a,B3: $o,C: $o] :
( Y
!= ( produc1797438980157902144od_o_o @ A3 @ ( product_Pair_o_o @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_18_prod__cases3,axiom,
! [Y: produc628930716821598713od_o_o] :
~ ! [A3: a,B3: a,C: product_prod_o_o] :
( Y
!= ( produc580260797871235561od_o_o @ A3 @ ( produc1797438980157902144od_o_o @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_19_prod__cases3,axiom,
! [Y: produc4044097585999906000od_a_a] :
~ ! [A3: a,B3: a,C: a] :
( Y
!= ( produc431845341423274048od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_20_prod__cases4,axiom,
! [Y: produc628930716821598713od_o_o] :
~ ! [A3: a,B3: a,C: $o,D: $o] :
( Y
!= ( produc580260797871235561od_o_o @ A3 @ ( produc1797438980157902144od_o_o @ B3 @ ( product_Pair_o_o @ C @ D ) ) ) ) ).
% prod_cases4
thf(fact_21_prod__induct3,axiom,
! [P: produc4033586494401613520od_o_o > $o,X: produc4033586494401613520od_o_o] :
( ! [A3: a,B3: $o,C: $o] : ( P @ ( produc1797438980157902144od_o_o @ A3 @ ( product_Pair_o_o @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_22_prod__induct3,axiom,
! [P: produc628930716821598713od_o_o > $o,X: produc628930716821598713od_o_o] :
( ! [A3: a,B3: a,C: product_prod_o_o] : ( P @ ( produc580260797871235561od_o_o @ A3 @ ( produc1797438980157902144od_o_o @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_23_prod__induct3,axiom,
! [P: produc4044097585999906000od_a_a > $o,X: produc4044097585999906000od_a_a] :
( ! [A3: a,B3: a,C: a] : ( P @ ( produc431845341423274048od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_24_prod__induct4,axiom,
! [P: produc628930716821598713od_o_o > $o,X: produc628930716821598713od_o_o] :
( ! [A3: a,B3: a,C: $o,D: $o] : ( P @ ( produc580260797871235561od_o_o @ A3 @ ( produc1797438980157902144od_o_o @ B3 @ ( product_Pair_o_o @ C @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_25_successively_Ocases,axiom,
! [X: produc6822914451196094351od_a_a] :
( ! [P2: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o] :
( X
!= ( produc9098471519666438399od_a_a @ P2 @ nil_Pr7291147438282410198od_a_a ) )
=> ( ! [P2: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X2: produc4044097585999906000od_a_a] :
( X
!= ( produc9098471519666438399od_a_a @ P2 @ ( cons_P64011158483510406od_a_a @ X2 @ nil_Pr7291147438282410198od_a_a ) ) )
=> ~ ! [P2: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X2: produc4044097585999906000od_a_a,Y3: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] :
( X
!= ( produc9098471519666438399od_a_a @ P2 @ ( cons_P64011158483510406od_a_a @ X2 @ ( cons_P64011158483510406od_a_a @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_26_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P2: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
=> ( ! [P2: nat > nat > $o,X2: nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ nil_nat ) ) )
=> ~ ! [P2: nat > nat > $o,X2: nat,Y3: nat,Xs: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_27_successively_Ocases,axiom,
! [X: produc5795555978541011586list_f] :
( ! [P2: f > f > $o] :
( X
!= ( produc8519404107117762418list_f @ P2 @ nil_f ) )
=> ( ! [P2: f > f > $o,X2: f] :
( X
!= ( produc8519404107117762418list_f @ P2 @ ( cons_f @ X2 @ nil_f ) ) )
=> ~ ! [P2: f > f > $o,X2: f,Y3: f,Xs: list_f] :
( X
!= ( produc8519404107117762418list_f @ P2 @ ( cons_f @ X2 @ ( cons_f @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_28_sorted__wrt_Ocases,axiom,
! [X: produc6822914451196094351od_a_a] :
( ! [P2: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o] :
( X
!= ( produc9098471519666438399od_a_a @ P2 @ nil_Pr7291147438282410198od_a_a ) )
=> ~ ! [P2: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X2: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( X
!= ( produc9098471519666438399od_a_a @ P2 @ ( cons_P64011158483510406od_a_a @ X2 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_29_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P2: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
=> ~ ! [P2: nat > nat > $o,X2: nat,Ys: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_30_sorted__wrt_Ocases,axiom,
! [X: produc5795555978541011586list_f] :
( ! [P2: f > f > $o] :
( X
!= ( produc8519404107117762418list_f @ P2 @ nil_f ) )
=> ~ ! [P2: f > f > $o,X2: f,Ys: list_f] :
( X
!= ( produc8519404107117762418list_f @ P2 @ ( cons_f @ X2 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_31_shuffles_Ocases,axiom,
! [X: produc3323165008400776615od_a_a] :
( ! [Ys: list_P1802437025701573206od_a_a] :
( X
!= ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Ys ) )
=> ( ! [Xs: list_P1802437025701573206od_a_a] :
( X
!= ( produc4694146025100153495od_a_a @ Xs @ nil_Pr7291147438282410198od_a_a ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( X
!= ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_32_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
=> ( ! [Xs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
=> ~ ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_33_shuffles_Ocases,axiom,
! [X: produc1604256039088728679list_f] :
( ! [Ys: list_f] :
( X
!= ( produc8144748701140958423list_f @ nil_f @ Ys ) )
=> ( ! [Xs: list_f] :
( X
!= ( produc8144748701140958423list_f @ Xs @ nil_f ) )
=> ~ ! [X2: f,Xs: list_f,Y3: f,Ys: list_f] :
( X
!= ( produc8144748701140958423list_f @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_34_Pair__inject,axiom,
! [A: $o,B: $o,A2: $o,B2: $o] :
( ( ( product_Pair_o_o @ A @ B )
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B = ~ B2 ) ) ) ).
% Pair_inject
thf(fact_35_Pair__inject,axiom,
! [A: a,B: product_prod_o_o,A2: a,B2: product_prod_o_o] :
( ( ( produc1797438980157902144od_o_o @ A @ B )
= ( produc1797438980157902144od_o_o @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_36_Pair__inject,axiom,
! [A: a,B: produc4033586494401613520od_o_o,A2: a,B2: produc4033586494401613520od_o_o] :
( ( ( produc580260797871235561od_o_o @ A @ B )
= ( produc580260797871235561od_o_o @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_37_Pair__inject,axiom,
! [A: a,B: product_prod_a_a,A2: a,B2: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ A @ B )
= ( produc431845341423274048od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_38_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_39_prod__cases,axiom,
! [P: product_prod_o_o > $o,P3: product_prod_o_o] :
( ! [A3: $o,B3: $o] : ( P @ ( product_Pair_o_o @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_40_prod__cases,axiom,
! [P: produc4033586494401613520od_o_o > $o,P3: produc4033586494401613520od_o_o] :
( ! [A3: a,B3: product_prod_o_o] : ( P @ ( produc1797438980157902144od_o_o @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_41_prod__cases,axiom,
! [P: produc628930716821598713od_o_o > $o,P3: produc628930716821598713od_o_o] :
( ! [A3: a,B3: produc4033586494401613520od_o_o] : ( P @ ( produc580260797871235561od_o_o @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_42_prod__cases,axiom,
! [P: produc4044097585999906000od_a_a > $o,P3: produc4044097585999906000od_a_a] :
( ! [A3: a,B3: product_prod_a_a] : ( P @ ( produc431845341423274048od_a_a @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_43_prod__cases,axiom,
! [P: product_prod_a_a > $o,P3: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_44_surj__pair,axiom,
! [P3: product_prod_o_o] :
? [X2: $o,Y3: $o] :
( P3
= ( product_Pair_o_o @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_45_surj__pair,axiom,
! [P3: produc4033586494401613520od_o_o] :
? [X2: a,Y3: product_prod_o_o] :
( P3
= ( produc1797438980157902144od_o_o @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_46_surj__pair,axiom,
! [P3: produc628930716821598713od_o_o] :
? [X2: a,Y3: produc4033586494401613520od_o_o] :
( P3
= ( produc580260797871235561od_o_o @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_47_surj__pair,axiom,
! [P3: produc4044097585999906000od_a_a] :
? [X2: a,Y3: product_prod_a_a] :
( P3
= ( produc431845341423274048od_a_a @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_48_surj__pair,axiom,
! [P3: product_prod_a_a] :
? [X2: a,Y3: a] :
( P3
= ( product_Pair_a_a @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_49_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_o_o] :
~ ! [A3: $o,B3: $o] :
( Y
!= ( product_Pair_o_o @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_50_old_Oprod_Oexhaust,axiom,
! [Y: produc4033586494401613520od_o_o] :
~ ! [A3: a,B3: product_prod_o_o] :
( Y
!= ( produc1797438980157902144od_o_o @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_51_old_Oprod_Oexhaust,axiom,
! [Y: produc628930716821598713od_o_o] :
~ ! [A3: a,B3: produc4033586494401613520od_o_o] :
( Y
!= ( produc580260797871235561od_o_o @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
! [Y: produc4044097585999906000od_a_a] :
~ ! [A3: a,B3: product_prod_a_a] :
( Y
!= ( produc431845341423274048od_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_53_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A3: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_54_not__Cons__self2,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( cons_P64011158483510406od_a_a @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_55_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_56_not__Cons__self2,axiom,
! [X: f,Xs2: list_f] :
( ( cons_f @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_57_list__nonempty__induct,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: list_P1802437025701573206od_a_a > $o] :
( ( Xs2 != nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: produc4044097585999906000od_a_a] : ( P @ ( cons_P64011158483510406od_a_a @ X2 @ nil_Pr7291147438282410198od_a_a ) )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] :
( ( Xs != nil_Pr7291147438282410198od_a_a )
=> ( ( P @ Xs )
=> ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_58_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_59_list__nonempty__induct,axiom,
! [Xs2: list_f,P: list_f > $o] :
( ( Xs2 != nil_f )
=> ( ! [X2: f] : ( P @ ( cons_f @ X2 @ nil_f ) )
=> ( ! [X2: f,Xs: list_f] :
( ( Xs != nil_f )
=> ( ( P @ Xs )
=> ( P @ ( cons_f @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_60_list__induct2_H,axiom,
! [P: list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a > $o,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( P @ nil_Pr7291147438282410198od_a_a @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] : ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ nil_Pr7291147438282410198od_a_a )
=> ( ! [Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] : ( P @ nil_Pr7291147438282410198od_a_a @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_61_list__induct2_H,axiom,
! [P: list_P1802437025701573206od_a_a > list_nat > $o,Xs2: list_P1802437025701573206od_a_a,Ys2: list_nat] :
( ( P @ nil_Pr7291147438282410198od_a_a @ nil_nat )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] : ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_Pr7291147438282410198od_a_a @ ( cons_nat @ Y3 @ Ys ) )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Y3: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_62_list__induct2_H,axiom,
! [P: list_P1802437025701573206od_a_a > list_f > $o,Xs2: list_P1802437025701573206od_a_a,Ys2: list_f] :
( ( P @ nil_Pr7291147438282410198od_a_a @ nil_f )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] : ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ nil_f )
=> ( ! [Y3: f,Ys: list_f] : ( P @ nil_Pr7291147438282410198od_a_a @ ( cons_f @ Y3 @ Ys ) )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Y3: f,Ys: list_f] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_63_list__induct2_H,axiom,
! [P: list_nat > list_P1802437025701573206od_a_a > $o,Xs2: list_nat,Ys2: list_P1802437025701573206od_a_a] :
( ( P @ nil_nat @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_Pr7291147438282410198od_a_a )
=> ( ! [Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] : ( P @ nil_nat @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_64_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_65_list__induct2_H,axiom,
! [P: list_nat > list_f > $o,Xs2: list_nat,Ys2: list_f] :
( ( P @ nil_nat @ nil_f )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_f )
=> ( ! [Y3: f,Ys: list_f] : ( P @ nil_nat @ ( cons_f @ Y3 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_66_list__induct2_H,axiom,
! [P: list_f > list_P1802437025701573206od_a_a > $o,Xs2: list_f,Ys2: list_P1802437025701573206od_a_a] :
( ( P @ nil_f @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: f,Xs: list_f] : ( P @ ( cons_f @ X2 @ Xs ) @ nil_Pr7291147438282410198od_a_a )
=> ( ! [Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] : ( P @ nil_f @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ( ! [X2: f,Xs: list_f,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_67_list__induct2_H,axiom,
! [P: list_f > list_nat > $o,Xs2: list_f,Ys2: list_nat] :
( ( P @ nil_f @ nil_nat )
=> ( ! [X2: f,Xs: list_f] : ( P @ ( cons_f @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_f @ ( cons_nat @ Y3 @ Ys ) )
=> ( ! [X2: f,Xs: list_f,Y3: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_68_list__induct2_H,axiom,
! [P: list_f > list_f > $o,Xs2: list_f,Ys2: list_f] :
( ( P @ nil_f @ nil_f )
=> ( ! [X2: f,Xs: list_f] : ( P @ ( cons_f @ X2 @ Xs ) @ nil_f )
=> ( ! [Y3: f,Ys: list_f] : ( P @ nil_f @ ( cons_f @ Y3 @ Ys ) )
=> ( ! [X2: f,Xs: list_f,Y3: f,Ys: list_f] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_69_neq__Nil__conv,axiom,
! [Xs2: list_P1802437025701573206od_a_a] :
( ( Xs2 != nil_Pr7291147438282410198od_a_a )
= ( ? [Y4: produc4044097585999906000od_a_a,Ys3: list_P1802437025701573206od_a_a] :
( Xs2
= ( cons_P64011158483510406od_a_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_70_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_71_neq__Nil__conv,axiom,
! [Xs2: list_f] :
( ( Xs2 != nil_f )
= ( ? [Y4: f,Ys3: list_f] :
( Xs2
= ( cons_f @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_72_remdups__adj_Ocases,axiom,
! [X: list_P1802437025701573206od_a_a] :
( ( X != nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: produc4044097585999906000od_a_a] :
( X
!= ( cons_P64011158483510406od_a_a @ X2 @ nil_Pr7291147438282410198od_a_a ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] :
( X
!= ( cons_P64011158483510406od_a_a @ X2 @ ( cons_P64011158483510406od_a_a @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_73_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y3: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_74_remdups__adj_Ocases,axiom,
! [X: list_f] :
( ( X != nil_f )
=> ( ! [X2: f] :
( X
!= ( cons_f @ X2 @ nil_f ) )
=> ~ ! [X2: f,Y3: f,Xs: list_f] :
( X
!= ( cons_f @ X2 @ ( cons_f @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_75_transpose_Ocases,axiom,
! [X: list_l4888837856047780700od_a_a] :
( ( X != nil_li6406708984265833308od_a_a )
=> ( ! [Xss: list_l4888837856047780700od_a_a] :
( X
!= ( cons_l7647087712702168844od_a_a @ nil_Pr7291147438282410198od_a_a @ Xss ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Xss: list_l4888837856047780700od_a_a] :
( X
!= ( cons_l7647087712702168844od_a_a @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_76_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_77_transpose_Ocases,axiom,
! [X: list_list_f] :
( ( X != nil_list_f )
=> ( ! [Xss: list_list_f] :
( X
!= ( cons_list_f @ nil_f @ Xss ) )
=> ~ ! [X2: f,Xs: list_f,Xss: list_list_f] :
( X
!= ( cons_list_f @ ( cons_f @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_78_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_79_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_81_list_Oexhaust,axiom,
! [Y: list_P1802437025701573206od_a_a] :
( ( Y != nil_Pr7291147438282410198od_a_a )
=> ~ ! [X212: produc4044097585999906000od_a_a,X222: list_P1802437025701573206od_a_a] :
( Y
!= ( cons_P64011158483510406od_a_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_82_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_83_list_Oexhaust,axiom,
! [Y: list_f] :
( ( Y != nil_f )
=> ~ ! [X212: f,X222: list_f] :
( Y
!= ( cons_f @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_84_list_OdiscI,axiom,
! [List: list_P1802437025701573206od_a_a,X21: produc4044097585999906000od_a_a,X22: list_P1802437025701573206od_a_a] :
( ( List
= ( cons_P64011158483510406od_a_a @ X21 @ X22 ) )
=> ( List != nil_Pr7291147438282410198od_a_a ) ) ).
% list.discI
thf(fact_85_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_86_list_OdiscI,axiom,
! [List: list_f,X21: f,X22: list_f] :
( ( List
= ( cons_f @ X21 @ X22 ) )
=> ( List != nil_f ) ) ).
% list.discI
thf(fact_87_list_Odistinct_I1_J,axiom,
! [X21: produc4044097585999906000od_a_a,X22: list_P1802437025701573206od_a_a] :
( nil_Pr7291147438282410198od_a_a
!= ( cons_P64011158483510406od_a_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_88_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_89_list_Odistinct_I1_J,axiom,
! [X21: f,X22: list_f] :
( nil_f
!= ( cons_f @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_90_subset__eq__mset__impl_Ocases,axiom,
! [X: produc3323165008400776615od_a_a] :
( ! [Ys: list_P1802437025701573206od_a_a] :
( X
!= ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Ys ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Ys: list_P1802437025701573206od_a_a] :
( X
!= ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_91_subset__eq__mset__impl_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
=> ~ ! [X2: nat,Xs: list_nat,Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_92_subset__eq__mset__impl_Ocases,axiom,
! [X: produc1604256039088728679list_f] :
( ! [Ys: list_f] :
( X
!= ( produc8144748701140958423list_f @ nil_f @ Ys ) )
=> ~ ! [X2: f,Xs: list_f,Ys: list_f] :
( X
!= ( produc8144748701140958423list_f @ ( cons_f @ X2 @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_93_insert__Nil,axiom,
! [X: produc4044097585999906000od_a_a] :
( ( insert5320946937069198828od_a_a @ X @ nil_Pr7291147438282410198od_a_a )
= ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) ) ).
% insert_Nil
thf(fact_94_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_95_insert__Nil,axiom,
! [X: f] :
( ( insert_f @ X @ nil_f )
= ( cons_f @ X @ nil_f ) ) ).
% insert_Nil
thf(fact_96_list__ex1__simps_I1_J,axiom,
! [P: produc4044097585999906000od_a_a > $o] :
~ ( list_e867735100643366937od_a_a @ P @ nil_Pr7291147438282410198od_a_a ) ).
% list_ex1_simps(1)
thf(fact_97_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_98_list__ex1__simps_I1_J,axiom,
! [P: f > $o] :
~ ( list_ex1_f @ P @ nil_f ) ).
% list_ex1_simps(1)
thf(fact_99_product_Osimps_I1_J,axiom,
! [Uu: list_P1396940483166286381od_a_a] :
( ( produc6674783723117535214od_a_a @ nil_a @ Uu )
= nil_Pr7291147438282410198od_a_a ) ).
% product.simps(1)
thf(fact_100_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
! [Eval: ( $o > $o ) > produc4044097585999906000od_a_a > $o,Enc_False: produc4044097585999906000od_a_a,Enc_True: produc4044097585999906000od_a_a,Enc_pos: $o > produc4044097585999906000od_a_a,Enc_neg: $o > produc4044097585999906000od_a_a,Enc_different: $o > $o > produc4044097585999906000od_a_a,Enc_equiv_and_not: $o > $o > $o > produc4044097585999906000od_a_a,Enc_equiv_ite: $o > $o > $o > $o > produc4044097585999906000od_a_a,Enc_ite: $o > $o > $o > produc4044097585999906000od_a_a,Enc_impl: $o > produc4044097585999906000od_a_a > produc4044097585999906000od_a_a,Enc_var_impl: $o > $o > produc4044097585999906000od_a_a,Enc_not_and: $o > $o > produc4044097585999906000od_a_a,Enc_not_all: list_o > produc4044097585999906000od_a_a,Enc_conj: list_P1802437025701573206od_a_a > produc4044097585999906000od_a_a,X: $o,Zero2: $o,One2: $o] :
( ( multis2077179667994930308od_a_a @ Eval @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj )
=> ( ( multis1321987043328851531od_a_a @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
= ( produc5440741947628353512od_o_o @ ( cons_P64011158483510406od_a_a @ ( Enc_different @ Zero2 @ X ) @ nil_Pr7291147438282410198od_a_a ) @ ( product_Pair_o_o @ Zero2 @ X ) ) ) ) ).
% encoder.encode_sum_0_1_main.simps(1)
thf(fact_101_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
! [Eval: ( $o > $o ) > nat > $o,Enc_False: nat,Enc_True: nat,Enc_pos: $o > nat,Enc_neg: $o > nat,Enc_different: $o > $o > nat,Enc_equiv_and_not: $o > $o > $o > nat,Enc_equiv_ite: $o > $o > $o > $o > nat,Enc_ite: $o > $o > $o > nat,Enc_impl: $o > nat > nat,Enc_var_impl: $o > $o > nat,Enc_not_and: $o > $o > nat,Enc_not_all: list_o > nat,Enc_conj: list_nat > nat,X: $o,Zero2: $o,One2: $o] :
( ( multis8460249006151941916_o_nat @ Eval @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj )
=> ( ( multis4961962968097414037_o_nat @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
= ( produc2647278605670320110od_o_o @ ( cons_nat @ ( Enc_different @ Zero2 @ X ) @ nil_nat ) @ ( product_Pair_o_o @ Zero2 @ X ) ) ) ) ).
% encoder.encode_sum_0_1_main.simps(1)
thf(fact_102_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
! [Eval: ( $o > $o ) > f > $o,Enc_False: f,Enc_True: f,Enc_pos: $o > f,Enc_neg: $o > f,Enc_different: $o > $o > f,Enc_equiv_and_not: $o > $o > $o > f,Enc_equiv_ite: $o > $o > $o > $o > f,Enc_ite: $o > $o > $o > f,Enc_impl: $o > f > f,Enc_var_impl: $o > $o > f,Enc_not_and: $o > $o > f,Enc_not_all: list_o > f,Enc_conj: list_f > f,X: $o,Zero2: $o,One2: $o] :
( ( multis2173133845132587703er_o_f @ Eval @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj )
=> ( ( multis2292260455496811134in_o_f @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
= ( produc6447895559475361781od_o_o @ ( cons_f @ ( Enc_different @ Zero2 @ X ) @ nil_f ) @ ( product_Pair_o_o @ Zero2 @ X ) ) ) ) ).
% encoder.encode_sum_0_1_main.simps(1)
thf(fact_103_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
! [Eval: ( a > $o ) > produc4044097585999906000od_a_a > $o,Enc_False: produc4044097585999906000od_a_a,Enc_True: produc4044097585999906000od_a_a,Enc_pos: a > produc4044097585999906000od_a_a,Enc_neg: a > produc4044097585999906000od_a_a,Enc_different: a > a > produc4044097585999906000od_a_a,Enc_equiv_and_not: a > a > a > produc4044097585999906000od_a_a,Enc_equiv_ite: a > a > a > a > produc4044097585999906000od_a_a,Enc_ite: a > a > a > produc4044097585999906000od_a_a,Enc_impl: a > produc4044097585999906000od_a_a > produc4044097585999906000od_a_a,Enc_var_impl: a > a > produc4044097585999906000od_a_a,Enc_not_and: a > a > produc4044097585999906000od_a_a,Enc_not_all: list_a > produc4044097585999906000od_a_a,Enc_conj: list_P1802437025701573206od_a_a > produc4044097585999906000od_a_a,X: a,Zero2: a,One2: a] :
( ( multis7616222132842180638od_a_a @ Eval @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj )
=> ( ( multis3004827603150622309od_a_a @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Zero2 @ One2 ) ) @ nil_Pr7291147438282410198od_a_a ) )
= ( produc2288087392375622376od_a_a @ ( cons_P64011158483510406od_a_a @ ( Enc_different @ Zero2 @ X ) @ nil_Pr7291147438282410198od_a_a ) @ ( product_Pair_a_a @ Zero2 @ X ) ) ) ) ).
% encoder.encode_sum_0_1_main.simps(1)
thf(fact_104_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
! [Eval: ( a > $o ) > nat > $o,Enc_False: nat,Enc_True: nat,Enc_pos: a > nat,Enc_neg: a > nat,Enc_different: a > a > nat,Enc_equiv_and_not: a > a > a > nat,Enc_equiv_ite: a > a > a > a > nat,Enc_ite: a > a > a > nat,Enc_impl: a > nat > nat,Enc_var_impl: a > a > nat,Enc_not_and: a > a > nat,Enc_not_all: list_a > nat,Enc_conj: list_nat > nat,X: a,Zero2: a,One2: a] :
( ( multis5206763771688736002_a_nat @ Eval @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj )
=> ( ( multis3220821980000666875_a_nat @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Zero2 @ One2 ) ) @ nil_Pr7291147438282410198od_a_a ) )
= ( produc6897089307146163438od_a_a @ ( cons_nat @ ( Enc_different @ Zero2 @ X ) @ nil_nat ) @ ( product_Pair_a_a @ Zero2 @ X ) ) ) ) ).
% encoder.encode_sum_0_1_main.simps(1)
thf(fact_105_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
! [Eval: ( a > $o ) > f > $o,Enc_False: f,Enc_True: f,Enc_pos: a > f,Enc_neg: a > f,Enc_different: a > a > f,Enc_equiv_and_not: a > a > a > f,Enc_equiv_ite: a > a > a > a > f,Enc_ite: a > a > a > f,Enc_impl: a > f > f,Enc_var_impl: a > a > f,Enc_not_and: a > a > f,Enc_not_all: list_a > f,Enc_conj: list_f > f,X: a,Zero2: a,One2: a] :
( ( multis2105231276199332433er_a_f @ Eval @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj )
=> ( ( multis3280010013254100632in_a_f @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Zero2 @ One2 ) ) @ nil_Pr7291147438282410198od_a_a ) )
= ( produc3886013187752893941od_a_a @ ( cons_f @ ( Enc_different @ Zero2 @ X ) @ nil_f ) @ ( product_Pair_a_a @ Zero2 @ X ) ) ) ) ).
% encoder.encode_sum_0_1_main.simps(1)
thf(fact_106_map__tailrec__rev_Oelims,axiom,
! [X: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a,Xa: list_P1802437025701573206od_a_a,Xb: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( ( map_ta5110329159104336885od_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Pr7291147438282410198od_a_a )
=> ( Y != Xb ) )
=> ~ ! [A3: produc4044097585999906000od_a_a,As: list_P1802437025701573206od_a_a] :
( ( Xa
= ( cons_P64011158483510406od_a_a @ A3 @ As ) )
=> ( Y
!= ( map_ta5110329159104336885od_a_a @ X @ As @ ( cons_P64011158483510406od_a_a @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_107_map__tailrec__rev_Oelims,axiom,
! [X: produc4044097585999906000od_a_a > nat,Xa: list_P1802437025701573206od_a_a,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7175420146098051947_a_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Pr7291147438282410198od_a_a )
=> ( Y != Xb ) )
=> ~ ! [A3: produc4044097585999906000od_a_a,As: list_P1802437025701573206od_a_a] :
( ( Xa
= ( cons_P64011158483510406od_a_a @ A3 @ As ) )
=> ( Y
!= ( map_ta7175420146098051947_a_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_108_map__tailrec__rev_Oelims,axiom,
! [X: produc4044097585999906000od_a_a > f,Xa: list_P1802437025701573206od_a_a,Xb: list_f,Y: list_f] :
( ( ( map_ta908609173253229608_a_a_f @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Pr7291147438282410198od_a_a )
=> ( Y != Xb ) )
=> ~ ! [A3: produc4044097585999906000od_a_a,As: list_P1802437025701573206od_a_a] :
( ( Xa
= ( cons_P64011158483510406od_a_a @ A3 @ As ) )
=> ( Y
!= ( map_ta908609173253229608_a_a_f @ X @ As @ ( cons_f @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_109_map__tailrec__rev_Oelims,axiom,
! [X: nat > produc4044097585999906000od_a_a,Xa: list_nat,Xb: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( ( map_ta354121929265743561od_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( Y
!= ( map_ta354121929265743561od_a_a @ X @ As @ ( cons_P64011158483510406od_a_a @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_110_map__tailrec__rev_Oelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( Y
!= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_111_map__tailrec__rev_Oelims,axiom,
! [X: nat > f,Xa: list_nat,Xb: list_f,Y: list_f] :
( ( ( map_ta3519391893248468732_nat_f @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( Y
!= ( map_ta3519391893248468732_nat_f @ X @ As @ ( cons_f @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_112_map__tailrec__rev_Oelims,axiom,
! [X: f > produc4044097585999906000od_a_a,Xa: list_f,Xb: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( ( map_ta2155695636830030466od_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_f )
=> ( Y != Xb ) )
=> ~ ! [A3: f,As: list_f] :
( ( Xa
= ( cons_f @ A3 @ As ) )
=> ( Y
!= ( map_ta2155695636830030466od_a_a @ X @ As @ ( cons_P64011158483510406od_a_a @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_113_map__tailrec__rev_Oelims,axiom,
! [X: f > nat,Xa: list_f,Xb: list_nat,Y: list_nat] :
( ( ( map_ta5664682036852154782_f_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_f )
=> ( Y != Xb ) )
=> ~ ! [A3: f,As: list_f] :
( ( Xa
= ( cons_f @ A3 @ As ) )
=> ( Y
!= ( map_ta5664682036852154782_f_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_114_map__tailrec__rev_Oelims,axiom,
! [X: f > f,Xa: list_f,Xb: list_f,Y: list_f] :
( ( ( map_tailrec_rev_f_f @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_f )
=> ( Y != Xb ) )
=> ~ ! [A3: f,As: list_f] :
( ( Xa
= ( cons_f @ A3 @ As ) )
=> ( Y
!= ( map_tailrec_rev_f_f @ X @ As @ ( cons_f @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_115_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
~ ( lexord7296994292201227036od_a_a @ Less @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ nil_Pr7291147438282410198od_a_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_116_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: nat > nat > $o,X: nat,Xs2: list_nat] :
~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs2 ) @ nil_nat ) ).
% ord.lexordp_eq_simps(3)
thf(fact_117_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: f > f > $o,X: f,Xs2: list_f] :
~ ( lexordp_eq_f @ Less @ ( cons_f @ X @ Xs2 ) @ nil_f ) ).
% ord.lexordp_eq_simps(3)
thf(fact_118_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( lexord7296994292201227036od_a_a @ Less @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexord7296994292201227036od_a_a @ Less @ Xs2 @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_119_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: nat > nat > $o,X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_120_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: f > f > $o,X: f,Xs2: list_f,Y: f,Ys2: list_f] :
( ( lexordp_eq_f @ Less @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_f @ Less @ Xs2 @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_121_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,Xs2: list_P1802437025701573206od_a_a] :
( ( lexord7296994292201227036od_a_a @ Less @ Xs2 @ nil_Pr7291147438282410198od_a_a )
= ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_122_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs2: list_nat] :
( ( lexordp_eq_nat @ Less @ Xs2 @ nil_nat )
= ( Xs2 = nil_nat ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_123_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: f > f > $o,Xs2: list_f] :
( ( lexordp_eq_f @ Less @ Xs2 @ nil_f )
= ( Xs2 = nil_f ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_124_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,Ys2: list_P1802437025701573206od_a_a] : ( lexord7296994292201227036od_a_a @ Less @ nil_Pr7291147438282410198od_a_a @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_125_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys2: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_126_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: f > f > $o,Ys2: list_f] : ( lexordp_eq_f @ Less @ nil_f @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_127_encoder_Oencode__sum__0__1__main_Ocong,axiom,
multis3280010013254100632in_a_f = multis3280010013254100632in_a_f ).
% encoder.encode_sum_0_1_main.cong
thf(fact_128_ord_Olexordp__eq_OCons,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X: produc4044097585999906000od_a_a,Y: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( Less @ X @ Y )
=> ( lexord7296994292201227036od_a_a @ Less @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_129_ord_Olexordp__eq_OCons,axiom,
! [Less: nat > nat > $o,X: nat,Y: nat,Xs2: list_nat,Ys2: list_nat] :
( ( Less @ X @ Y )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_130_ord_Olexordp__eq_OCons,axiom,
! [Less: f > f > $o,X: f,Y: f,Xs2: list_f,Ys2: list_f] :
( ( Less @ X @ Y )
=> ( lexordp_eq_f @ Less @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_131_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X: produc4044097585999906000od_a_a,Y: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexord7296994292201227036od_a_a @ Less @ Xs2 @ Ys2 )
=> ( lexord7296994292201227036od_a_a @ Less @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_132_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: nat > nat > $o,X: nat,Y: nat,Xs2: list_nat,Ys2: list_nat] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_133_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: f > f > $o,X: f,Y: f,Xs2: list_f,Ys2: list_f] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_f @ Less @ Xs2 @ Ys2 )
=> ( lexordp_eq_f @ Less @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_134_ord_Olexordp__eq_ONil,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,Ys2: list_P1802437025701573206od_a_a] : ( lexord7296994292201227036od_a_a @ Less @ nil_Pr7291147438282410198od_a_a @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_135_ord_Olexordp__eq_ONil,axiom,
! [Less: nat > nat > $o,Ys2: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_136_ord_Olexordp__eq_ONil,axiom,
! [Less: f > f > $o,Ys2: list_f] : ( lexordp_eq_f @ Less @ nil_f @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_137_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a,A: produc4044097585999906000od_a_a,As2: list_P1802437025701573206od_a_a,Bs: list_P1802437025701573206od_a_a] :
( ( map_ta5110329159104336885od_a_a @ F @ ( cons_P64011158483510406od_a_a @ A @ As2 ) @ Bs )
= ( map_ta5110329159104336885od_a_a @ F @ As2 @ ( cons_P64011158483510406od_a_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_138_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: produc4044097585999906000od_a_a > nat,A: produc4044097585999906000od_a_a,As2: list_P1802437025701573206od_a_a,Bs: list_nat] :
( ( map_ta7175420146098051947_a_nat @ F @ ( cons_P64011158483510406od_a_a @ A @ As2 ) @ Bs )
= ( map_ta7175420146098051947_a_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_139_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: produc4044097585999906000od_a_a > f,A: produc4044097585999906000od_a_a,As2: list_P1802437025701573206od_a_a,Bs: list_f] :
( ( map_ta908609173253229608_a_a_f @ F @ ( cons_P64011158483510406od_a_a @ A @ As2 ) @ Bs )
= ( map_ta908609173253229608_a_a_f @ F @ As2 @ ( cons_f @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_140_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > produc4044097585999906000od_a_a,A: nat,As2: list_nat,Bs: list_P1802437025701573206od_a_a] :
( ( map_ta354121929265743561od_a_a @ F @ ( cons_nat @ A @ As2 ) @ Bs )
= ( map_ta354121929265743561od_a_a @ F @ As2 @ ( cons_P64011158483510406od_a_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_141_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A: nat,As2: list_nat,Bs: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A @ As2 ) @ Bs )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_142_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > f,A: nat,As2: list_nat,Bs: list_f] :
( ( map_ta3519391893248468732_nat_f @ F @ ( cons_nat @ A @ As2 ) @ Bs )
= ( map_ta3519391893248468732_nat_f @ F @ As2 @ ( cons_f @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_143_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: f > produc4044097585999906000od_a_a,A: f,As2: list_f,Bs: list_P1802437025701573206od_a_a] :
( ( map_ta2155695636830030466od_a_a @ F @ ( cons_f @ A @ As2 ) @ Bs )
= ( map_ta2155695636830030466od_a_a @ F @ As2 @ ( cons_P64011158483510406od_a_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_144_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: f > nat,A: f,As2: list_f,Bs: list_nat] :
( ( map_ta5664682036852154782_f_nat @ F @ ( cons_f @ A @ As2 ) @ Bs )
= ( map_ta5664682036852154782_f_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_145_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: f > f,A: f,As2: list_f,Bs: list_f] :
( ( map_tailrec_rev_f_f @ F @ ( cons_f @ A @ As2 ) @ Bs )
= ( map_tailrec_rev_f_f @ F @ As2 @ ( cons_f @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_146_ord_Olexordp__eq_Ocases,axiom,
! [Less: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,A1: list_P1802437025701573206od_a_a,A22: list_P1802437025701573206od_a_a] :
( ( lexord7296994292201227036od_a_a @ Less @ A1 @ A22 )
=> ( ( A1 != nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: produc4044097585999906000od_a_a] :
( ? [Xs: list_P1802437025701573206od_a_a] :
( A1
= ( cons_P64011158483510406od_a_a @ X2 @ Xs ) )
=> ! [Y3: produc4044097585999906000od_a_a] :
( ? [Ys: list_P1802437025701573206od_a_a] :
( A22
= ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ~ ( Less @ X2 @ Y3 ) ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X2 @ Xs ) )
=> ! [Ys: list_P1802437025701573206od_a_a] :
( ( A22
= ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ( ~ ( Less @ X2 @ Y3 )
=> ( ~ ( Less @ Y3 @ X2 )
=> ~ ( lexord7296994292201227036od_a_a @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_147_ord_Olexordp__eq_Ocases,axiom,
! [Less: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( lexordp_eq_nat @ Less @ A1 @ A22 )
=> ( ( A1 != nil_nat )
=> ( ! [X2: nat] :
( ? [Xs: list_nat] :
( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Y3: nat] :
( ? [Ys: list_nat] :
( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ~ ( Less @ X2 @ Y3 ) ) )
=> ~ ! [X2: nat,Y3: nat,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ~ ( Less @ X2 @ Y3 )
=> ( ~ ( Less @ Y3 @ X2 )
=> ~ ( lexordp_eq_nat @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_148_ord_Olexordp__eq_Ocases,axiom,
! [Less: f > f > $o,A1: list_f,A22: list_f] :
( ( lexordp_eq_f @ Less @ A1 @ A22 )
=> ( ( A1 != nil_f )
=> ( ! [X2: f] :
( ? [Xs: list_f] :
( A1
= ( cons_f @ X2 @ Xs ) )
=> ! [Y3: f] :
( ? [Ys: list_f] :
( A22
= ( cons_f @ Y3 @ Ys ) )
=> ~ ( Less @ X2 @ Y3 ) ) )
=> ~ ! [X2: f,Y3: f,Xs: list_f] :
( ( A1
= ( cons_f @ X2 @ Xs ) )
=> ! [Ys: list_f] :
( ( A22
= ( cons_f @ Y3 @ Ys ) )
=> ( ~ ( Less @ X2 @ Y3 )
=> ( ~ ( Less @ Y3 @ X2 )
=> ~ ( lexordp_eq_f @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_149_ord_Olexordp__eq_Osimps,axiom,
( lexord7296994292201227036od_a_a
= ( ^ [Less2: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,A12: list_P1802437025701573206od_a_a,A23: list_P1802437025701573206od_a_a] :
( ? [Ys3: list_P1802437025701573206od_a_a] :
( ( A12 = nil_Pr7291147438282410198od_a_a )
& ( A23 = Ys3 ) )
| ? [X3: produc4044097585999906000od_a_a,Y4: produc4044097585999906000od_a_a,Xs3: list_P1802437025701573206od_a_a,Ys3: list_P1802437025701573206od_a_a] :
( ( A12
= ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) )
& ( A23
= ( cons_P64011158483510406od_a_a @ Y4 @ Ys3 ) )
& ( Less2 @ X3 @ Y4 ) )
| ? [X3: produc4044097585999906000od_a_a,Y4: produc4044097585999906000od_a_a,Xs3: list_P1802437025701573206od_a_a,Ys3: list_P1802437025701573206od_a_a] :
( ( A12
= ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) )
& ( A23
= ( cons_P64011158483510406od_a_a @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y4 )
& ~ ( Less2 @ Y4 @ X3 )
& ( lexord7296994292201227036od_a_a @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_150_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_nat
= ( ^ [Less2: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ? [Ys3: list_nat] :
( ( A12 = nil_nat )
& ( A23 = Ys3 ) )
| ? [X3: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys3 ) )
& ( Less2 @ X3 @ Y4 ) )
| ? [X3: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y4 )
& ~ ( Less2 @ Y4 @ X3 )
& ( lexordp_eq_nat @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_151_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_f
= ( ^ [Less2: f > f > $o,A12: list_f,A23: list_f] :
( ? [Ys3: list_f] :
( ( A12 = nil_f )
& ( A23 = Ys3 ) )
| ? [X3: f,Y4: f,Xs3: list_f,Ys3: list_f] :
( ( A12
= ( cons_f @ X3 @ Xs3 ) )
& ( A23
= ( cons_f @ Y4 @ Ys3 ) )
& ( Less2 @ X3 @ Y4 ) )
| ? [X3: f,Y4: f,Xs3: list_f,Ys3: list_f] :
( ( A12
= ( cons_f @ X3 @ Xs3 ) )
& ( A23
= ( cons_f @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y4 )
& ~ ( Less2 @ Y4 @ X3 )
& ( lexordp_eq_f @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_152_product__lists_Osimps_I1_J,axiom,
( ( produc4796932526259527359od_a_a @ nil_li6406708984265833308od_a_a )
= ( cons_l7647087712702168844od_a_a @ nil_Pr7291147438282410198od_a_a @ nil_li6406708984265833308od_a_a ) ) ).
% product_lists.simps(1)
thf(fact_153_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_154_product__lists_Osimps_I1_J,axiom,
( ( product_lists_f @ nil_list_f )
= ( cons_list_f @ nil_f @ nil_list_f ) ) ).
% product_lists.simps(1)
thf(fact_155_subseqs_Osimps_I1_J,axiom,
( ( subseq1910477207492193829od_a_a @ nil_Pr7291147438282410198od_a_a )
= ( cons_l7647087712702168844od_a_a @ nil_Pr7291147438282410198od_a_a @ nil_li6406708984265833308od_a_a ) ) ).
% subseqs.simps(1)
thf(fact_156_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_157_subseqs_Osimps_I1_J,axiom,
( ( subseqs_f @ nil_f )
= ( cons_list_f @ nil_f @ nil_list_f ) ) ).
% subseqs.simps(1)
thf(fact_158_map__tailrec__rev_Opelims,axiom,
! [X: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a,Xa: list_P1802437025701573206od_a_a,Xb: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( ( map_ta5110329159104336885od_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P1059454084221188526od_a_a @ map_ta4054191778087535172od_a_a @ ( produc5651543778595181225od_a_a @ X @ ( produc4694146025100153495od_a_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Pr7291147438282410198od_a_a )
=> ( ( Y = Xb )
=> ~ ( accp_P1059454084221188526od_a_a @ map_ta4054191778087535172od_a_a @ ( produc5651543778595181225od_a_a @ X @ ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Xb ) ) ) ) )
=> ~ ! [A3: produc4044097585999906000od_a_a,As: list_P1802437025701573206od_a_a] :
( ( Xa
= ( cons_P64011158483510406od_a_a @ A3 @ As ) )
=> ( ( Y
= ( map_ta5110329159104336885od_a_a @ X @ As @ ( cons_P64011158483510406od_a_a @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P1059454084221188526od_a_a @ map_ta4054191778087535172od_a_a @ ( produc5651543778595181225od_a_a @ X @ ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_159_map__tailrec__rev_Opelims,axiom,
! [X: produc4044097585999906000od_a_a > nat,Xa: list_P1802437025701573206od_a_a,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7175420146098051947_a_nat @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P4316615423486620892st_nat @ map_ta9079712231957281116_a_nat @ ( produc3469052808376550559st_nat @ X @ ( produc4321862821133114975st_nat @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Pr7291147438282410198od_a_a )
=> ( ( Y = Xb )
=> ~ ( accp_P4316615423486620892st_nat @ map_ta9079712231957281116_a_nat @ ( produc3469052808376550559st_nat @ X @ ( produc4321862821133114975st_nat @ nil_Pr7291147438282410198od_a_a @ Xb ) ) ) ) )
=> ~ ! [A3: produc4044097585999906000od_a_a,As: list_P1802437025701573206od_a_a] :
( ( Xa
= ( cons_P64011158483510406od_a_a @ A3 @ As ) )
=> ( ( Y
= ( map_ta7175420146098051947_a_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P4316615423486620892st_nat @ map_ta9079712231957281116_a_nat @ ( produc3469052808376550559st_nat @ X @ ( produc4321862821133114975st_nat @ ( cons_P64011158483510406od_a_a @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_160_map__tailrec__rev_Opelims,axiom,
! [X: produc4044097585999906000od_a_a > f,Xa: list_P1802437025701573206od_a_a,Xb: list_f,Y: list_f] :
( ( ( map_ta908609173253229608_a_a_f @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P2150597548507289454list_f @ map_ta1801900064162990199_a_a_f @ ( produc1474026983010990569list_f @ X @ ( produc6118079160660574026list_f @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Pr7291147438282410198od_a_a )
=> ( ( Y = Xb )
=> ~ ( accp_P2150597548507289454list_f @ map_ta1801900064162990199_a_a_f @ ( produc1474026983010990569list_f @ X @ ( produc6118079160660574026list_f @ nil_Pr7291147438282410198od_a_a @ Xb ) ) ) ) )
=> ~ ! [A3: produc4044097585999906000od_a_a,As: list_P1802437025701573206od_a_a] :
( ( Xa
= ( cons_P64011158483510406od_a_a @ A3 @ As ) )
=> ( ( Y
= ( map_ta908609173253229608_a_a_f @ X @ As @ ( cons_f @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P2150597548507289454list_f @ map_ta1801900064162990199_a_a_f @ ( produc1474026983010990569list_f @ X @ ( produc6118079160660574026list_f @ ( cons_P64011158483510406od_a_a @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_161_map__tailrec__rev_Opelims,axiom,
! [X: nat > produc4044097585999906000od_a_a,Xa: list_nat,Xb: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( ( map_ta354121929265743561od_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P4649787823440830728od_a_a @ map_ta2258414015124972730od_a_a @ ( produc2329199598056461259od_a_a @ X @ ( produc5825293536187362205od_a_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_nat )
=> ( ( Y = Xb )
=> ~ ( accp_P4649787823440830728od_a_a @ map_ta2258414015124972730od_a_a @ ( produc2329199598056461259od_a_a @ X @ ( produc5825293536187362205od_a_a @ nil_nat @ Xb ) ) ) ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( ( Y
= ( map_ta354121929265743561od_a_a @ X @ As @ ( cons_P64011158483510406od_a_a @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P4649787823440830728od_a_a @ map_ta2258414015124972730od_a_a @ ( produc2329199598056461259od_a_a @ X @ ( produc5825293536187362205od_a_a @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_162_map__tailrec__rev_Opelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_nat )
=> ( ( Y = Xb )
=> ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( ( Y
= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_163_map__tailrec__rev_Opelims,axiom,
! [X: nat > f,Xa: list_nat,Xb: list_f,Y: list_f] :
( ( ( map_ta3519391893248468732_nat_f @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P7918123743290328942list_f @ map_ta2206423409835128557_nat_f @ ( produc383970973803451313list_f @ X @ ( produc7723716031568168016list_f @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_nat )
=> ( ( Y = Xb )
=> ~ ( accp_P7918123743290328942list_f @ map_ta2206423409835128557_nat_f @ ( produc383970973803451313list_f @ X @ ( produc7723716031568168016list_f @ nil_nat @ Xb ) ) ) ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( ( Y
= ( map_ta3519391893248468732_nat_f @ X @ As @ ( cons_f @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P7918123743290328942list_f @ map_ta2206423409835128557_nat_f @ ( produc383970973803451313list_f @ X @ ( produc7723716031568168016list_f @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_164_map__tailrec__rev_Opelims,axiom,
! [X: f > produc4044097585999906000od_a_a,Xa: list_f,Xb: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( ( map_ta2155695636830030466od_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P953168501168768366od_a_a @ map_ta3048986527739791057od_a_a @ ( produc5566396455418049769od_a_a @ X @ ( produc2139721702810641956od_a_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_f )
=> ( ( Y = Xb )
=> ~ ( accp_P953168501168768366od_a_a @ map_ta3048986527739791057od_a_a @ ( produc5566396455418049769od_a_a @ X @ ( produc2139721702810641956od_a_a @ nil_f @ Xb ) ) ) ) )
=> ~ ! [A3: f,As: list_f] :
( ( Xa
= ( cons_f @ A3 @ As ) )
=> ( ( Y
= ( map_ta2155695636830030466od_a_a @ X @ As @ ( cons_P64011158483510406od_a_a @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P953168501168768366od_a_a @ map_ta3048986527739791057od_a_a @ ( produc5566396455418049769od_a_a @ X @ ( produc2139721702810641956od_a_a @ ( cons_f @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_165_map__tailrec__rev_Opelims,axiom,
! [X: f > nat,Xa: list_f,Xb: list_nat,Y: list_nat] :
( ( ( map_ta5664682036852154782_f_nat @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P180089516854618306st_nat @ map_ta4351713553438814607_f_nat @ ( produc3088929835406758405st_nat @ X @ ( produc8127447161900435346st_nat @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_f )
=> ( ( Y = Xb )
=> ~ ( accp_P180089516854618306st_nat @ map_ta4351713553438814607_f_nat @ ( produc3088929835406758405st_nat @ X @ ( produc8127447161900435346st_nat @ nil_f @ Xb ) ) ) ) )
=> ~ ! [A3: f,As: list_f] :
( ( Xa
= ( cons_f @ A3 @ As ) )
=> ( ( Y
= ( map_ta5664682036852154782_f_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P180089516854618306st_nat @ map_ta4351713553438814607_f_nat @ ( produc3088929835406758405st_nat @ X @ ( produc8127447161900435346st_nat @ ( cons_f @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_166_map__tailrec__rev_Opelims,axiom,
! [X: f > f,Xa: list_f,Xb: list_f,Y: list_f] :
( ( ( map_tailrec_rev_f_f @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P5019072637223936814list_f @ map_ta3902997236269430532el_f_f @ ( produc6058385586266807337list_f @ X @ ( produc8144748701140958423list_f @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_f )
=> ( ( Y = Xb )
=> ~ ( accp_P5019072637223936814list_f @ map_ta3902997236269430532el_f_f @ ( produc6058385586266807337list_f @ X @ ( produc8144748701140958423list_f @ nil_f @ Xb ) ) ) ) )
=> ~ ! [A3: f,As: list_f] :
( ( Xa
= ( cons_f @ A3 @ As ) )
=> ( ( Y
= ( map_tailrec_rev_f_f @ X @ As @ ( cons_f @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P5019072637223936814list_f @ map_ta3902997236269430532el_f_f @ ( produc6058385586266807337list_f @ X @ ( produc8144748701140958423list_f @ ( cons_f @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_167_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: nat,Y3: nat,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_168_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_f,R: set_Pr4548512049067879902_nat_f] :
( ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ A1 @ A22 ) @ ( listrel_nat_f @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_f ) )
=> ~ ! [X2: nat,Y3: f,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys: list_f] :
( ( A22
= ( cons_f @ Y3 @ Ys ) )
=> ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X2 @ Y3 ) @ R )
=> ~ ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ Xs @ Ys ) @ ( listrel_nat_f @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_169_listrel_Ocases,axiom,
! [A1: list_f,A22: list_nat,R: set_Pr5245412377734449720_f_nat] :
( ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ A1 @ A22 ) @ ( listrel_f_nat @ R ) )
=> ( ( ( A1 = nil_f )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: f,Y3: nat,Xs: list_f] :
( ( A1
= ( cons_f @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y3 ) @ R )
=> ~ ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ Xs @ Ys ) @ ( listrel_f_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_170_listrel_Ocases,axiom,
! [A1: list_f,A22: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ A1 @ A22 ) @ ( listrel_f_f @ R ) )
=> ( ( ( A1 = nil_f )
=> ( A22 != nil_f ) )
=> ~ ! [X2: f,Y3: f,Xs: list_f] :
( ( A1
= ( cons_f @ X2 @ Xs ) )
=> ! [Ys: list_f] :
( ( A22
= ( cons_f @ Y3 @ Ys ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ R )
=> ~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys ) @ ( listrel_f_f @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_171_listrel_Ocases,axiom,
! [A1: list_o,A22: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ A1 @ A22 ) @ ( listrel_o_o @ R ) )
=> ( ( ( A1 = nil_o )
=> ( A22 != nil_o ) )
=> ~ ! [X2: $o,Y3: $o,Xs: list_o] :
( ( A1
= ( cons_o @ X2 @ Xs ) )
=> ! [Ys: list_o] :
( ( A22
= ( cons_o @ Y3 @ Ys ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel_o_o @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_172_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X2: a,Y3: a,Xs: list_a] :
( ( A1
= ( cons_a @ X2 @ Xs ) )
=> ! [Ys: list_a] :
( ( A22
= ( cons_a @ Y3 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_173_listrel_Ocases,axiom,
! [A1: list_a,A22: list_P4002435161011370285od_o_o,R: set_Pr6507870997995571760od_o_o] :
( ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ A1 @ A22 ) @ ( listre5995639776835297690od_o_o @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_Product_prod_o_o ) )
=> ~ ! [X2: a,Y3: product_prod_o_o,Xs: list_a] :
( ( A1
= ( cons_a @ X2 @ Xs ) )
=> ! [Ys: list_P4002435161011370285od_o_o] :
( ( A22
= ( cons_P8766293264717362397od_o_o @ Y3 @ Ys ) )
=> ( ( member1089989612134903033od_o_o @ ( produc1797438980157902144od_o_o @ X2 @ Y3 ) @ R )
=> ~ ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ Xs @ Ys ) @ ( listre5995639776835297690od_o_o @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_174_listrel_Ocases,axiom,
! [A1: list_a,A22: list_P1396940483166286381od_a_a,R: set_Pr5530083903271594800od_a_a] :
( ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ A1 @ A22 ) @ ( listre367282320202154650od_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_Product_prod_a_a ) )
=> ~ ! [X2: a,Y3: product_prod_a_a,Xs: list_a] :
( ( A1
= ( cons_a @ X2 @ Xs ) )
=> ! [Ys: list_P1396940483166286381od_a_a] :
( ( A22
= ( cons_P7316939126706565853od_a_a @ Y3 @ Ys ) )
=> ( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X2 @ Y3 ) @ R )
=> ~ ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ Xs @ Ys ) @ ( listre367282320202154650od_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_175_listrel_Ocases,axiom,
! [A1: list_P1802437025701573206od_a_a,A22: list_nat,R: set_Pr8612406289294286213_a_nat] :
( ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ A1 @ A22 ) @ ( listre6748024528684829295_a_nat @ R ) )
=> ( ( ( A1 = nil_Pr7291147438282410198od_a_a )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: nat,Xs: list_P1802437025701573206od_a_a] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member1741560536349867494_a_nat @ ( produc8961142355356622153_a_nat @ X2 @ Y3 ) @ R )
=> ~ ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ Xs @ Ys ) @ ( listre6748024528684829295_a_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_176_listrel_Ocases,axiom,
! [A1: list_P1802437025701573206od_a_a,A22: list_f,R: set_Pr4596002078154142650_a_a_f] :
( ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ A1 @ A22 ) @ ( listre3440917786431297060_a_a_f @ R ) )
=> ( ( ( A1 = nil_Pr7291147438282410198od_a_a )
=> ( A22 != nil_f ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: f,Xs: list_P1802437025701573206od_a_a] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X2 @ Xs ) )
=> ! [Ys: list_f] :
( ( A22
= ( cons_f @ Y3 @ Ys ) )
=> ( ( member6782030652459103107_a_a_f @ ( produc4616610431800165834_a_a_f @ X2 @ Y3 ) @ R )
=> ~ ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ Xs @ Ys ) @ ( listre3440917786431297060_a_a_f @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_177_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_nat ) )
| ? [X3: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_178_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_f,R: set_Pr4548512049067879902_nat_f] :
( ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ A1 @ A22 ) @ ( listrel_nat_f @ R ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_f ) )
| ? [X3: nat,Y4: f,Xs3: list_nat,Ys3: list_f] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_f @ Y4 @ Ys3 ) )
& ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X3 @ Y4 ) @ R )
& ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ Xs3 @ Ys3 ) @ ( listrel_nat_f @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_179_listrel_Osimps,axiom,
! [A1: list_f,A22: list_nat,R: set_Pr5245412377734449720_f_nat] :
( ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ A1 @ A22 ) @ ( listrel_f_nat @ R ) )
= ( ( ( A1 = nil_f )
& ( A22 = nil_nat ) )
| ? [X3: f,Y4: nat,Xs3: list_f,Ys3: list_nat] :
( ( A1
= ( cons_f @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X3 @ Y4 ) @ R )
& ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ Xs3 @ Ys3 ) @ ( listrel_f_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_180_listrel_Osimps,axiom,
! [A1: list_f,A22: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ A1 @ A22 ) @ ( listrel_f_f @ R ) )
= ( ( ( A1 = nil_f )
& ( A22 = nil_f ) )
| ? [X3: f,Y4: f,Xs3: list_f,Ys3: list_f] :
( ( A1
= ( cons_f @ X3 @ Xs3 ) )
& ( A22
= ( cons_f @ Y4 @ Ys3 ) )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y4 ) @ R )
& ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs3 @ Ys3 ) @ ( listrel_f_f @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_181_listrel_Osimps,axiom,
! [A1: list_o,A22: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ A1 @ A22 ) @ ( listrel_o_o @ R ) )
= ( ( ( A1 = nil_o )
& ( A22 = nil_o ) )
| ? [X3: $o,Y4: $o,Xs3: list_o,Ys3: list_o] :
( ( A1
= ( cons_o @ X3 @ Xs3 ) )
& ( A22
= ( cons_o @ Y4 @ Ys3 ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R )
& ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs3 @ Ys3 ) @ ( listrel_o_o @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_182_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X3: a,Y4: a,Xs3: list_a,Ys3: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs3 ) )
& ( A22
= ( cons_a @ Y4 @ Ys3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_183_listrel_Osimps,axiom,
! [A1: list_a,A22: list_P4002435161011370285od_o_o,R: set_Pr6507870997995571760od_o_o] :
( ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ A1 @ A22 ) @ ( listre5995639776835297690od_o_o @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_Product_prod_o_o ) )
| ? [X3: a,Y4: product_prod_o_o,Xs3: list_a,Ys3: list_P4002435161011370285od_o_o] :
( ( A1
= ( cons_a @ X3 @ Xs3 ) )
& ( A22
= ( cons_P8766293264717362397od_o_o @ Y4 @ Ys3 ) )
& ( member1089989612134903033od_o_o @ ( produc1797438980157902144od_o_o @ X3 @ Y4 ) @ R )
& ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ Xs3 @ Ys3 ) @ ( listre5995639776835297690od_o_o @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_184_listrel_Osimps,axiom,
! [A1: list_a,A22: list_P1396940483166286381od_a_a,R: set_Pr5530083903271594800od_a_a] :
( ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ A1 @ A22 ) @ ( listre367282320202154650od_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_Product_prod_a_a ) )
| ? [X3: a,Y4: product_prod_a_a,Xs3: list_a,Ys3: list_P1396940483166286381od_a_a] :
( ( A1
= ( cons_a @ X3 @ Xs3 ) )
& ( A22
= ( cons_P7316939126706565853od_a_a @ Y4 @ Ys3 ) )
& ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y4 ) @ R )
& ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ Xs3 @ Ys3 ) @ ( listre367282320202154650od_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_185_listrel_Osimps,axiom,
! [A1: list_P1802437025701573206od_a_a,A22: list_nat,R: set_Pr8612406289294286213_a_nat] :
( ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ A1 @ A22 ) @ ( listre6748024528684829295_a_nat @ R ) )
= ( ( ( A1 = nil_Pr7291147438282410198od_a_a )
& ( A22 = nil_nat ) )
| ? [X3: produc4044097585999906000od_a_a,Y4: nat,Xs3: list_P1802437025701573206od_a_a,Ys3: list_nat] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member1741560536349867494_a_nat @ ( produc8961142355356622153_a_nat @ X3 @ Y4 ) @ R )
& ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ Xs3 @ Ys3 ) @ ( listre6748024528684829295_a_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_186_listrel_Osimps,axiom,
! [A1: list_P1802437025701573206od_a_a,A22: list_f,R: set_Pr4596002078154142650_a_a_f] :
( ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ A1 @ A22 ) @ ( listre3440917786431297060_a_a_f @ R ) )
= ( ( ( A1 = nil_Pr7291147438282410198od_a_a )
& ( A22 = nil_f ) )
| ? [X3: produc4044097585999906000od_a_a,Y4: f,Xs3: list_P1802437025701573206od_a_a,Ys3: list_f] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) )
& ( A22
= ( cons_f @ Y4 @ Ys3 ) )
& ( member6782030652459103107_a_a_f @ ( produc4616610431800165834_a_a_f @ X3 @ Y4 ) @ R )
& ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ Xs3 @ Ys3 ) @ ( listre3440917786431297060_a_a_f @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_187_lexord__Nil__left,axiom,
! [Y: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Y ) @ ( lexord2630910672304566467od_a_a @ R ) )
= ( ? [A5: produc4044097585999906000od_a_a,X3: list_P1802437025701573206od_a_a] :
( Y
= ( cons_P64011158483510406od_a_a @ A5 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_188_lexord__Nil__left,axiom,
! [Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R ) )
= ( ? [A5: nat,X3: list_nat] :
( Y
= ( cons_nat @ A5 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_189_lexord__Nil__left,axiom,
! [Y: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ nil_f @ Y ) @ ( lexord_f @ R ) )
= ( ? [A5: f,X3: list_f] :
( Y
= ( cons_f @ A5 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_190_lexord__cons__cons,axiom,
! [A: produc4044097585999906000od_a_a,X: list_P1802437025701573206od_a_a,B: produc4044097585999906000od_a_a,Y: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ A @ X ) @ ( cons_P64011158483510406od_a_a @ B @ Y ) ) @ ( lexord2630910672304566467od_a_a @ R ) )
= ( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ X @ Y ) @ ( lexord2630910672304566467od_a_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_191_lexord__cons__cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
| ( ( A = B )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_192_lexord__cons__cons,axiom,
! [A: f,X: list_f,B: f,Y: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ A @ X ) @ ( cons_f @ B @ Y ) ) @ ( lexord_f @ R ) )
= ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
| ( ( A = B )
& ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ X @ Y ) @ ( lexord_f @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_193_lexord__cons__cons,axiom,
! [A: $o,X: list_o,B: $o,Y: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ A @ X ) @ ( cons_o @ B @ Y ) ) @ ( lexord_o @ R ) )
= ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
| ( ( A = B )
& ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ X @ Y ) @ ( lexord_o @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_194_lexord__cons__cons,axiom,
! [A: a,X: list_a,B: a,Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ A @ X ) @ ( cons_a @ B @ Y ) ) @ ( lexord_a @ R ) )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_195_Cons__listrel1__Cons,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) @ ( listre3467974182297698991od_a_a @ R ) )
= ( ( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Ys2 ) @ ( listre3467974182297698991od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_196_Cons__listrel1__Cons,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_197_Cons__listrel1__Cons,axiom,
! [X: f,Xs2: list_f,Y: f,Ys2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) @ ( listrel1_f @ R ) )
= ( ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( listrel1_f @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_198_Cons__listrel1__Cons,axiom,
! [X: $o,Xs2: list_o,Y: $o,Ys2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs2 ) @ ( cons_o @ Y @ Ys2 ) ) @ ( listrel1_o @ R ) )
= ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_199_Cons__listrel1__Cons,axiom,
! [X: a,Xs2: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel1_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_200_encodeGammaCond_Ocases,axiom,
! [X: produc628930716821598713od_o_o] :
( ! [Gam: a,Eps: a] :
( X
!= ( produc580260797871235561od_o_o @ Gam @ ( produc1797438980157902144od_o_o @ Eps @ ( product_Pair_o_o @ $true @ $true ) ) ) )
=> ( ! [Gam: a,Eps: a] :
( X
!= ( produc580260797871235561od_o_o @ Gam @ ( produc1797438980157902144od_o_o @ Eps @ ( product_Pair_o_o @ $false @ $false ) ) ) )
=> ( ! [Gam: a,Eps: a] :
( X
!= ( produc580260797871235561od_o_o @ Gam @ ( produc1797438980157902144od_o_o @ Eps @ ( product_Pair_o_o @ $false @ $true ) ) ) )
=> ~ ! [Gam: a,Eps: a] :
( X
!= ( produc580260797871235561od_o_o @ Gam @ ( produc1797438980157902144od_o_o @ Eps @ ( product_Pair_o_o @ $true @ $false ) ) ) ) ) ) ) ).
% encodeGammaCond.cases
thf(fact_201_listrel1I2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a,X: produc4044097585999906000od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Ys2 ) @ ( listre3467974182297698991od_a_a @ R ) )
=> ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ X @ Ys2 ) ) @ ( listre3467974182297698991od_a_a @ R ) ) ) ).
% listrel1I2
thf(fact_202_listrel1I2,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ X @ Ys2 ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I2
thf(fact_203_listrel1I2,axiom,
! [Xs2: list_f,Ys2: list_f,R: set_Product_prod_f_f,X: f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( listrel1_f @ R ) )
=> ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ X @ Ys2 ) ) @ ( listrel1_f @ R ) ) ) ).
% listrel1I2
thf(fact_204_not__Nil__listrel1,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Xs2 ) @ ( listre3467974182297698991od_a_a @ R ) ) ).
% not_Nil_listrel1
thf(fact_205_not__Nil__listrel1,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel1_nat @ R ) ) ).
% not_Nil_listrel1
thf(fact_206_not__Nil__listrel1,axiom,
! [Xs2: list_f,R: set_Product_prod_f_f] :
~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ nil_f @ Xs2 ) @ ( listrel1_f @ R ) ) ).
% not_Nil_listrel1
thf(fact_207_not__listrel1__Nil,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ nil_Pr7291147438282410198od_a_a ) @ ( listre3467974182297698991od_a_a @ R ) ) ).
% not_listrel1_Nil
thf(fact_208_not__listrel1__Nil,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel1_nat @ R ) ) ).
% not_listrel1_Nil
thf(fact_209_not__listrel1__Nil,axiom,
! [Xs2: list_f,R: set_Product_prod_f_f] :
~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ nil_f ) @ ( listrel1_f @ R ) ) ).
% not_listrel1_Nil
thf(fact_210_lexord__irreflexive,axiom,
! [R: set_Product_prod_o_o,Xs2: list_o] :
( ! [X2: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
=> ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Xs2 ) @ ( lexord_o @ R ) ) ) ).
% lexord_irreflexive
thf(fact_211_lexord__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Xs2 ) @ ( lexord_a @ R ) ) ) ).
% lexord_irreflexive
thf(fact_212_lexord__linear,axiom,
! [R: set_Product_prod_o_o,X: list_o,Y: list_o] :
( ! [A3: $o,B3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ A3 ) @ R ) )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ X @ Y ) @ ( lexord_o @ R ) )
| ( X = Y )
| ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Y @ X ) @ ( lexord_o @ R ) ) ) ) ).
% lexord_linear
thf(fact_213_lexord__linear,axiom,
! [R: set_Product_prod_a_a,X: list_a,Y: list_a] :
( ! [A3: a,B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ R ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) )
| ( X = Y )
| ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ X ) @ ( lexord_a @ R ) ) ) ) ).
% lexord_linear
thf(fact_214_listrel_ONil,axiom,
! [R: set_Pr9202094101234783879od_a_a] : ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ nil_Pr7291147438282410198od_a_a ) @ ( listre7590761571950618097od_a_a @ R ) ) ).
% listrel.Nil
thf(fact_215_listrel_ONil,axiom,
! [R: set_Pr8612406289294286213_a_nat] : ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ nil_Pr7291147438282410198od_a_a @ nil_nat ) @ ( listre6748024528684829295_a_nat @ R ) ) ).
% listrel.Nil
thf(fact_216_listrel_ONil,axiom,
! [R: set_Pr4596002078154142650_a_a_f] : ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ nil_Pr7291147438282410198od_a_a @ nil_f ) @ ( listre3440917786431297060_a_a_f @ R ) ) ).
% listrel.Nil
thf(fact_217_listrel_ONil,axiom,
! [R: set_Pr7697616133272258731od_a_a] : ( member3102152822625457154od_a_a @ ( produc5825293536187362205od_a_a @ nil_nat @ nil_Pr7291147438282410198od_a_a ) @ ( listre9150098348707296717od_a_a @ R ) ) ).
% listrel.Nil
thf(fact_218_listrel_ONil,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).
% listrel.Nil
thf(fact_219_listrel_ONil,axiom,
! [R: set_Pr4548512049067879902_nat_f] : ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ nil_nat @ nil_f ) @ ( listrel_nat_f @ R ) ) ).
% listrel.Nil
thf(fact_220_listrel_ONil,axiom,
! [R: set_Pr160775897027298964od_a_a] : ( member2314694291444880989od_a_a @ ( produc2139721702810641956od_a_a @ nil_f @ nil_Pr7291147438282410198od_a_a ) @ ( listre4688004250008097918od_a_a @ R ) ) ).
% listrel.Nil
thf(fact_221_listrel_ONil,axiom,
! [R: set_Pr5245412377734449720_f_nat] : ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ nil_f @ nil_nat ) @ ( listrel_f_nat @ R ) ) ).
% listrel.Nil
thf(fact_222_listrel_ONil,axiom,
! [R: set_Product_prod_f_f] : ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ nil_f @ nil_f ) @ ( listrel_f_f @ R ) ) ).
% listrel.Nil
thf(fact_223_listrel__Nil1,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Xs2 ) @ ( listre7590761571950618097od_a_a @ R ) )
=> ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% listrel_Nil1
thf(fact_224_listrel__Nil1,axiom,
! [Xs2: list_nat,R: set_Pr8612406289294286213_a_nat] :
( ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ nil_Pr7291147438282410198od_a_a @ Xs2 ) @ ( listre6748024528684829295_a_nat @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil1
thf(fact_225_listrel__Nil1,axiom,
! [Xs2: list_f,R: set_Pr4596002078154142650_a_a_f] :
( ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ nil_Pr7291147438282410198od_a_a @ Xs2 ) @ ( listre3440917786431297060_a_a_f @ R ) )
=> ( Xs2 = nil_f ) ) ).
% listrel_Nil1
thf(fact_226_listrel__Nil1,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr7697616133272258731od_a_a] :
( ( member3102152822625457154od_a_a @ ( produc5825293536187362205od_a_a @ nil_nat @ Xs2 ) @ ( listre9150098348707296717od_a_a @ R ) )
=> ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% listrel_Nil1
thf(fact_227_listrel__Nil1,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel_nat_nat @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil1
thf(fact_228_listrel__Nil1,axiom,
! [Xs2: list_f,R: set_Pr4548512049067879902_nat_f] :
( ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ nil_nat @ Xs2 ) @ ( listrel_nat_f @ R ) )
=> ( Xs2 = nil_f ) ) ).
% listrel_Nil1
thf(fact_229_listrel__Nil1,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr160775897027298964od_a_a] :
( ( member2314694291444880989od_a_a @ ( produc2139721702810641956od_a_a @ nil_f @ Xs2 ) @ ( listre4688004250008097918od_a_a @ R ) )
=> ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% listrel_Nil1
thf(fact_230_listrel__Nil1,axiom,
! [Xs2: list_nat,R: set_Pr5245412377734449720_f_nat] :
( ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ nil_f @ Xs2 ) @ ( listrel_f_nat @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil1
thf(fact_231_listrel__Nil1,axiom,
! [Xs2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ nil_f @ Xs2 ) @ ( listrel_f_f @ R ) )
=> ( Xs2 = nil_f ) ) ).
% listrel_Nil1
thf(fact_232_listrel__Nil2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ nil_Pr7291147438282410198od_a_a ) @ ( listre7590761571950618097od_a_a @ R ) )
=> ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% listrel_Nil2
thf(fact_233_listrel__Nil2,axiom,
! [Xs2: list_nat,R: set_Pr7697616133272258731od_a_a] :
( ( member3102152822625457154od_a_a @ ( produc5825293536187362205od_a_a @ Xs2 @ nil_Pr7291147438282410198od_a_a ) @ ( listre9150098348707296717od_a_a @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil2
thf(fact_234_listrel__Nil2,axiom,
! [Xs2: list_f,R: set_Pr160775897027298964od_a_a] :
( ( member2314694291444880989od_a_a @ ( produc2139721702810641956od_a_a @ Xs2 @ nil_Pr7291147438282410198od_a_a ) @ ( listre4688004250008097918od_a_a @ R ) )
=> ( Xs2 = nil_f ) ) ).
% listrel_Nil2
thf(fact_235_listrel__Nil2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr8612406289294286213_a_nat] :
( ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ Xs2 @ nil_nat ) @ ( listre6748024528684829295_a_nat @ R ) )
=> ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% listrel_Nil2
thf(fact_236_listrel__Nil2,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel_nat_nat @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil2
thf(fact_237_listrel__Nil2,axiom,
! [Xs2: list_f,R: set_Pr5245412377734449720_f_nat] :
( ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ Xs2 @ nil_nat ) @ ( listrel_f_nat @ R ) )
=> ( Xs2 = nil_f ) ) ).
% listrel_Nil2
thf(fact_238_listrel__Nil2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr4596002078154142650_a_a_f] :
( ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ Xs2 @ nil_f ) @ ( listre3440917786431297060_a_a_f @ R ) )
=> ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% listrel_Nil2
thf(fact_239_listrel__Nil2,axiom,
! [Xs2: list_nat,R: set_Pr4548512049067879902_nat_f] :
( ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ Xs2 @ nil_f ) @ ( listrel_nat_f @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil2
thf(fact_240_listrel__Nil2,axiom,
! [Xs2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ nil_f ) @ ( listrel_f_f @ R ) )
=> ( Xs2 = nil_f ) ) ).
% listrel_Nil2
thf(fact_241_lexord__Nil__right,axiom,
! [X: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) @ ( lexord2630910672304566467od_a_a @ R ) ) ).
% lexord_Nil_right
thf(fact_242_lexord__Nil__right,axiom,
! [X: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ nil_nat ) @ ( lexord_nat @ R ) ) ).
% lexord_Nil_right
thf(fact_243_lexord__Nil__right,axiom,
! [X: list_f,R: set_Product_prod_f_f] :
~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ X @ nil_f ) @ ( lexord_f @ R ) ) ).
% lexord_Nil_right
thf(fact_244_Cons__listrel1E2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) @ ( listre3467974182297698991od_a_a @ R ) )
=> ( ! [X2: produc4044097585999906000od_a_a] :
( ( Xs2
= ( cons_P64011158483510406od_a_a @ X2 @ Ys2 ) )
=> ~ ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X2 @ Y ) @ R ) )
=> ~ ! [Zs: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( cons_P64011158483510406od_a_a @ Y @ Zs ) )
=> ~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Zs @ Ys2 ) @ ( listre3467974182297698991od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_245_Cons__listrel1E2,axiom,
! [Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X2: nat] :
( ( Xs2
= ( cons_nat @ X2 @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R ) )
=> ~ ! [Zs: list_nat] :
( ( Xs2
= ( cons_nat @ Y @ Zs ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_246_Cons__listrel1E2,axiom,
! [Xs2: list_f,Y: f,Ys2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ ( cons_f @ Y @ Ys2 ) ) @ ( listrel1_f @ R ) )
=> ( ! [X2: f] :
( ( Xs2
= ( cons_f @ X2 @ Ys2 ) )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y ) @ R ) )
=> ~ ! [Zs: list_f] :
( ( Xs2
= ( cons_f @ Y @ Zs ) )
=> ~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Zs @ Ys2 ) @ ( listrel1_f @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_247_Cons__listrel1E2,axiom,
! [Xs2: list_o,Y: $o,Ys2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ ( cons_o @ Y @ Ys2 ) ) @ ( listrel1_o @ R ) )
=> ( ! [X2: $o] :
( ( Xs2
= ( cons_o @ X2 @ Ys2 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y ) @ R ) )
=> ~ ! [Zs: list_o] :
( ( Xs2
= ( cons_o @ Y @ Zs ) )
=> ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Zs @ Ys2 ) @ ( listrel1_o @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_248_Cons__listrel1E2,axiom,
! [Xs2: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel1_a @ R ) )
=> ( ! [X2: a] :
( ( Xs2
= ( cons_a @ X2 @ Ys2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) )
=> ~ ! [Zs: list_a] :
( ( Xs2
= ( cons_a @ Y @ Zs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_249_Cons__listrel1E1,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ Ys2 ) @ ( listre3467974182297698991od_a_a @ R ) )
=> ( ! [Y3: produc4044097585999906000od_a_a] :
( ( Ys2
= ( cons_P64011158483510406od_a_a @ Y3 @ Xs2 ) )
=> ~ ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_P1802437025701573206od_a_a] :
( ( Ys2
= ( cons_P64011158483510406od_a_a @ X @ Zs ) )
=> ~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Zs ) @ ( listre3467974182297698991od_a_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_250_Cons__listrel1E1,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( ! [Y3: nat] :
( ( Ys2
= ( cons_nat @ Y3 @ Xs2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Zs ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_251_Cons__listrel1E1,axiom,
! [X: f,Xs2: list_f,Ys2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X @ Xs2 ) @ Ys2 ) @ ( listrel1_f @ R ) )
=> ( ! [Y3: f] :
( ( Ys2
= ( cons_f @ Y3 @ Xs2 ) )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_f] :
( ( Ys2
= ( cons_f @ X @ Zs ) )
=> ~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Zs ) @ ( listrel1_f @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_252_Cons__listrel1E1,axiom,
! [X: $o,Xs2: list_o,Ys2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs2 ) @ Ys2 ) @ ( listrel1_o @ R ) )
=> ( ! [Y3: $o] :
( ( Ys2
= ( cons_o @ Y3 @ Xs2 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_o] :
( ( Ys2
= ( cons_o @ X @ Zs ) )
=> ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Zs ) @ ( listrel1_o @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_253_Cons__listrel1E1,axiom,
! [X: a,Xs2: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ Ys2 ) @ ( listrel1_a @ R ) )
=> ( ! [Y3: a] :
( ( Ys2
= ( cons_a @ Y3 @ Xs2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_a] :
( ( Ys2
= ( cons_a @ X @ Zs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Zs ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_254_listrel1I1,axiom,
! [X: produc4044097585999906000od_a_a,Y: produc4044097585999906000od_a_a,R: set_Pr9202094101234783879od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X @ Y ) @ R )
=> ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Xs2 ) ) @ ( listre3467974182297698991od_a_a @ R ) ) ) ).
% listrel1I1
thf(fact_255_listrel1I1,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Xs2 ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_256_listrel1I1,axiom,
! [X: f,Y: f,R: set_Product_prod_f_f,Xs2: list_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ R )
=> ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Xs2 ) ) @ ( listrel1_f @ R ) ) ) ).
% listrel1I1
thf(fact_257_listrel1I1,axiom,
! [X: $o,Y: $o,R: set_Product_prod_o_o,Xs2: list_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs2 ) @ ( cons_o @ Y @ Xs2 ) ) @ ( listrel1_o @ R ) ) ) ).
% listrel1I1
thf(fact_258_listrel1I1,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Xs2 ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I1
thf(fact_259_listrel__Cons2,axiom,
! [Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( Xs2
= ( cons_nat @ X2 @ Xs ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_260_listrel__Cons2,axiom,
! [Xs2: list_f,Y: nat,Ys2: list_nat,R: set_Pr5245412377734449720_f_nat] :
( ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_f_nat @ R ) )
=> ~ ! [X2: f,Xs: list_f] :
( ( Xs2
= ( cons_f @ X2 @ Xs ) )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y ) @ R )
=> ~ ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ Xs @ Ys2 ) @ ( listrel_f_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_261_listrel__Cons2,axiom,
! [Xs2: list_nat,Y: f,Ys2: list_f,R: set_Pr4548512049067879902_nat_f] :
( ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ Xs2 @ ( cons_f @ Y @ Ys2 ) ) @ ( listrel_nat_f @ R ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( Xs2
= ( cons_nat @ X2 @ Xs ) )
=> ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X2 @ Y ) @ R )
=> ~ ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ Xs @ Ys2 ) @ ( listrel_nat_f @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_262_listrel__Cons2,axiom,
! [Xs2: list_f,Y: f,Ys2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ ( cons_f @ Y @ Ys2 ) ) @ ( listrel_f_f @ R ) )
=> ~ ! [X2: f,Xs: list_f] :
( ( Xs2
= ( cons_f @ X2 @ Xs ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y ) @ R )
=> ~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys2 ) @ ( listrel_f_f @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_263_listrel__Cons2,axiom,
! [Xs2: list_o,Y: $o,Ys2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ ( cons_o @ Y @ Ys2 ) ) @ ( listrel_o_o @ R ) )
=> ~ ! [X2: $o,Xs: list_o] :
( ( Xs2
= ( cons_o @ X2 @ Xs ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y ) @ R )
=> ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys2 ) @ ( listrel_o_o @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_264_listrel__Cons2,axiom,
! [Xs2: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_a_a @ R ) )
=> ~ ! [X2: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X2 @ Xs ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_265_listrel__Cons2,axiom,
! [Xs2: list_a,Y: product_prod_o_o,Ys2: list_P4002435161011370285od_o_o,R: set_Pr6507870997995571760od_o_o] :
( ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ Xs2 @ ( cons_P8766293264717362397od_o_o @ Y @ Ys2 ) ) @ ( listre5995639776835297690od_o_o @ R ) )
=> ~ ! [X2: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X2 @ Xs ) )
=> ( ( member1089989612134903033od_o_o @ ( produc1797438980157902144od_o_o @ X2 @ Y ) @ R )
=> ~ ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ Xs @ Ys2 ) @ ( listre5995639776835297690od_o_o @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_266_listrel__Cons2,axiom,
! [Xs2: list_a,Y: product_prod_a_a,Ys2: list_P1396940483166286381od_a_a,R: set_Pr5530083903271594800od_a_a] :
( ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ Xs2 @ ( cons_P7316939126706565853od_a_a @ Y @ Ys2 ) ) @ ( listre367282320202154650od_a_a @ R ) )
=> ~ ! [X2: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X2 @ Xs ) )
=> ( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X2 @ Y ) @ R )
=> ~ ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ Xs @ Ys2 ) @ ( listre367282320202154650od_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_267_listrel__Cons2,axiom,
! [Xs2: list_nat,Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr7697616133272258731od_a_a] :
( ( member3102152822625457154od_a_a @ ( produc5825293536187362205od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) @ ( listre9150098348707296717od_a_a @ R ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( Xs2
= ( cons_nat @ X2 @ Xs ) )
=> ( ( member4271028325217606668od_a_a @ ( produc2139844138524313767od_a_a @ X2 @ Y ) @ R )
=> ~ ( member3102152822625457154od_a_a @ ( produc5825293536187362205od_a_a @ Xs @ Ys2 ) @ ( listre9150098348707296717od_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_268_listrel__Cons2,axiom,
! [Xs2: list_f,Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr160775897027298964od_a_a] :
( ( member2314694291444880989od_a_a @ ( produc2139721702810641956od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) @ ( listre4688004250008097918od_a_a @ R ) )
=> ~ ! [X2: f,Xs: list_f] :
( ( Xs2
= ( cons_f @ X2 @ Xs ) )
=> ( ( member8950279100819641181od_a_a @ ( produc5863696895376966692od_a_a @ X2 @ Y ) @ R )
=> ~ ( member2314694291444880989od_a_a @ ( produc2139721702810641956od_a_a @ Xs @ Ys2 ) @ ( listre4688004250008097918od_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_269_listrel__Cons1,axiom,
! [Y: nat,Ys2: list_nat,Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [Y3: nat,Ys: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_270_listrel__Cons1,axiom,
! [Y: nat,Ys2: list_nat,Xs2: list_f,R: set_Pr4548512049067879902_nat_f] :
( ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ ( cons_nat @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_nat_f @ R ) )
=> ~ ! [Y3: f,Ys: list_f] :
( ( Xs2
= ( cons_f @ Y3 @ Ys ) )
=> ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ Y @ Y3 ) @ R )
=> ~ ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ Ys2 @ Ys ) @ ( listrel_nat_f @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_271_listrel__Cons1,axiom,
! [Y: f,Ys2: list_f,Xs2: list_nat,R: set_Pr5245412377734449720_f_nat] :
( ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ ( cons_f @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_f_nat @ R ) )
=> ~ ! [Y3: nat,Ys: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ Y @ Y3 ) @ R )
=> ~ ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ Ys2 @ Ys ) @ ( listrel_f_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_272_listrel__Cons1,axiom,
! [Y: f,Ys2: list_f,Xs2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_f_f @ R ) )
=> ~ ! [Y3: f,Ys: list_f] :
( ( Xs2
= ( cons_f @ Y3 @ Ys ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y @ Y3 ) @ R )
=> ~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Ys2 @ Ys ) @ ( listrel_f_f @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_273_listrel__Cons1,axiom,
! [Y: $o,Ys2: list_o,Xs2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_o_o @ R ) )
=> ~ ! [Y3: $o,Ys: list_o] :
( ( Xs2
= ( cons_o @ Y3 @ Ys ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Y3 ) @ R )
=> ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Ys ) @ ( listrel_o_o @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_274_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_a_a @ R ) )
=> ~ ! [Y3: a,Ys: list_a] :
( ( Xs2
= ( cons_a @ Y3 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y3 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_275_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs2: list_P4002435161011370285od_o_o,R: set_Pr6507870997995571760od_o_o] :
( ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ ( cons_a @ Y @ Ys2 ) @ Xs2 ) @ ( listre5995639776835297690od_o_o @ R ) )
=> ~ ! [Y3: product_prod_o_o,Ys: list_P4002435161011370285od_o_o] :
( ( Xs2
= ( cons_P8766293264717362397od_o_o @ Y3 @ Ys ) )
=> ( ( member1089989612134903033od_o_o @ ( produc1797438980157902144od_o_o @ Y @ Y3 ) @ R )
=> ~ ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ Ys2 @ Ys ) @ ( listre5995639776835297690od_o_o @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_276_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs2: list_P1396940483166286381od_a_a,R: set_Pr5530083903271594800od_a_a] :
( ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ ( cons_a @ Y @ Ys2 ) @ Xs2 ) @ ( listre367282320202154650od_a_a @ R ) )
=> ~ ! [Y3: product_prod_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( Xs2
= ( cons_P7316939126706565853od_a_a @ Y3 @ Ys ) )
=> ( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ Y @ Y3 ) @ R )
=> ~ ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ Ys2 @ Ys ) @ ( listre367282320202154650od_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_277_listrel__Cons1,axiom,
! [Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,Xs2: list_nat,R: set_Pr8612406289294286213_a_nat] :
( ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) @ Xs2 ) @ ( listre6748024528684829295_a_nat @ R ) )
=> ~ ! [Y3: nat,Ys: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member1741560536349867494_a_nat @ ( produc8961142355356622153_a_nat @ Y @ Y3 ) @ R )
=> ~ ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ Ys2 @ Ys ) @ ( listre6748024528684829295_a_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_278_listrel__Cons1,axiom,
! [Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,Xs2: list_f,R: set_Pr4596002078154142650_a_a_f] :
( ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) @ Xs2 ) @ ( listre3440917786431297060_a_a_f @ R ) )
=> ~ ! [Y3: f,Ys: list_f] :
( ( Xs2
= ( cons_f @ Y3 @ Ys ) )
=> ( ( member6782030652459103107_a_a_f @ ( produc4616610431800165834_a_a_f @ Y @ Y3 ) @ R )
=> ~ ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ Ys2 @ Ys ) @ ( listre3440917786431297060_a_a_f @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_279_listrel_OCons,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_280_listrel_OCons,axiom,
! [X: nat,Y: f,R: set_Pr4548512049067879902_nat_f,Xs2: list_nat,Ys2: list_f] :
( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X @ Y ) @ R )
=> ( ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ Xs2 @ Ys2 ) @ ( listrel_nat_f @ R ) )
=> ( member6287320593503803189list_f @ ( produc7723716031568168016list_f @ ( cons_nat @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) @ ( listrel_nat_f @ R ) ) ) ) ).
% listrel.Cons
thf(fact_281_listrel_OCons,axiom,
! [X: f,Y: nat,R: set_Pr5245412377734449720_f_nat,Xs2: list_f,Ys2: list_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y ) @ R )
=> ( ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ Xs2 @ Ys2 ) @ ( listrel_f_nat @ R ) )
=> ( member8185636152533575727st_nat @ ( produc8127447161900435346st_nat @ ( cons_f @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_f_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_282_listrel_OCons,axiom,
! [X: f,Y: f,R: set_Product_prod_f_f,Xs2: list_f,Ys2: list_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ R )
=> ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( listrel_f_f @ R ) )
=> ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) @ ( listrel_f_f @ R ) ) ) ) ).
% listrel.Cons
thf(fact_283_listrel_OCons,axiom,
! [X: $o,Y: $o,R: set_Product_prod_o_o,Xs2: list_o,Ys2: list_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel_o_o @ R ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs2 ) @ ( cons_o @ Y @ Ys2 ) ) @ ( listrel_o_o @ R ) ) ) ) ).
% listrel.Cons
thf(fact_284_listrel_OCons,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs2: list_a,Ys2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_285_listrel_OCons,axiom,
! [X: a,Y: product_prod_o_o,R: set_Pr6507870997995571760od_o_o,Xs2: list_a,Ys2: list_P4002435161011370285od_o_o] :
( ( member1089989612134903033od_o_o @ ( produc1797438980157902144od_o_o @ X @ Y ) @ R )
=> ( ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ Xs2 @ Ys2 ) @ ( listre5995639776835297690od_o_o @ R ) )
=> ( member5940269043467357561od_o_o @ ( produc8860646620457897152od_o_o @ ( cons_a @ X @ Xs2 ) @ ( cons_P8766293264717362397od_o_o @ Y @ Ys2 ) ) @ ( listre5995639776835297690od_o_o @ R ) ) ) ) ).
% listrel.Cons
thf(fact_286_listrel_OCons,axiom,
! [X: a,Y: product_prod_a_a,R: set_Pr5530083903271594800od_a_a,Xs2: list_a,Ys2: list_P1396940483166286381od_a_a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X @ Y ) @ R )
=> ( ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ Xs2 @ Ys2 ) @ ( listre367282320202154650od_a_a @ R ) )
=> ( member4220520676121331065od_a_a @ ( produc5139040788227789504od_a_a @ ( cons_a @ X @ Xs2 ) @ ( cons_P7316939126706565853od_a_a @ Y @ Ys2 ) ) @ ( listre367282320202154650od_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_287_listrel_OCons,axiom,
! [X: produc4044097585999906000od_a_a,Y: nat,R: set_Pr8612406289294286213_a_nat,Xs2: list_P1802437025701573206od_a_a,Ys2: list_nat] :
( ( member1741560536349867494_a_nat @ ( produc8961142355356622153_a_nat @ X @ Y ) @ R )
=> ( ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ Xs2 @ Ys2 ) @ ( listre6748024528684829295_a_nat @ R ) )
=> ( member1707075787549946620st_nat @ ( produc4321862821133114975st_nat @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listre6748024528684829295_a_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_288_listrel_OCons,axiom,
! [X: produc4044097585999906000od_a_a,Y: f,R: set_Pr4596002078154142650_a_a_f,Xs2: list_P1802437025701573206od_a_a,Ys2: list_f] :
( ( member6782030652459103107_a_a_f @ ( produc4616610431800165834_a_a_f @ X @ Y ) @ R )
=> ( ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ Xs2 @ Ys2 ) @ ( listre3440917786431297060_a_a_f @ R ) )
=> ( member4906821000576044547list_f @ ( produc6118079160660574026list_f @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) @ ( listre3440917786431297060_a_a_f @ R ) ) ) ) ).
% listrel.Cons
thf(fact_289_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,Y: produc4044097585999906000od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) ) @ ( append4725183380742660459od_a_a @ Ys2 @ ( cons_P64011158483510406od_a_a @ Y @ nil_Pr7291147438282410198od_a_a ) ) ) @ ( listre3467974182297698991od_a_a @ R ) )
= ( ( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Ys2 ) @ ( listre3467974182297698991od_a_a @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_290_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_291_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_f,X: f,Ys2: list_f,Y: f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( append_f @ Xs2 @ ( cons_f @ X @ nil_f ) ) @ ( append_f @ Ys2 @ ( cons_f @ Y @ nil_f ) ) ) @ ( listrel1_f @ R ) )
= ( ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( listrel1_f @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_292_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_o,X: $o,Ys2: list_o,Y: $o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ ( cons_o @ X @ nil_o ) ) @ ( append_o @ Ys2 @ ( cons_o @ Y @ nil_o ) ) ) @ ( listrel1_o @ R ) )
= ( ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_293_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_a,X: a,Ys2: list_a,Y: a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_294_lexord__append__left__rightI,axiom,
! [A: produc4044097585999906000od_a_a,B: produc4044097585999906000od_a_a,R: set_Pr9202094101234783879od_a_a,U: list_P1802437025701573206od_a_a,X: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ A @ B ) @ R )
=> ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( append4725183380742660459od_a_a @ U @ ( cons_P64011158483510406od_a_a @ A @ X ) ) @ ( append4725183380742660459od_a_a @ U @ ( cons_P64011158483510406od_a_a @ B @ Y ) ) ) @ ( lexord2630910672304566467od_a_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_295_lexord__append__left__rightI,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X: list_nat,Y: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_296_lexord__append__left__rightI,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,U: list_f,X: list_f,Y: list_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
=> ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( append_f @ U @ ( cons_f @ A @ X ) ) @ ( append_f @ U @ ( cons_f @ B @ Y ) ) ) @ ( lexord_f @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_297_lexord__append__left__rightI,axiom,
! [A: $o,B: $o,R: set_Product_prod_o_o,U: list_o,X: list_o,Y: list_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ U @ ( cons_o @ A @ X ) ) @ ( append_o @ U @ ( cons_o @ B @ Y ) ) ) @ ( lexord_o @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_298_lexord__append__left__rightI,axiom,
! [A: a,B: a,R: set_Product_prod_a_a,U: list_a,X: list_a,Y: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ ( cons_a @ A @ X ) ) @ ( append_a @ U @ ( cons_a @ B @ Y ) ) ) @ ( lexord_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_299_listrel1I,axiom,
! [X: produc4044097585999906000od_a_a,Y: produc4044097585999906000od_a_a,R: set_Pr9202094101234783879od_a_a,Xs2: list_P1802437025701573206od_a_a,Us: list_P1802437025701573206od_a_a,Vs: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X @ Y ) @ R )
=> ( ( Xs2
= ( append4725183380742660459od_a_a @ Us @ ( cons_P64011158483510406od_a_a @ X @ Vs ) ) )
=> ( ( Ys2
= ( append4725183380742660459od_a_a @ Us @ ( cons_P64011158483510406od_a_a @ Y @ Vs ) ) )
=> ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Ys2 ) @ ( listre3467974182297698991od_a_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_300_listrel1I,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Us: list_nat,Vs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( Xs2
= ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_301_listrel1I,axiom,
! [X: f,Y: f,R: set_Product_prod_f_f,Xs2: list_f,Us: list_f,Vs: list_f,Ys2: list_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ R )
=> ( ( Xs2
= ( append_f @ Us @ ( cons_f @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_f @ Us @ ( cons_f @ Y @ Vs ) ) )
=> ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( listrel1_f @ R ) ) ) ) ) ).
% listrel1I
thf(fact_302_listrel1I,axiom,
! [X: $o,Y: $o,R: set_Product_prod_o_o,Xs2: list_o,Us: list_o,Vs: list_o,Ys2: list_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
=> ( ( Xs2
= ( append_o @ Us @ ( cons_o @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_o @ Us @ ( cons_o @ Y @ Vs ) ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) ) ) ) ) ).
% listrel1I
thf(fact_303_listrel1I,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs2: list_a,Us: list_a,Vs: list_a,Ys2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( Xs2
= ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_304_listrel1E,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Ys2 ) @ ( listre3467974182297698991od_a_a @ R ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: produc4044097585999906000od_a_a] :
( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X2 @ Y3 ) @ R )
=> ! [Us2: list_P1802437025701573206od_a_a,Vs2: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Us2 @ ( cons_P64011158483510406od_a_a @ X2 @ Vs2 ) ) )
=> ( Ys2
!= ( append4725183380742660459od_a_a @ Us2 @ ( cons_P64011158483510406od_a_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_305_listrel1E,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ~ ! [X2: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ! [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ ( cons_nat @ X2 @ Vs2 ) ) )
=> ( Ys2
!= ( append_nat @ Us2 @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_306_listrel1E,axiom,
! [Xs2: list_f,Ys2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( listrel1_f @ R ) )
=> ~ ! [X2: f,Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ R )
=> ! [Us2: list_f,Vs2: list_f] :
( ( Xs2
= ( append_f @ Us2 @ ( cons_f @ X2 @ Vs2 ) ) )
=> ( Ys2
!= ( append_f @ Us2 @ ( cons_f @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_307_listrel1E,axiom,
! [Xs2: list_o,Ys2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) )
=> ~ ! [X2: $o,Y3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ! [Us2: list_o,Vs2: list_o] :
( ( Xs2
= ( append_o @ Us2 @ ( cons_o @ X2 @ Vs2 ) ) )
=> ( Ys2
!= ( append_o @ Us2 @ ( cons_o @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_308_listrel1E,axiom,
! [Xs2: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ R ) )
=> ~ ! [X2: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R )
=> ! [Us2: list_a,Vs2: list_a] :
( ( Xs2
= ( append_a @ Us2 @ ( cons_a @ X2 @ Vs2 ) ) )
=> ( Ys2
!= ( append_a @ Us2 @ ( cons_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_309_Nil__lenlex__iff1,axiom,
! [Ns: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Ns ) @ ( lenlex2249140905286214619od_a_a @ R ) )
= ( Ns != nil_Pr7291147438282410198od_a_a ) ) ).
% Nil_lenlex_iff1
thf(fact_310_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_311_Nil__lenlex__iff1,axiom,
! [Ns: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ nil_f @ Ns ) @ ( lenlex_f @ R ) )
= ( Ns != nil_f ) ) ).
% Nil_lenlex_iff1
thf(fact_312_formula__of__lit_Ocases,axiom,
! [X: product_prod_o_o] :
( ! [X2: $o] :
( X
!= ( product_Pair_o_o @ X2 @ $true ) )
=> ~ ! [X2: $o] :
( X
!= ( product_Pair_o_o @ X2 @ $false ) ) ) ).
% formula_of_lit.cases
thf(fact_313_partition_Osimps_I1_J,axiom,
! [P: produc4044097585999906000od_a_a > $o] :
( ( partit8336039082511689039od_a_a @ P @ nil_Pr7291147438282410198od_a_a )
= ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ nil_Pr7291147438282410198od_a_a ) ) ).
% partition.simps(1)
thf(fact_314_partition_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( partition_nat @ P @ nil_nat )
= ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).
% partition.simps(1)
thf(fact_315_partition_Osimps_I1_J,axiom,
! [P: f > $o] :
( ( partition_f @ P @ nil_f )
= ( produc8144748701140958423list_f @ nil_f @ nil_f ) ) ).
% partition.simps(1)
thf(fact_316_bind__simps_I1_J,axiom,
! [F: produc4044097585999906000od_a_a > list_P1802437025701573206od_a_a] :
( ( bind_P6033372204491099015od_a_a @ nil_Pr7291147438282410198od_a_a @ F )
= nil_Pr7291147438282410198od_a_a ) ).
% bind_simps(1)
thf(fact_317_bind__simps_I1_J,axiom,
! [F: produc4044097585999906000od_a_a > list_nat] :
( ( bind_P5255732232670723929_a_nat @ nil_Pr7291147438282410198od_a_a @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_318_bind__simps_I1_J,axiom,
! [F: produc4044097585999906000od_a_a > list_f] :
( ( bind_P5070077749997897658_a_a_f @ nil_Pr7291147438282410198od_a_a @ F )
= nil_f ) ).
% bind_simps(1)
thf(fact_319_bind__simps_I1_J,axiom,
! [F: nat > list_P1802437025701573206od_a_a] :
( ( bind_n7657806052693191351od_a_a @ nil_nat @ F )
= nil_Pr7291147438282410198od_a_a ) ).
% bind_simps(1)
thf(fact_320_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_321_bind__simps_I1_J,axiom,
! [F: nat > list_f] :
( ( bind_nat_f @ nil_nat @ F )
= nil_f ) ).
% bind_simps(1)
thf(fact_322_bind__simps_I1_J,axiom,
! [F: f > list_P1802437025701573206od_a_a] :
( ( bind_f6317164213574698516od_a_a @ nil_f @ F )
= nil_Pr7291147438282410198od_a_a ) ).
% bind_simps(1)
thf(fact_323_bind__simps_I1_J,axiom,
! [F: f > list_nat] :
( ( bind_f_nat @ nil_f @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_324_bind__simps_I1_J,axiom,
! [F: f > list_f] :
( ( bind_f_f @ nil_f @ F )
= nil_f ) ).
% bind_simps(1)
thf(fact_325_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C2: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
= ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_326_append__assoc,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ( append_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Zs2 )
= ( append_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_327_append__same__eq,axiom,
! [Ys2: list_nat,Xs2: list_nat,Zs2: list_nat] :
( ( ( append_nat @ Ys2 @ Xs2 )
= ( append_nat @ Zs2 @ Xs2 ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_328_same__append__eq,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Xs2 @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_329_append_Oright__neutral,axiom,
! [A: list_P1802437025701573206od_a_a] :
( ( append4725183380742660459od_a_a @ A @ nil_Pr7291147438282410198od_a_a )
= A ) ).
% append.right_neutral
thf(fact_330_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_331_append_Oright__neutral,axiom,
! [A: list_f] :
( ( append_f @ A @ nil_f )
= A ) ).
% append.right_neutral
thf(fact_332_append__Nil2,axiom,
! [Xs2: list_P1802437025701573206od_a_a] :
( ( append4725183380742660459od_a_a @ Xs2 @ nil_Pr7291147438282410198od_a_a )
= Xs2 ) ).
% append_Nil2
thf(fact_333_append__Nil2,axiom,
! [Xs2: list_nat] :
( ( append_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_334_append__Nil2,axiom,
! [Xs2: list_f] :
( ( append_f @ Xs2 @ nil_f )
= Xs2 ) ).
% append_Nil2
thf(fact_335_append__self__conv,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( ( append4725183380742660459od_a_a @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_Pr7291147438282410198od_a_a ) ) ).
% append_self_conv
thf(fact_336_append__self__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_337_append__self__conv,axiom,
! [Xs2: list_f,Ys2: list_f] :
( ( ( append_f @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_f ) ) ).
% append_self_conv
thf(fact_338_self__append__conv,axiom,
! [Y: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( Y
= ( append4725183380742660459od_a_a @ Y @ Ys2 ) )
= ( Ys2 = nil_Pr7291147438282410198od_a_a ) ) ).
% self_append_conv
thf(fact_339_self__append__conv,axiom,
! [Y: list_nat,Ys2: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_340_self__append__conv,axiom,
! [Y: list_f,Ys2: list_f] :
( ( Y
= ( append_f @ Y @ Ys2 ) )
= ( Ys2 = nil_f ) ) ).
% self_append_conv
thf(fact_341_append__self__conv2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( ( append4725183380742660459od_a_a @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% append_self_conv2
thf(fact_342_append__self__conv2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_nat ) ) ).
% append_self_conv2
thf(fact_343_append__self__conv2,axiom,
! [Xs2: list_f,Ys2: list_f] :
( ( ( append_f @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_f ) ) ).
% append_self_conv2
thf(fact_344_self__append__conv2,axiom,
! [Y: list_P1802437025701573206od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( Y
= ( append4725183380742660459od_a_a @ Xs2 @ Y ) )
= ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% self_append_conv2
thf(fact_345_self__append__conv2,axiom,
! [Y: list_nat,Xs2: list_nat] :
( ( Y
= ( append_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_nat ) ) ).
% self_append_conv2
thf(fact_346_self__append__conv2,axiom,
! [Y: list_f,Xs2: list_f] :
( ( Y
= ( append_f @ Xs2 @ Y ) )
= ( Xs2 = nil_f ) ) ).
% self_append_conv2
thf(fact_347_Nil__is__append__conv,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( nil_Pr7291147438282410198od_a_a
= ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_Pr7291147438282410198od_a_a )
& ( Ys2 = nil_Pr7291147438282410198od_a_a ) ) ) ).
% Nil_is_append_conv
thf(fact_348_Nil__is__append__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( nil_nat
= ( append_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_349_Nil__is__append__conv,axiom,
! [Xs2: list_f,Ys2: list_f] :
( ( nil_f
= ( append_f @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_f )
& ( Ys2 = nil_f ) ) ) ).
% Nil_is_append_conv
thf(fact_350_append__is__Nil__conv,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( ( append4725183380742660459od_a_a @ Xs2 @ Ys2 )
= nil_Pr7291147438282410198od_a_a )
= ( ( Xs2 = nil_Pr7291147438282410198od_a_a )
& ( Ys2 = nil_Pr7291147438282410198od_a_a ) ) ) ).
% append_is_Nil_conv
thf(fact_351_append__is__Nil__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= nil_nat )
= ( ( Xs2 = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_352_append__is__Nil__conv,axiom,
! [Xs2: list_f,Ys2: list_f] :
( ( ( append_f @ Xs2 @ Ys2 )
= nil_f )
= ( ( Xs2 = nil_f )
& ( Ys2 = nil_f ) ) ) ).
% append_is_Nil_conv
thf(fact_353_append1__eq__conv,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,Y: produc4044097585999906000od_a_a] :
( ( ( append4725183380742660459od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) )
= ( append4725183380742660459od_a_a @ Ys2 @ ( cons_P64011158483510406od_a_a @ Y @ nil_Pr7291147438282410198od_a_a ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_354_append1__eq__conv,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_355_append1__eq__conv,axiom,
! [Xs2: list_f,X: f,Ys2: list_f,Y: f] :
( ( ( append_f @ Xs2 @ ( cons_f @ X @ nil_f ) )
= ( append_f @ Ys2 @ ( cons_f @ Y @ nil_f ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_356_bind__simps_I2_J,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,F: produc4044097585999906000od_a_a > list_nat] :
( ( bind_P5255732232670723929_a_nat @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_P5255732232670723929_a_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_357_bind__simps_I2_J,axiom,
! [X: nat,Xs2: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_358_bind__simps_I2_J,axiom,
! [X: f,Xs2: list_f,F: f > list_nat] :
( ( bind_f_nat @ ( cons_f @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_f_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_359_append__eq__appendI,axiom,
! [Xs2: list_nat,Xs1: list_nat,Zs2: list_nat,Ys2: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs2 @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_360_append__eq__append__conv2,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Zs2 @ Ts ) )
= ( ? [Us3: list_nat] :
( ( ( Xs2
= ( append_nat @ Zs2 @ Us3 ) )
& ( ( append_nat @ Us3 @ Ys2 )
= Ts ) )
| ( ( ( append_nat @ Xs2 @ Us3 )
= Zs2 )
& ( Ys2
= ( append_nat @ Us3 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_361_append__Cons,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( append4725183380742660459od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ Ys2 )
= ( cons_P64011158483510406od_a_a @ X @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_362_append__Cons,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs2 ) @ Ys2 )
= ( cons_nat @ X @ ( append_nat @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_363_append__Cons,axiom,
! [X: f,Xs2: list_f,Ys2: list_f] :
( ( append_f @ ( cons_f @ X @ Xs2 ) @ Ys2 )
= ( cons_f @ X @ ( append_f @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_364_Cons__eq__appendI,axiom,
! [X: produc4044097585999906000od_a_a,Xs1: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,Xs2: list_P1802437025701573206od_a_a,Zs2: list_P1802437025701573206od_a_a] :
( ( ( cons_P64011158483510406od_a_a @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append4725183380742660459od_a_a @ Xs1 @ Zs2 ) )
=> ( ( cons_P64011158483510406od_a_a @ X @ Xs2 )
= ( append4725183380742660459od_a_a @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_365_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys2: list_nat,Xs2: list_nat,Zs2: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_nat @ Xs1 @ Zs2 ) )
=> ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_366_Cons__eq__appendI,axiom,
! [X: f,Xs1: list_f,Ys2: list_f,Xs2: list_f,Zs2: list_f] :
( ( ( cons_f @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_f @ Xs1 @ Zs2 ) )
=> ( ( cons_f @ X @ Xs2 )
= ( append_f @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_367_append__Nil,axiom,
! [Ys2: list_P1802437025701573206od_a_a] :
( ( append4725183380742660459od_a_a @ nil_Pr7291147438282410198od_a_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_368_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_369_append__Nil,axiom,
! [Ys2: list_f] :
( ( append_f @ nil_f @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_370_append_Oleft__neutral,axiom,
! [A: list_P1802437025701573206od_a_a] :
( ( append4725183380742660459od_a_a @ nil_Pr7291147438282410198od_a_a @ A )
= A ) ).
% append.left_neutral
thf(fact_371_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_372_append_Oleft__neutral,axiom,
! [A: list_f] :
( ( append_f @ nil_f @ A )
= A ) ).
% append.left_neutral
thf(fact_373_eq__Nil__appendI,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append4725183380742660459od_a_a @ nil_Pr7291147438282410198od_a_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_374_eq__Nil__appendI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_375_eq__Nil__appendI,axiom,
! [Xs2: list_f,Ys2: list_f] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_f @ nil_f @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_376_ord_Olexordp__eq__pref,axiom,
! [Less: nat > nat > $o,U: list_nat,V2: list_nat] : ( lexordp_eq_nat @ Less @ U @ ( append_nat @ U @ V2 ) ) ).
% ord.lexordp_eq_pref
thf(fact_377_listrelp_OCons,axiom,
! [R: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,X: produc4044097585999906000od_a_a,Y: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( R @ X @ Y )
=> ( ( listre2671164784574791231od_a_a @ R @ Xs2 @ Ys2 )
=> ( listre2671164784574791231od_a_a @ R @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_378_listrelp_OCons,axiom,
! [R: produc4044097585999906000od_a_a > nat > $o,X: produc4044097585999906000od_a_a,Y: nat,Xs2: list_P1802437025701573206od_a_a,Ys2: list_nat] :
( ( R @ X @ Y )
=> ( ( listre3835667834449976993_a_nat @ R @ Xs2 @ Ys2 )
=> ( listre3835667834449976993_a_nat @ R @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_379_listrelp_OCons,axiom,
! [R: produc4044097585999906000od_a_a > f > $o,X: produc4044097585999906000od_a_a,Y: f,Xs2: list_P1802437025701573206od_a_a,Ys2: list_f] :
( ( R @ X @ Y )
=> ( ( listre673953193114217074_a_a_f @ R @ Xs2 @ Ys2 )
=> ( listre673953193114217074_a_a_f @ R @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_380_listrelp_OCons,axiom,
! [R: nat > produc4044097585999906000od_a_a > $o,X: nat,Y: produc4044097585999906000od_a_a,Xs2: list_nat,Ys2: list_P1802437025701573206od_a_a] :
( ( R @ X @ Y )
=> ( ( listre6237741654472444415od_a_a @ R @ Xs2 @ Ys2 )
=> ( listre6237741654472444415od_a_a @ R @ ( cons_nat @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_381_listrelp_OCons,axiom,
! [R: nat > nat > $o,X: nat,Y: nat,Xs2: list_nat,Ys2: list_nat] :
( ( R @ X @ Y )
=> ( ( listrelp_nat_nat @ R @ Xs2 @ Ys2 )
=> ( listrelp_nat_nat @ R @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_382_listrelp_OCons,axiom,
! [R: nat > f > $o,X: nat,Y: f,Xs2: list_nat,Ys2: list_f] :
( ( R @ X @ Y )
=> ( ( listrelp_nat_f @ R @ Xs2 @ Ys2 )
=> ( listrelp_nat_f @ R @ ( cons_nat @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_383_listrelp_OCons,axiom,
! [R: f > produc4044097585999906000od_a_a > $o,X: f,Y: produc4044097585999906000od_a_a,Xs2: list_f,Ys2: list_P1802437025701573206od_a_a] :
( ( R @ X @ Y )
=> ( ( listre1921039656691017932od_a_a @ R @ Xs2 @ Ys2 )
=> ( listre1921039656691017932od_a_a @ R @ ( cons_f @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_384_listrelp_OCons,axiom,
! [R: f > nat > $o,X: f,Y: nat,Xs2: list_f,Ys2: list_nat] :
( ( R @ X @ Y )
=> ( ( listrelp_f_nat @ R @ Xs2 @ Ys2 )
=> ( listrelp_f_nat @ R @ ( cons_f @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_385_listrelp_OCons,axiom,
! [R: f > f > $o,X: f,Y: f,Xs2: list_f,Ys2: list_f] :
( ( R @ X @ Y )
=> ( ( listrelp_f_f @ R @ Xs2 @ Ys2 )
=> ( listrelp_f_f @ R @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_386_listrelp_ONil,axiom,
! [R: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o] : ( listre2671164784574791231od_a_a @ R @ nil_Pr7291147438282410198od_a_a @ nil_Pr7291147438282410198od_a_a ) ).
% listrelp.Nil
thf(fact_387_listrelp_ONil,axiom,
! [R: produc4044097585999906000od_a_a > nat > $o] : ( listre3835667834449976993_a_nat @ R @ nil_Pr7291147438282410198od_a_a @ nil_nat ) ).
% listrelp.Nil
thf(fact_388_listrelp_ONil,axiom,
! [R: produc4044097585999906000od_a_a > f > $o] : ( listre673953193114217074_a_a_f @ R @ nil_Pr7291147438282410198od_a_a @ nil_f ) ).
% listrelp.Nil
thf(fact_389_listrelp_ONil,axiom,
! [R: nat > produc4044097585999906000od_a_a > $o] : ( listre6237741654472444415od_a_a @ R @ nil_nat @ nil_Pr7291147438282410198od_a_a ) ).
% listrelp.Nil
thf(fact_390_listrelp_ONil,axiom,
! [R: nat > nat > $o] : ( listrelp_nat_nat @ R @ nil_nat @ nil_nat ) ).
% listrelp.Nil
thf(fact_391_listrelp_ONil,axiom,
! [R: nat > f > $o] : ( listrelp_nat_f @ R @ nil_nat @ nil_f ) ).
% listrelp.Nil
thf(fact_392_listrelp_ONil,axiom,
! [R: f > produc4044097585999906000od_a_a > $o] : ( listre1921039656691017932od_a_a @ R @ nil_f @ nil_Pr7291147438282410198od_a_a ) ).
% listrelp.Nil
thf(fact_393_listrelp_ONil,axiom,
! [R: f > nat > $o] : ( listrelp_f_nat @ R @ nil_f @ nil_nat ) ).
% listrelp.Nil
thf(fact_394_listrelp_ONil,axiom,
! [R: f > f > $o] : ( listrelp_f_f @ R @ nil_f @ nil_f ) ).
% listrelp.Nil
thf(fact_395_rev__induct,axiom,
! [P: list_P1802437025701573206od_a_a > $o,Xs2: list_P1802437025701573206od_a_a] :
( ( P @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] :
( ( P @ Xs )
=> ( P @ ( append4725183380742660459od_a_a @ Xs @ ( cons_P64011158483510406od_a_a @ X2 @ nil_Pr7291147438282410198od_a_a ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_396_rev__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] :
( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_397_rev__induct,axiom,
! [P: list_f > $o,Xs2: list_f] :
( ( P @ nil_f )
=> ( ! [X2: f,Xs: list_f] :
( ( P @ Xs )
=> ( P @ ( append_f @ Xs @ ( cons_f @ X2 @ nil_f ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_398_rev__exhaust,axiom,
! [Xs2: list_P1802437025701573206od_a_a] :
( ( Xs2 != nil_Pr7291147438282410198od_a_a )
=> ~ ! [Ys: list_P1802437025701573206od_a_a,Y3: produc4044097585999906000od_a_a] :
( Xs2
!= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ Y3 @ nil_Pr7291147438282410198od_a_a ) ) ) ) ).
% rev_exhaust
thf(fact_399_rev__exhaust,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ~ ! [Ys: list_nat,Y3: nat] :
( Xs2
!= ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_400_rev__exhaust,axiom,
! [Xs2: list_f] :
( ( Xs2 != nil_f )
=> ~ ! [Ys: list_f,Y3: f] :
( Xs2
!= ( append_f @ Ys @ ( cons_f @ Y3 @ nil_f ) ) ) ) ).
% rev_exhaust
thf(fact_401_Cons__eq__append__conv,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,Zs2: list_P1802437025701573206od_a_a] :
( ( ( cons_P64011158483510406od_a_a @ X @ Xs2 )
= ( append4725183380742660459od_a_a @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_Pr7291147438282410198od_a_a )
& ( ( cons_P64011158483510406od_a_a @ X @ Xs2 )
= Zs2 ) )
| ? [Ys4: list_P1802437025701573206od_a_a] :
( ( ( cons_P64011158483510406od_a_a @ X @ Ys4 )
= Ys2 )
& ( Xs2
= ( append4725183380742660459od_a_a @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_402_Cons__eq__append__conv,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_nat )
& ( ( cons_nat @ X @ Xs2 )
= Zs2 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys2 )
& ( Xs2
= ( append_nat @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_403_Cons__eq__append__conv,axiom,
! [X: f,Xs2: list_f,Ys2: list_f,Zs2: list_f] :
( ( ( cons_f @ X @ Xs2 )
= ( append_f @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_f )
& ( ( cons_f @ X @ Xs2 )
= Zs2 ) )
| ? [Ys4: list_f] :
( ( ( cons_f @ X @ Ys4 )
= Ys2 )
& ( Xs2
= ( append_f @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_404_append__eq__Cons__conv,axiom,
! [Ys2: list_P1802437025701573206od_a_a,Zs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( ( append4725183380742660459od_a_a @ Ys2 @ Zs2 )
= ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_Pr7291147438282410198od_a_a )
& ( Zs2
= ( cons_P64011158483510406od_a_a @ X @ Xs2 ) ) )
| ? [Ys4: list_P1802437025701573206od_a_a] :
( ( Ys2
= ( cons_P64011158483510406od_a_a @ X @ Ys4 ) )
& ( ( append4725183380742660459od_a_a @ Ys4 @ Zs2 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_405_append__eq__Cons__conv,axiom,
! [Ys2: list_nat,Zs2: list_nat,X: nat,Xs2: list_nat] :
( ( ( append_nat @ Ys2 @ Zs2 )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_nat )
& ( Zs2
= ( cons_nat @ X @ Xs2 ) ) )
| ? [Ys4: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs2 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_406_append__eq__Cons__conv,axiom,
! [Ys2: list_f,Zs2: list_f,X: f,Xs2: list_f] :
( ( ( append_f @ Ys2 @ Zs2 )
= ( cons_f @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_f )
& ( Zs2
= ( cons_f @ X @ Xs2 ) ) )
| ? [Ys4: list_f] :
( ( Ys2
= ( cons_f @ X @ Ys4 ) )
& ( ( append_f @ Ys4 @ Zs2 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_407_rev__nonempty__induct,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: list_P1802437025701573206od_a_a > $o] :
( ( Xs2 != nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: produc4044097585999906000od_a_a] : ( P @ ( cons_P64011158483510406od_a_a @ X2 @ nil_Pr7291147438282410198od_a_a ) )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] :
( ( Xs != nil_Pr7291147438282410198od_a_a )
=> ( ( P @ Xs )
=> ( P @ ( append4725183380742660459od_a_a @ Xs @ ( cons_P64011158483510406od_a_a @ X2 @ nil_Pr7291147438282410198od_a_a ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_408_rev__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_409_rev__nonempty__induct,axiom,
! [Xs2: list_f,P: list_f > $o] :
( ( Xs2 != nil_f )
=> ( ! [X2: f] : ( P @ ( cons_f @ X2 @ nil_f ) )
=> ( ! [X2: f,Xs: list_f] :
( ( Xs != nil_f )
=> ( ( P @ Xs )
=> ( P @ ( append_f @ Xs @ ( cons_f @ X2 @ nil_f ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_410_append__listrel1I,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Us ) @ ( append_nat @ Ys2 @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).
% append_listrel1I
thf(fact_411_lexord__append__leftI,axiom,
! [U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat,X: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V2 ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_leftI
thf(fact_412_lenlex__irreflexive,axiom,
! [R: set_Product_prod_o_o,Xs2: list_o] :
( ! [X2: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
=> ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Xs2 ) @ ( lenlex_o @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_413_lenlex__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Xs2 ) @ ( lenlex_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_414_Nil__lenlex__iff2,axiom,
! [Ns: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Ns @ nil_Pr7291147438282410198od_a_a ) @ ( lenlex2249140905286214619od_a_a @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_415_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_416_Nil__lenlex__iff2,axiom,
! [Ns: list_f,R: set_Product_prod_f_f] :
~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Ns @ nil_f ) @ ( lenlex_f @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_417_lexord__append__leftD,axiom,
! [X: list_nat,U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V2 ) ) @ ( lexord_nat @ R ) )
=> ( ! [A3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_418_lexord__append__leftD,axiom,
! [X: list_o,U: list_o,V2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ X @ U ) @ ( append_o @ X @ V2 ) ) @ ( lexord_o @ R ) )
=> ( ! [A3: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A3 ) @ R )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ U @ V2 ) @ ( lexord_o @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_419_lexord__append__leftD,axiom,
! [X: list_a,U: list_a,V2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ X @ U ) @ ( append_a @ X @ V2 ) ) @ ( lexord_a @ R ) )
=> ( ! [A3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U @ V2 ) @ ( lexord_a @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_420_lexord__append__rightI,axiom,
! [Y: list_P1802437025701573206od_a_a,X: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ? [B4: produc4044097585999906000od_a_a,Z: list_P1802437025701573206od_a_a] :
( Y
= ( cons_P64011158483510406od_a_a @ B4 @ Z ) )
=> ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ X @ ( append4725183380742660459od_a_a @ X @ Y ) ) @ ( lexord2630910672304566467od_a_a @ R ) ) ) ).
% lexord_append_rightI
thf(fact_421_lexord__append__rightI,axiom,
! [Y: list_nat,X: list_nat,R: set_Pr1261947904930325089at_nat] :
( ? [B4: nat,Z: list_nat] :
( Y
= ( cons_nat @ B4 @ Z ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ ( append_nat @ X @ Y ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_rightI
thf(fact_422_lexord__append__rightI,axiom,
! [Y: list_f,X: list_f,R: set_Product_prod_f_f] :
( ? [B4: f,Z: list_f] :
( Y
= ( cons_f @ B4 @ Z ) )
=> ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ X @ ( append_f @ X @ Y ) ) @ ( lexord_f @ R ) ) ) ).
% lexord_append_rightI
thf(fact_423_listrelp_Ocases,axiom,
! [R: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,A1: list_P1802437025701573206od_a_a,A22: list_P1802437025701573206od_a_a] :
( ( listre2671164784574791231od_a_a @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Pr7291147438282410198od_a_a )
=> ( A22 != nil_Pr7291147438282410198od_a_a ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X2 @ Xs ) )
=> ! [Ys: list_P1802437025701573206od_a_a] :
( ( A22
= ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listre2671164784574791231od_a_a @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_424_listrelp_Ocases,axiom,
! [R: produc4044097585999906000od_a_a > nat > $o,A1: list_P1802437025701573206od_a_a,A22: list_nat] :
( ( listre3835667834449976993_a_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Pr7291147438282410198od_a_a )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: nat,Xs: list_P1802437025701573206od_a_a] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listre3835667834449976993_a_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_425_listrelp_Ocases,axiom,
! [R: produc4044097585999906000od_a_a > f > $o,A1: list_P1802437025701573206od_a_a,A22: list_f] :
( ( listre673953193114217074_a_a_f @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Pr7291147438282410198od_a_a )
=> ( A22 != nil_f ) )
=> ~ ! [X2: produc4044097585999906000od_a_a,Y3: f,Xs: list_P1802437025701573206od_a_a] :
( ( A1
= ( cons_P64011158483510406od_a_a @ X2 @ Xs ) )
=> ! [Ys: list_f] :
( ( A22
= ( cons_f @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listre673953193114217074_a_a_f @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_426_listrelp_Ocases,axiom,
! [R: nat > produc4044097585999906000od_a_a > $o,A1: list_nat,A22: list_P1802437025701573206od_a_a] :
( ( listre6237741654472444415od_a_a @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_Pr7291147438282410198od_a_a ) )
=> ~ ! [X2: nat,Y3: produc4044097585999906000od_a_a,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys: list_P1802437025701573206od_a_a] :
( ( A22
= ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listre6237741654472444415od_a_a @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_427_listrelp_Ocases,axiom,
! [R: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( listrelp_nat_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: nat,Y3: nat,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listrelp_nat_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_428_listrelp_Ocases,axiom,
! [R: nat > f > $o,A1: list_nat,A22: list_f] :
( ( listrelp_nat_f @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_f ) )
=> ~ ! [X2: nat,Y3: f,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys: list_f] :
( ( A22
= ( cons_f @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listrelp_nat_f @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_429_listrelp_Ocases,axiom,
! [R: f > produc4044097585999906000od_a_a > $o,A1: list_f,A22: list_P1802437025701573206od_a_a] :
( ( listre1921039656691017932od_a_a @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_f )
=> ( A22 != nil_Pr7291147438282410198od_a_a ) )
=> ~ ! [X2: f,Y3: produc4044097585999906000od_a_a,Xs: list_f] :
( ( A1
= ( cons_f @ X2 @ Xs ) )
=> ! [Ys: list_P1802437025701573206od_a_a] :
( ( A22
= ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listre1921039656691017932od_a_a @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_430_listrelp_Ocases,axiom,
! [R: f > nat > $o,A1: list_f,A22: list_nat] :
( ( listrelp_f_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_f )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: f,Y3: nat,Xs: list_f] :
( ( A1
= ( cons_f @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listrelp_f_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_431_listrelp_Ocases,axiom,
! [R: f > f > $o,A1: list_f,A22: list_f] :
( ( listrelp_f_f @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_f )
=> ( A22 != nil_f ) )
=> ~ ! [X2: f,Y3: f,Xs: list_f] :
( ( A1
= ( cons_f @ X2 @ Xs ) )
=> ! [Ys: list_f] :
( ( A22
= ( cons_f @ Y3 @ Ys ) )
=> ( ( R @ X2 @ Y3 )
=> ~ ( listrelp_f_f @ R @ Xs @ Ys ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_432_listrelp_Osimps,axiom,
( listre2671164784574791231od_a_a
= ( ^ [R2: produc4044097585999906000od_a_a > produc4044097585999906000od_a_a > $o,A12: list_P1802437025701573206od_a_a,A23: list_P1802437025701573206od_a_a] :
( ( ( A12 = nil_Pr7291147438282410198od_a_a )
& ( A23 = nil_Pr7291147438282410198od_a_a ) )
| ? [X3: produc4044097585999906000od_a_a,Y4: produc4044097585999906000od_a_a,Xs3: list_P1802437025701573206od_a_a,Ys3: list_P1802437025701573206od_a_a] :
( ( A12
= ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) )
& ( A23
= ( cons_P64011158483510406od_a_a @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listre2671164784574791231od_a_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_433_listrelp_Osimps,axiom,
( listre3835667834449976993_a_nat
= ( ^ [R2: produc4044097585999906000od_a_a > nat > $o,A12: list_P1802437025701573206od_a_a,A23: list_nat] :
( ( ( A12 = nil_Pr7291147438282410198od_a_a )
& ( A23 = nil_nat ) )
| ? [X3: produc4044097585999906000od_a_a,Y4: nat,Xs3: list_P1802437025701573206od_a_a,Ys3: list_nat] :
( ( A12
= ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listre3835667834449976993_a_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_434_listrelp_Osimps,axiom,
( listre673953193114217074_a_a_f
= ( ^ [R2: produc4044097585999906000od_a_a > f > $o,A12: list_P1802437025701573206od_a_a,A23: list_f] :
( ( ( A12 = nil_Pr7291147438282410198od_a_a )
& ( A23 = nil_f ) )
| ? [X3: produc4044097585999906000od_a_a,Y4: f,Xs3: list_P1802437025701573206od_a_a,Ys3: list_f] :
( ( A12
= ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) )
& ( A23
= ( cons_f @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listre673953193114217074_a_a_f @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_435_listrelp_Osimps,axiom,
( listre6237741654472444415od_a_a
= ( ^ [R2: nat > produc4044097585999906000od_a_a > $o,A12: list_nat,A23: list_P1802437025701573206od_a_a] :
( ( ( A12 = nil_nat )
& ( A23 = nil_Pr7291147438282410198od_a_a ) )
| ? [X3: nat,Y4: produc4044097585999906000od_a_a,Xs3: list_nat,Ys3: list_P1802437025701573206od_a_a] :
( ( A12
= ( cons_nat @ X3 @ Xs3 ) )
& ( A23
= ( cons_P64011158483510406od_a_a @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listre6237741654472444415od_a_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_436_listrelp_Osimps,axiom,
( listrelp_nat_nat
= ( ^ [R2: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ( ( A12 = nil_nat )
& ( A23 = nil_nat ) )
| ? [X3: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listrelp_nat_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_437_listrelp_Osimps,axiom,
( listrelp_nat_f
= ( ^ [R2: nat > f > $o,A12: list_nat,A23: list_f] :
( ( ( A12 = nil_nat )
& ( A23 = nil_f ) )
| ? [X3: nat,Y4: f,Xs3: list_nat,Ys3: list_f] :
( ( A12
= ( cons_nat @ X3 @ Xs3 ) )
& ( A23
= ( cons_f @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listrelp_nat_f @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_438_listrelp_Osimps,axiom,
( listre1921039656691017932od_a_a
= ( ^ [R2: f > produc4044097585999906000od_a_a > $o,A12: list_f,A23: list_P1802437025701573206od_a_a] :
( ( ( A12 = nil_f )
& ( A23 = nil_Pr7291147438282410198od_a_a ) )
| ? [X3: f,Y4: produc4044097585999906000od_a_a,Xs3: list_f,Ys3: list_P1802437025701573206od_a_a] :
( ( A12
= ( cons_f @ X3 @ Xs3 ) )
& ( A23
= ( cons_P64011158483510406od_a_a @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listre1921039656691017932od_a_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_439_listrelp_Osimps,axiom,
( listrelp_f_nat
= ( ^ [R2: f > nat > $o,A12: list_f,A23: list_nat] :
( ( ( A12 = nil_f )
& ( A23 = nil_nat ) )
| ? [X3: f,Y4: nat,Xs3: list_f,Ys3: list_nat] :
( ( A12
= ( cons_f @ X3 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listrelp_f_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_440_listrelp_Osimps,axiom,
( listrelp_f_f
= ( ^ [R2: f > f > $o,A12: list_f,A23: list_f] :
( ( ( A12 = nil_f )
& ( A23 = nil_f ) )
| ? [X3: f,Y4: f,Xs3: list_f,Ys3: list_f] :
( ( A12
= ( cons_f @ X3 @ Xs3 ) )
& ( A23
= ( cons_f @ Y4 @ Ys3 ) )
& ( R2 @ X3 @ Y4 )
& ( listrelp_f_f @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_441_maps__simps_I1_J,axiom,
! [F: produc4044097585999906000od_a_a > list_nat,X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( maps_P8591894958123311391_a_nat @ F @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_P8591894958123311391_a_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_442_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs2: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_443_maps__simps_I1_J,axiom,
! [F: f > list_nat,X: f,Xs2: list_f] :
( ( maps_f_nat @ F @ ( cons_f @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_f_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_444_concat__eq__append__conv,axiom,
! [Xss2: list_l4888837856047780700od_a_a,Ys2: list_P1802437025701573206od_a_a,Zs2: list_P1802437025701573206od_a_a] :
( ( ( concat5889154033839980561od_a_a @ Xss2 )
= ( append4725183380742660459od_a_a @ Ys2 @ Zs2 ) )
= ( ( ( Xss2 = nil_li6406708984265833308od_a_a )
=> ( ( Ys2 = nil_Pr7291147438282410198od_a_a )
& ( Zs2 = nil_Pr7291147438282410198od_a_a ) ) )
& ( ( Xss2 != nil_li6406708984265833308od_a_a )
=> ? [Xss1: list_l4888837856047780700od_a_a,Xs3: list_P1802437025701573206od_a_a,Xs4: list_P1802437025701573206od_a_a,Xss22: list_l4888837856047780700od_a_a] :
( ( Xss2
= ( append2032168774138627441od_a_a @ Xss1 @ ( cons_l7647087712702168844od_a_a @ ( append4725183380742660459od_a_a @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append4725183380742660459od_a_a @ ( concat5889154033839980561od_a_a @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append4725183380742660459od_a_a @ Xs4 @ ( concat5889154033839980561od_a_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_445_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs2: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs2 ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys2 = nil_nat )
& ( Zs2 = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs3: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_446_concat__eq__append__conv,axiom,
! [Xss2: list_list_f,Ys2: list_f,Zs2: list_f] :
( ( ( concat_f @ Xss2 )
= ( append_f @ Ys2 @ Zs2 ) )
= ( ( ( Xss2 = nil_list_f )
=> ( ( Ys2 = nil_f )
& ( Zs2 = nil_f ) ) )
& ( ( Xss2 != nil_list_f )
=> ? [Xss1: list_list_f,Xs3: list_f,Xs4: list_f,Xss22: list_list_f] :
( ( Xss2
= ( append_list_f @ Xss1 @ ( cons_list_f @ ( append_f @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append_f @ ( concat_f @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append_f @ Xs4 @ ( concat_f @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_447_butlast__snoc,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a] :
( ( butlas456703966728320546od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_448_butlast__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_449_butlast__snoc,axiom,
! [Xs2: list_f,X: f] :
( ( butlast_f @ ( append_f @ Xs2 @ ( cons_f @ X @ nil_f ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_450_lexord__same__pref__iff,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,Zs2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) @ ( append4725183380742660459od_a_a @ Xs2 @ Zs2 ) ) @ ( lexord2630910672304566467od_a_a @ R ) )
= ( ? [X3: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X3 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X3 @ X3 ) @ R ) )
| ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Ys2 @ Zs2 ) @ ( lexord2630910672304566467od_a_a @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_451_lexord__same__pref__iff,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs2 ) ) @ ( lexord_nat @ R ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R ) )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs2 ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_452_lexord__same__pref__iff,axiom,
! [Xs2: list_f,Ys2: list_f,Zs2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( append_f @ Xs2 @ Ys2 ) @ ( append_f @ Xs2 @ Zs2 ) ) @ ( lexord_f @ R ) )
= ( ? [X3: f] :
( ( member_f @ X3 @ ( set_f2 @ Xs2 ) )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ X3 ) @ R ) )
| ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Ys2 @ Zs2 ) @ ( lexord_f @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_453_lexord__same__pref__iff,axiom,
! [Xs2: list_o,Ys2: list_o,Zs2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ Ys2 ) @ ( append_o @ Xs2 @ Zs2 ) ) @ ( lexord_o @ R ) )
= ( ? [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R ) )
| ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs2 ) @ ( lexord_o @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_454_lexord__same__pref__iff,axiom,
! [Xs2: list_a,Ys2: list_a,Zs2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Ys2 ) @ ( append_a @ Xs2 @ Zs2 ) ) @ ( lexord_a @ R ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R ) )
| ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Zs2 ) @ ( lexord_a @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_455_last__snoc,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a] :
( ( last_P6719674514767459439od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) ) )
= X ) ).
% last_snoc
thf(fact_456_last__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_457_last__snoc,axiom,
! [Xs2: list_f,X: f] :
( ( last_f @ ( append_f @ Xs2 @ ( cons_f @ X @ nil_f ) ) )
= X ) ).
% last_snoc
thf(fact_458_rotate1_Osimps_I2_J,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( rotate3742350382119052963od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= ( append4725183380742660459od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) ) ) ).
% rotate1.simps(2)
thf(fact_459_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_460_rotate1_Osimps_I2_J,axiom,
! [X: f,Xs2: list_f] :
( ( rotate1_f @ ( cons_f @ X @ Xs2 ) )
= ( append_f @ Xs2 @ ( cons_f @ X @ nil_f ) ) ) ).
% rotate1.simps(2)
thf(fact_461_lex__append__left__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs2 ) ) @ ( lex_nat @ R ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs2 ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_462_lex__append__left__iff,axiom,
! [R: set_Product_prod_o_o,Xs2: list_o,Ys2: list_o,Zs2: list_o] :
( ! [X2: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ Ys2 ) @ ( append_o @ Xs2 @ Zs2 ) ) @ ( lex_o @ R ) )
= ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs2 ) @ ( lex_o @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_463_lex__append__left__iff,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a,Ys2: list_a,Zs2: list_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Ys2 ) @ ( append_a @ Xs2 @ Zs2 ) ) @ ( lex_a @ R ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Zs2 ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_464_lex__append__leftD,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs2 ) ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs2 ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_465_lex__append__leftD,axiom,
! [R: set_Product_prod_o_o,Xs2: list_o,Ys2: list_o,Zs2: list_o] :
( ! [X2: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ Ys2 ) @ ( append_o @ Xs2 @ Zs2 ) ) @ ( lex_o @ R ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs2 ) @ ( lex_o @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_466_lex__append__leftD,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a,Ys2: list_a,Zs2: list_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Ys2 ) @ ( append_a @ Xs2 @ Zs2 ) ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Zs2 ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_467_SuccD,axiom,
! [K: a,Kl: set_list_a,Kl2: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
=> ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_468_SuccD,axiom,
! [K: produc4044097585999906000od_a_a,Kl: set_li6330610657153406774od_a_a,Kl2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ K @ ( bNF_Gr428374559055706347od_a_a @ Kl @ Kl2 ) )
=> ( member7908416913644183295od_a_a @ ( append4725183380742660459od_a_a @ Kl2 @ ( cons_P64011158483510406od_a_a @ K @ nil_Pr7291147438282410198od_a_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_469_SuccD,axiom,
! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
=> ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_470_SuccD,axiom,
! [K: f,Kl: set_list_f,Kl2: list_f] :
( ( member_f @ K @ ( bNF_Greatest_Succ_f @ Kl @ Kl2 ) )
=> ( member_list_f @ ( append_f @ Kl2 @ ( cons_f @ K @ nil_f ) ) @ Kl ) ) ).
% SuccD
thf(fact_471_concat__eq__Nil__conv,axiom,
! [Xss2: list_l4888837856047780700od_a_a] :
( ( ( concat5889154033839980561od_a_a @ Xss2 )
= nil_Pr7291147438282410198od_a_a )
= ( ! [X3: list_P1802437025701573206od_a_a] :
( ( member7908416913644183295od_a_a @ X3 @ ( set_li8573657874255358827od_a_a @ Xss2 ) )
=> ( X3 = nil_Pr7291147438282410198od_a_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_472_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_473_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_f] :
( ( ( concat_f @ Xss2 )
= nil_f )
= ( ! [X3: list_f] :
( ( member_list_f @ X3 @ ( set_list_f2 @ Xss2 ) )
=> ( X3 = nil_f ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_474_Nil__eq__concat__conv,axiom,
! [Xss2: list_l4888837856047780700od_a_a] :
( ( nil_Pr7291147438282410198od_a_a
= ( concat5889154033839980561od_a_a @ Xss2 ) )
= ( ! [X3: list_P1802437025701573206od_a_a] :
( ( member7908416913644183295od_a_a @ X3 @ ( set_li8573657874255358827od_a_a @ Xss2 ) )
=> ( X3 = nil_Pr7291147438282410198od_a_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_475_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_476_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_f] :
( ( nil_f
= ( concat_f @ Xss2 ) )
= ( ! [X3: list_f] :
( ( member_list_f @ X3 @ ( set_list_f2 @ Xss2 ) )
=> ( X3 = nil_f ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_477_rotate1__is__Nil__conv,axiom,
! [Xs2: list_P1802437025701573206od_a_a] :
( ( ( rotate3742350382119052963od_a_a @ Xs2 )
= nil_Pr7291147438282410198od_a_a )
= ( Xs2 = nil_Pr7291147438282410198od_a_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_478_rotate1__is__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( ( rotate1_nat @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_479_rotate1__is__Nil__conv,axiom,
! [Xs2: list_f] :
( ( ( rotate1_f @ Xs2 )
= nil_f )
= ( Xs2 = nil_f ) ) ).
% rotate1_is_Nil_conv
thf(fact_480_set__rotate1,axiom,
! [Xs2: list_P1802437025701573206od_a_a] :
( ( set_Pr7008919305446870117od_a_a @ ( rotate3742350382119052963od_a_a @ Xs2 ) )
= ( set_Pr7008919305446870117od_a_a @ Xs2 ) ) ).
% set_rotate1
thf(fact_481_set__rotate1,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_482_set__rotate1,axiom,
! [Xs2: list_f] :
( ( set_f2 @ ( rotate1_f @ Xs2 ) )
= ( set_f2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_483_set__rotate1,axiom,
! [Xs2: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs2 ) )
= ( set_a2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_484_in__set__insert,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ( ( insert5320946937069198828od_a_a @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_485_in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_486_in__set__insert,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ( ( insert_f @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_487_in__set__insert,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_488_last__appendL,axiom,
! [Ys2: list_P1802437025701573206od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( Ys2 = nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) )
= ( last_P6719674514767459439od_a_a @ Xs2 ) ) ) ).
% last_appendL
thf(fact_489_last__appendL,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_appendL
thf(fact_490_last__appendL,axiom,
! [Ys2: list_f,Xs2: list_f] :
( ( Ys2 = nil_f )
=> ( ( last_f @ ( append_f @ Xs2 @ Ys2 ) )
= ( last_f @ Xs2 ) ) ) ).
% last_appendL
thf(fact_491_last__appendR,axiom,
! [Ys2: list_P1802437025701573206od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( Ys2 != nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) )
= ( last_P6719674514767459439od_a_a @ Ys2 ) ) ) ).
% last_appendR
thf(fact_492_last__appendR,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_493_last__appendR,axiom,
! [Ys2: list_f,Xs2: list_f] :
( ( Ys2 != nil_f )
=> ( ( last_f @ ( append_f @ Xs2 @ Ys2 ) )
= ( last_f @ Ys2 ) ) ) ).
% last_appendR
thf(fact_494_not__in__set__insert,axiom,
! [X: a,Xs2: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X @ Xs2 )
= ( cons_a @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_495_not__in__set__insert,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ( ( insert5320946937069198828od_a_a @ X @ Xs2 )
= ( cons_P64011158483510406od_a_a @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_496_not__in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= ( cons_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_497_not__in__set__insert,axiom,
! [X: f,Xs2: list_f] :
( ~ ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ( ( insert_f @ X @ Xs2 )
= ( cons_f @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_498_concat__append,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( concat_nat @ Xs2 ) @ ( concat_nat @ Ys2 ) ) ) ).
% concat_append
thf(fact_499_append__butlast__last__id,axiom,
! [Xs2: list_P1802437025701573206od_a_a] :
( ( Xs2 != nil_Pr7291147438282410198od_a_a )
=> ( ( append4725183380742660459od_a_a @ ( butlas456703966728320546od_a_a @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ ( last_P6719674514767459439od_a_a @ Xs2 ) @ nil_Pr7291147438282410198od_a_a ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_500_append__butlast__last__id,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs2 ) @ ( cons_nat @ ( last_nat @ Xs2 ) @ nil_nat ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_501_append__butlast__last__id,axiom,
! [Xs2: list_f] :
( ( Xs2 != nil_f )
=> ( ( append_f @ ( butlast_f @ Xs2 ) @ ( cons_f @ ( last_f @ Xs2 ) @ nil_f ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_502_in__set__butlast__appendI,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ ( butlas456703966728320546od_a_a @ Xs2 ) ) )
| ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ ( butlas456703966728320546od_a_a @ Ys2 ) ) ) )
=> ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ ( butlas456703966728320546od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_503_in__set__butlast__appendI,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys2 ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_504_in__set__butlast__appendI,axiom,
! [X: f,Xs2: list_f,Ys2: list_f] :
( ( ( member_f @ X @ ( set_f2 @ ( butlast_f @ Xs2 ) ) )
| ( member_f @ X @ ( set_f2 @ ( butlast_f @ Ys2 ) ) ) )
=> ( member_f @ X @ ( set_f2 @ ( butlast_f @ ( append_f @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_505_in__set__butlast__appendI,axiom,
! [X: a,Xs2: list_a,Ys2: list_a] :
( ( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs2 ) ) )
| ( member_a @ X @ ( set_a2 @ ( butlast_a @ Ys2 ) ) ) )
=> ( member_a @ X @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_506_in__set__butlastD,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ ( butlas456703966728320546od_a_a @ Xs2 ) ) )
=> ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_507_in__set__butlastD,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_508_in__set__butlastD,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ ( butlast_f @ Xs2 ) ) )
=> ( member_f @ X @ ( set_f2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_509_in__set__butlastD,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs2 ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_510_last__in__set,axiom,
! [As2: list_P1802437025701573206od_a_a] :
( ( As2 != nil_Pr7291147438282410198od_a_a )
=> ( member3071122053849602553od_a_a @ ( last_P6719674514767459439od_a_a @ As2 ) @ ( set_Pr7008919305446870117od_a_a @ As2 ) ) ) ).
% last_in_set
thf(fact_511_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_512_last__in__set,axiom,
! [As2: list_f] :
( ( As2 != nil_f )
=> ( member_f @ ( last_f @ As2 ) @ ( set_f2 @ As2 ) ) ) ).
% last_in_set
thf(fact_513_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_514_set__ConsD,axiom,
! [Y: a,X: a,Xs2: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_515_set__ConsD,axiom,
! [Y: produc4044097585999906000od_a_a,X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ Y @ ( set_Pr7008919305446870117od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member3071122053849602553od_a_a @ Y @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_516_set__ConsD,axiom,
! [Y: nat,X: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_517_set__ConsD,axiom,
! [Y: f,X: f,Xs2: list_f] :
( ( member_f @ Y @ ( set_f2 @ ( cons_f @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_f @ Y @ ( set_f2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_518_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z2: list_a] :
( A
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_519_list_Oset__cases,axiom,
! [E: produc4044097585999906000od_a_a,A: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ E @ ( set_Pr7008919305446870117od_a_a @ A ) )
=> ( ! [Z2: list_P1802437025701573206od_a_a] :
( A
!= ( cons_P64011158483510406od_a_a @ E @ Z2 ) )
=> ~ ! [Z1: produc4044097585999906000od_a_a,Z2: list_P1802437025701573206od_a_a] :
( ( A
= ( cons_P64011158483510406od_a_a @ Z1 @ Z2 ) )
=> ~ ( member3071122053849602553od_a_a @ E @ ( set_Pr7008919305446870117od_a_a @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_520_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_521_list_Oset__cases,axiom,
! [E: f,A: list_f] :
( ( member_f @ E @ ( set_f2 @ A ) )
=> ( ! [Z2: list_f] :
( A
!= ( cons_f @ E @ Z2 ) )
=> ~ ! [Z1: f,Z2: list_f] :
( ( A
= ( cons_f @ Z1 @ Z2 ) )
=> ~ ( member_f @ E @ ( set_f2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_522_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_523_list_Oset__intros_I1_J,axiom,
! [X21: produc4044097585999906000od_a_a,X22: list_P1802437025701573206od_a_a] : ( member3071122053849602553od_a_a @ X21 @ ( set_Pr7008919305446870117od_a_a @ ( cons_P64011158483510406od_a_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_524_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_525_list_Oset__intros_I1_J,axiom,
! [X21: f,X22: list_f] : ( member_f @ X21 @ ( set_f2 @ ( cons_f @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_526_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_527_list_Oset__intros_I2_J,axiom,
! [Y: produc4044097585999906000od_a_a,X22: list_P1802437025701573206od_a_a,X21: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y @ ( set_Pr7008919305446870117od_a_a @ X22 ) )
=> ( member3071122053849602553od_a_a @ Y @ ( set_Pr7008919305446870117od_a_a @ ( cons_P64011158483510406od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_528_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_529_list_Oset__intros_I2_J,axiom,
! [Y: f,X22: list_f,X21: f] :
( ( member_f @ Y @ ( set_f2 @ X22 ) )
=> ( member_f @ Y @ ( set_f2 @ ( cons_f @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_530_butlast_Osimps_I1_J,axiom,
( ( butlas456703966728320546od_a_a @ nil_Pr7291147438282410198od_a_a )
= nil_Pr7291147438282410198od_a_a ) ).
% butlast.simps(1)
thf(fact_531_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_532_butlast_Osimps_I1_J,axiom,
( ( butlast_f @ nil_f )
= nil_f ) ).
% butlast.simps(1)
thf(fact_533_rotate1_Osimps_I1_J,axiom,
( ( rotate3742350382119052963od_a_a @ nil_Pr7291147438282410198od_a_a )
= nil_Pr7291147438282410198od_a_a ) ).
% rotate1.simps(1)
thf(fact_534_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_535_rotate1_Osimps_I1_J,axiom,
( ( rotate1_f @ nil_f )
= nil_f ) ).
% rotate1.simps(1)
thf(fact_536_snoc__eq__iff__butlast,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( ( append4725183380742660459od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) )
= Ys2 )
= ( ( Ys2 != nil_Pr7291147438282410198od_a_a )
& ( ( butlas456703966728320546od_a_a @ Ys2 )
= Xs2 )
& ( ( last_P6719674514767459439od_a_a @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_537_snoc__eq__iff__butlast,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= Ys2 )
= ( ( Ys2 != nil_nat )
& ( ( butlast_nat @ Ys2 )
= Xs2 )
& ( ( last_nat @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_538_snoc__eq__iff__butlast,axiom,
! [Xs2: list_f,X: f,Ys2: list_f] :
( ( ( append_f @ Xs2 @ ( cons_f @ X @ nil_f ) )
= Ys2 )
= ( ( Ys2 != nil_f )
& ( ( butlast_f @ Ys2 )
= Xs2 )
& ( ( last_f @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_539_list__ex1__iff,axiom,
( list_e867735100643366937od_a_a
= ( ^ [P4: produc4044097585999906000od_a_a > $o,Xs3: list_P1802437025701573206od_a_a] :
? [X3: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X3 @ ( set_Pr7008919305446870117od_a_a @ Xs3 ) )
& ( P4 @ X3 )
& ! [Y4: produc4044097585999906000od_a_a] :
( ( ( member3071122053849602553od_a_a @ Y4 @ ( set_Pr7008919305446870117od_a_a @ Xs3 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_540_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P4: nat > $o,Xs3: list_nat] :
? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs3 ) )
& ( P4 @ X3 )
& ! [Y4: nat] :
( ( ( member_nat @ Y4 @ ( set_nat2 @ Xs3 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_541_list__ex1__iff,axiom,
( list_ex1_f
= ( ^ [P4: f > $o,Xs3: list_f] :
? [X3: f] :
( ( member_f @ X3 @ ( set_f2 @ Xs3 ) )
& ( P4 @ X3 )
& ! [Y4: f] :
( ( ( member_f @ Y4 @ ( set_f2 @ Xs3 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_542_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P4: a > $o,Xs3: list_a] :
? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs3 ) )
& ( P4 @ X3 )
& ! [Y4: a] :
( ( ( member_a @ Y4 @ ( set_a2 @ Xs3 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_543_split__list__first__prop__iff,axiom,
! [Xs2: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_544_split__list__first__prop__iff,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ( ? [X3: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X3 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_P1802437025701573206od_a_a,X3: produc4044097585999906000od_a_a] :
( ? [Zs3: list_P1802437025701573206od_a_a] :
( Xs2
= ( append4725183380742660459od_a_a @ Ys3 @ ( cons_P64011158483510406od_a_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y4 @ ( set_Pr7008919305446870117od_a_a @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_545_split__list__first__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_546_split__list__first__prop__iff,axiom,
! [Xs2: list_f,P: f > $o] :
( ( ? [X3: f] :
( ( member_f @ X3 @ ( set_f2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_f,X3: f] :
( ? [Zs3: list_f] :
( Xs2
= ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: f] :
( ( member_f @ Y4 @ ( set_f2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_547_split__list__last__prop__iff,axiom,
! [Xs2: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_548_split__list__last__prop__iff,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ( ? [X3: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X3 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_P1802437025701573206od_a_a,X3: produc4044097585999906000od_a_a,Zs3: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ys3 @ ( cons_P64011158483510406od_a_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y4 @ ( set_Pr7008919305446870117od_a_a @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_549_split__list__last__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_550_split__list__last__prop__iff,axiom,
! [Xs2: list_f,P: f > $o] :
( ( ? [X3: f] :
( ( member_f @ X3 @ ( set_f2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_f,X3: f,Zs3: list_f] :
( ( Xs2
= ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: f] :
( ( member_f @ Y4 @ ( set_f2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_551_in__set__conv__decomp__first,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_552_in__set__conv__decomp__first,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
= ( ? [Ys3: list_P1802437025701573206od_a_a,Zs3: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ys3 @ ( cons_P64011158483510406od_a_a @ X @ Zs3 ) ) )
& ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_553_in__set__conv__decomp__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_554_in__set__conv__decomp__first,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ Xs2 ) )
= ( ? [Ys3: list_f,Zs3: list_f] :
( ( Xs2
= ( append_f @ Ys3 @ ( cons_f @ X @ Zs3 ) ) )
& ~ ( member_f @ X @ ( set_f2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_555_in__set__conv__decomp__last,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_556_in__set__conv__decomp__last,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
= ( ? [Ys3: list_P1802437025701573206od_a_a,Zs3: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ys3 @ ( cons_P64011158483510406od_a_a @ X @ Zs3 ) ) )
& ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_557_in__set__conv__decomp__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_558_in__set__conv__decomp__last,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ Xs2 ) )
= ( ? [Ys3: list_f,Zs3: list_f] :
( ( Xs2
= ( append_f @ Ys3 @ ( cons_f @ X @ Zs3 ) ) )
& ~ ( member_f @ X @ ( set_f2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_559_split__list__first__propE,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_a,X2: a] :
( ? [Zs: list_a] :
( Xs2
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_560_split__list__first__propE,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ? [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_P1802437025701573206od_a_a,X2: produc4044097585999906000od_a_a] :
( ? [Zs: list_P1802437025701573206od_a_a] :
( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Xa2 @ ( set_Pr7008919305446870117od_a_a @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_561_split__list__first__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_562_split__list__first__propE,axiom,
! [Xs2: list_f,P: f > $o] :
( ? [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_f,X2: f] :
( ? [Zs: list_f] :
( Xs2
= ( append_f @ Ys @ ( cons_f @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: f] :
( ( member_f @ Xa2 @ ( set_f2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_563_split__list__last__propE,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_a,X2: a,Zs: list_a] :
( ( Xs2
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_564_split__list__last__propE,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ? [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_P1802437025701573206od_a_a,X2: produc4044097585999906000od_a_a,Zs: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Xa2 @ ( set_Pr7008919305446870117od_a_a @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_565_split__list__last__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_nat,X2: nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_566_split__list__last__propE,axiom,
! [Xs2: list_f,P: f > $o] :
( ? [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_f,X2: f,Zs: list_f] :
( ( Xs2
= ( append_f @ Ys @ ( cons_f @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: f] :
( ( member_f @ Xa2 @ ( set_f2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_567_split__list__first__prop,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_a,X2: a] :
( ? [Zs: list_a] :
( Xs2
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_568_split__list__first__prop,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ? [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_P1802437025701573206od_a_a,X2: produc4044097585999906000od_a_a] :
( ? [Zs: list_P1802437025701573206od_a_a] :
( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Xa2 @ ( set_Pr7008919305446870117od_a_a @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_569_split__list__first__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_570_split__list__first__prop,axiom,
! [Xs2: list_f,P: f > $o] :
( ? [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_f,X2: f] :
( ? [Zs: list_f] :
( Xs2
= ( append_f @ Ys @ ( cons_f @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: f] :
( ( member_f @ Xa2 @ ( set_f2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_571_split__list__last__prop,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_a,X2: a,Zs: list_a] :
( ( Xs2
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_572_split__list__last__prop,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ? [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_P1802437025701573206od_a_a,X2: produc4044097585999906000od_a_a,Zs: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Xa2 @ ( set_Pr7008919305446870117od_a_a @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_573_split__list__last__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_nat,X2: nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_574_split__list__last__prop,axiom,
! [Xs2: list_f,P: f > $o] :
( ? [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_f,X2: f,Zs: list_f] :
( ( Xs2
= ( append_f @ Ys @ ( cons_f @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa2: f] :
( ( member_f @ Xa2 @ ( set_f2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_575_in__set__conv__decomp,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_576_in__set__conv__decomp,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
= ( ? [Ys3: list_P1802437025701573206od_a_a,Zs3: list_P1802437025701573206od_a_a] :
( Xs2
= ( append4725183380742660459od_a_a @ Ys3 @ ( cons_P64011158483510406od_a_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_577_in__set__conv__decomp,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_578_in__set__conv__decomp,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ Xs2 ) )
= ( ? [Ys3: list_f,Zs3: list_f] :
( Xs2
= ( append_f @ Ys3 @ ( cons_f @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_579_append__Cons__eq__iff,axiom,
! [X: a,Xs2: list_a,Ys2: list_a,Xs5: list_a,Ys5: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys2 ) )
=> ( ( ( append_a @ Xs2 @ ( cons_a @ X @ Ys2 ) )
= ( append_a @ Xs5 @ ( cons_a @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_580_append__Cons__eq__iff,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,Xs5: list_P1802437025701573206od_a_a,Ys5: list_P1802437025701573206od_a_a] :
( ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ( ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Ys2 ) )
=> ( ( ( append4725183380742660459od_a_a @ Xs2 @ ( cons_P64011158483510406od_a_a @ X @ Ys2 ) )
= ( append4725183380742660459od_a_a @ Xs5 @ ( cons_P64011158483510406od_a_a @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_581_append__Cons__eq__iff,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,Xs5: list_nat,Ys5: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys2 ) )
= ( append_nat @ Xs5 @ ( cons_nat @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_582_append__Cons__eq__iff,axiom,
! [X: f,Xs2: list_f,Ys2: list_f,Xs5: list_f,Ys5: list_f] :
( ~ ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ( ~ ( member_f @ X @ ( set_f2 @ Ys2 ) )
=> ( ( ( append_f @ Xs2 @ ( cons_f @ X @ Ys2 ) )
= ( append_f @ Xs5 @ ( cons_f @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_583_split__list__propE,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_a,X2: a] :
( ? [Zs: list_a] :
( Xs2
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_584_split__list__propE,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ? [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_P1802437025701573206od_a_a,X2: produc4044097585999906000od_a_a] :
( ? [Zs: list_P1802437025701573206od_a_a] :
( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X2 @ Zs ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_585_split__list__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_586_split__list__propE,axiom,
! [Xs2: list_f,P: f > $o] :
( ? [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_f,X2: f] :
( ? [Zs: list_f] :
( Xs2
= ( append_f @ Ys @ ( cons_f @ X2 @ Zs ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_587_split__list__first,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ? [Ys: list_a,Zs: list_a] :
( ( Xs2
= ( append_a @ Ys @ ( cons_a @ X @ Zs ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_588_split__list__first,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ? [Ys: list_P1802437025701573206od_a_a,Zs: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X @ Zs ) ) )
& ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Ys ) ) ) ) ).
% split_list_first
thf(fact_589_split__list__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_590_split__list__first,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ? [Ys: list_f,Zs: list_f] :
( ( Xs2
= ( append_f @ Ys @ ( cons_f @ X @ Zs ) ) )
& ~ ( member_f @ X @ ( set_f2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_591_split__list__prop,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_a,X2: a] :
( ? [Zs: list_a] :
( Xs2
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_592_split__list__prop,axiom,
! [Xs2: list_P1802437025701573206od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ? [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_P1802437025701573206od_a_a,X2: produc4044097585999906000od_a_a] :
( ? [Zs: list_P1802437025701573206od_a_a] :
( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X2 @ Zs ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_593_split__list__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_594_split__list__prop,axiom,
! [Xs2: list_f,P: f > $o] :
( ? [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_f,X2: f] :
( ? [Zs: list_f] :
( Xs2
= ( append_f @ Ys @ ( cons_f @ X2 @ Zs ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_595_split__list__last,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ? [Ys: list_a,Zs: list_a] :
( ( Xs2
= ( append_a @ Ys @ ( cons_a @ X @ Zs ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_596_split__list__last,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ? [Ys: list_P1802437025701573206od_a_a,Zs: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X @ Zs ) ) )
& ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Zs ) ) ) ) ).
% split_list_last
thf(fact_597_split__list__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_598_split__list__last,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ? [Ys: list_f,Zs: list_f] :
( ( Xs2
= ( append_f @ Ys @ ( cons_f @ X @ Zs ) ) )
& ~ ( member_f @ X @ ( set_f2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_599_split__list,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ? [Ys: list_a,Zs: list_a] :
( Xs2
= ( append_a @ Ys @ ( cons_a @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_600_split__list,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ? [Ys: list_P1802437025701573206od_a_a,Zs: list_P1802437025701573206od_a_a] :
( Xs2
= ( append4725183380742660459od_a_a @ Ys @ ( cons_P64011158483510406od_a_a @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_601_split__list,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_602_split__list,axiom,
! [X: f,Xs2: list_f] :
( ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ? [Ys: list_f,Zs: list_f] :
( Xs2
= ( append_f @ Ys @ ( cons_f @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_603_last__ConsR,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a] :
( ( Xs2 != nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= ( last_P6719674514767459439od_a_a @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_604_last__ConsR,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_605_last__ConsR,axiom,
! [Xs2: list_f,X: f] :
( ( Xs2 != nil_f )
=> ( ( last_f @ ( cons_f @ X @ Xs2 ) )
= ( last_f @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_606_last__ConsL,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a] :
( ( Xs2 = nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_607_last__ConsL,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_608_last__ConsL,axiom,
! [Xs2: list_f,X: f] :
( ( Xs2 = nil_f )
=> ( ( last_f @ ( cons_f @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_609_last_Osimps,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a] :
( ( ( Xs2 = nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= ( last_P6719674514767459439od_a_a @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_610_last_Osimps,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_611_last_Osimps,axiom,
! [Xs2: list_f,X: f] :
( ( ( Xs2 = nil_f )
=> ( ( last_f @ ( cons_f @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_f )
=> ( ( last_f @ ( cons_f @ X @ Xs2 ) )
= ( last_f @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_612_longest__common__suffix,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
? [Ss: list_P1802437025701573206od_a_a,Xs6: list_P1802437025701573206od_a_a,Ys6: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Xs6 @ Ss ) )
& ( Ys2
= ( append4725183380742660459od_a_a @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_Pr7291147438282410198od_a_a )
| ( Ys6 = nil_Pr7291147438282410198od_a_a )
| ( ( last_P6719674514767459439od_a_a @ Xs6 )
!= ( last_P6719674514767459439od_a_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_613_longest__common__suffix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
? [Ss: list_nat,Xs6: list_nat,Ys6: list_nat] :
( ( Xs2
= ( append_nat @ Xs6 @ Ss ) )
& ( Ys2
= ( append_nat @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( last_nat @ Xs6 )
!= ( last_nat @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_614_longest__common__suffix,axiom,
! [Xs2: list_f,Ys2: list_f] :
? [Ss: list_f,Xs6: list_f,Ys6: list_f] :
( ( Xs2
= ( append_f @ Xs6 @ Ss ) )
& ( Ys2
= ( append_f @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_f )
| ( Ys6 = nil_f )
| ( ( last_f @ Xs6 )
!= ( last_f @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_615_last__append,axiom,
! [Ys2: list_P1802437025701573206od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( ( Ys2 = nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) )
= ( last_P6719674514767459439od_a_a @ Xs2 ) ) )
& ( ( Ys2 != nil_Pr7291147438282410198od_a_a )
=> ( ( last_P6719674514767459439od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) )
= ( last_P6719674514767459439od_a_a @ Ys2 ) ) ) ) ).
% last_append
thf(fact_616_last__append,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Xs2 ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_617_last__append,axiom,
! [Ys2: list_f,Xs2: list_f] :
( ( ( Ys2 = nil_f )
=> ( ( last_f @ ( append_f @ Xs2 @ Ys2 ) )
= ( last_f @ Xs2 ) ) )
& ( ( Ys2 != nil_f )
=> ( ( last_f @ ( append_f @ Xs2 @ Ys2 ) )
= ( last_f @ Ys2 ) ) ) ) ).
% last_append
thf(fact_618_concat_Osimps_I1_J,axiom,
( ( concat5889154033839980561od_a_a @ nil_li6406708984265833308od_a_a )
= nil_Pr7291147438282410198od_a_a ) ).
% concat.simps(1)
thf(fact_619_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_620_concat_Osimps_I1_J,axiom,
( ( concat_f @ nil_list_f )
= nil_f ) ).
% concat.simps(1)
thf(fact_621_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs3: list_a] : ( if_list_a @ ( member_a @ X3 @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_622_List_Oinsert__def,axiom,
( insert5320946937069198828od_a_a
= ( ^ [X3: produc4044097585999906000od_a_a,Xs3: list_P1802437025701573206od_a_a] : ( if_lis3736317622110187548od_a_a @ ( member3071122053849602553od_a_a @ X3 @ ( set_Pr7008919305446870117od_a_a @ Xs3 ) ) @ Xs3 @ ( cons_P64011158483510406od_a_a @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_623_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X3 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_624_List_Oinsert__def,axiom,
( insert_f
= ( ^ [X3: f,Xs3: list_f] : ( if_list_f @ ( member_f @ X3 @ ( set_f2 @ Xs3 ) ) @ Xs3 @ ( cons_f @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_625_Cons__in__subseqsD,axiom,
! [Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( member7908416913644183295od_a_a @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) @ ( set_li8573657874255358827od_a_a @ ( subseq1910477207492193829od_a_a @ Xs2 ) ) )
=> ( member7908416913644183295od_a_a @ Ys2 @ ( set_li8573657874255358827od_a_a @ ( subseq1910477207492193829od_a_a @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_626_Cons__in__subseqsD,axiom,
! [Y: nat,Ys2: list_nat,Xs2: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys2 ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_627_Cons__in__subseqsD,axiom,
! [Y: f,Ys2: list_f,Xs2: list_f] :
( ( member_list_f @ ( cons_f @ Y @ Ys2 ) @ ( set_list_f2 @ ( subseqs_f @ Xs2 ) ) )
=> ( member_list_f @ Ys2 @ ( set_list_f2 @ ( subseqs_f @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_628_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_nat @ X @ ( concat_nat @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_629_butlast_Osimps_I2_J,axiom,
! [Xs2: list_P1802437025701573206od_a_a,X: produc4044097585999906000od_a_a] :
( ( ( Xs2 = nil_Pr7291147438282410198od_a_a )
=> ( ( butlas456703966728320546od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= nil_Pr7291147438282410198od_a_a ) )
& ( ( Xs2 != nil_Pr7291147438282410198od_a_a )
=> ( ( butlas456703966728320546od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) )
= ( cons_P64011158483510406od_a_a @ X @ ( butlas456703966728320546od_a_a @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_630_butlast_Osimps_I2_J,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= nil_nat ) )
& ( ( Xs2 != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_631_butlast_Osimps_I2_J,axiom,
! [Xs2: list_f,X: f] :
( ( ( Xs2 = nil_f )
=> ( ( butlast_f @ ( cons_f @ X @ Xs2 ) )
= nil_f ) )
& ( ( Xs2 != nil_f )
=> ( ( butlast_f @ ( cons_f @ X @ Xs2 ) )
= ( cons_f @ X @ ( butlast_f @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_632_butlast__append,axiom,
! [Ys2: list_P1802437025701573206od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( ( Ys2 = nil_Pr7291147438282410198od_a_a )
=> ( ( butlas456703966728320546od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) )
= ( butlas456703966728320546od_a_a @ Xs2 ) ) )
& ( ( Ys2 != nil_Pr7291147438282410198od_a_a )
=> ( ( butlas456703966728320546od_a_a @ ( append4725183380742660459od_a_a @ Xs2 @ Ys2 ) )
= ( append4725183380742660459od_a_a @ Xs2 @ ( butlas456703966728320546od_a_a @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_633_butlast__append,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( butlast_nat @ Xs2 ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ Xs2 @ ( butlast_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_634_butlast__append,axiom,
! [Ys2: list_f,Xs2: list_f] :
( ( ( Ys2 = nil_f )
=> ( ( butlast_f @ ( append_f @ Xs2 @ Ys2 ) )
= ( butlast_f @ Xs2 ) ) )
& ( ( Ys2 != nil_f )
=> ( ( butlast_f @ ( append_f @ Xs2 @ Ys2 ) )
= ( append_f @ Xs2 @ ( butlast_f @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_635_partition__P,axiom,
! [P: produc4044097585999906000od_a_a > $o,Xs2: list_P1802437025701573206od_a_a,Yes: list_P1802437025701573206od_a_a,No: list_P1802437025701573206od_a_a] :
( ( ( partit8336039082511689039od_a_a @ P @ Xs2 )
= ( produc4694146025100153495od_a_a @ Yes @ No ) )
=> ( ! [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X4 @ ( set_Pr7008919305446870117od_a_a @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_636_partition__P,axiom,
! [P: nat > $o,Xs2: list_nat,Yes: list_nat,No: list_nat] :
( ( ( partition_nat @ P @ Xs2 )
= ( produc2694037385005941721st_nat @ Yes @ No ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_637_partition__P,axiom,
! [P: f > $o,Xs2: list_f,Yes: list_f,No: list_f] :
( ( ( partition_f @ P @ Xs2 )
= ( produc8144748701140958423list_f @ Yes @ No ) )
=> ( ! [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_638_partition__P,axiom,
! [P: a > $o,Xs2: list_a,Yes: list_a,No: list_a] :
( ( ( partition_a @ P @ Xs2 )
= ( produc6837034575241423639list_a @ Yes @ No ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_639_maps__simps_I2_J,axiom,
! [F: produc4044097585999906000od_a_a > list_P1802437025701573206od_a_a] :
( ( maps_P8107709928678292289od_a_a @ F @ nil_Pr7291147438282410198od_a_a )
= nil_Pr7291147438282410198od_a_a ) ).
% maps_simps(2)
thf(fact_640_maps__simps_I2_J,axiom,
! [F: produc4044097585999906000od_a_a > list_nat] :
( ( maps_P8591894958123311391_a_nat @ F @ nil_Pr7291147438282410198od_a_a )
= nil_nat ) ).
% maps_simps(2)
thf(fact_641_maps__simps_I2_J,axiom,
! [F: produc4044097585999906000od_a_a > list_f] :
( ( maps_P2924313656325622644_a_a_f @ F @ nil_Pr7291147438282410198od_a_a )
= nil_f ) ).
% maps_simps(2)
thf(fact_642_maps__simps_I2_J,axiom,
! [F: nat > list_P1802437025701573206od_a_a] :
( ( maps_n1770596741291003005od_a_a @ F @ nil_nat )
= nil_Pr7291147438282410198od_a_a ) ).
% maps_simps(2)
thf(fact_643_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_644_maps__simps_I2_J,axiom,
! [F: nat > list_f] :
( ( maps_nat_f @ F @ nil_nat )
= nil_f ) ).
% maps_simps(2)
thf(fact_645_maps__simps_I2_J,axiom,
! [F: f > list_P1802437025701573206od_a_a] :
( ( maps_f4171400119902423502od_a_a @ F @ nil_f )
= nil_Pr7291147438282410198od_a_a ) ).
% maps_simps(2)
thf(fact_646_maps__simps_I2_J,axiom,
! [F: f > list_nat] :
( ( maps_f_nat @ F @ nil_f )
= nil_nat ) ).
% maps_simps(2)
thf(fact_647_maps__simps_I2_J,axiom,
! [F: f > list_f] :
( ( maps_f_f @ F @ nil_f )
= nil_f ) ).
% maps_simps(2)
thf(fact_648_Nil__notin__lex,axiom,
! [Ys2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ nil_Pr7291147438282410198od_a_a @ Ys2 ) @ ( lex_Pr4501241201491742522od_a_a @ R ) ) ).
% Nil_notin_lex
thf(fact_649_Nil__notin__lex,axiom,
! [Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) @ ( lex_nat @ R ) ) ).
% Nil_notin_lex
thf(fact_650_Nil__notin__lex,axiom,
! [Ys2: list_f,R: set_Product_prod_f_f] :
~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ nil_f @ Ys2 ) @ ( lex_f @ R ) ) ).
% Nil_notin_lex
thf(fact_651_Nil2__notin__lex,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
~ ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ nil_Pr7291147438282410198od_a_a ) @ ( lex_Pr4501241201491742522od_a_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_652_Nil2__notin__lex,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( lex_nat @ R ) ) ).
% Nil2_notin_lex
thf(fact_653_Nil2__notin__lex,axiom,
! [Xs2: list_f,R: set_Product_prod_f_f] :
~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ nil_f ) @ ( lex_f @ R ) ) ).
% Nil2_notin_lex
thf(fact_654_lex__append__leftI,axiom,
! [Ys2: list_nat,Zs2: list_nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs2 ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs2 ) ) @ ( lex_nat @ R ) ) ) ).
% lex_append_leftI
thf(fact_655_lexord__partial__trans,axiom,
! [Xs2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a,Ys2: list_P1802437025701573206od_a_a,Zs2: list_P1802437025701573206od_a_a] :
( ! [X2: produc4044097585999906000od_a_a,Y3: produc4044097585999906000od_a_a,Z3: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X2 @ Y3 ) @ R )
=> ( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ Y3 @ Z3 ) @ R )
=> ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X2 @ Z3 ) @ R ) ) ) )
=> ( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Ys2 ) @ ( lexord2630910672304566467od_a_a @ R ) )
=> ( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Ys2 @ Zs2 ) @ ( lexord2630910672304566467od_a_a @ R ) )
=> ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Zs2 ) @ ( lexord2630910672304566467od_a_a @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_656_lexord__partial__trans,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat,Ys2: list_nat,Zs2: list_nat] :
( ! [X2: nat,Y3: nat,Z3: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z3 ) @ R ) ) ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lexord_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs2 ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs2 ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_657_lexord__partial__trans,axiom,
! [Xs2: list_f,R: set_Product_prod_f_f,Ys2: list_f,Zs2: list_f] :
( ! [X2: f,Y3: f,Z3: f] :
( ( member_f @ X2 @ ( set_f2 @ Xs2 ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ R )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Z3 ) @ R ) ) ) )
=> ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( lexord_f @ R ) )
=> ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Ys2 @ Zs2 ) @ ( lexord_f @ R ) )
=> ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Zs2 ) @ ( lexord_f @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_658_lexord__partial__trans,axiom,
! [Xs2: list_o,R: set_Product_prod_o_o,Ys2: list_o,Zs2: list_o] :
( ! [X2: $o,Y3: $o,Z3: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z3 ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Z3 ) @ R ) ) ) )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( lexord_o @ R ) )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs2 ) @ ( lexord_o @ R ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Zs2 ) @ ( lexord_o @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_659_lexord__partial__trans,axiom,
! [Xs2: list_a,R: set_Product_prod_a_a,Ys2: list_a,Zs2: list_a] :
( ! [X2: a,Y3: a,Z3: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z3 ) @ R ) ) ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( lexord_a @ R ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Zs2 ) @ ( lexord_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Zs2 ) @ ( lexord_a @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_660_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs2: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs2 ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs: list_nat,Xs6: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs @ Xs6 ) @ Xss23 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs ) )
& ( Zs2
= ( append_nat @ Xs6 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_661_SuccI,axiom,
! [Kl2: list_a,K: a,Kl: set_list_a] :
( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_662_SuccI,axiom,
! [Kl2: list_P1802437025701573206od_a_a,K: produc4044097585999906000od_a_a,Kl: set_li6330610657153406774od_a_a] :
( ( member7908416913644183295od_a_a @ ( append4725183380742660459od_a_a @ Kl2 @ ( cons_P64011158483510406od_a_a @ K @ nil_Pr7291147438282410198od_a_a ) ) @ Kl )
=> ( member3071122053849602553od_a_a @ K @ ( bNF_Gr428374559055706347od_a_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_663_SuccI,axiom,
! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_664_SuccI,axiom,
! [Kl2: list_f,K: f,Kl: set_list_f] :
( ( member_list_f @ ( append_f @ Kl2 @ ( cons_f @ K @ nil_f ) ) @ Kl )
=> ( member_f @ K @ ( bNF_Greatest_Succ_f @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_665_can__select__set__list__ex1,axiom,
! [P: produc4044097585999906000od_a_a > $o,A4: list_P1802437025701573206od_a_a] :
( ( can_se4212505445558963533od_a_a @ P @ ( set_Pr7008919305446870117od_a_a @ A4 ) )
= ( list_e867735100643366937od_a_a @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_666_can__select__set__list__ex1,axiom,
! [P: nat > $o,A4: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A4 ) )
= ( list_ex1_nat @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_667_can__select__set__list__ex1,axiom,
! [P: f > $o,A4: list_f] :
( ( can_select_f @ P @ ( set_f2 @ A4 ) )
= ( list_ex1_f @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_668_can__select__set__list__ex1,axiom,
! [P: a > $o,A4: list_a] :
( ( can_select_a @ P @ ( set_a2 @ A4 ) )
= ( list_ex1_a @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_669_the__elem__set,axiom,
! [X: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% the_elem_set
thf(fact_670_the__elem__set,axiom,
! [X: produc4044097585999906000od_a_a] :
( ( the_el7097569866588581061od_a_a @ ( set_Pr7008919305446870117od_a_a @ ( cons_P64011158483510406od_a_a @ X @ nil_Pr7291147438282410198od_a_a ) ) )
= X ) ).
% the_elem_set
thf(fact_671_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_672_the__elem__set,axiom,
! [X: f] :
( ( the_elem_f @ ( set_f2 @ ( cons_f @ X @ nil_f ) ) )
= X ) ).
% the_elem_set
thf(fact_673_empty__Shift,axiom,
! [Kl: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_674_empty__Shift,axiom,
! [Kl: set_li6330610657153406774od_a_a,K: produc4044097585999906000od_a_a] :
( ( member7908416913644183295od_a_a @ nil_Pr7291147438282410198od_a_a @ Kl )
=> ( ( member3071122053849602553od_a_a @ K @ ( bNF_Gr428374559055706347od_a_a @ Kl @ nil_Pr7291147438282410198od_a_a ) )
=> ( member7908416913644183295od_a_a @ nil_Pr7291147438282410198od_a_a @ ( bNF_Gr2674349488622646127od_a_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_675_empty__Shift,axiom,
! [Kl: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_676_empty__Shift,axiom,
! [Kl: set_list_f,K: f] :
( ( member_list_f @ nil_f @ Kl )
=> ( ( member_f @ K @ ( bNF_Greatest_Succ_f @ Kl @ nil_f ) )
=> ( member_list_f @ nil_f @ ( bNF_Greatest_Shift_f @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_677_Succ__Shift,axiom,
! [Kl: set_li6330610657153406774od_a_a,K: produc4044097585999906000od_a_a,Kl2: list_P1802437025701573206od_a_a] :
( ( bNF_Gr428374559055706347od_a_a @ ( bNF_Gr2674349488622646127od_a_a @ Kl @ K ) @ Kl2 )
= ( bNF_Gr428374559055706347od_a_a @ Kl @ ( cons_P64011158483510406od_a_a @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_678_Succ__Shift,axiom,
! [Kl: set_list_nat,K: nat,Kl2: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl @ ( cons_nat @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_679_Succ__Shift,axiom,
! [Kl: set_list_f,K: f,Kl2: list_f] :
( ( bNF_Greatest_Succ_f @ ( bNF_Greatest_Shift_f @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ_f @ Kl @ ( cons_f @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_680_Cons__in__lex,axiom,
! [X: f,Xs2: list_f,Y: f,Ys2: list_f,R: set_Product_prod_f_f] :
( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X @ Xs2 ) @ ( cons_f @ Y @ Ys2 ) ) @ ( lex_f @ R ) )
= ( ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ R )
& ( ( size_size_list_f @ Xs2 )
= ( size_size_list_f @ Ys2 ) ) )
| ( ( X = Y )
& ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs2 @ Ys2 ) @ ( lex_f @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_681_Cons__in__lex,axiom,
! [X: $o,Xs2: list_o,Y: $o,Ys2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs2 ) @ ( cons_o @ Y @ Ys2 ) ) @ ( lex_o @ R ) )
= ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
& ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys2 ) ) )
| ( ( X = Y )
& ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( lex_o @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_682_Cons__in__lex,axiom,
! [X: a,Xs2: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_683_Cons__in__lex,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_684_Cons__in__lex,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Y: produc4044097585999906000od_a_a,Ys2: list_P1802437025701573206od_a_a,R: set_Pr9202094101234783879od_a_a] :
( ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ ( cons_P64011158483510406od_a_a @ X @ Xs2 ) @ ( cons_P64011158483510406od_a_a @ Y @ Ys2 ) ) @ ( lex_Pr4501241201491742522od_a_a @ R ) )
= ( ( ( member1309718761806480592od_a_a @ ( produc8914748963271895447od_a_a @ X @ Y ) @ R )
& ( ( size_s6425312265427517634od_a_a @ Xs2 )
= ( size_s6425312265427517634od_a_a @ Ys2 ) ) )
| ( ( X = Y )
& ( member1628670812943486672od_a_a @ ( produc4694146025100153495od_a_a @ Xs2 @ Ys2 ) @ ( lex_Pr4501241201491742522od_a_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_685_concat__conv__foldr,axiom,
( concat5889154033839980561od_a_a
= ( ^ [Xss3: list_l4888837856047780700od_a_a] : ( foldr_1618125860701030823od_a_a @ append4725183380742660459od_a_a @ Xss3 @ nil_Pr7291147438282410198od_a_a ) ) ) ).
% concat_conv_foldr
thf(fact_686_concat__conv__foldr,axiom,
( concat_nat
= ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).
% concat_conv_foldr
thf(fact_687_concat__conv__foldr,axiom,
( concat_f
= ( ^ [Xss3: list_list_f] : ( foldr_list_f_list_f @ append_f @ Xss3 @ nil_f ) ) ) ).
% concat_conv_foldr
thf(fact_688_remove1__split,axiom,
! [A: a,Xs2: list_a,Ys2: list_a] :
( ( member_a @ A @ ( set_a2 @ Xs2 ) )
=> ( ( ( remove1_a @ A @ Xs2 )
= Ys2 )
= ( ? [Ls: list_a,Rs: list_a] :
( ( Xs2
= ( append_a @ Ls @ ( cons_a @ A @ Rs ) ) )
& ~ ( member_a @ A @ ( set_a2 @ Ls ) )
& ( Ys2
= ( append_a @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_689_remove1__split,axiom,
! [A: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( member3071122053849602553od_a_a @ A @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ( ( ( remove4341252465481845868od_a_a @ A @ Xs2 )
= Ys2 )
= ( ? [Ls: list_P1802437025701573206od_a_a,Rs: list_P1802437025701573206od_a_a] :
( ( Xs2
= ( append4725183380742660459od_a_a @ Ls @ ( cons_P64011158483510406od_a_a @ A @ Rs ) ) )
& ~ ( member3071122053849602553od_a_a @ A @ ( set_Pr7008919305446870117od_a_a @ Ls ) )
& ( Ys2
= ( append4725183380742660459od_a_a @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_690_remove1__split,axiom,
! [A: nat,Xs2: list_nat,Ys2: list_nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( ( remove1_nat @ A @ Xs2 )
= Ys2 )
= ( ? [Ls: list_nat,Rs: list_nat] :
( ( Xs2
= ( append_nat @ Ls @ ( cons_nat @ A @ Rs ) ) )
& ~ ( member_nat @ A @ ( set_nat2 @ Ls ) )
& ( Ys2
= ( append_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_691_remove1__split,axiom,
! [A: f,Xs2: list_f,Ys2: list_f] :
( ( member_f @ A @ ( set_f2 @ Xs2 ) )
=> ( ( ( remove1_f @ A @ Xs2 )
= Ys2 )
= ( ? [Ls: list_f,Rs: list_f] :
( ( Xs2
= ( append_f @ Ls @ ( cons_f @ A @ Rs ) ) )
& ~ ( member_f @ A @ ( set_f2 @ Ls ) )
& ( Ys2
= ( append_f @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_692_append__eq__append__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs2 @ Us )
= ( append_nat @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_693_append__eq__append__conv,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,Us: list_P1802437025701573206od_a_a,Vs: list_P1802437025701573206od_a_a] :
( ( ( ( size_s6425312265427517634od_a_a @ Xs2 )
= ( size_s6425312265427517634od_a_a @ Ys2 ) )
| ( ( size_s6425312265427517634od_a_a @ Us )
= ( size_s6425312265427517634od_a_a @ Vs ) ) )
=> ( ( ( append4725183380742660459od_a_a @ Xs2 @ Us )
= ( append4725183380742660459od_a_a @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_694_in__set__remove1,axiom,
! [A: produc4044097585999906000od_a_a,B: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( A != B )
=> ( ( member3071122053849602553od_a_a @ A @ ( set_Pr7008919305446870117od_a_a @ ( remove4341252465481845868od_a_a @ B @ Xs2 ) ) )
= ( member3071122053849602553od_a_a @ A @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_695_in__set__remove1,axiom,
! [A: nat,B: nat,Xs2: list_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs2 ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_696_in__set__remove1,axiom,
! [A: f,B: f,Xs2: list_f] :
( ( A != B )
=> ( ( member_f @ A @ ( set_f2 @ ( remove1_f @ B @ Xs2 ) ) )
= ( member_f @ A @ ( set_f2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_697_in__set__remove1,axiom,
! [A: a,B: a,Xs2: list_a] :
( ( A != B )
=> ( ( member_a @ A @ ( set_a2 @ ( remove1_a @ B @ Xs2 ) ) )
= ( member_a @ A @ ( set_a2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_698_length__rotate1,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_699_length__rotate1,axiom,
! [Xs2: list_P1802437025701573206od_a_a] :
( ( size_s6425312265427517634od_a_a @ ( rotate3742350382119052963od_a_a @ Xs2 ) )
= ( size_s6425312265427517634od_a_a @ Xs2 ) ) ).
% length_rotate1
thf(fact_700_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_701_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_P1802437025701573206od_a_a] :
( ( size_s6425312265427517634od_a_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_702_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_703_neq__if__length__neq,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a] :
( ( ( size_s6425312265427517634od_a_a @ Xs2 )
!= ( size_s6425312265427517634od_a_a @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_704_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_705_in__set__product__lists__length,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Xss2: list_l4888837856047780700od_a_a] :
( ( member7908416913644183295od_a_a @ Xs2 @ ( set_li8573657874255358827od_a_a @ ( produc4796932526259527359od_a_a @ Xss2 ) ) )
=> ( ( size_s6425312265427517634od_a_a @ Xs2 )
= ( size_s1994944411682077640od_a_a @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_706_can__select__def,axiom,
( can_select_a
= ( ^ [P4: a > $o,A6: set_a] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( P4 @ X3 )
& ! [Y4: a] :
( ( ( member_a @ Y4 @ A6 )
& ( P4 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_707_remove1_Osimps_I2_J,axiom,
! [X: produc4044097585999906000od_a_a,Y: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ( ( X = Y )
=> ( ( remove4341252465481845868od_a_a @ X @ ( cons_P64011158483510406od_a_a @ Y @ Xs2 ) )
= Xs2 ) )
& ( ( X != Y )
=> ( ( remove4341252465481845868od_a_a @ X @ ( cons_P64011158483510406od_a_a @ Y @ Xs2 ) )
= ( cons_P64011158483510406od_a_a @ Y @ ( remove4341252465481845868od_a_a @ X @ Xs2 ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_708_remove1_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs2: list_nat] :
( ( ( X = Y )
=> ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs2 ) )
= Xs2 ) )
& ( ( X != Y )
=> ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs2 ) )
= ( cons_nat @ Y @ ( remove1_nat @ X @ Xs2 ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_709_remove1_Osimps_I2_J,axiom,
! [X: f,Y: f,Xs2: list_f] :
( ( ( X = Y )
=> ( ( remove1_f @ X @ ( cons_f @ Y @ Xs2 ) )
= Xs2 ) )
& ( ( X != Y )
=> ( ( remove1_f @ X @ ( cons_f @ Y @ Xs2 ) )
= ( cons_f @ Y @ ( remove1_f @ X @ Xs2 ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_710_remove1_Osimps_I1_J,axiom,
! [X: produc4044097585999906000od_a_a] :
( ( remove4341252465481845868od_a_a @ X @ nil_Pr7291147438282410198od_a_a )
= nil_Pr7291147438282410198od_a_a ) ).
% remove1.simps(1)
thf(fact_711_remove1_Osimps_I1_J,axiom,
! [X: nat] :
( ( remove1_nat @ X @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_712_remove1_Osimps_I1_J,axiom,
! [X: f] :
( ( remove1_f @ X @ nil_f )
= nil_f ) ).
% remove1.simps(1)
thf(fact_713_remove1__idem,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a] :
( ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ( ( remove4341252465481845868od_a_a @ X @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_714_remove1__idem,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( remove1_nat @ X @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_715_remove1__idem,axiom,
! [X: f,Xs2: list_f] :
( ~ ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ( ( remove1_f @ X @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_716_remove1__idem,axiom,
! [X: a,Xs2: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ( remove1_a @ X @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_717_notin__set__remove1,axiom,
! [X: produc4044097585999906000od_a_a,Xs2: list_P1802437025701573206od_a_a,Y: produc4044097585999906000od_a_a] :
( ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ Xs2 ) )
=> ~ ( member3071122053849602553od_a_a @ X @ ( set_Pr7008919305446870117od_a_a @ ( remove4341252465481845868od_a_a @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_718_notin__set__remove1,axiom,
! [X: nat,Xs2: list_nat,Y: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( member_nat @ X @ ( set_nat2 @ ( remove1_nat @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_719_notin__set__remove1,axiom,
! [X: f,Xs2: list_f,Y: f] :
( ~ ( member_f @ X @ ( set_f2 @ Xs2 ) )
=> ~ ( member_f @ X @ ( set_f2 @ ( remove1_f @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_720_notin__set__remove1,axiom,
! [X: a,Xs2: list_a,Y: a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ~ ( member_a @ X @ ( set_a2 @ ( remove1_a @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_721_list__induct2,axiom,
! [Xs2: list_f,Ys2: list_f,P: list_f > list_f > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( P @ nil_f @ nil_f )
=> ( ! [X2: f,Xs: list_f,Y3: f,Ys: list_f] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_722_list__induct2,axiom,
! [Xs2: list_f,Ys2: list_nat,P: list_f > list_nat > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_f @ nil_nat )
=> ( ! [X2: f,Xs: list_f,Y3: nat,Ys: list_nat] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_723_list__induct2,axiom,
! [Xs2: list_f,Ys2: list_P1802437025701573206od_a_a,P: list_f > list_P1802437025701573206od_a_a > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_s6425312265427517634od_a_a @ Ys2 ) )
=> ( ( P @ nil_f @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: f,Xs: list_f,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( ( ( size_size_list_f @ Xs )
= ( size_s6425312265427517634od_a_a @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_724_list__induct2,axiom,
! [Xs2: list_nat,Ys2: list_f,P: list_nat > list_f > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_f )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_725_list__induct2,axiom,
! [Xs2: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_726_list__induct2,axiom,
! [Xs2: list_nat,Ys2: list_P1802437025701573206od_a_a,P: list_nat > list_P1802437025701573206od_a_a > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s6425312265427517634od_a_a @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: nat,Xs: list_nat,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_s6425312265427517634od_a_a @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_727_list__induct2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_f,P: list_P1802437025701573206od_a_a > list_f > $o] :
( ( ( size_s6425312265427517634od_a_a @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( P @ nil_Pr7291147438282410198od_a_a @ nil_f )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Y3: f,Ys: list_f] :
( ( ( size_s6425312265427517634od_a_a @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_728_list__induct2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_nat,P: list_P1802437025701573206od_a_a > list_nat > $o] :
( ( ( size_s6425312265427517634od_a_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_Pr7291147438282410198od_a_a @ nil_nat )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Y3: nat,Ys: list_nat] :
( ( ( size_s6425312265427517634od_a_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_729_list__induct2,axiom,
! [Xs2: list_P1802437025701573206od_a_a,Ys2: list_P1802437025701573206od_a_a,P: list_P1802437025701573206od_a_a > list_P1802437025701573206od_a_a > $o] :
( ( ( size_s6425312265427517634od_a_a @ Xs2 )
= ( size_s6425312265427517634od_a_a @ Ys2 ) )
=> ( ( P @ nil_Pr7291147438282410198od_a_a @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: produc4044097585999906000od_a_a,Xs: list_P1802437025701573206od_a_a,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a] :
( ( ( size_s6425312265427517634od_a_a @ Xs )
= ( size_s6425312265427517634od_a_a @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_P64011158483510406od_a_a @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_730_list__induct3,axiom,
! [Xs2: list_f,Ys2: list_f,Zs2: list_f,P: list_f > list_f > list_f > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( P @ nil_f @ nil_f @ nil_f )
=> ( ! [X2: f,Xs: list_f,Y3: f,Ys: list_f,Z3: f,Zs: list_f] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_731_list__induct3,axiom,
! [Xs2: list_f,Ys2: list_f,Zs2: list_nat,P: list_f > list_f > list_nat > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_f @ nil_f @ nil_nat )
=> ( ! [X2: f,Xs: list_f,Y3: f,Ys: list_f,Z3: nat,Zs: list_nat] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_732_list__induct3,axiom,
! [Xs2: list_f,Ys2: list_nat,Zs2: list_f,P: list_f > list_nat > list_f > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( P @ nil_f @ nil_nat @ nil_f )
=> ( ! [X2: f,Xs: list_f,Y3: nat,Ys: list_nat,Z3: f,Zs: list_f] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_733_list__induct3,axiom,
! [Xs2: list_f,Ys2: list_nat,Zs2: list_nat,P: list_f > list_nat > list_nat > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_f @ nil_nat @ nil_nat )
=> ( ! [X2: f,Xs: list_f,Y3: nat,Ys: list_nat,Z3: nat,Zs: list_nat] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_734_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_f,Zs2: list_f,P: list_nat > list_f > list_f > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_f @ nil_f )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f,Z3: f,Zs: list_f] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_735_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_f,Zs2: list_nat,P: list_nat > list_f > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_f @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f,Z3: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_736_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_f,P: list_nat > list_nat > list_f > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_f )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z3: f,Zs: list_f] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_737_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z3: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_738_list__induct3,axiom,
! [Xs2: list_f,Ys2: list_f,Zs2: list_P1802437025701573206od_a_a,P: list_f > list_f > list_P1802437025701573206od_a_a > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_s6425312265427517634od_a_a @ Zs2 ) )
=> ( ( P @ nil_f @ nil_f @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: f,Xs: list_f,Y3: f,Ys: list_f,Z3: produc4044097585999906000od_a_a,Zs: list_P1802437025701573206od_a_a] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_s6425312265427517634od_a_a @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_P64011158483510406od_a_a @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_739_list__induct3,axiom,
! [Xs2: list_f,Ys2: list_nat,Zs2: list_P1802437025701573206od_a_a,P: list_f > list_nat > list_P1802437025701573206od_a_a > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s6425312265427517634od_a_a @ Zs2 ) )
=> ( ( P @ nil_f @ nil_nat @ nil_Pr7291147438282410198od_a_a )
=> ( ! [X2: f,Xs: list_f,Y3: nat,Ys: list_nat,Z3: produc4044097585999906000od_a_a,Zs: list_P1802437025701573206od_a_a] :
( ( ( size_size_list_f @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s6425312265427517634od_a_a @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_P64011158483510406od_a_a @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_740_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_f,Zs2: list_f,Ws: list_f,P: list_nat > list_f > list_f > list_f > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( ( size_size_list_f @ Zs2 )
= ( size_size_list_f @ Ws ) )
=> ( ( P @ nil_nat @ nil_f @ nil_f @ nil_f )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f,Z3: f,Zs: list_f,W: f,Ws2: list_f] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( ( size_size_list_f @ Zs )
= ( size_size_list_f @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) @ ( cons_f @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_741_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_f,Zs2: list_f,Ws: list_nat,P: list_nat > list_f > list_f > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( ( size_size_list_f @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_f @ nil_f @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f,Z3: f,Zs: list_f,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( ( size_size_list_f @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_742_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_f,Zs2: list_nat,Ws: list_f,P: list_nat > list_f > list_nat > list_f > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_f @ Ws ) )
=> ( ( P @ nil_nat @ nil_f @ nil_nat @ nil_f )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f,Z3: nat,Zs: list_nat,W: f,Ws2: list_f] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_f @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_f @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_743_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_f,Zs2: list_nat,Ws: list_nat,P: list_nat > list_f > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_f @ Ys2 ) )
=> ( ( ( size_size_list_f @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_f @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y3: f,Ys: list_f,Z3: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_f @ Ys ) )
=> ( ( ( size_size_list_f @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_744_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_f,Ws: list_f,P: list_nat > list_nat > list_f > list_f > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( ( size_size_list_f @ Zs2 )
= ( size_size_list_f @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_f @ nil_f )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z3: f,Zs: list_f,W: f,Ws2: list_f] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( ( size_size_list_f @ Zs )
= ( size_size_list_f @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) @ ( cons_f @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_745_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_f,Ws: list_nat,P: list_nat > list_nat > list_f > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( ( size_size_list_f @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_f @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z3: f,Zs: list_f,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( ( size_size_list_f @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_746_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_nat,Ws: list_f,P: list_nat > list_nat > list_nat > list_f > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_f @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_f )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z3: nat,Zs: list_nat,W: f,Ws2: list_f] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_f @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_f @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_747_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs2: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z3: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_748_list__induct4,axiom,
! [Xs2: list_f,Ys2: list_P1802437025701573206od_a_a,Zs2: list_f,Ws: list_f,P: list_f > list_P1802437025701573206od_a_a > list_f > list_f > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_s6425312265427517634od_a_a @ Ys2 ) )
=> ( ( ( size_s6425312265427517634od_a_a @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( ( size_size_list_f @ Zs2 )
= ( size_size_list_f @ Ws ) )
=> ( ( P @ nil_f @ nil_Pr7291147438282410198od_a_a @ nil_f @ nil_f )
=> ( ! [X2: f,Xs: list_f,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a,Z3: f,Zs: list_f,W: f,Ws2: list_f] :
( ( ( size_size_list_f @ Xs )
= ( size_s6425312265427517634od_a_a @ Ys ) )
=> ( ( ( size_s6425312265427517634od_a_a @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( ( size_size_list_f @ Zs )
= ( size_size_list_f @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) @ ( cons_f @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_749_list__induct4,axiom,
! [Xs2: list_f,Ys2: list_P1802437025701573206od_a_a,Zs2: list_f,Ws: list_nat,P: list_f > list_P1802437025701573206od_a_a > list_f > list_nat > $o] :
( ( ( size_size_list_f @ Xs2 )
= ( size_s6425312265427517634od_a_a @ Ys2 ) )
=> ( ( ( size_s6425312265427517634od_a_a @ Ys2 )
= ( size_size_list_f @ Zs2 ) )
=> ( ( ( size_size_list_f @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_f @ nil_Pr7291147438282410198od_a_a @ nil_f @ nil_nat )
=> ( ! [X2: f,Xs: list_f,Y3: produc4044097585999906000od_a_a,Ys: list_P1802437025701573206od_a_a,Z3: f,Zs: list_f,W: nat,Ws2: list_nat] :
( ( ( size_size_list_f @ Xs )
= ( size_s6425312265427517634od_a_a @ Ys ) )
=> ( ( ( size_s6425312265427517634od_a_a @ Ys )
= ( size_size_list_f @ Zs ) )
=> ( ( ( size_size_list_f @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_f @ X2 @ Xs ) @ ( cons_P64011158483510406od_a_a @ Y3 @ Ys ) @ ( cons_f @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_750_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_751_assms_I3_J,axiom,
distinct_a @ ( map_Pr4634343309551623282_a_a_a @ produc6516300539023753772od_a_a @ vars ) ).
% assms(3)
thf(fact_752_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_753_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_754_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_755_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_756_assms_I2_J,axiom,
! [I: nat,X: a,Ze: a,On: a] :
( ( ord_less_nat @ I @ ( size_s6425312265427517634od_a_a @ vars ) )
=> ( ( ( nth_Pr797287453650735447od_a_a @ vars @ I )
= ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Ze @ On ) ) )
=> ( ( ( alpha @ Ze )
= ( ~ ? [X3: a] :
( ( member_a @ X3 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ ( drop_P3280791721155218966od_a_a @ I @ vars ) ) ) )
& ( alpha @ X3 ) ) ) )
& ( ( alpha @ On )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ ( drop_P3280791721155218966od_a_a @ I @ vars ) ) ) )
& ( alpha @ X3 )
& ! [Y4: a] :
( ( ( member_a @ Y4 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ ( drop_P3280791721155218966od_a_a @ I @ vars ) ) ) )
& ( alpha @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ) ) ) ).
% assms(2)
thf(fact_757_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_758_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_759_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_760_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% drop_upt
thf(fact_761_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_762_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_763_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_764_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_765_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_766_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_767_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_768_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M2 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_769_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_770_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_771_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_772_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_773_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_774_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_775_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_776_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J2 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_777_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_778_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_779_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_780_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs2: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_781_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_782_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_783_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_784_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_785_sorted__list__of__set__range,axiom,
! [M: nat,N: nat] :
( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sorted_list_of_set_range
thf(fact_786_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_787_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_788_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I2: nat,J2: nat] : ( set_nat2 @ ( upt @ I2 @ J2 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_789_encode__sum__0__1__main,axiom,
! [Vars: list_P1802437025701573206od_a_a,Conds: list_f,Zero2: a,One2: a,Prop: $o,Alpha: a > $o] :
( ( ( multis3280010013254100632in_a_f @ enc_different @ enc_equiv_and_not @ enc_equiv_ite @ Vars )
= ( produc3886013187752893941od_a_a @ Conds @ ( product_Pair_a_a @ Zero2 @ One2 ) ) )
=> ( ( Prop
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6425312265427517634od_a_a @ Vars ) )
=> ! [X2: a,Ze2: a,On2: a,Re: list_P1802437025701573206od_a_a] :
( ( ( drop_P3280791721155218966od_a_a @ I3 @ Vars )
= ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Ze2 @ On2 ) ) @ Re ) )
=> ( ( ( Alpha @ Ze2 )
= ( ~ ? [Y4: a] :
( ( member_a @ Y4 @ ( insert_a2 @ X2 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ Re ) ) ) )
& ( Alpha @ Y4 ) ) ) )
& ( ( Alpha @ On2 )
= ( ? [Y4: a] :
( ( member_a @ Y4 @ ( insert_a2 @ X2 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ Re ) ) ) )
& ( Alpha @ Y4 )
& ! [Z4: a] :
( ( ( member_a @ Z4 @ ( insert_a2 @ X2 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ Re ) ) ) )
& ( Alpha @ Z4 ) )
=> ( Z4 = Y4 ) ) ) ) ) ) ) ) )
=> ( ( ~ Prop
=> ! [X2: f] :
( ( member_f @ X2 @ ( set_f2 @ Conds ) )
=> ( eval @ Alpha @ X2 ) ) )
=> ( ( distinct_a @ ( map_Pr4634343309551623282_a_a_a @ produc6516300539023753772od_a_a @ Vars ) )
=> ( ( Vars != nil_Pr7291147438282410198od_a_a )
=> ( ! [X4: f] :
( ( member_f @ X4 @ ( set_f2 @ Conds ) )
=> ( eval @ Alpha @ X4 ) )
& ( ( Alpha @ Zero2 )
= ( ~ ? [X3: a] :
( ( member_a @ X3 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ Vars ) ) )
& ( Alpha @ X3 ) ) ) )
& ( ( Alpha @ One2 )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ Vars ) ) )
& ( Alpha @ X3 )
& ! [Y4: a] :
( ( ( member_a @ Y4 @ ( image_1366754389297715763_a_a_a @ produc6516300539023753772od_a_a @ ( set_Pr7008919305446870117od_a_a @ Vars ) ) )
& ( Alpha @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ) ) ) ) ) ) ).
% encode_sum_0_1_main
thf(fact_790_encode__sum__0__1__main_Osimps_I1_J,axiom,
! [X: a,Zero2: a,One2: a] :
( ( multis3280010013254100632in_a_f @ enc_different @ enc_equiv_and_not @ enc_equiv_ite @ ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Zero2 @ One2 ) ) @ nil_Pr7291147438282410198od_a_a ) )
= ( produc3886013187752893941od_a_a @ ( cons_f @ ( enc_different @ Zero2 @ X ) @ nil_f ) @ ( product_Pair_a_a @ Zero2 @ X ) ) ) ).
% encode_sum_0_1_main.simps(1)
thf(fact_791_enc__sound_I5_J,axiom,
! [Alpha: a > $o,X: a,Y: a] :
( ( eval @ Alpha @ ( enc_different @ X @ Y ) )
= ( ( Alpha @ X )
!= ( Alpha @ Y ) ) ) ).
% enc_sound(5)
thf(fact_792_enc__sound_I6_J,axiom,
! [Alpha: a > $o,X: a,Y: a,Z5: a] :
( ( eval @ Alpha @ ( enc_equiv_and_not @ X @ Y @ Z5 ) )
= ( ( Alpha @ X )
= ( ( Alpha @ Y )
& ~ ( Alpha @ Z5 ) ) ) ) ).
% enc_sound(6)
thf(fact_793_enc__sound_I7_J,axiom,
! [Alpha: a > $o,X: a,Y: a,Z5: a,U: a] :
( ( eval @ Alpha @ ( enc_equiv_ite @ X @ Y @ Z5 @ U ) )
= ( ( Alpha @ X )
= ( ( ( Alpha @ Y )
=> ( Alpha @ Z5 ) )
& ( ~ ( Alpha @ Y )
=> ( Alpha @ U ) ) ) ) ) ).
% enc_sound(7)
thf(fact_794_enc__sound_I13_J,axiom,
! [Alpha: a > $o,Fs: list_f] :
( ( eval @ Alpha @ ( enc_conj @ Fs ) )
= ( ! [X3: f] :
( ( member_f @ X3 @ ( set_f2 @ Fs ) )
=> ( eval @ Alpha @ X3 ) ) ) ) ).
% enc_sound(13)
thf(fact_795_enc__sound_I12_J,axiom,
! [Alpha: a > $o,Xs2: list_a] :
( ( eval @ Alpha @ ( enc_not_all @ Xs2 ) )
= ( ~ ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ( Alpha @ X3 ) ) ) ) ).
% enc_sound(12)
thf(fact_796_enc__sound_I9_J,axiom,
! [Alpha: a > $o,X: a,F: f] :
( ( eval @ Alpha @ ( enc_impl @ X @ F ) )
= ( ( Alpha @ X )
=> ( eval @ Alpha @ F ) ) ) ).
% enc_sound(9)
thf(fact_797_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_798_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_799_encode__sum__0__1__main_Osimps_I2_J,axiom,
! [X: a,Zero2: a,One2: a,V2: produc4044097585999906000od_a_a,Va2: list_P1802437025701573206od_a_a] :
( ( multis3280010013254100632in_a_f @ enc_different @ enc_equiv_and_not @ enc_equiv_ite @ ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Zero2 @ One2 ) ) @ ( cons_P64011158483510406od_a_a @ V2 @ Va2 ) ) )
= ( produc7623036209211823483od_a_a
@ ^ [Conds2: list_f] :
( produc3537265031573258201od_a_a
@ ^ [Fzero: a,Fone: a] : ( produc3886013187752893941od_a_a @ ( cons_f @ ( enc_equiv_and_not @ Zero2 @ Fzero @ X ) @ ( cons_f @ ( enc_equiv_ite @ One2 @ X @ Fzero @ Fone ) @ Conds2 ) ) @ ( product_Pair_a_a @ Zero2 @ One2 ) ) )
@ ( multis3280010013254100632in_a_f @ enc_different @ enc_equiv_and_not @ enc_equiv_ite @ ( cons_P64011158483510406od_a_a @ V2 @ Va2 ) ) ) ) ).
% encode_sum_0_1_main.simps(2)
% Helper facts (9)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__f_J_T,axiom,
! [X: list_f,Y: list_f] :
( ( if_list_f @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__f_J_T,axiom,
! [X: list_f,Y: list_f] :
( ( if_list_f @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_T,axiom,
! [X: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( if_lis3736317622110187548od_a_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_T,axiom,
! [X: list_P1802437025701573206od_a_a,Y: list_P1802437025701573206od_a_a] :
( ( if_lis3736317622110187548od_a_a @ $true @ X @ Y )
= X ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,hypothesis,
! [X4: a,Ze3: a,On3: a] :
( ( vars
= ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X4 @ ( product_Pair_a_a @ Ze3 @ On3 ) ) @ nil_Pr7291147438282410198od_a_a ) )
=> thesis ) ).
thf(conj_2,hypothesis,
! [X4: a,Ze3: a,On3: a,V3: produc4044097585999906000od_a_a,Vs3: list_P1802437025701573206od_a_a] :
( ( vars
= ( cons_P64011158483510406od_a_a @ ( produc431845341423274048od_a_a @ X4 @ ( product_Pair_a_a @ Ze3 @ On3 ) ) @ ( cons_P64011158483510406od_a_a @ V3 @ Vs3 ) ) )
=> thesis ) ).
thf(conj_3,conjecture,
thesis ).
%------------------------------------------------------------------------------