TPTP Problem File: SLH0132^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00069_002724__11680948_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1770 ( 675 unt; 489 typ; 0 def)
% Number of atoms : 3773 (2231 equ; 0 cnn)
% Maximal formula atoms : 19 ( 2 avg)
% Number of connectives : 12857 ( 687 ~; 92 |; 417 &;10235 @)
% ( 0 <=>;1426 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 80 ( 79 usr)
% Number of type conns : 1287 (1287 >; 0 *; 0 +; 0 <<)
% Number of symbols : 413 ( 410 usr; 34 con; 0-4 aty)
% Number of variables : 4084 ( 109 ^;3689 !; 286 ?;4084 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:56:49.184
%------------------------------------------------------------------------------
% Could-be-implicit typings (79)
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% Explicit typings (410)
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_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_Otimes__class_Otimes_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__List__Olist_I_Eo_J,type,
if_list_o: $o > list_o > list_o > list_o ).
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__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__List__Olist_I_Eo_J,type,
lattic7994203477452000287list_o: set_list_o > list_o ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001_062_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001_Eo,type,
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thf(sy_c_List_Oappend_001t__List__Olist_I_Eo_J,type,
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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__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_List_Olist_Otl_001_Eo,type,
tl_o: list_o > list_o ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist__ex1_001_062_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Nat__Onat_J,type,
list_e357688041540588142_o_nat: ( ( option_list_o > nat ) > $o ) > list_o8358219866161907755_o_nat > $o ).
thf(sy_c_List_Olist__ex1_001_Eo,type,
list_ex1_o: ( $o > $o ) > list_o > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_I_Eo_J,type,
list_ex1_list_o: ( list_o > $o ) > list_list_o > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__update_001_Eo,type,
list_update_o: list_o > nat > $o > list_o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
list_u6420874592553512018list_o: list_option_list_o > nat > option_list_o > list_option_list_o ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
list_u1500079835982728340al_nat: list_P189569681052899986al_nat > nat > produc9155354469731034754al_nat > list_P189569681052899986al_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
list_u2015267059982357653list_o: list_P1348966709184090669list_o > nat > produc4882884732533091879list_o > list_P1348966709184090669list_o ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
list_u873804745024221131al_nat: list_P6834414599653733731al_nat > nat > produc3741383161447143261al_nat > list_P6834414599653733731al_nat ).
thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
list_update_real: list_real > nat > real > list_real ).
thf(sy_c_List_Olistrel1_001_Eo,type,
listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel1_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
listre3104272764119459907list_o: set_Pr7497825696620840711list_o > set_Pr1386088649869593607list_o ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
listre1017866332378306906al_nat: set_Pr836851361167288120al_nat > set_Pr3318565038780223662al_nat ).
thf(sy_c_List_Olistrel_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
listre2892261910510432881list_o: set_Pr7497825696620840711list_o > set_Pr1386088649869593607list_o ).
thf(sy_c_List_Olistrel_001t__Real__Oreal_001t__Nat__Onat,type,
listrel_real_nat: set_Pr3510011417693777981al_nat > set_Pr6676954283489005405st_nat ).
thf(sy_c_List_Olists_001_Eo,type,
lists_o: set_o > set_list_o ).
thf(sy_c_List_Olists_001t__Nat__Onat,type,
lists_nat: set_nat > set_list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev__rel_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta8615873517111064934at_nat: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001_Eo,type,
maps_nat_o: ( nat > list_o ) > list_nat > list_o ).
thf(sy_c_List_Omeasures_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
measur3775467863237363684list_o: list_o8358219866161907755_o_nat > set_Pr7497825696620840711list_o ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001_Eo,type,
nth_o: list_o > nat > $o ).
thf(sy_c_List_Onth_001t__List__Olist_I_Eo_J,type,
nth_list_o: list_list_o > nat > list_o ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
nth_option_list_o: list_option_list_o > nat > option_list_o ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
nth_Pr7063540399719084603al_nat: list_P189569681052899986al_nat > nat > produc9155354469731034754al_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
nth_Pr5324855644858119342list_o: list_P1348966709184090669list_o > nat > produc4882884732533091879list_o ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
nth_Pr3792769368220640996al_nat: list_P6834414599653733731al_nat > nat > produc3741383161447143261al_nat ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
produc7250786521479175942al_nat: list_nat > list_P6834414599653733731al_nat > list_P189569681052899986al_nat ).
thf(sy_c_List_Oproduct_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
produc2445308772338649029list_o: list_option_list_o > list_option_list_o > list_P1348966709184090669list_o ).
thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
product_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oremdups__adj_001_Eo,type,
remdups_adj_o: list_o > list_o ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj__rel_001t__Nat__Onat,type,
remdups_adj_rel_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oremove1_001_Eo,type,
remove1_o: $o > list_o > list_o ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001_Eo,type,
removeAll_o: $o > list_o > list_o ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oreplicate_001_Eo,type,
replicate_o: nat > $o > list_o ).
thf(sy_c_List_Oreplicate_001t__List__Olist_I_Eo_J,type,
replicate_list_o: nat > list_o > list_list_o ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
replic6237453613039860388list_o: nat > option_list_o > list_option_list_o ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
replic2733456754749247170al_nat: nat > produc9155354469731034754al_nat > list_P189569681052899986al_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
replic6149313115625166311list_o: nat > produc4882884732533091879list_o > list_P1348966709184090669list_o ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
replic1728136009727115037al_nat: nat > produc3741383161447143261al_nat > list_P6834414599653733731al_nat ).
thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
replicate_real: nat > real > list_real ).
thf(sy_c_List_Orev_001_Eo,type,
rev_o: list_o > list_o ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
rev_Pr6102188148953555047at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Orotate1_001_Eo,type,
rotate1_o: list_o > list_o ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate_001_Eo,type,
rotate_o: nat > list_o > list_o ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oshuffles_001t__Nat__Onat,type,
shuffles_nat: list_nat > list_nat > set_list_nat ).
thf(sy_c_List_Oshuffles__rel_001t__Nat__Onat,type,
shuffles_rel_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).
thf(sy_c_List_Osorted__wrt_001_Eo,type,
sorted_wrt_o: ( $o > $o > $o ) > list_o > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osplice_001t__Nat__Onat,type,
splice_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Osplice__rel_001t__Nat__Onat,type,
splice_rel_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Otake_001_Eo,type,
take_o: nat > list_o > list_o ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
take_option_list_o: nat > list_option_list_o > list_option_list_o ).
thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
take_P5981256949253159243al_nat: nat > list_P6834414599653733731al_nat > list_P6834414599653733731al_nat ).
thf(sy_c_List_Othose_001_062_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Nat__Onat_J,type,
those_7536171816368882977_o_nat: list_o1613249628871224945_o_nat > option1302615508826106225_o_nat ).
thf(sy_c_List_Othose_001_Eo,type,
those_o: list_option_o > option_list_o ).
thf(sy_c_List_Othose_001t__List__Olist_I_Eo_J,type,
those_list_o: list_option_list_o > option_list_list_o ).
thf(sy_c_List_Othose_001t__List__Olist_It__Nat__Onat_J,type,
those_list_nat: list_option_list_nat > option_list_list_nat ).
thf(sy_c_List_Othose_001t__Nat__Onat,type,
those_nat: list_option_nat > option_list_nat ).
thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
zip_na1967359233052233492al_nat: list_nat > list_P6834414599653733731al_nat > list_P189569681052899986al_nat ).
thf(sy_c_List_Ozip_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
zip_op2665896746229601079list_o: list_option_list_o > list_option_list_o > list_P1348966709184090669list_o ).
thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
size_s2710708370519433104list_o: list_list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
size_s5120752914790010966list_o: list_option_list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
size_s1877336372972134351al_nat: list_P6834414599653733731al_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
size_s9086679804311813456list_o: option_list_o > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Option_Obind_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
bind_list_o_list_o: option_list_o > ( list_o > option_list_o ) > option_list_o ).
thf(sy_c_Option_Obind_001t__List__Olist_I_Eo_J_001t__Nat__Onat,type,
bind_list_o_nat2: option_list_o > ( list_o > option_nat ) > option_nat ).
thf(sy_c_Option_Obind_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
bind_nat_list_o2: option_nat > ( nat > option_list_o ) > option_list_o ).
thf(sy_c_Option_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat2: option_nat > ( nat > option_nat ) > option_nat ).
thf(sy_c_Option_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__Product____Type__Oprod_I_Eo_Mt__List__Olist_I_Eo_J_J_J,type,
bind_n3968521476251553702list_o: option_nat > ( nat > option2540781896007660796list_o ) > option2540781896007660796list_o ).
thf(sy_c_Option_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
bind_n8583154692918325046st_nat: option_nat > ( nat > option7375638066962775414st_nat ) > option7375638066962775414st_nat ).
thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__Product____Type__Oprod_I_Eo_Mt__List__Olist_I_Eo_J_J_J_001t__List__Olist_I_Eo_J,type,
bind_P2403420563366411686list_o: option2540781896007660796list_o > ( produc6173137527213542198list_o > option_list_o ) > option_list_o ).
thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__Product____Type__Oprod_I_Eo_Mt__List__Olist_I_Eo_J_J_J_001t__Nat__Onat,type,
bind_P7999860724539673800_o_nat: option2540781896007660796list_o > ( produc6173137527213542198list_o > option_nat ) > option_nat ).
thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__Product____Type__Oprod_I_Eo_Mt__List__Olist_I_Eo_J_J_J_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__Product____Type__Oprod_I_Eo_Mt__List__Olist_I_Eo_J_J_J,type,
bind_P4009872011549414462list_o: option2540781896007660796list_o > ( produc6173137527213542198list_o > option2540781896007660796list_o ) > option2540781896007660796list_o ).
thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J_001t__List__Olist_I_Eo_J,type,
bind_P2299425358426170424list_o: option7375638066962775414st_nat > ( produc8196726482729697190st_nat > option_list_o ) > option_list_o ).
thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
bind_P6985806841975176630at_nat: option7375638066962775414st_nat > ( produc8196726482729697190st_nat > option_nat ) > option_nat ).
thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
bind_P6234047912424235660st_nat: option7375638066962775414st_nat > ( produc8196726482729697190st_nat > option7375638066962775414st_nat ) > option7375638066962775414st_nat ).
thf(sy_c_Option_Ocombine__options_001t__List__Olist_I_Eo_J,type,
combin4983522934621174429list_o: ( list_o > list_o > list_o ) > option_list_o > option_list_o > option_list_o ).
thf(sy_c_Option_Ocombine__options_001t__Nat__Onat,type,
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produc4490577844368043572al_nat: nat > produc3741383161447143261al_nat > produc9155354469731034754al_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
produc5745850523778858007list_o: option_list_o > option_list_o > produc4882884732533091879list_o ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J_001t__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J,type,
produc1623406484878470137st_nat: option_list_nat > option_list_nat > produc7162494062886987457st_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_001t__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
produc1805441192919750679list_o: set_option_list_o > set_option_list_o > produc1290640719454552615list_o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
set_fo5235541981465435595al_nat: ( nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat ) > nat > nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
set_fo6615314570929945908al_nat: produc1019696013813152916al_nat > produc1019696013813152916al_nat > $o ).
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_Sublist_OLongest__common__prefix_001_Eo,type,
longes2916941589570675010efix_o: set_list_o > list_o ).
thf(sy_c_Sublist_OLongest__common__prefix_001t__Nat__Onat,type,
longes514542611558403238ix_nat: set_list_nat > list_nat ).
thf(sy_c_Sublist_Olongest__common__prefix_001_Eo,type,
longes218265413398645026efix_o: list_o > list_o > list_o ).
thf(sy_c_Sublist_Olongest__common__prefix_001t__Nat__Onat,type,
longes266370323089874118ix_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Sublist_Olongest__common__prefix__rel_001t__Nat__Onat,type,
longes4198368034990555989el_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).
thf(sy_c_Sublist_Oprefix_001_062_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Nat__Onat_J,type,
prefix2577173861203319680_o_nat: list_o8358219866161907755_o_nat > list_o8358219866161907755_o_nat > $o ).
thf(sy_c_Sublist_Oprefix_001_Eo,type,
prefix_o: list_o > list_o > $o ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_I_Eo_J,type,
prefix_list_o: list_list_o > list_list_o > $o ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__Nat__Onat_J,type,
prefix_list_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
prefix_option_list_o: list_option_list_o > list_option_list_o > $o ).
thf(sy_c_Sublist_Oprefix_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
prefix3150859221070158812at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Real__Oreal,type,
prefix_real: list_real > list_real > $o ).
thf(sy_c_Sublist_Oprefixes_001_Eo,type,
prefixes_o: list_o > list_list_o ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Ostrict__prefix_001_Eo,type,
strict_prefix_o: list_o > list_o > $o ).
thf(sy_c_Sublist_Ostrict__prefix_001t__Nat__Onat,type,
strict_prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J_J,type,
accp_P1387443933282790731al_nat: ( produc1019696013813152916al_nat > produc1019696013813152916al_nat > $o ) > produc1019696013813152916al_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
accp_P3924000266242761678st_nat: ( produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o ) > produc1616951275169580055st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
accp_P8037286306265792042st_nat: ( produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ) > produc1828647624359046049st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
accp_P4987914878654715440list_o: ( produc4882884732533091879list_o > produc4882884732533091879list_o > $o ) > produc4882884732533091879list_o > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J_Mt__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J_J,type,
accp_P1545021376356841034st_nat: ( produc7162494062886987457st_nat > produc7162494062886987457st_nat > $o ) > produc7162494062886987457st_nat > $o ).
thf(sy_c_Wellfounded_Omax__ext_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
max_ex20081290693658143list_o: set_Pr7497825696620840711list_o > set_Pr5484948088385368839list_o ).
thf(sy_c_Wellfounded_Omeasure_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
measur8478699559534842796list_o: ( option_list_o > nat ) > set_Pr7497825696620840711list_o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
member_option_list_o: option_list_o > set_option_list_o > $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_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
member1290970816896746895al_nat: produc9220417599568932344al_nat > set_Pr3318565038780223662al_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_Mt__List__Olist_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_J,type,
member2844550799419677264list_o: produc389223140111462439list_o > set_Pr1386088649869593607list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Real__Oreal_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7421688103010841766st_nat: produc8740824290856567677st_nat > set_Pr6676954283489005405st_nat > $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_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
member7408976046919118105al_nat: produc9155354469731034754al_nat > set_Pr836851361167288120al_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
member1589324699396745552list_o: produc4882884732533091879list_o > set_Pr7497825696620840711list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_Mt__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_J,type,
member6168996299790611792list_o: produc1290640719454552615list_o > set_Pr5484948088385368839list_o > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_x,type,
x: option_list_o ).
thf(sy_v_y,type,
y: option_list_o ).
thf(sy_v_z,type,
z: option_list_o ).
% Relevant facts (1273)
thf(fact_0_assms,axiom,
prefix454693708527911765comp_o @ x @ ( prefix5314359684614007693append @ y @ z ) ).
% assms
thf(fact_1_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X: option_list_o,Y: option_list_o] : ( prefix454693708527911765comp_o @ Y @ X ) ) ) ).
% opt_comp_sym
thf(fact_2_opt__comp__append,axiom,
! [X2: option_list_o,Y2: option_list_o,Z: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X2 @ Y2 ) @ Z )
=> ( prefix454693708527911765comp_o @ X2 @ Z ) ) ).
% opt_comp_append
thf(fact_3_opt__comp__def,axiom,
( prefix454693708527911765comp_o
= ( ^ [X: option_list_o,Y: option_list_o] :
( ( prefix8824957607401505554efix_o @ X @ Y )
| ( prefix8824957607401505554efix_o @ Y @ X ) ) ) ) ).
% opt_comp_def
thf(fact_4_opt__append_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
( ( prefix5314359684614007693append @ Uu @ none_list_o )
= none_list_o ) ).
% opt_append.simps(3)
thf(fact_5_opt__append_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
( ( prefix5314359684614007693append @ none_list_o @ Uv )
= none_list_o ) ).
% opt_append.simps(2)
thf(fact_6_opt__prefix_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
~ ( prefix8824957607401505554efix_o @ Uu @ none_list_o ) ).
% opt_prefix.simps(3)
thf(fact_7_opt__prefix_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
~ ( prefix8824957607401505554efix_o @ none_list_o @ Uv ) ).
% opt_prefix.simps(2)
thf(fact_8_opt__append_Osimps_I1_J,axiom,
! [X2: list_o,Y2: list_o] :
( ( prefix5314359684614007693append @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) )
= ( some_list_o @ ( append_o @ X2 @ Y2 ) ) ) ).
% opt_append.simps(1)
thf(fact_9_opt__prefix_Oelims_I2_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat] :
( ( prefix9059114240934526678ix_nat @ X2 @ Xa )
=> ~ ! [X3: list_nat] :
( ( X2
= ( some_list_nat @ X3 ) )
=> ! [Y3: list_nat] :
( ( Xa
= ( some_list_nat @ Y3 ) )
=> ~ ( prefix_nat @ X3 @ Y3 ) ) ) ) ).
% opt_prefix.elims(2)
thf(fact_10_opt__prefix_Oelims_I2_J,axiom,
! [X2: option_list_o,Xa: option_list_o] :
( ( prefix8824957607401505554efix_o @ X2 @ Xa )
=> ~ ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ~ ( prefix_o @ X3 @ Y3 ) ) ) ) ).
% opt_prefix.elims(2)
thf(fact_11_opt__prefix_Osimps_I1_J,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( prefix9059114240934526678ix_nat @ ( some_list_nat @ X2 ) @ ( some_list_nat @ Y2 ) )
= ( prefix_nat @ X2 @ Y2 ) ) ).
% opt_prefix.simps(1)
thf(fact_12_opt__prefix_Osimps_I1_J,axiom,
! [X2: list_o,Y2: list_o] :
( ( prefix8824957607401505554efix_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) )
= ( prefix_o @ X2 @ Y2 ) ) ).
% opt_prefix.simps(1)
thf(fact_13_opt__append_Oelims,axiom,
! [X2: option_list_o,Xa: option_list_o,Y2: option_list_o] :
( ( ( prefix5314359684614007693append @ X2 @ Xa )
= Y2 )
=> ( ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ( Y2
!= ( some_list_o @ ( append_o @ X3 @ Y3 ) ) ) ) )
=> ( ( ( X2 = none_list_o )
=> ( Y2 != none_list_o ) )
=> ~ ( ( Xa = none_list_o )
=> ( Y2 != none_list_o ) ) ) ) ) ).
% opt_append.elims
thf(fact_14_opt__prefix_Oelims_I3_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat] :
( ~ ( prefix9059114240934526678ix_nat @ X2 @ Xa )
=> ( ! [X3: list_nat] :
( ( X2
= ( some_list_nat @ X3 ) )
=> ! [Y3: list_nat] :
( ( Xa
= ( some_list_nat @ Y3 ) )
=> ( prefix_nat @ X3 @ Y3 ) ) )
=> ( ( X2 != none_list_nat )
=> ( Xa = none_list_nat ) ) ) ) ).
% opt_prefix.elims(3)
thf(fact_15_opt__prefix_Oelims_I3_J,axiom,
! [X2: option_list_o,Xa: option_list_o] :
( ~ ( prefix8824957607401505554efix_o @ X2 @ Xa )
=> ( ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ( prefix_o @ X3 @ Y3 ) ) )
=> ( ( X2 != none_list_o )
=> ( Xa = none_list_o ) ) ) ) ).
% opt_prefix.elims(3)
thf(fact_16_opt__prefix_Oelims_I1_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat,Y2: $o] :
( ( ( prefix9059114240934526678ix_nat @ X2 @ Xa )
= Y2 )
=> ( ! [X3: list_nat] :
( ( X2
= ( some_list_nat @ X3 ) )
=> ! [Y3: list_nat] :
( ( Xa
= ( some_list_nat @ Y3 ) )
=> ( Y2
= ( ~ ( prefix_nat @ X3 @ Y3 ) ) ) ) )
=> ( ( ( X2 = none_list_nat )
=> Y2 )
=> ~ ( ( Xa = none_list_nat )
=> Y2 ) ) ) ) ).
% opt_prefix.elims(1)
thf(fact_17_opt__prefix_Oelims_I1_J,axiom,
! [X2: option_list_o,Xa: option_list_o,Y2: $o] :
( ( ( prefix8824957607401505554efix_o @ X2 @ Xa )
= Y2 )
=> ( ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ( Y2
= ( ~ ( prefix_o @ X3 @ Y3 ) ) ) ) )
=> ( ( ( X2 = none_list_o )
=> Y2 )
=> ~ ( ( Xa = none_list_o )
=> Y2 ) ) ) ) ).
% opt_prefix.elims(1)
thf(fact_18_not__Some__eq,axiom,
! [X2: option7375638066962775414st_nat] :
( ( ! [Y: produc8196726482729697190st_nat] :
( X2
!= ( some_P617598486967069697st_nat @ Y ) ) )
= ( X2 = none_P7817216376769613445st_nat ) ) ).
% not_Some_eq
thf(fact_19_not__Some__eq,axiom,
! [X2: option2540781896007660796list_o] :
( ( ! [Y: produc6173137527213542198list_o] :
( X2
!= ( some_P3766925538565183643list_o @ Y ) ) )
= ( X2 = none_P8458051792225729303list_o ) ) ).
% not_Some_eq
thf(fact_20_not__Some__eq,axiom,
! [X2: option_nat] :
( ( ! [Y: nat] :
( X2
!= ( some_nat @ Y ) ) )
= ( X2 = none_nat ) ) ).
% not_Some_eq
thf(fact_21_not__Some__eq,axiom,
! [X2: option_list_o] :
( ( ! [Y: list_o] :
( X2
!= ( some_list_o @ Y ) ) )
= ( X2 = none_list_o ) ) ).
% not_Some_eq
thf(fact_22_not__None__eq,axiom,
! [X2: option7375638066962775414st_nat] :
( ( X2 != none_P7817216376769613445st_nat )
= ( ? [Y: produc8196726482729697190st_nat] :
( X2
= ( some_P617598486967069697st_nat @ Y ) ) ) ) ).
% not_None_eq
thf(fact_23_not__None__eq,axiom,
! [X2: option2540781896007660796list_o] :
( ( X2 != none_P8458051792225729303list_o )
= ( ? [Y: produc6173137527213542198list_o] :
( X2
= ( some_P3766925538565183643list_o @ Y ) ) ) ) ).
% not_None_eq
thf(fact_24_not__None__eq,axiom,
! [X2: option_nat] :
( ( X2 != none_nat )
= ( ? [Y: nat] :
( X2
= ( some_nat @ Y ) ) ) ) ).
% not_None_eq
thf(fact_25_not__None__eq,axiom,
! [X2: option_list_o] :
( ( X2 != none_list_o )
= ( ? [Y: list_o] :
( X2
= ( some_list_o @ Y ) ) ) ) ).
% not_None_eq
thf(fact_26_same__prefix__prefix,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( prefix3150859221070158812at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ ( append985823374593552924at_nat @ Xs @ Zs ) )
= ( prefix3150859221070158812at_nat @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_27_same__prefix__prefix,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ( prefix_option_list_o @ ( append_option_list_o @ Xs @ Ys ) @ ( append_option_list_o @ Xs @ Zs ) )
= ( prefix_option_list_o @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_28_same__prefix__prefix,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ ( append_list_o @ Xs @ Ys ) @ ( append_list_o @ Xs @ Zs ) )
= ( prefix_list_o @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_29_same__prefix__prefix,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys ) @ ( append_list_nat @ Xs @ Zs ) )
= ( prefix_list_nat @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_30_same__prefix__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) )
= ( prefix_nat @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_31_same__prefix__prefix,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Xs @ Zs ) )
= ( prefix_o @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_32_option_Oinject,axiom,
! [X22: produc8196726482729697190st_nat,Y22: produc8196726482729697190st_nat] :
( ( ( some_P617598486967069697st_nat @ X22 )
= ( some_P617598486967069697st_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_33_option_Oinject,axiom,
! [X22: produc6173137527213542198list_o,Y22: produc6173137527213542198list_o] :
( ( ( some_P3766925538565183643list_o @ X22 )
= ( some_P3766925538565183643list_o @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_34_option_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( some_nat @ X22 )
= ( some_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_35_option_Oinject,axiom,
! [X22: list_o,Y22: list_o] :
( ( ( some_list_o @ X22 )
= ( some_list_o @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_36_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_37_prefix__order_Odual__order_Orefl,axiom,
! [A: list_o] : ( prefix_o @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_38_prefix__order_Oorder__refl,axiom,
! [X2: list_nat] : ( prefix_nat @ X2 @ X2 ) ).
% prefix_order.order_refl
thf(fact_39_prefix__order_Oorder__refl,axiom,
! [X2: list_o] : ( prefix_o @ X2 @ X2 ) ).
% prefix_order.order_refl
thf(fact_40_same__append__eq,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= ( append985823374593552924at_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_41_same__append__eq,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ( ( append_option_list_o @ Xs @ Ys )
= ( append_option_list_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_42_same__append__eq,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_43_same__append__eq,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_44_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_45_same__append__eq,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( append_o @ Xs @ Ys )
= ( append_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_46_append__same__eq,axiom,
! [Ys: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Ys @ Xs )
= ( append985823374593552924at_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_47_append__same__eq,axiom,
! [Ys: list_option_list_o,Xs: list_option_list_o,Zs: list_option_list_o] :
( ( ( append_option_list_o @ Ys @ Xs )
= ( append_option_list_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_48_append__same__eq,axiom,
! [Ys: list_list_o,Xs: list_list_o,Zs: list_list_o] :
( ( ( append_list_o @ Ys @ Xs )
= ( append_list_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_49_append__same__eq,axiom,
! [Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Ys @ Xs )
= ( append_list_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_50_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_51_append__same__eq,axiom,
! [Ys: list_o,Xs: list_o,Zs: list_o] :
( ( ( append_o @ Ys @ Xs )
= ( append_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_52_append__assoc,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ Zs )
= ( append985823374593552924at_nat @ Xs @ ( append985823374593552924at_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_53_append__assoc,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ( append_option_list_o @ ( append_option_list_o @ Xs @ Ys ) @ Zs )
= ( append_option_list_o @ Xs @ ( append_option_list_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_54_append__assoc,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( append_list_o @ ( append_list_o @ Xs @ Ys ) @ Zs )
= ( append_list_o @ Xs @ ( append_list_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_55_append__assoc,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs @ Ys ) @ Zs )
= ( append_list_nat @ Xs @ ( append_list_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_56_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_57_append__assoc,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( append_o @ ( append_o @ Xs @ Ys ) @ Zs )
= ( append_o @ Xs @ ( append_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_58_append_Oassoc,axiom,
! [A: list_P6011104703257516679at_nat,B: list_P6011104703257516679at_nat,C: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( append985823374593552924at_nat @ A @ B ) @ C )
= ( append985823374593552924at_nat @ A @ ( append985823374593552924at_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_59_append_Oassoc,axiom,
! [A: list_option_list_o,B: list_option_list_o,C: list_option_list_o] :
( ( append_option_list_o @ ( append_option_list_o @ A @ B ) @ C )
= ( append_option_list_o @ A @ ( append_option_list_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_60_append_Oassoc,axiom,
! [A: list_list_o,B: list_list_o,C: list_list_o] :
( ( append_list_o @ ( append_list_o @ A @ B ) @ C )
= ( append_list_o @ A @ ( append_list_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_61_append_Oassoc,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ A @ B ) @ C )
= ( append_list_nat @ A @ ( append_list_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_62_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C )
= ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_63_append_Oassoc,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( append_o @ ( append_o @ A @ B ) @ C )
= ( append_o @ A @ ( append_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_64_combine__options__cases,axiom,
! [X2: option_list_o,P: option_list_o > option_list_o > $o,Y2: option_list_o] :
( ( ( X2 = none_list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: list_o,B2: list_o] :
( ( X2
= ( some_list_o @ A2 ) )
=> ( ( Y2
= ( some_list_o @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_65_combine__options__cases,axiom,
! [X2: option_nat,P: option_nat > option_nat > $o,Y2: option_nat] :
( ( ( X2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( X2
= ( some_nat @ A2 ) )
=> ( ( Y2
= ( some_nat @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_66_combine__options__cases,axiom,
! [X2: option_list_o,P: option_list_o > option_nat > $o,Y2: option_nat] :
( ( ( X2 = none_list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: list_o,B2: nat] :
( ( X2
= ( some_list_o @ A2 ) )
=> ( ( Y2
= ( some_nat @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_67_combine__options__cases,axiom,
! [X2: option_nat,P: option_nat > option_list_o > $o,Y2: option_list_o] :
( ( ( X2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: nat,B2: list_o] :
( ( X2
= ( some_nat @ A2 ) )
=> ( ( Y2
= ( some_list_o @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_68_combine__options__cases,axiom,
! [X2: option7375638066962775414st_nat,P: option7375638066962775414st_nat > option_nat > $o,Y2: option_nat] :
( ( ( X2 = none_P7817216376769613445st_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: produc8196726482729697190st_nat,B2: nat] :
( ( X2
= ( some_P617598486967069697st_nat @ A2 ) )
=> ( ( Y2
= ( some_nat @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_69_combine__options__cases,axiom,
! [X2: option2540781896007660796list_o,P: option2540781896007660796list_o > option_nat > $o,Y2: option_nat] :
( ( ( X2 = none_P8458051792225729303list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: produc6173137527213542198list_o,B2: nat] :
( ( X2
= ( some_P3766925538565183643list_o @ A2 ) )
=> ( ( Y2
= ( some_nat @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_70_combine__options__cases,axiom,
! [X2: option_nat,P: option_nat > option7375638066962775414st_nat > $o,Y2: option7375638066962775414st_nat] :
( ( ( X2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_P7817216376769613445st_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: nat,B2: produc8196726482729697190st_nat] :
( ( X2
= ( some_nat @ A2 ) )
=> ( ( Y2
= ( some_P617598486967069697st_nat @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_71_combine__options__cases,axiom,
! [X2: option_nat,P: option_nat > option2540781896007660796list_o > $o,Y2: option2540781896007660796list_o] :
( ( ( X2 = none_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_P8458051792225729303list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: nat,B2: produc6173137527213542198list_o] :
( ( X2
= ( some_nat @ A2 ) )
=> ( ( Y2
= ( some_P3766925538565183643list_o @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_72_combine__options__cases,axiom,
! [X2: option_list_o,P: option_list_o > option7375638066962775414st_nat > $o,Y2: option7375638066962775414st_nat] :
( ( ( X2 = none_list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_P7817216376769613445st_nat )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: list_o,B2: produc8196726482729697190st_nat] :
( ( X2
= ( some_list_o @ A2 ) )
=> ( ( Y2
= ( some_P617598486967069697st_nat @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_73_combine__options__cases,axiom,
! [X2: option_list_o,P: option_list_o > option2540781896007660796list_o > $o,Y2: option2540781896007660796list_o] :
( ( ( X2 = none_list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_P8458051792225729303list_o )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A2: list_o,B2: produc6173137527213542198list_o] :
( ( X2
= ( some_list_o @ A2 ) )
=> ( ( Y2
= ( some_P3766925538565183643list_o @ B2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_74_split__option__all,axiom,
( ( ^ [P2: option7375638066962775414st_nat > $o] :
! [X4: option7375638066962775414st_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option7375638066962775414st_nat > $o] :
( ( P3 @ none_P7817216376769613445st_nat )
& ! [X: produc8196726482729697190st_nat] : ( P3 @ ( some_P617598486967069697st_nat @ X ) ) ) ) ) ).
% split_option_all
thf(fact_75_split__option__all,axiom,
( ( ^ [P2: option2540781896007660796list_o > $o] :
! [X4: option2540781896007660796list_o] : ( P2 @ X4 ) )
= ( ^ [P3: option2540781896007660796list_o > $o] :
( ( P3 @ none_P8458051792225729303list_o )
& ! [X: produc6173137527213542198list_o] : ( P3 @ ( some_P3766925538565183643list_o @ X ) ) ) ) ) ).
% split_option_all
thf(fact_76_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% split_option_all
thf(fact_77_split__option__all,axiom,
( ( ^ [P2: option_list_o > $o] :
! [X4: option_list_o] : ( P2 @ X4 ) )
= ( ^ [P3: option_list_o > $o] :
( ( P3 @ none_list_o )
& ! [X: list_o] : ( P3 @ ( some_list_o @ X ) ) ) ) ) ).
% split_option_all
thf(fact_78_append__eq__appendI,axiom,
! [Xs: list_P6011104703257516679at_nat,Xs1: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Us: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append985823374593552924at_nat @ Xs1 @ Us ) )
=> ( ( append985823374593552924at_nat @ Xs @ Ys )
= ( append985823374593552924at_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_79_append__eq__appendI,axiom,
! [Xs: list_option_list_o,Xs1: list_option_list_o,Zs: list_option_list_o,Ys: list_option_list_o,Us: list_option_list_o] :
( ( ( append_option_list_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_option_list_o @ Xs1 @ Us ) )
=> ( ( append_option_list_o @ Xs @ Ys )
= ( append_option_list_o @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_80_append__eq__appendI,axiom,
! [Xs: list_list_o,Xs1: list_list_o,Zs: list_list_o,Ys: list_list_o,Us: list_list_o] :
( ( ( append_list_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_o @ Xs1 @ Us ) )
=> ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_81_append__eq__appendI,axiom,
! [Xs: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys: list_list_nat,Us: list_list_nat] :
( ( ( append_list_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_nat @ Xs1 @ Us ) )
=> ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_82_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_83_append__eq__appendI,axiom,
! [Xs: list_o,Xs1: list_o,Zs: list_o,Ys: list_o,Us: list_o] :
( ( ( append_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_o @ Xs1 @ Us ) )
=> ( ( append_o @ Xs @ Ys )
= ( append_o @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_84_append__eq__append__conv2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ts: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= ( append985823374593552924at_nat @ Zs @ Ts ) )
= ( ? [Us2: list_P6011104703257516679at_nat] :
( ( ( Xs
= ( append985823374593552924at_nat @ Zs @ Us2 ) )
& ( ( append985823374593552924at_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append985823374593552924at_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append985823374593552924at_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_85_append__eq__append__conv2,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o,Ts: list_option_list_o] :
( ( ( append_option_list_o @ Xs @ Ys )
= ( append_option_list_o @ Zs @ Ts ) )
= ( ? [Us2: list_option_list_o] :
( ( ( Xs
= ( append_option_list_o @ Zs @ Us2 ) )
& ( ( append_option_list_o @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_option_list_o @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_option_list_o @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_86_append__eq__append__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o,Ts: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs @ Ts ) )
= ( ? [Us2: list_list_o] :
( ( ( Xs
= ( append_list_o @ Zs @ Us2 ) )
& ( ( append_list_o @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_o @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_list_o @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_87_append__eq__append__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Zs @ Ts ) )
= ( ? [Us2: list_list_nat] :
( ( ( Xs
= ( append_list_nat @ Zs @ Us2 ) )
& ( ( append_list_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_88_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_89_append__eq__append__conv2,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o,Ts: list_o] :
( ( ( append_o @ Xs @ Ys )
= ( append_o @ Zs @ Ts ) )
= ( ? [Us2: list_o] :
( ( ( Xs
= ( append_o @ Zs @ Us2 ) )
& ( ( append_o @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_o @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_o @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_90_prefix__order_Otrans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_91_prefix__order_Otrans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( prefix_o @ A @ B )
=> ( ( prefix_o @ B @ C )
=> ( prefix_o @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_92_prefix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: list_nat,B3: list_nat] :
( ( prefix_nat @ A3 @ B3 )
& ( prefix_nat @ B3 @ A3 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_93_prefix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_o,Z2: list_o] : ( Y4 = Z2 ) )
= ( ^ [A3: list_o,B3: list_o] :
( ( prefix_o @ A3 @ B3 )
& ( prefix_o @ B3 @ A3 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_94_prefix__order_Oantisym,axiom,
! [A: list_nat,B: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_95_prefix__order_Oantisym,axiom,
! [A: list_o,B: list_o] :
( ( prefix_o @ A @ B )
=> ( ( prefix_o @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_96_prefix__order_Oeq__refl,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( X2 = Y2 )
=> ( prefix_nat @ X2 @ Y2 ) ) ).
% prefix_order.eq_refl
thf(fact_97_prefix__order_Oeq__refl,axiom,
! [X2: list_o,Y2: list_o] :
( ( X2 = Y2 )
=> ( prefix_o @ X2 @ Y2 ) ) ).
% prefix_order.eq_refl
thf(fact_98_prefix__order_Oorder__trans,axiom,
! [X2: list_nat,Y2: list_nat,Z: list_nat] :
( ( prefix_nat @ X2 @ Y2 )
=> ( ( prefix_nat @ Y2 @ Z )
=> ( prefix_nat @ X2 @ Z ) ) ) ).
% prefix_order.order_trans
thf(fact_99_prefix__order_Oorder__trans,axiom,
! [X2: list_o,Y2: list_o,Z: list_o] :
( ( prefix_o @ X2 @ Y2 )
=> ( ( prefix_o @ Y2 @ Z )
=> ( prefix_o @ X2 @ Z ) ) ) ).
% prefix_order.order_trans
thf(fact_100_prefix__order_Oantisym__conv,axiom,
! [Y2: list_nat,X2: list_nat] :
( ( prefix_nat @ Y2 @ X2 )
=> ( ( prefix_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% prefix_order.antisym_conv
thf(fact_101_prefix__order_Oantisym__conv,axiom,
! [Y2: list_o,X2: list_o] :
( ( prefix_o @ Y2 @ X2 )
=> ( ( prefix_o @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% prefix_order.antisym_conv
thf(fact_102_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) )
= ( ^ [X: list_nat,Y: list_nat] :
( ( prefix_nat @ X @ Y )
& ( prefix_nat @ Y @ X ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_103_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_o,Z2: list_o] : ( Y4 = Z2 ) )
= ( ^ [X: list_o,Y: list_o] :
( ( prefix_o @ X @ Y )
& ( prefix_o @ Y @ X ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_104_prefix__order_Oorder__antisym,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( prefix_nat @ X2 @ Y2 )
=> ( ( prefix_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% prefix_order.order_antisym
thf(fact_105_prefix__order_Oorder__antisym,axiom,
! [X2: list_o,Y2: list_o] :
( ( prefix_o @ X2 @ Y2 )
=> ( ( prefix_o @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% prefix_order.order_antisym
thf(fact_106_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_107_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( A = B )
=> ( ( prefix_o @ B @ C )
=> ( prefix_o @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_108_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( B = C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_109_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( prefix_o @ A @ B )
=> ( ( B = C )
=> ( prefix_o @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_110_prefix__order_Odual__order_Otrans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ C @ B )
=> ( prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_111_prefix__order_Odual__order_Otrans,axiom,
! [B: list_o,A: list_o,C: list_o] :
( ( prefix_o @ B @ A )
=> ( ( prefix_o @ C @ B )
=> ( prefix_o @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_112_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: list_nat,B3: list_nat] :
( ( prefix_nat @ B3 @ A3 )
& ( prefix_nat @ A3 @ B3 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_113_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_o,Z2: list_o] : ( Y4 = Z2 ) )
= ( ^ [A3: list_o,B3: list_o] :
( ( prefix_o @ B3 @ A3 )
& ( prefix_o @ A3 @ B3 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_114_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_nat,A: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_115_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_o,A: list_o] :
( ( prefix_o @ B @ A )
=> ( ( prefix_o @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_116_prefix__same__cases,axiom,
! [Xs_1: list_nat,Ys: list_nat,Xs_2: list_nat] :
( ( prefix_nat @ Xs_1 @ Ys )
=> ( ( prefix_nat @ Xs_2 @ Ys )
=> ( ( prefix_nat @ Xs_1 @ Xs_2 )
| ( prefix_nat @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_117_prefix__same__cases,axiom,
! [Xs_1: list_o,Ys: list_o,Xs_2: list_o] :
( ( prefix_o @ Xs_1 @ Ys )
=> ( ( prefix_o @ Xs_2 @ Ys )
=> ( ( prefix_o @ Xs_1 @ Xs_2 )
| ( prefix_o @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_118_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_119_mem__Collect__eq,axiom,
! [A: list_o,P: list_o > $o] :
( ( member_list_o @ A @ ( collect_list_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A4: set_list_o] :
( ( collect_list_o
@ ^ [X: list_o] : ( member_list_o @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_125_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_126_Collect__mem__eq,axiom,
! [A4: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_128_prefixE,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( prefix3150859221070158812at_nat @ Xs @ Ys )
=> ~ ! [Zs2: list_P6011104703257516679at_nat] :
( Ys
!= ( append985823374593552924at_nat @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_129_prefixE,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( prefix_option_list_o @ Xs @ Ys )
=> ~ ! [Zs2: list_option_list_o] :
( Ys
!= ( append_option_list_o @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_130_prefixE,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( prefix_list_o @ Xs @ Ys )
=> ~ ! [Zs2: list_list_o] :
( Ys
!= ( append_list_o @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_131_prefixE,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys )
=> ~ ! [Zs2: list_list_nat] :
( Ys
!= ( append_list_nat @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_132_prefixE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ~ ! [Zs2: list_nat] :
( Ys
!= ( append_nat @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_133_prefixE,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ~ ! [Zs2: list_o] :
( Ys
!= ( append_o @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_134_prefixI,axiom,
! [Ys: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( Ys
= ( append985823374593552924at_nat @ Xs @ Zs ) )
=> ( prefix3150859221070158812at_nat @ Xs @ Ys ) ) ).
% prefixI
thf(fact_135_prefixI,axiom,
! [Ys: list_option_list_o,Xs: list_option_list_o,Zs: list_option_list_o] :
( ( Ys
= ( append_option_list_o @ Xs @ Zs ) )
=> ( prefix_option_list_o @ Xs @ Ys ) ) ).
% prefixI
thf(fact_136_prefixI,axiom,
! [Ys: list_list_o,Xs: list_list_o,Zs: list_list_o] :
( ( Ys
= ( append_list_o @ Xs @ Zs ) )
=> ( prefix_list_o @ Xs @ Ys ) ) ).
% prefixI
thf(fact_137_prefixI,axiom,
! [Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( Ys
= ( append_list_nat @ Xs @ Zs ) )
=> ( prefix_list_nat @ Xs @ Ys ) ) ).
% prefixI
thf(fact_138_prefixI,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( Ys
= ( append_nat @ Xs @ Zs ) )
=> ( prefix_nat @ Xs @ Ys ) ) ).
% prefixI
thf(fact_139_prefixI,axiom,
! [Ys: list_o,Xs: list_o,Zs: list_o] :
( ( Ys
= ( append_o @ Xs @ Zs ) )
=> ( prefix_o @ Xs @ Ys ) ) ).
% prefixI
thf(fact_140_prefix__def,axiom,
( prefix3150859221070158812at_nat
= ( ^ [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
? [Zs3: list_P6011104703257516679at_nat] :
( Ys2
= ( append985823374593552924at_nat @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_141_prefix__def,axiom,
( prefix_option_list_o
= ( ^ [Xs2: list_option_list_o,Ys2: list_option_list_o] :
? [Zs3: list_option_list_o] :
( Ys2
= ( append_option_list_o @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_142_prefix__def,axiom,
( prefix_list_o
= ( ^ [Xs2: list_list_o,Ys2: list_list_o] :
? [Zs3: list_list_o] :
( Ys2
= ( append_list_o @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_143_prefix__def,axiom,
( prefix_list_nat
= ( ^ [Xs2: list_list_nat,Ys2: list_list_nat] :
? [Zs3: list_list_nat] :
( Ys2
= ( append_list_nat @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_144_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
? [Zs3: list_nat] :
( Ys2
= ( append_nat @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_145_prefix__def,axiom,
( prefix_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
? [Zs3: list_o] :
( Ys2
= ( append_o @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_146_prefix__append,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( prefix3150859221070158812at_nat @ Xs @ ( append985823374593552924at_nat @ Ys @ Zs ) )
= ( ( prefix3150859221070158812at_nat @ Xs @ Ys )
| ? [Us2: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Ys @ Us2 ) )
& ( prefix3150859221070158812at_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_147_prefix__append,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ( prefix_option_list_o @ Xs @ ( append_option_list_o @ Ys @ Zs ) )
= ( ( prefix_option_list_o @ Xs @ Ys )
| ? [Us2: list_option_list_o] :
( ( Xs
= ( append_option_list_o @ Ys @ Us2 ) )
& ( prefix_option_list_o @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_148_prefix__append,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ Xs @ ( append_list_o @ Ys @ Zs ) )
= ( ( prefix_list_o @ Xs @ Ys )
| ? [Us2: list_list_o] :
( ( Xs
= ( append_list_o @ Ys @ Us2 ) )
& ( prefix_list_o @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_149_prefix__append,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys @ Zs ) )
= ( ( prefix_list_nat @ Xs @ Ys )
| ? [Us2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ Us2 ) )
& ( prefix_list_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_150_prefix__append,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ Zs ) )
= ( ( prefix_nat @ Xs @ Ys )
| ? [Us2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ Us2 ) )
& ( prefix_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_151_prefix__append,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ Xs @ ( append_o @ Ys @ Zs ) )
= ( ( prefix_o @ Xs @ Ys )
| ? [Us2: list_o] :
( ( Xs
= ( append_o @ Ys @ Us2 ) )
& ( prefix_o @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_152_prefix__prefix,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( prefix3150859221070158812at_nat @ Xs @ Ys )
=> ( prefix3150859221070158812at_nat @ Xs @ ( append985823374593552924at_nat @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_153_prefix__prefix,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ( prefix_option_list_o @ Xs @ Ys )
=> ( prefix_option_list_o @ Xs @ ( append_option_list_o @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_154_prefix__prefix,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ Xs @ Ys )
=> ( prefix_list_o @ Xs @ ( append_list_o @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_155_prefix__prefix,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys )
=> ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_156_prefix__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( prefix_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_157_prefix__prefix,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( prefix_o @ Xs @ ( append_o @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_158_append__prefixD,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( prefix3150859221070158812at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ Zs )
=> ( prefix3150859221070158812at_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_159_append__prefixD,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ( prefix_option_list_o @ ( append_option_list_o @ Xs @ Ys ) @ Zs )
=> ( prefix_option_list_o @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_160_append__prefixD,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ ( append_list_o @ Xs @ Ys ) @ Zs )
=> ( prefix_list_o @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_161_append__prefixD,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys ) @ Zs )
=> ( prefix_list_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_162_append__prefixD,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
=> ( prefix_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_163_append__prefixD,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ Zs )
=> ( prefix_o @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_164_option_Odistinct_I1_J,axiom,
! [X22: produc8196726482729697190st_nat] :
( none_P7817216376769613445st_nat
!= ( some_P617598486967069697st_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_165_option_Odistinct_I1_J,axiom,
! [X22: produc6173137527213542198list_o] :
( none_P8458051792225729303list_o
!= ( some_P3766925538565183643list_o @ X22 ) ) ).
% option.distinct(1)
thf(fact_166_option_Odistinct_I1_J,axiom,
! [X22: nat] :
( none_nat
!= ( some_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_167_option_Odistinct_I1_J,axiom,
! [X22: list_o] :
( none_list_o
!= ( some_list_o @ X22 ) ) ).
% option.distinct(1)
thf(fact_168_option_OdiscI,axiom,
! [Option: option7375638066962775414st_nat,X22: produc8196726482729697190st_nat] :
( ( Option
= ( some_P617598486967069697st_nat @ X22 ) )
=> ( Option != none_P7817216376769613445st_nat ) ) ).
% option.discI
thf(fact_169_option_OdiscI,axiom,
! [Option: option2540781896007660796list_o,X22: produc6173137527213542198list_o] :
( ( Option
= ( some_P3766925538565183643list_o @ X22 ) )
=> ( Option != none_P8458051792225729303list_o ) ) ).
% option.discI
thf(fact_170_option_OdiscI,axiom,
! [Option: option_nat,X22: nat] :
( ( Option
= ( some_nat @ X22 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_171_option_OdiscI,axiom,
! [Option: option_list_o,X22: list_o] :
( ( Option
= ( some_list_o @ X22 ) )
=> ( Option != none_list_o ) ) ).
% option.discI
thf(fact_172_option_Oexhaust,axiom,
! [Y2: option7375638066962775414st_nat] :
( ( Y2 != none_P7817216376769613445st_nat )
=> ~ ! [X23: produc8196726482729697190st_nat] :
( Y2
!= ( some_P617598486967069697st_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_173_option_Oexhaust,axiom,
! [Y2: option2540781896007660796list_o] :
( ( Y2 != none_P8458051792225729303list_o )
=> ~ ! [X23: produc6173137527213542198list_o] :
( Y2
!= ( some_P3766925538565183643list_o @ X23 ) ) ) ).
% option.exhaust
thf(fact_174_option_Oexhaust,axiom,
! [Y2: option_nat] :
( ( Y2 != none_nat )
=> ~ ! [X23: nat] :
( Y2
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_175_option_Oexhaust,axiom,
! [Y2: option_list_o] :
( ( Y2 != none_list_o )
=> ~ ! [X23: list_o] :
( Y2
!= ( some_list_o @ X23 ) ) ) ).
% option.exhaust
thf(fact_176_split__option__ex,axiom,
( ( ^ [P2: option7375638066962775414st_nat > $o] :
? [X4: option7375638066962775414st_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option7375638066962775414st_nat > $o] :
( ( P3 @ none_P7817216376769613445st_nat )
| ? [X: produc8196726482729697190st_nat] : ( P3 @ ( some_P617598486967069697st_nat @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_177_split__option__ex,axiom,
( ( ^ [P2: option2540781896007660796list_o > $o] :
? [X4: option2540781896007660796list_o] : ( P2 @ X4 ) )
= ( ^ [P3: option2540781896007660796list_o > $o] :
( ( P3 @ none_P8458051792225729303list_o )
| ? [X: produc6173137527213542198list_o] : ( P3 @ ( some_P3766925538565183643list_o @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_178_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_179_split__option__ex,axiom,
( ( ^ [P2: option_list_o > $o] :
? [X4: option_list_o] : ( P2 @ X4 ) )
= ( ^ [P3: option_list_o > $o] :
( ( P3 @ none_list_o )
| ? [X: list_o] : ( P3 @ ( some_list_o @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_180_option_Ocollapse,axiom,
! [Option: option7375638066962775414st_nat] :
( ( Option != none_P7817216376769613445st_nat )
=> ( ( some_P617598486967069697st_nat @ ( the_Pr5986943824442002450st_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_181_option_Ocollapse,axiom,
! [Option: option2540781896007660796list_o] :
( ( Option != none_P8458051792225729303list_o )
=> ( ( some_P3766925538565183643list_o @ ( the_Pr4888239224808105418list_o @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_182_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( ( some_nat @ ( the_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_183_option_Ocollapse,axiom,
! [Option: option_list_o] :
( ( Option != none_list_o )
=> ( ( some_list_o @ ( the_list_o @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_184_same__prefix__nil,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( prefix3150859221070158812at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_Pr5478986624290739719at_nat ) ) ).
% same_prefix_nil
thf(fact_185_same__prefix__nil,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( prefix_option_list_o @ ( append_option_list_o @ Xs @ Ys ) @ Xs )
= ( Ys = nil_option_list_o ) ) ).
% same_prefix_nil
thf(fact_186_same__prefix__nil,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( prefix_list_o @ ( append_list_o @ Xs @ Ys ) @ Xs )
= ( Ys = nil_list_o ) ) ).
% same_prefix_nil
thf(fact_187_same__prefix__nil,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_list_nat ) ) ).
% same_prefix_nil
thf(fact_188_same__prefix__nil,axiom,
! [Xs: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat] :
( ( prefix2577173861203319680_o_nat @ ( append2548158351732429248_o_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_op686717097995950635_o_nat ) ) ).
% same_prefix_nil
thf(fact_189_same__prefix__nil,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_nat ) ) ).
% same_prefix_nil
thf(fact_190_same__prefix__nil,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ Xs )
= ( Ys = nil_o ) ) ).
% same_prefix_nil
thf(fact_191_opt__append_Ocases,axiom,
! [X2: produc4882884732533091879list_o] :
( ! [X3: list_o,Y3: list_o] :
( X2
!= ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) )
=> ( ! [Uv2: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ none_list_o @ Uv2 ) )
=> ~ ! [Uu2: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ Uu2 @ none_list_o ) ) ) ) ).
% opt_append.cases
thf(fact_192_bind__split,axiom,
! [P: option_list_o > $o,M: option7375638066962775414st_nat,F: produc8196726482729697190st_nat > option_list_o] :
( ( P @ ( bind_P2299425358426170424list_o @ M @ F ) )
= ( ( ( M = none_P7817216376769613445st_nat )
=> ( P @ none_list_o ) )
& ! [V: produc8196726482729697190st_nat] :
( ( M
= ( some_P617598486967069697st_nat @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_193_bind__split,axiom,
! [P: option_list_o > $o,M: option2540781896007660796list_o,F: produc6173137527213542198list_o > option_list_o] :
( ( P @ ( bind_P2403420563366411686list_o @ M @ F ) )
= ( ( ( M = none_P8458051792225729303list_o )
=> ( P @ none_list_o ) )
& ! [V: produc6173137527213542198list_o] :
( ( M
= ( some_P3766925538565183643list_o @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_194_bind__split,axiom,
! [P: option_list_o > $o,M: option_nat,F: nat > option_list_o] :
( ( P @ ( bind_nat_list_o2 @ M @ F ) )
= ( ( ( M = none_nat )
=> ( P @ none_list_o ) )
& ! [V: nat] :
( ( M
= ( some_nat @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_195_bind__split,axiom,
! [P: option_list_o > $o,M: option_list_o,F: list_o > option_list_o] :
( ( P @ ( bind_list_o_list_o @ M @ F ) )
= ( ( ( M = none_list_o )
=> ( P @ none_list_o ) )
& ! [V: list_o] :
( ( M
= ( some_list_o @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_196_bind__split__asm,axiom,
! [P: option_list_o > $o,M: option7375638066962775414st_nat,F: produc8196726482729697190st_nat > option_list_o] :
( ( P @ ( bind_P2299425358426170424list_o @ M @ F ) )
= ( ~ ( ( ( M = none_P7817216376769613445st_nat )
& ~ ( P @ none_list_o ) )
| ? [X: produc8196726482729697190st_nat] :
( ( M
= ( some_P617598486967069697st_nat @ X ) )
& ~ ( P @ ( F @ X ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_197_bind__split__asm,axiom,
! [P: option_list_o > $o,M: option2540781896007660796list_o,F: produc6173137527213542198list_o > option_list_o] :
( ( P @ ( bind_P2403420563366411686list_o @ M @ F ) )
= ( ~ ( ( ( M = none_P8458051792225729303list_o )
& ~ ( P @ none_list_o ) )
| ? [X: produc6173137527213542198list_o] :
( ( M
= ( some_P3766925538565183643list_o @ X ) )
& ~ ( P @ ( F @ X ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_198_bind__split__asm,axiom,
! [P: option_list_o > $o,M: option_nat,F: nat > option_list_o] :
( ( P @ ( bind_nat_list_o2 @ M @ F ) )
= ( ~ ( ( ( M = none_nat )
& ~ ( P @ none_list_o ) )
| ? [X: nat] :
( ( M
= ( some_nat @ X ) )
& ~ ( P @ ( F @ X ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_199_bind__split__asm,axiom,
! [P: option_list_o > $o,M: option_list_o,F: list_o > option_list_o] :
( ( P @ ( bind_list_o_list_o @ M @ F ) )
= ( ~ ( ( ( M = none_list_o )
& ~ ( P @ none_list_o ) )
| ? [X: list_o] :
( ( M
= ( some_list_o @ X ) )
& ~ ( P @ ( F @ X ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_200_option_Oexhaust__sel,axiom,
! [Option: option7375638066962775414st_nat] :
( ( Option != none_P7817216376769613445st_nat )
=> ( Option
= ( some_P617598486967069697st_nat @ ( the_Pr5986943824442002450st_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_201_option_Oexhaust__sel,axiom,
! [Option: option2540781896007660796list_o] :
( ( Option != none_P8458051792225729303list_o )
=> ( Option
= ( some_P3766925538565183643list_o @ ( the_Pr4888239224808105418list_o @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_202_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option
= ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_203_option_Oexhaust__sel,axiom,
! [Option: option_list_o] :
( ( Option != none_list_o )
=> ( Option
= ( some_list_o @ ( the_list_o @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_204_opt__prefix_Ocases,axiom,
! [X2: produc4882884732533091879list_o] :
( ! [X3: list_o,Y3: list_o] :
( X2
!= ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) )
=> ( ! [Uv2: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ none_list_o @ Uv2 ) )
=> ~ ! [Uu2: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ Uu2 @ none_list_o ) ) ) ) ).
% opt_prefix.cases
thf(fact_205_combine__options__simps_I3_J,axiom,
! [F: produc8196726482729697190st_nat > produc8196726482729697190st_nat > produc8196726482729697190st_nat,A: produc8196726482729697190st_nat,B: produc8196726482729697190st_nat] :
( ( combin3299571334482030951st_nat @ F @ ( some_P617598486967069697st_nat @ A ) @ ( some_P617598486967069697st_nat @ B ) )
= ( some_P617598486967069697st_nat @ ( F @ A @ B ) ) ) ).
% combine_options_simps(3)
thf(fact_206_combine__options__simps_I3_J,axiom,
! [F: produc6173137527213542198list_o > produc6173137527213542198list_o > produc6173137527213542198list_o,A: produc6173137527213542198list_o,B: produc6173137527213542198list_o] :
( ( combin7320422588414878389list_o @ F @ ( some_P3766925538565183643list_o @ A ) @ ( some_P3766925538565183643list_o @ B ) )
= ( some_P3766925538565183643list_o @ ( F @ A @ B ) ) ) ).
% combine_options_simps(3)
thf(fact_207_combine__options__simps_I3_J,axiom,
! [F: nat > nat > nat,A: nat,B: nat] :
( ( combine_options_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
= ( some_nat @ ( F @ A @ B ) ) ) ).
% combine_options_simps(3)
thf(fact_208_combine__options__simps_I3_J,axiom,
! [F: list_o > list_o > list_o,A: list_o,B: list_o] :
( ( combin4983522934621174429list_o @ F @ ( some_list_o @ A ) @ ( some_list_o @ B ) )
= ( some_list_o @ ( F @ A @ B ) ) ) ).
% combine_options_simps(3)
thf(fact_209_combine__options__simps_I2_J,axiom,
! [F: list_o > list_o > list_o,X2: option_list_o] :
( ( combin4983522934621174429list_o @ F @ X2 @ none_list_o )
= X2 ) ).
% combine_options_simps(2)
thf(fact_210_append__is__Nil__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
& ( Ys = nil_Pr5478986624290739719at_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_211_append__is__Nil__conv,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( ( append_option_list_o @ Xs @ Ys )
= nil_option_list_o )
= ( ( Xs = nil_option_list_o )
& ( Ys = nil_option_list_o ) ) ) ).
% append_is_Nil_conv
thf(fact_212_append__is__Nil__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= nil_list_o )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% append_is_Nil_conv
thf(fact_213_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_214_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_215_append__is__Nil__conv,axiom,
! [Xs: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat] :
( ( ( append2548158351732429248_o_nat @ Xs @ Ys )
= nil_op686717097995950635_o_nat )
= ( ( Xs = nil_op686717097995950635_o_nat )
& ( Ys = nil_op686717097995950635_o_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_216_append__is__Nil__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= nil_o )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% append_is_Nil_conv
thf(fact_217_Nil__is__append__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( append985823374593552924at_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
& ( Ys = nil_Pr5478986624290739719at_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_218_Nil__is__append__conv,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( nil_option_list_o
= ( append_option_list_o @ Xs @ Ys ) )
= ( ( Xs = nil_option_list_o )
& ( Ys = nil_option_list_o ) ) ) ).
% Nil_is_append_conv
thf(fact_219_Nil__is__append__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( nil_list_o
= ( append_list_o @ Xs @ Ys ) )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% Nil_is_append_conv
thf(fact_220_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys ) )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_221_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_222_Nil__is__append__conv,axiom,
! [Xs: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat] :
( ( nil_op686717097995950635_o_nat
= ( append2548158351732429248_o_nat @ Xs @ Ys ) )
= ( ( Xs = nil_op686717097995950635_o_nat )
& ( Ys = nil_op686717097995950635_o_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_223_Nil__is__append__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( nil_o
= ( append_o @ Xs @ Ys ) )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% Nil_is_append_conv
thf(fact_224_self__append__conv2,axiom,
! [Y2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat] :
( ( Y2
= ( append985823374593552924at_nat @ Xs @ Y2 ) )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv2
thf(fact_225_self__append__conv2,axiom,
! [Y2: list_option_list_o,Xs: list_option_list_o] :
( ( Y2
= ( append_option_list_o @ Xs @ Y2 ) )
= ( Xs = nil_option_list_o ) ) ).
% self_append_conv2
thf(fact_226_self__append__conv2,axiom,
! [Y2: list_list_o,Xs: list_list_o] :
( ( Y2
= ( append_list_o @ Xs @ Y2 ) )
= ( Xs = nil_list_o ) ) ).
% self_append_conv2
thf(fact_227_self__append__conv2,axiom,
! [Y2: list_list_nat,Xs: list_list_nat] :
( ( Y2
= ( append_list_nat @ Xs @ Y2 ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_228_self__append__conv2,axiom,
! [Y2: list_nat,Xs: list_nat] :
( ( Y2
= ( append_nat @ Xs @ Y2 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_229_self__append__conv2,axiom,
! [Y2: list_o8358219866161907755_o_nat,Xs: list_o8358219866161907755_o_nat] :
( ( Y2
= ( append2548158351732429248_o_nat @ Xs @ Y2 ) )
= ( Xs = nil_op686717097995950635_o_nat ) ) ).
% self_append_conv2
thf(fact_230_self__append__conv2,axiom,
! [Y2: list_o,Xs: list_o] :
( ( Y2
= ( append_o @ Xs @ Y2 ) )
= ( Xs = nil_o ) ) ).
% self_append_conv2
thf(fact_231_append__self__conv2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv2
thf(fact_232_append__self__conv2,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( ( append_option_list_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_option_list_o ) ) ).
% append_self_conv2
thf(fact_233_append__self__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_o ) ) ).
% append_self_conv2
thf(fact_234_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_235_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_236_append__self__conv2,axiom,
! [Xs: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat] :
( ( ( append2548158351732429248_o_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_op686717097995950635_o_nat ) ) ).
% append_self_conv2
thf(fact_237_append__self__conv2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_o ) ) ).
% append_self_conv2
thf(fact_238_self__append__conv,axiom,
! [Y2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( Y2
= ( append985823374593552924at_nat @ Y2 @ Ys ) )
= ( Ys = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv
thf(fact_239_self__append__conv,axiom,
! [Y2: list_option_list_o,Ys: list_option_list_o] :
( ( Y2
= ( append_option_list_o @ Y2 @ Ys ) )
= ( Ys = nil_option_list_o ) ) ).
% self_append_conv
thf(fact_240_self__append__conv,axiom,
! [Y2: list_list_o,Ys: list_list_o] :
( ( Y2
= ( append_list_o @ Y2 @ Ys ) )
= ( Ys = nil_list_o ) ) ).
% self_append_conv
thf(fact_241_self__append__conv,axiom,
! [Y2: list_list_nat,Ys: list_list_nat] :
( ( Y2
= ( append_list_nat @ Y2 @ Ys ) )
= ( Ys = nil_list_nat ) ) ).
% self_append_conv
thf(fact_242_self__append__conv,axiom,
! [Y2: list_nat,Ys: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_243_self__append__conv,axiom,
! [Y2: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat] :
( ( Y2
= ( append2548158351732429248_o_nat @ Y2 @ Ys ) )
= ( Ys = nil_op686717097995950635_o_nat ) ) ).
% self_append_conv
thf(fact_244_self__append__conv,axiom,
! [Y2: list_o,Ys: list_o] :
( ( Y2
= ( append_o @ Y2 @ Ys ) )
= ( Ys = nil_o ) ) ).
% self_append_conv
thf(fact_245_append__self__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv
thf(fact_246_append__self__conv,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( ( append_option_list_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_option_list_o ) ) ).
% append_self_conv
thf(fact_247_append__self__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_o ) ) ).
% append_self_conv
thf(fact_248_append__self__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_nat ) ) ).
% append_self_conv
thf(fact_249_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_250_append__self__conv,axiom,
! [Xs: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat] :
( ( ( append2548158351732429248_o_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_op686717097995950635_o_nat ) ) ).
% append_self_conv
thf(fact_251_append__self__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_o ) ) ).
% append_self_conv
thf(fact_252_append__Nil2,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ Xs @ nil_Pr5478986624290739719at_nat )
= Xs ) ).
% append_Nil2
thf(fact_253_append__Nil2,axiom,
! [Xs: list_option_list_o] :
( ( append_option_list_o @ Xs @ nil_option_list_o )
= Xs ) ).
% append_Nil2
thf(fact_254_append__Nil2,axiom,
! [Xs: list_list_o] :
( ( append_list_o @ Xs @ nil_list_o )
= Xs ) ).
% append_Nil2
thf(fact_255_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_256_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_257_append__Nil2,axiom,
! [Xs: list_o8358219866161907755_o_nat] :
( ( append2548158351732429248_o_nat @ Xs @ nil_op686717097995950635_o_nat )
= Xs ) ).
% append_Nil2
thf(fact_258_append__Nil2,axiom,
! [Xs: list_o] :
( ( append_o @ Xs @ nil_o )
= Xs ) ).
% append_Nil2
thf(fact_259_append_Oright__neutral,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ A @ nil_Pr5478986624290739719at_nat )
= A ) ).
% append.right_neutral
thf(fact_260_append_Oright__neutral,axiom,
! [A: list_option_list_o] :
( ( append_option_list_o @ A @ nil_option_list_o )
= A ) ).
% append.right_neutral
thf(fact_261_append_Oright__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ A @ nil_list_o )
= A ) ).
% append.right_neutral
thf(fact_262_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_263_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_264_append_Oright__neutral,axiom,
! [A: list_o8358219866161907755_o_nat] :
( ( append2548158351732429248_o_nat @ A @ nil_op686717097995950635_o_nat )
= A ) ).
% append.right_neutral
thf(fact_265_append_Oright__neutral,axiom,
! [A: list_o] :
( ( append_o @ A @ nil_o )
= A ) ).
% append.right_neutral
thf(fact_266_prefix__Nil,axiom,
! [Xs: list_list_o] :
( ( prefix_list_o @ Xs @ nil_list_o )
= ( Xs = nil_list_o ) ) ).
% prefix_Nil
thf(fact_267_prefix__Nil,axiom,
! [Xs: list_list_nat] :
( ( prefix_list_nat @ Xs @ nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% prefix_Nil
thf(fact_268_prefix__Nil,axiom,
! [Xs: list_o8358219866161907755_o_nat] :
( ( prefix2577173861203319680_o_nat @ Xs @ nil_op686717097995950635_o_nat )
= ( Xs = nil_op686717097995950635_o_nat ) ) ).
% prefix_Nil
thf(fact_269_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_270_prefix__Nil,axiom,
! [Xs: list_o] :
( ( prefix_o @ Xs @ nil_o )
= ( Xs = nil_o ) ) ).
% prefix_Nil
thf(fact_271_prefix__bot_Oextremum__unique,axiom,
! [A: list_list_o] :
( ( prefix_list_o @ A @ nil_list_o )
= ( A = nil_list_o ) ) ).
% prefix_bot.extremum_unique
thf(fact_272_prefix__bot_Oextremum__unique,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
= ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_273_prefix__bot_Oextremum__unique,axiom,
! [A: list_o8358219866161907755_o_nat] :
( ( prefix2577173861203319680_o_nat @ A @ nil_op686717097995950635_o_nat )
= ( A = nil_op686717097995950635_o_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_274_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_275_prefix__bot_Oextremum__unique,axiom,
! [A: list_o] :
( ( prefix_o @ A @ nil_o )
= ( A = nil_o ) ) ).
% prefix_bot.extremum_unique
thf(fact_276_prefix__code_I1_J,axiom,
! [Xs: list_list_o] : ( prefix_list_o @ nil_list_o @ Xs ) ).
% prefix_code(1)
thf(fact_277_prefix__code_I1_J,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% prefix_code(1)
thf(fact_278_prefix__code_I1_J,axiom,
! [Xs: list_o8358219866161907755_o_nat] : ( prefix2577173861203319680_o_nat @ nil_op686717097995950635_o_nat @ Xs ) ).
% prefix_code(1)
thf(fact_279_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_280_prefix__code_I1_J,axiom,
! [Xs: list_o] : ( prefix_o @ nil_o @ Xs ) ).
% prefix_code(1)
thf(fact_281_bind__runit,axiom,
! [X2: option7375638066962775414st_nat] :
( ( bind_P6234047912424235660st_nat @ X2 @ some_P617598486967069697st_nat )
= X2 ) ).
% bind_runit
thf(fact_282_bind__runit,axiom,
! [X2: option2540781896007660796list_o] :
( ( bind_P4009872011549414462list_o @ X2 @ some_P3766925538565183643list_o )
= X2 ) ).
% bind_runit
thf(fact_283_bind__runit,axiom,
! [X2: option_nat] :
( ( bind_nat_nat2 @ X2 @ some_nat )
= X2 ) ).
% bind_runit
thf(fact_284_bind__runit,axiom,
! [X2: option_list_o] :
( ( bind_list_o_list_o @ X2 @ some_list_o )
= X2 ) ).
% bind_runit
thf(fact_285_combine__options__simps_I1_J,axiom,
! [F: list_o > list_o > list_o,Y2: option_list_o] :
( ( combin4983522934621174429list_o @ F @ none_list_o @ Y2 )
= Y2 ) ).
% combine_options_simps(1)
thf(fact_286_bind__option__cong__code,axiom,
! [X2: option_list_o,Y2: option_list_o,F: list_o > option_list_o] :
( ( X2 = Y2 )
=> ( ( bind_list_o_list_o @ X2 @ F )
= ( bind_list_o_list_o @ Y2 @ F ) ) ) ).
% bind_option_cong_code
thf(fact_287_combine__options__assoc,axiom,
! [F: list_o > list_o > list_o,X2: option_list_o,Y2: option_list_o,Z: option_list_o] :
( ! [X3: list_o,Y3: list_o,Z3: list_o] :
( ( F @ ( F @ X3 @ Y3 ) @ Z3 )
= ( F @ X3 @ ( F @ Y3 @ Z3 ) ) )
=> ( ( combin4983522934621174429list_o @ F @ ( combin4983522934621174429list_o @ F @ X2 @ Y2 ) @ Z )
= ( combin4983522934621174429list_o @ F @ X2 @ ( combin4983522934621174429list_o @ F @ Y2 @ Z ) ) ) ) ).
% combine_options_assoc
thf(fact_288_combine__options__commute,axiom,
! [F: list_o > list_o > list_o,X2: option_list_o,Y2: option_list_o] :
( ! [X3: list_o,Y3: list_o] :
( ( F @ X3 @ Y3 )
= ( F @ Y3 @ X3 ) )
=> ( ( combin4983522934621174429list_o @ F @ X2 @ Y2 )
= ( combin4983522934621174429list_o @ F @ Y2 @ X2 ) ) ) ).
% combine_options_commute
thf(fact_289_combine__options__left__commute,axiom,
! [F: list_o > list_o > list_o,Y2: option_list_o,X2: option_list_o,Z: option_list_o] :
( ! [X3: list_o,Y3: list_o] :
( ( F @ X3 @ Y3 )
= ( F @ Y3 @ X3 ) )
=> ( ! [X3: list_o,Y3: list_o,Z3: list_o] :
( ( F @ ( F @ X3 @ Y3 ) @ Z3 )
= ( F @ X3 @ ( F @ Y3 @ Z3 ) ) )
=> ( ( combin4983522934621174429list_o @ F @ Y2 @ ( combin4983522934621174429list_o @ F @ X2 @ Z ) )
= ( combin4983522934621174429list_o @ F @ X2 @ ( combin4983522934621174429list_o @ F @ Y2 @ Z ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_290_bind__eq__None__conv,axiom,
! [A: option_list_o,F: list_o > option_list_o] :
( ( ( bind_list_o_list_o @ A @ F )
= none_list_o )
= ( ( A = none_list_o )
| ( ( F @ ( the_list_o @ A ) )
= none_list_o ) ) ) ).
% bind_eq_None_conv
thf(fact_291_eq__Nil__appendI,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_292_eq__Nil__appendI,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_option_list_o @ nil_option_list_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_293_eq__Nil__appendI,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_o @ nil_list_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_294_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_295_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_296_eq__Nil__appendI,axiom,
! [Xs: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append2548158351732429248_o_nat @ nil_op686717097995950635_o_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_297_eq__Nil__appendI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_o @ nil_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_298_append_Oleft__neutral,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_299_append_Oleft__neutral,axiom,
! [A: list_option_list_o] :
( ( append_option_list_o @ nil_option_list_o @ A )
= A ) ).
% append.left_neutral
thf(fact_300_append_Oleft__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ nil_list_o @ A )
= A ) ).
% append.left_neutral
thf(fact_301_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_302_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_303_append_Oleft__neutral,axiom,
! [A: list_o8358219866161907755_o_nat] :
( ( append2548158351732429248_o_nat @ nil_op686717097995950635_o_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_304_append_Oleft__neutral,axiom,
! [A: list_o] :
( ( append_o @ nil_o @ A )
= A ) ).
% append.left_neutral
thf(fact_305_append__Nil,axiom,
! [Ys: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_306_append__Nil,axiom,
! [Ys: list_option_list_o] :
( ( append_option_list_o @ nil_option_list_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_307_append__Nil,axiom,
! [Ys: list_list_o] :
( ( append_list_o @ nil_list_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_308_append__Nil,axiom,
! [Ys: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_309_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_310_append__Nil,axiom,
! [Ys: list_o8358219866161907755_o_nat] :
( ( append2548158351732429248_o_nat @ nil_op686717097995950635_o_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_311_append__Nil,axiom,
! [Ys: list_o] :
( ( append_o @ nil_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_312_Nil__prefix,axiom,
! [Xs: list_list_o] : ( prefix_list_o @ nil_list_o @ Xs ) ).
% Nil_prefix
thf(fact_313_Nil__prefix,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% Nil_prefix
thf(fact_314_Nil__prefix,axiom,
! [Xs: list_o8358219866161907755_o_nat] : ( prefix2577173861203319680_o_nat @ nil_op686717097995950635_o_nat @ Xs ) ).
% Nil_prefix
thf(fact_315_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_316_Nil__prefix,axiom,
! [Xs: list_o] : ( prefix_o @ nil_o @ Xs ) ).
% Nil_prefix
thf(fact_317_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_list_o] :
( ( prefix_list_o @ A @ nil_list_o )
=> ( A = nil_list_o ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_318_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
=> ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_319_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_o8358219866161907755_o_nat] :
( ( prefix2577173861203319680_o_nat @ A @ nil_op686717097995950635_o_nat )
=> ( A = nil_op686717097995950635_o_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_320_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_321_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_o] :
( ( prefix_o @ A @ nil_o )
=> ( A = nil_o ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_322_prefix__bot_Obot__least,axiom,
! [A: list_list_o] : ( prefix_list_o @ nil_list_o @ A ) ).
% prefix_bot.bot_least
thf(fact_323_prefix__bot_Obot__least,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_324_prefix__bot_Obot__least,axiom,
! [A: list_o8358219866161907755_o_nat] : ( prefix2577173861203319680_o_nat @ nil_op686717097995950635_o_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_325_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_326_prefix__bot_Obot__least,axiom,
! [A: list_o] : ( prefix_o @ nil_o @ A ) ).
% prefix_bot.bot_least
thf(fact_327_option_Oexpand,axiom,
! [Option: option_list_o,Option2: option_list_o] :
( ( ( Option = none_list_o )
= ( Option2 = none_list_o ) )
=> ( ( ( Option != none_list_o )
=> ( ( Option2 != none_list_o )
=> ( ( the_list_o @ Option )
= ( the_list_o @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_328_option_Osel,axiom,
! [X22: produc8196726482729697190st_nat] :
( ( the_Pr5986943824442002450st_nat @ ( some_P617598486967069697st_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_329_option_Osel,axiom,
! [X22: produc6173137527213542198list_o] :
( ( the_Pr4888239224808105418list_o @ ( some_P3766925538565183643list_o @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_330_option_Osel,axiom,
! [X22: nat] :
( ( the_nat @ ( some_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_331_option_Osel,axiom,
! [X22: list_o] :
( ( the_list_o @ ( some_list_o @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_332_bind__eq__Some__conv,axiom,
! [F: option_list_o,G: list_o > option_list_o,X2: list_o] :
( ( ( bind_list_o_list_o @ F @ G )
= ( some_list_o @ X2 ) )
= ( ? [Y: list_o] :
( ( F
= ( some_list_o @ Y ) )
& ( ( G @ Y )
= ( some_list_o @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_333_bind__eq__Some__conv,axiom,
! [F: option_nat,G: nat > option_nat,X2: nat] :
( ( ( bind_nat_nat2 @ F @ G )
= ( some_nat @ X2 ) )
= ( ? [Y: nat] :
( ( F
= ( some_nat @ Y ) )
& ( ( G @ Y )
= ( some_nat @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_334_bind__eq__Some__conv,axiom,
! [F: option_nat,G: nat > option_list_o,X2: list_o] :
( ( ( bind_nat_list_o2 @ F @ G )
= ( some_list_o @ X2 ) )
= ( ? [Y: nat] :
( ( F
= ( some_nat @ Y ) )
& ( ( G @ Y )
= ( some_list_o @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_335_bind__eq__Some__conv,axiom,
! [F: option_list_o,G: list_o > option_nat,X2: nat] :
( ( ( bind_list_o_nat2 @ F @ G )
= ( some_nat @ X2 ) )
= ( ? [Y: list_o] :
( ( F
= ( some_list_o @ Y ) )
& ( ( G @ Y )
= ( some_nat @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_336_bind__eq__Some__conv,axiom,
! [F: option_nat,G: nat > option7375638066962775414st_nat,X2: produc8196726482729697190st_nat] :
( ( ( bind_n8583154692918325046st_nat @ F @ G )
= ( some_P617598486967069697st_nat @ X2 ) )
= ( ? [Y: nat] :
( ( F
= ( some_nat @ Y ) )
& ( ( G @ Y )
= ( some_P617598486967069697st_nat @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_337_bind__eq__Some__conv,axiom,
! [F: option_nat,G: nat > option2540781896007660796list_o,X2: produc6173137527213542198list_o] :
( ( ( bind_n3968521476251553702list_o @ F @ G )
= ( some_P3766925538565183643list_o @ X2 ) )
= ( ? [Y: nat] :
( ( F
= ( some_nat @ Y ) )
& ( ( G @ Y )
= ( some_P3766925538565183643list_o @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_338_bind__eq__Some__conv,axiom,
! [F: option7375638066962775414st_nat,G: produc8196726482729697190st_nat > option_nat,X2: nat] :
( ( ( bind_P6985806841975176630at_nat @ F @ G )
= ( some_nat @ X2 ) )
= ( ? [Y: produc8196726482729697190st_nat] :
( ( F
= ( some_P617598486967069697st_nat @ Y ) )
& ( ( G @ Y )
= ( some_nat @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_339_bind__eq__Some__conv,axiom,
! [F: option2540781896007660796list_o,G: produc6173137527213542198list_o > option_nat,X2: nat] :
( ( ( bind_P7999860724539673800_o_nat @ F @ G )
= ( some_nat @ X2 ) )
= ( ? [Y: produc6173137527213542198list_o] :
( ( F
= ( some_P3766925538565183643list_o @ Y ) )
& ( ( G @ Y )
= ( some_nat @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_340_bind__eq__Some__conv,axiom,
! [F: option7375638066962775414st_nat,G: produc8196726482729697190st_nat > option_list_o,X2: list_o] :
( ( ( bind_P2299425358426170424list_o @ F @ G )
= ( some_list_o @ X2 ) )
= ( ? [Y: produc8196726482729697190st_nat] :
( ( F
= ( some_P617598486967069697st_nat @ Y ) )
& ( ( G @ Y )
= ( some_list_o @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_341_bind__eq__Some__conv,axiom,
! [F: option2540781896007660796list_o,G: produc6173137527213542198list_o > option_list_o,X2: list_o] :
( ( ( bind_P2403420563366411686list_o @ F @ G )
= ( some_list_o @ X2 ) )
= ( ? [Y: produc6173137527213542198list_o] :
( ( F
= ( some_P3766925538565183643list_o @ Y ) )
& ( ( G @ Y )
= ( some_list_o @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_342_Option_Obind__cong,axiom,
! [X2: option_list_o,Y2: option_list_o,F: list_o > option_list_o,G: list_o > option_list_o] :
( ( X2 = Y2 )
=> ( ! [A2: list_o] :
( ( Y2
= ( some_list_o @ A2 ) )
=> ( ( F @ A2 )
= ( G @ A2 ) ) )
=> ( ( bind_list_o_list_o @ X2 @ F )
= ( bind_list_o_list_o @ Y2 @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_343_bind_Obind__lzero,axiom,
! [F: list_o > option_list_o] :
( ( bind_list_o_list_o @ none_list_o @ F )
= none_list_o ) ).
% bind.bind_lzero
thf(fact_344_bind_Obind__lunit,axiom,
! [X2: list_o,F: list_o > option_list_o] :
( ( bind_list_o_list_o @ ( some_list_o @ X2 ) @ F )
= ( F @ X2 ) ) ).
% bind.bind_lunit
thf(fact_345_old_Oprod_Oinject,axiom,
! [A: nat,B: list_nat,A5: nat,B4: list_nat] :
( ( ( produc8282810413953273033st_nat @ A @ B )
= ( produc8282810413953273033st_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_346_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A5: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_347_old_Oprod_Oinject,axiom,
! [A: nat > nat,B: produc1828647624359046049st_nat,A5: nat > nat,B4: produc1828647624359046049st_nat] :
( ( ( produc4626581765195395529st_nat @ A @ B )
= ( produc4626581765195395529st_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_348_old_Oprod_Oinject,axiom,
! [A: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B: produc5963376359100343987al_nat,A5: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B4: produc5963376359100343987al_nat] :
( ( ( produc5211167162158505542al_nat @ A @ B )
= ( produc5211167162158505542al_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_349_old_Oprod_Oinject,axiom,
! [A: nat > nat > $o,B: list_nat,A5: nat > nat > $o,B4: list_nat] :
( ( ( produc4727192421694094319st_nat @ A @ B )
= ( produc4727192421694094319st_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_350_old_Oprod_Oinject,axiom,
! [A: option_list_o,B: option_list_o,A5: option_list_o,B4: option_list_o] :
( ( ( produc5745850523778858007list_o @ A @ B )
= ( produc5745850523778858007list_o @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_351_old_Oprod_Oinject,axiom,
! [A: nat,B: produc3741383161447143261al_nat,A5: nat,B4: produc3741383161447143261al_nat] :
( ( ( produc4490577844368043572al_nat @ A @ B )
= ( produc4490577844368043572al_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_352_old_Oprod_Oinject,axiom,
! [A: real,B: nat,A5: real,B4: nat] :
( ( ( produc3181502643871035669al_nat @ A @ B )
= ( produc3181502643871035669al_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_353_prod_Oinject,axiom,
! [X1: nat,X22: list_nat,Y1: nat,Y22: list_nat] :
( ( ( produc8282810413953273033st_nat @ X1 @ X22 )
= ( produc8282810413953273033st_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_354_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_355_prod_Oinject,axiom,
! [X1: nat > nat,X22: produc1828647624359046049st_nat,Y1: nat > nat,Y22: produc1828647624359046049st_nat] :
( ( ( produc4626581765195395529st_nat @ X1 @ X22 )
= ( produc4626581765195395529st_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_356_prod_Oinject,axiom,
! [X1: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,X22: produc5963376359100343987al_nat,Y1: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,Y22: produc5963376359100343987al_nat] :
( ( ( produc5211167162158505542al_nat @ X1 @ X22 )
= ( produc5211167162158505542al_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_357_prod_Oinject,axiom,
! [X1: nat > nat > $o,X22: list_nat,Y1: nat > nat > $o,Y22: list_nat] :
( ( ( produc4727192421694094319st_nat @ X1 @ X22 )
= ( produc4727192421694094319st_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_358_prod_Oinject,axiom,
! [X1: option_list_o,X22: option_list_o,Y1: option_list_o,Y22: option_list_o] :
( ( ( produc5745850523778858007list_o @ X1 @ X22 )
= ( produc5745850523778858007list_o @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_359_prod_Oinject,axiom,
! [X1: nat,X22: produc3741383161447143261al_nat,Y1: nat,Y22: produc3741383161447143261al_nat] :
( ( ( produc4490577844368043572al_nat @ X1 @ X22 )
= ( produc4490577844368043572al_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_360_prod_Oinject,axiom,
! [X1: real,X22: nat,Y1: real,Y22: nat] :
( ( ( produc3181502643871035669al_nat @ X1 @ X22 )
= ( produc3181502643871035669al_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_361_those_Osimps_I1_J,axiom,
( ( those_list_o @ nil_option_list_o )
= ( some_list_list_o @ nil_list_o ) ) ).
% those.simps(1)
thf(fact_362_those_Osimps_I1_J,axiom,
( ( those_list_nat @ nil_option_list_nat )
= ( some_list_list_nat @ nil_list_nat ) ) ).
% those.simps(1)
thf(fact_363_those_Osimps_I1_J,axiom,
( ( those_nat @ nil_option_nat )
= ( some_list_nat @ nil_nat ) ) ).
% those.simps(1)
thf(fact_364_those_Osimps_I1_J,axiom,
( ( those_7536171816368882977_o_nat @ nil_op1278377377907025649_o_nat )
= ( some_l3920453119453148048_o_nat @ nil_op686717097995950635_o_nat ) ) ).
% those.simps(1)
thf(fact_365_those_Osimps_I1_J,axiom,
( ( those_o @ nil_option_o )
= ( some_list_o @ nil_o ) ) ).
% those.simps(1)
thf(fact_366_opt__append_Opelims,axiom,
! [X2: option_list_o,Xa: option_list_o,Y2: option_list_o] :
( ( ( prefix5314359684614007693append @ X2 @ Xa )
= Y2 )
=> ( ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ X2 @ Xa ) )
=> ( ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ( ( Y2
= ( some_list_o @ ( append_o @ X3 @ Y3 ) ) )
=> ~ ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) ) ) ) )
=> ( ( ( X2 = none_list_o )
=> ( ( Y2 = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ none_list_o @ Xa ) ) ) )
=> ~ ( ( Xa = none_list_o )
=> ( ( Y2 = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ X2 @ none_list_o ) ) ) ) ) ) ) ) ).
% opt_append.pelims
thf(fact_367_prefix__bot_Omin__bot2,axiom,
! [X2: list_list_o] :
( ( min_list_list_o @ prefix_list_o @ X2 @ nil_list_o )
= nil_list_o ) ).
% prefix_bot.min_bot2
thf(fact_368_prefix__bot_Omin__bot2,axiom,
! [X2: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ X2 @ nil_list_nat )
= nil_list_nat ) ).
% prefix_bot.min_bot2
thf(fact_369_prefix__bot_Omin__bot2,axiom,
! [X2: list_o8358219866161907755_o_nat] :
( ( min_li4529822302168730582_o_nat @ prefix2577173861203319680_o_nat @ X2 @ nil_op686717097995950635_o_nat )
= nil_op686717097995950635_o_nat ) ).
% prefix_bot.min_bot2
thf(fact_370_prefix__bot_Omin__bot2,axiom,
! [X2: list_nat] :
( ( min_list_nat @ prefix_nat @ X2 @ nil_nat )
= nil_nat ) ).
% prefix_bot.min_bot2
thf(fact_371_prefix__bot_Omin__bot2,axiom,
! [X2: list_o] :
( ( min_list_o @ prefix_o @ X2 @ nil_o )
= nil_o ) ).
% prefix_bot.min_bot2
thf(fact_372_prefix__bot_Omax__bot2,axiom,
! [X2: list_list_o] :
( ( max_list_list_o @ prefix_list_o @ X2 @ nil_list_o )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_373_prefix__bot_Omax__bot2,axiom,
! [X2: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ X2 @ nil_list_nat )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_374_prefix__bot_Omax__bot2,axiom,
! [X2: list_o8358219866161907755_o_nat] :
( ( max_li6474167285248177476_o_nat @ prefix2577173861203319680_o_nat @ X2 @ nil_op686717097995950635_o_nat )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_375_prefix__bot_Omax__bot2,axiom,
! [X2: list_nat] :
( ( max_list_nat @ prefix_nat @ X2 @ nil_nat )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_376_prefix__bot_Omax__bot2,axiom,
! [X2: list_o] :
( ( max_list_o @ prefix_o @ X2 @ nil_o )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_377_prefix__bot_Omin__bot,axiom,
! [X2: list_list_o] :
( ( min_list_list_o @ prefix_list_o @ nil_list_o @ X2 )
= nil_list_o ) ).
% prefix_bot.min_bot
thf(fact_378_prefix__bot_Omin__bot,axiom,
! [X2: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ nil_list_nat @ X2 )
= nil_list_nat ) ).
% prefix_bot.min_bot
thf(fact_379_prefix__bot_Omin__bot,axiom,
! [X2: list_o8358219866161907755_o_nat] :
( ( min_li4529822302168730582_o_nat @ prefix2577173861203319680_o_nat @ nil_op686717097995950635_o_nat @ X2 )
= nil_op686717097995950635_o_nat ) ).
% prefix_bot.min_bot
thf(fact_380_prefix__bot_Omin__bot,axiom,
! [X2: list_nat] :
( ( min_list_nat @ prefix_nat @ nil_nat @ X2 )
= nil_nat ) ).
% prefix_bot.min_bot
thf(fact_381_prefix__bot_Omin__bot,axiom,
! [X2: list_o] :
( ( min_list_o @ prefix_o @ nil_o @ X2 )
= nil_o ) ).
% prefix_bot.min_bot
thf(fact_382_prefix__bot_Omax__bot,axiom,
! [X2: list_list_o] :
( ( max_list_list_o @ prefix_list_o @ nil_list_o @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_383_prefix__bot_Omax__bot,axiom,
! [X2: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ nil_list_nat @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_384_prefix__bot_Omax__bot,axiom,
! [X2: list_o8358219866161907755_o_nat] :
( ( max_li6474167285248177476_o_nat @ prefix2577173861203319680_o_nat @ nil_op686717097995950635_o_nat @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_385_prefix__bot_Omax__bot,axiom,
! [X2: list_nat] :
( ( max_list_nat @ prefix_nat @ nil_nat @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_386_prefix__bot_Omax__bot,axiom,
! [X2: list_o] :
( ( max_list_o @ prefix_o @ nil_o @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_387_prefix__order_Omax__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= B ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= A ) ) ) ).
% prefix_order.max_def
thf(fact_388_prefix__order_Omax__def,axiom,
! [A: list_o,B: list_o] :
( ( ( prefix_o @ A @ B )
=> ( ( max_list_o @ prefix_o @ A @ B )
= B ) )
& ( ~ ( prefix_o @ A @ B )
=> ( ( max_list_o @ prefix_o @ A @ B )
= A ) ) ) ).
% prefix_order.max_def
thf(fact_389_prefix__order_Omin__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= A ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= B ) ) ) ).
% prefix_order.min_def
thf(fact_390_prefix__order_Omin__def,axiom,
! [A: list_o,B: list_o] :
( ( ( prefix_o @ A @ B )
=> ( ( min_list_o @ prefix_o @ A @ B )
= A ) )
& ( ~ ( prefix_o @ A @ B )
=> ( ( min_list_o @ prefix_o @ A @ B )
= B ) ) ) ).
% prefix_order.min_def
thf(fact_391_old_Oprod_Oexhaust,axiom,
! [Y2: produc4575160907756185873st_nat] :
~ ! [A2: nat,B2: list_nat] :
( Y2
!= ( produc8282810413953273033st_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_392_old_Oprod_Oexhaust,axiom,
! [Y2: product_prod_nat_nat] :
~ ! [A2: nat,B2: nat] :
( Y2
!= ( product_Pair_nat_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_393_old_Oprod_Oexhaust,axiom,
! [Y2: produc1616951275169580055st_nat] :
~ ! [A2: nat > nat,B2: produc1828647624359046049st_nat] :
( Y2
!= ( produc4626581765195395529st_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_394_old_Oprod_Oexhaust,axiom,
! [Y2: produc1019696013813152916al_nat] :
~ ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: produc5963376359100343987al_nat] :
( Y2
!= ( produc5211167162158505542al_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_395_old_Oprod_Oexhaust,axiom,
! [Y2: produc254973753779126261st_nat] :
~ ! [A2: nat > nat > $o,B2: list_nat] :
( Y2
!= ( produc4727192421694094319st_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_396_old_Oprod_Oexhaust,axiom,
! [Y2: produc4882884732533091879list_o] :
~ ! [A2: option_list_o,B2: option_list_o] :
( Y2
!= ( produc5745850523778858007list_o @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_397_old_Oprod_Oexhaust,axiom,
! [Y2: produc9155354469731034754al_nat] :
~ ! [A2: nat,B2: produc3741383161447143261al_nat] :
( Y2
!= ( produc4490577844368043572al_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_398_old_Oprod_Oexhaust,axiom,
! [Y2: produc3741383161447143261al_nat] :
~ ! [A2: real,B2: nat] :
( Y2
!= ( produc3181502643871035669al_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_399_surj__pair,axiom,
! [P4: produc4575160907756185873st_nat] :
? [X3: nat,Y3: list_nat] :
( P4
= ( produc8282810413953273033st_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_400_surj__pair,axiom,
! [P4: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P4
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_401_surj__pair,axiom,
! [P4: produc1616951275169580055st_nat] :
? [X3: nat > nat,Y3: produc1828647624359046049st_nat] :
( P4
= ( produc4626581765195395529st_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_402_surj__pair,axiom,
! [P4: produc1019696013813152916al_nat] :
? [X3: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,Y3: produc5963376359100343987al_nat] :
( P4
= ( produc5211167162158505542al_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_403_surj__pair,axiom,
! [P4: produc254973753779126261st_nat] :
? [X3: nat > nat > $o,Y3: list_nat] :
( P4
= ( produc4727192421694094319st_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_404_surj__pair,axiom,
! [P4: produc4882884732533091879list_o] :
? [X3: option_list_o,Y3: option_list_o] :
( P4
= ( produc5745850523778858007list_o @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_405_surj__pair,axiom,
! [P4: produc9155354469731034754al_nat] :
? [X3: nat,Y3: produc3741383161447143261al_nat] :
( P4
= ( produc4490577844368043572al_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_406_surj__pair,axiom,
! [P4: produc3741383161447143261al_nat] :
? [X3: real,Y3: nat] :
( P4
= ( produc3181502643871035669al_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_407_prod__cases,axiom,
! [P: produc4575160907756185873st_nat > $o,P4: produc4575160907756185873st_nat] :
( ! [A2: nat,B2: list_nat] : ( P @ ( produc8282810413953273033st_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_408_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
( ! [A2: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_409_prod__cases,axiom,
! [P: produc1616951275169580055st_nat > $o,P4: produc1616951275169580055st_nat] :
( ! [A2: nat > nat,B2: produc1828647624359046049st_nat] : ( P @ ( produc4626581765195395529st_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_410_prod__cases,axiom,
! [P: produc1019696013813152916al_nat > $o,P4: produc1019696013813152916al_nat] :
( ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: produc5963376359100343987al_nat] : ( P @ ( produc5211167162158505542al_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_411_prod__cases,axiom,
! [P: produc254973753779126261st_nat > $o,P4: produc254973753779126261st_nat] :
( ! [A2: nat > nat > $o,B2: list_nat] : ( P @ ( produc4727192421694094319st_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_412_prod__cases,axiom,
! [P: produc4882884732533091879list_o > $o,P4: produc4882884732533091879list_o] :
( ! [A2: option_list_o,B2: option_list_o] : ( P @ ( produc5745850523778858007list_o @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_413_prod__cases,axiom,
! [P: produc9155354469731034754al_nat > $o,P4: produc9155354469731034754al_nat] :
( ! [A2: nat,B2: produc3741383161447143261al_nat] : ( P @ ( produc4490577844368043572al_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_414_prod__cases,axiom,
! [P: produc3741383161447143261al_nat > $o,P4: produc3741383161447143261al_nat] :
( ! [A2: real,B2: nat] : ( P @ ( produc3181502643871035669al_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_415_Pair__inject,axiom,
! [A: nat,B: list_nat,A5: nat,B4: list_nat] :
( ( ( produc8282810413953273033st_nat @ A @ B )
= ( produc8282810413953273033st_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_416_Pair__inject,axiom,
! [A: nat,B: nat,A5: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_417_Pair__inject,axiom,
! [A: nat > nat,B: produc1828647624359046049st_nat,A5: nat > nat,B4: produc1828647624359046049st_nat] :
( ( ( produc4626581765195395529st_nat @ A @ B )
= ( produc4626581765195395529st_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_418_Pair__inject,axiom,
! [A: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B: produc5963376359100343987al_nat,A5: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B4: produc5963376359100343987al_nat] :
( ( ( produc5211167162158505542al_nat @ A @ B )
= ( produc5211167162158505542al_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_419_Pair__inject,axiom,
! [A: nat > nat > $o,B: list_nat,A5: nat > nat > $o,B4: list_nat] :
( ( ( produc4727192421694094319st_nat @ A @ B )
= ( produc4727192421694094319st_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_420_Pair__inject,axiom,
! [A: option_list_o,B: option_list_o,A5: option_list_o,B4: option_list_o] :
( ( ( produc5745850523778858007list_o @ A @ B )
= ( produc5745850523778858007list_o @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_421_Pair__inject,axiom,
! [A: nat,B: produc3741383161447143261al_nat,A5: nat,B4: produc3741383161447143261al_nat] :
( ( ( produc4490577844368043572al_nat @ A @ B )
= ( produc4490577844368043572al_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_422_Pair__inject,axiom,
! [A: real,B: nat,A5: real,B4: nat] :
( ( ( produc3181502643871035669al_nat @ A @ B )
= ( produc3181502643871035669al_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_423_prod__cases3,axiom,
! [Y2: produc1616951275169580055st_nat] :
~ ! [A2: nat > nat,B2: list_nat,C2: list_nat] :
( Y2
!= ( produc4626581765195395529st_nat @ A2 @ ( produc2694037385005941721st_nat @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_424_prod__cases3,axiom,
! [Y2: produc1019696013813152916al_nat] :
~ ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: nat,C2: produc9155354469731034754al_nat] :
( Y2
!= ( produc5211167162158505542al_nat @ A2 @ ( produc2879132834507146987al_nat @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_425_prod__cases3,axiom,
! [Y2: produc9155354469731034754al_nat] :
~ ! [A2: nat,B2: real,C2: nat] :
( Y2
!= ( produc4490577844368043572al_nat @ A2 @ ( produc3181502643871035669al_nat @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_426_prod__cases4,axiom,
! [Y2: produc1019696013813152916al_nat] :
~ ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: nat,C2: nat,D: produc3741383161447143261al_nat] :
( Y2
!= ( produc5211167162158505542al_nat @ A2 @ ( produc2879132834507146987al_nat @ B2 @ ( produc4490577844368043572al_nat @ C2 @ D ) ) ) ) ).
% prod_cases4
thf(fact_427_prod__cases5,axiom,
! [Y2: produc1019696013813152916al_nat] :
~ ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: nat,C2: nat,D: real,E: nat] :
( Y2
!= ( produc5211167162158505542al_nat @ A2 @ ( produc2879132834507146987al_nat @ B2 @ ( produc4490577844368043572al_nat @ C2 @ ( produc3181502643871035669al_nat @ D @ E ) ) ) ) ) ).
% prod_cases5
thf(fact_428_prod__induct3,axiom,
! [P: produc1616951275169580055st_nat > $o,X2: produc1616951275169580055st_nat] :
( ! [A2: nat > nat,B2: list_nat,C2: list_nat] : ( P @ ( produc4626581765195395529st_nat @ A2 @ ( produc2694037385005941721st_nat @ B2 @ C2 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_429_prod__induct3,axiom,
! [P: produc1019696013813152916al_nat > $o,X2: produc1019696013813152916al_nat] :
( ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: nat,C2: produc9155354469731034754al_nat] : ( P @ ( produc5211167162158505542al_nat @ A2 @ ( produc2879132834507146987al_nat @ B2 @ C2 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_430_prod__induct3,axiom,
! [P: produc9155354469731034754al_nat > $o,X2: produc9155354469731034754al_nat] :
( ! [A2: nat,B2: real,C2: nat] : ( P @ ( produc4490577844368043572al_nat @ A2 @ ( produc3181502643871035669al_nat @ B2 @ C2 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_431_prod__induct4,axiom,
! [P: produc1019696013813152916al_nat > $o,X2: produc1019696013813152916al_nat] :
( ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: nat,C2: nat,D: produc3741383161447143261al_nat] : ( P @ ( produc5211167162158505542al_nat @ A2 @ ( produc2879132834507146987al_nat @ B2 @ ( produc4490577844368043572al_nat @ C2 @ D ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_432_prod__induct5,axiom,
! [P: produc1019696013813152916al_nat > $o,X2: produc1019696013813152916al_nat] :
( ! [A2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,B2: nat,C2: nat,D: real,E: nat] : ( P @ ( produc5211167162158505542al_nat @ A2 @ ( produc2879132834507146987al_nat @ B2 @ ( produc4490577844368043572al_nat @ C2 @ ( produc3181502643871035669al_nat @ D @ E ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_433_opt__prefix_Opelims_I1_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat,Y2: $o] :
( ( ( prefix9059114240934526678ix_nat @ X2 @ Xa )
= Y2 )
=> ( ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ X2 @ Xa ) )
=> ( ! [X3: list_nat] :
( ( X2
= ( some_list_nat @ X3 ) )
=> ! [Y3: list_nat] :
( ( Xa
= ( some_list_nat @ Y3 ) )
=> ( ( Y2
= ( prefix_nat @ X3 @ Y3 ) )
=> ~ ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ ( some_list_nat @ X3 ) @ ( some_list_nat @ Y3 ) ) ) ) ) )
=> ( ( ( X2 = none_list_nat )
=> ( ~ Y2
=> ~ ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ none_list_nat @ Xa ) ) ) )
=> ~ ( ( Xa = none_list_nat )
=> ( ~ Y2
=> ~ ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ X2 @ none_list_nat ) ) ) ) ) ) ) ) ).
% opt_prefix.pelims(1)
thf(fact_434_opt__prefix_Opelims_I1_J,axiom,
! [X2: option_list_o,Xa: option_list_o,Y2: $o] :
( ( ( prefix8824957607401505554efix_o @ X2 @ Xa )
= Y2 )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X2 @ Xa ) )
=> ( ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ( ( Y2
= ( prefix_o @ X3 @ Y3 ) )
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) ) ) ) )
=> ( ( ( X2 = none_list_o )
=> ( ~ Y2
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ none_list_o @ Xa ) ) ) )
=> ~ ( ( Xa = none_list_o )
=> ( ~ Y2
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X2 @ none_list_o ) ) ) ) ) ) ) ) ).
% opt_prefix.pelims(1)
thf(fact_435_opt__prefix_Opelims_I3_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat] :
( ~ ( prefix9059114240934526678ix_nat @ X2 @ Xa )
=> ( ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ X2 @ Xa ) )
=> ( ! [X3: list_nat] :
( ( X2
= ( some_list_nat @ X3 ) )
=> ! [Y3: list_nat] :
( ( Xa
= ( some_list_nat @ Y3 ) )
=> ( ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ ( some_list_nat @ X3 ) @ ( some_list_nat @ Y3 ) ) )
=> ( prefix_nat @ X3 @ Y3 ) ) ) )
=> ( ( ( X2 = none_list_nat )
=> ~ ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ none_list_nat @ Xa ) ) )
=> ~ ( ( Xa = none_list_nat )
=> ~ ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ X2 @ none_list_nat ) ) ) ) ) ) ) ).
% opt_prefix.pelims(3)
thf(fact_436_opt__prefix_Opelims_I3_J,axiom,
! [X2: option_list_o,Xa: option_list_o] :
( ~ ( prefix8824957607401505554efix_o @ X2 @ Xa )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X2 @ Xa ) )
=> ( ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) )
=> ( prefix_o @ X3 @ Y3 ) ) ) )
=> ( ( ( X2 = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ none_list_o @ Xa ) ) )
=> ~ ( ( Xa = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X2 @ none_list_o ) ) ) ) ) ) ) ).
% opt_prefix.pelims(3)
thf(fact_437_opt__prefix_Opelims_I2_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat] :
( ( prefix9059114240934526678ix_nat @ X2 @ Xa )
=> ( ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ X2 @ Xa ) )
=> ~ ! [X3: list_nat] :
( ( X2
= ( some_list_nat @ X3 ) )
=> ! [Y3: list_nat] :
( ( Xa
= ( some_list_nat @ Y3 ) )
=> ( ( accp_P1545021376356841034st_nat @ prefix9175423978189927749el_nat @ ( produc1623406484878470137st_nat @ ( some_list_nat @ X3 ) @ ( some_list_nat @ Y3 ) ) )
=> ~ ( prefix_nat @ X3 @ Y3 ) ) ) ) ) ) ).
% opt_prefix.pelims(2)
thf(fact_438_opt__prefix_Opelims_I2_J,axiom,
! [X2: option_list_o,Xa: option_list_o] :
( ( prefix8824957607401505554efix_o @ X2 @ Xa )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X2 @ Xa ) )
=> ~ ! [X3: list_o] :
( ( X2
= ( some_list_o @ X3 ) )
=> ! [Y3: list_o] :
( ( Xa
= ( some_list_o @ Y3 ) )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) )
=> ~ ( prefix_o @ X3 @ Y3 ) ) ) ) ) ) ).
% opt_prefix.pelims(2)
thf(fact_439_ord_Omax__def,axiom,
( max_list_o
= ( ^ [Less_eq: list_o > list_o > $o,A3: list_o,B3: list_o] : ( if_list_o @ ( Less_eq @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% ord.max_def
thf(fact_440_ord_Omin__def,axiom,
( min_list_o
= ( ^ [Less_eq: list_o > list_o > $o,A3: list_o,B3: list_o] : ( if_list_o @ ( Less_eq @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% ord.min_def
thf(fact_441_ord_Omax_Ocong,axiom,
max_list_o = max_list_o ).
% ord.max.cong
thf(fact_442_ord_Omin_Ocong,axiom,
min_list_o = min_list_o ).
% ord.min.cong
thf(fact_443_list__ex1__simps_I1_J,axiom,
! [P: list_o > $o] :
~ ( list_ex1_list_o @ P @ nil_list_o ) ).
% list_ex1_simps(1)
thf(fact_444_list__ex1__simps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_445_list__ex1__simps_I1_J,axiom,
! [P: $o > $o] :
~ ( list_ex1_o @ P @ nil_o ) ).
% list_ex1_simps(1)
thf(fact_446_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_447_list__ex1__simps_I1_J,axiom,
! [P: ( option_list_o > nat ) > $o] :
~ ( list_e357688041540588142_o_nat @ P @ nil_op686717097995950635_o_nat ) ).
% list_ex1_simps(1)
thf(fact_448_prefix__snoc,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Y2: product_prod_nat_nat] :
( ( prefix3150859221070158812at_nat @ Xs @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y2 @ nil_Pr5478986624290739719at_nat ) ) )
= ( ( Xs
= ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y2 @ nil_Pr5478986624290739719at_nat ) ) )
| ( prefix3150859221070158812at_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_449_prefix__snoc,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Y2: option_list_o] :
( ( prefix_option_list_o @ Xs @ ( append_option_list_o @ Ys @ ( cons_option_list_o @ Y2 @ nil_option_list_o ) ) )
= ( ( Xs
= ( append_option_list_o @ Ys @ ( cons_option_list_o @ Y2 @ nil_option_list_o ) ) )
| ( prefix_option_list_o @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_450_prefix__snoc,axiom,
! [Xs: list_real,Ys: list_real,Y2: real] :
( ( prefix_real @ Xs @ ( append_real @ Ys @ ( cons_real @ Y2 @ nil_real ) ) )
= ( ( Xs
= ( append_real @ Ys @ ( cons_real @ Y2 @ nil_real ) ) )
| ( prefix_real @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_451_prefix__snoc,axiom,
! [Xs: list_list_o,Ys: list_list_o,Y2: list_o] :
( ( prefix_list_o @ Xs @ ( append_list_o @ Ys @ ( cons_list_o @ Y2 @ nil_list_o ) ) )
= ( ( Xs
= ( append_list_o @ Ys @ ( cons_list_o @ Y2 @ nil_list_o ) ) )
| ( prefix_list_o @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_452_prefix__snoc,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Y2: list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
= ( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
| ( prefix_list_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_453_prefix__snoc,axiom,
! [Xs: list_o8358219866161907755_o_nat,Ys: list_o8358219866161907755_o_nat,Y2: option_list_o > nat] :
( ( prefix2577173861203319680_o_nat @ Xs @ ( append2548158351732429248_o_nat @ Ys @ ( cons_o6089670980329342427_o_nat @ Y2 @ nil_op686717097995950635_o_nat ) ) )
= ( ( Xs
= ( append2548158351732429248_o_nat @ Ys @ ( cons_o6089670980329342427_o_nat @ Y2 @ nil_op686717097995950635_o_nat ) ) )
| ( prefix2577173861203319680_o_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_454_prefix__snoc,axiom,
! [Xs: list_nat,Ys: list_nat,Y2: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_455_prefix__snoc,axiom,
! [Xs: list_o,Ys: list_o,Y2: $o] :
( ( prefix_o @ Xs @ ( append_o @ Ys @ ( cons_o @ Y2 @ nil_o ) ) )
= ( ( Xs
= ( append_o @ Ys @ ( cons_o @ Y2 @ nil_o ) ) )
| ( prefix_o @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_456_is__none__bind,axiom,
! [F: option_list_o,G: list_o > option_list_o] :
( ( is_none_list_o @ ( bind_list_o_list_o @ F @ G ) )
= ( ( is_none_list_o @ F )
| ( is_none_list_o @ ( G @ ( the_list_o @ F ) ) ) ) ) ).
% is_none_bind
thf(fact_457_bind__simps_I1_J,axiom,
! [F: $o > list_o] :
( ( bind_o_o @ nil_o @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_458_bind__simps_I1_J,axiom,
! [F: $o > list_nat] :
( ( bind_o_nat @ nil_o @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_459_bind__simps_I1_J,axiom,
! [F: nat > list_o] :
( ( bind_nat_o @ nil_nat @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_460_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_461_bind__simps_I1_J,axiom,
! [F: list_o > list_o] :
( ( bind_list_o_o @ nil_list_o @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_462_bind__simps_I1_J,axiom,
! [F: list_o > list_nat] :
( ( bind_list_o_nat @ nil_list_o @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_463_bind__simps_I1_J,axiom,
! [F: list_nat > list_o] :
( ( bind_list_nat_o @ nil_list_nat @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_464_bind__simps_I1_J,axiom,
! [F: list_nat > list_nat] :
( ( bind_list_nat_nat @ nil_list_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_465_bind__simps_I1_J,axiom,
! [F: $o > list_list_o] :
( ( bind_o_list_o @ nil_o @ F )
= nil_list_o ) ).
% bind_simps(1)
thf(fact_466_bind__simps_I1_J,axiom,
! [F: $o > list_list_nat] :
( ( bind_o_list_nat @ nil_o @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_467_accp__induct__rule,axiom,
! [R: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o,A: produc1828647624359046049st_nat,P: produc1828647624359046049st_nat > $o] :
( ( accp_P8037286306265792042st_nat @ R @ A )
=> ( ! [X3: produc1828647624359046049st_nat] :
( ( accp_P8037286306265792042st_nat @ R @ X3 )
=> ( ! [Y5: produc1828647624359046049st_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_468_accp__induct__rule,axiom,
! [R: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o,A: produc1616951275169580055st_nat,P: produc1616951275169580055st_nat > $o] :
( ( accp_P3924000266242761678st_nat @ R @ A )
=> ( ! [X3: produc1616951275169580055st_nat] :
( ( accp_P3924000266242761678st_nat @ R @ X3 )
=> ( ! [Y5: produc1616951275169580055st_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_469_accp__induct__rule,axiom,
! [R: produc1019696013813152916al_nat > produc1019696013813152916al_nat > $o,A: produc1019696013813152916al_nat,P: produc1019696013813152916al_nat > $o] :
( ( accp_P1387443933282790731al_nat @ R @ A )
=> ( ! [X3: produc1019696013813152916al_nat] :
( ( accp_P1387443933282790731al_nat @ R @ X3 )
=> ( ! [Y5: produc1019696013813152916al_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_470_accp__induct__rule,axiom,
! [R: list_nat > list_nat > $o,A: list_nat,P: list_nat > $o] :
( ( accp_list_nat @ R @ A )
=> ( ! [X3: list_nat] :
( ( accp_list_nat @ R @ X3 )
=> ( ! [Y5: list_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_471_accp__induct__rule,axiom,
! [R: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,A: produc4882884732533091879list_o,P: produc4882884732533091879list_o > $o] :
( ( accp_P4987914878654715440list_o @ R @ A )
=> ( ! [X3: produc4882884732533091879list_o] :
( ( accp_P4987914878654715440list_o @ R @ X3 )
=> ( ! [Y5: produc4882884732533091879list_o] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_472_not__accp__down,axiom,
! [R2: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o,X2: produc1828647624359046049st_nat] :
( ~ ( accp_P8037286306265792042st_nat @ R2 @ X2 )
=> ~ ! [Z3: produc1828647624359046049st_nat] :
( ( R2 @ Z3 @ X2 )
=> ( accp_P8037286306265792042st_nat @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_473_not__accp__down,axiom,
! [R2: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o,X2: produc1616951275169580055st_nat] :
( ~ ( accp_P3924000266242761678st_nat @ R2 @ X2 )
=> ~ ! [Z3: produc1616951275169580055st_nat] :
( ( R2 @ Z3 @ X2 )
=> ( accp_P3924000266242761678st_nat @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_474_not__accp__down,axiom,
! [R2: produc1019696013813152916al_nat > produc1019696013813152916al_nat > $o,X2: produc1019696013813152916al_nat] :
( ~ ( accp_P1387443933282790731al_nat @ R2 @ X2 )
=> ~ ! [Z3: produc1019696013813152916al_nat] :
( ( R2 @ Z3 @ X2 )
=> ( accp_P1387443933282790731al_nat @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_475_not__accp__down,axiom,
! [R2: list_nat > list_nat > $o,X2: list_nat] :
( ~ ( accp_list_nat @ R2 @ X2 )
=> ~ ! [Z3: list_nat] :
( ( R2 @ Z3 @ X2 )
=> ( accp_list_nat @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_476_not__accp__down,axiom,
! [R2: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,X2: produc4882884732533091879list_o] :
( ~ ( accp_P4987914878654715440list_o @ R2 @ X2 )
=> ~ ! [Z3: produc4882884732533091879list_o] :
( ( R2 @ Z3 @ X2 )
=> ( accp_P4987914878654715440list_o @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_477_accp__downward,axiom,
! [R: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o,B: produc1828647624359046049st_nat,A: produc1828647624359046049st_nat] :
( ( accp_P8037286306265792042st_nat @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P8037286306265792042st_nat @ R @ A ) ) ) ).
% accp_downward
thf(fact_478_accp__downward,axiom,
! [R: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o,B: produc1616951275169580055st_nat,A: produc1616951275169580055st_nat] :
( ( accp_P3924000266242761678st_nat @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P3924000266242761678st_nat @ R @ A ) ) ) ).
% accp_downward
thf(fact_479_accp__downward,axiom,
! [R: produc1019696013813152916al_nat > produc1019696013813152916al_nat > $o,B: produc1019696013813152916al_nat,A: produc1019696013813152916al_nat] :
( ( accp_P1387443933282790731al_nat @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P1387443933282790731al_nat @ R @ A ) ) ) ).
% accp_downward
thf(fact_480_accp__downward,axiom,
! [R: list_nat > list_nat > $o,B: list_nat,A: list_nat] :
( ( accp_list_nat @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_list_nat @ R @ A ) ) ) ).
% accp_downward
thf(fact_481_accp__downward,axiom,
! [R: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,B: produc4882884732533091879list_o,A: produc4882884732533091879list_o] :
( ( accp_P4987914878654715440list_o @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P4987914878654715440list_o @ R @ A ) ) ) ).
% accp_downward
thf(fact_482_accp__induct,axiom,
! [R: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o,A: produc1828647624359046049st_nat,P: produc1828647624359046049st_nat > $o] :
( ( accp_P8037286306265792042st_nat @ R @ A )
=> ( ! [X3: produc1828647624359046049st_nat] :
( ( accp_P8037286306265792042st_nat @ R @ X3 )
=> ( ! [Y5: produc1828647624359046049st_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_483_accp__induct,axiom,
! [R: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o,A: produc1616951275169580055st_nat,P: produc1616951275169580055st_nat > $o] :
( ( accp_P3924000266242761678st_nat @ R @ A )
=> ( ! [X3: produc1616951275169580055st_nat] :
( ( accp_P3924000266242761678st_nat @ R @ X3 )
=> ( ! [Y5: produc1616951275169580055st_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_484_accp__induct,axiom,
! [R: produc1019696013813152916al_nat > produc1019696013813152916al_nat > $o,A: produc1019696013813152916al_nat,P: produc1019696013813152916al_nat > $o] :
( ( accp_P1387443933282790731al_nat @ R @ A )
=> ( ! [X3: produc1019696013813152916al_nat] :
( ( accp_P1387443933282790731al_nat @ R @ X3 )
=> ( ! [Y5: produc1019696013813152916al_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_485_accp__induct,axiom,
! [R: list_nat > list_nat > $o,A: list_nat,P: list_nat > $o] :
( ( accp_list_nat @ R @ A )
=> ( ! [X3: list_nat] :
( ( accp_list_nat @ R @ X3 )
=> ( ! [Y5: list_nat] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_486_accp__induct,axiom,
! [R: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,A: produc4882884732533091879list_o,P: produc4882884732533091879list_o > $o] :
( ( accp_P4987914878654715440list_o @ R @ A )
=> ( ! [X3: produc4882884732533091879list_o] :
( ( accp_P4987914878654715440list_o @ R @ X3 )
=> ( ! [Y5: produc4882884732533091879list_o] :
( ( R @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_487_accpI,axiom,
! [R: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o,X2: produc1828647624359046049st_nat] :
( ! [Y3: produc1828647624359046049st_nat] :
( ( R @ Y3 @ X2 )
=> ( accp_P8037286306265792042st_nat @ R @ Y3 ) )
=> ( accp_P8037286306265792042st_nat @ R @ X2 ) ) ).
% accpI
thf(fact_488_accpI,axiom,
! [R: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o,X2: produc1616951275169580055st_nat] :
( ! [Y3: produc1616951275169580055st_nat] :
( ( R @ Y3 @ X2 )
=> ( accp_P3924000266242761678st_nat @ R @ Y3 ) )
=> ( accp_P3924000266242761678st_nat @ R @ X2 ) ) ).
% accpI
thf(fact_489_accpI,axiom,
! [R: produc1019696013813152916al_nat > produc1019696013813152916al_nat > $o,X2: produc1019696013813152916al_nat] :
( ! [Y3: produc1019696013813152916al_nat] :
( ( R @ Y3 @ X2 )
=> ( accp_P1387443933282790731al_nat @ R @ Y3 ) )
=> ( accp_P1387443933282790731al_nat @ R @ X2 ) ) ).
% accpI
thf(fact_490_accpI,axiom,
! [R: list_nat > list_nat > $o,X2: list_nat] :
( ! [Y3: list_nat] :
( ( R @ Y3 @ X2 )
=> ( accp_list_nat @ R @ Y3 ) )
=> ( accp_list_nat @ R @ X2 ) ) ).
% accpI
thf(fact_491_accpI,axiom,
! [R: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,X2: produc4882884732533091879list_o] :
( ! [Y3: produc4882884732533091879list_o] :
( ( R @ Y3 @ X2 )
=> ( accp_P4987914878654715440list_o @ R @ Y3 ) )
=> ( accp_P4987914878654715440list_o @ R @ X2 ) ) ).
% accpI
thf(fact_492_list_Oinject,axiom,
! [X21: real,X222: list_real,Y21: real,Y222: list_real] :
( ( ( cons_real @ X21 @ X222 )
= ( cons_real @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_493_list_Oinject,axiom,
! [X21: list_o,X222: list_list_o,Y21: list_o,Y222: list_list_o] :
( ( ( cons_list_o @ X21 @ X222 )
= ( cons_list_o @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_494_list_Oinject,axiom,
! [X21: list_nat,X222: list_list_nat,Y21: list_nat,Y222: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X222 )
= ( cons_list_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_495_list_Oinject,axiom,
! [X21: $o,X222: list_o,Y21: $o,Y222: list_o] :
( ( ( cons_o @ X21 @ X222 )
= ( cons_o @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_496_list_Oinject,axiom,
! [X21: option_list_o > nat,X222: list_o8358219866161907755_o_nat,Y21: option_list_o > nat,Y222: list_o8358219866161907755_o_nat] :
( ( ( cons_o6089670980329342427_o_nat @ X21 @ X222 )
= ( cons_o6089670980329342427_o_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_497_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_498_Cons__prefix__Cons,axiom,
! [X2: real,Xs: list_real,Y2: real,Ys: list_real] :
( ( prefix_real @ ( cons_real @ X2 @ Xs ) @ ( cons_real @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix_real @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_499_Cons__prefix__Cons,axiom,
! [X2: list_o,Xs: list_list_o,Y2: list_o,Ys: list_list_o] :
( ( prefix_list_o @ ( cons_list_o @ X2 @ Xs ) @ ( cons_list_o @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix_list_o @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_500_Cons__prefix__Cons,axiom,
! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix_list_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_501_Cons__prefix__Cons,axiom,
! [X2: option_list_o > nat,Xs: list_o8358219866161907755_o_nat,Y2: option_list_o > nat,Ys: list_o8358219866161907755_o_nat] :
( ( prefix2577173861203319680_o_nat @ ( cons_o6089670980329342427_o_nat @ X2 @ Xs ) @ ( cons_o6089670980329342427_o_nat @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix2577173861203319680_o_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_502_Cons__prefix__Cons,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_503_Cons__prefix__Cons,axiom,
! [X2: $o,Xs: list_o,Y2: $o,Ys: list_o] :
( ( prefix_o @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix_o @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_504_is__none__code_I2_J,axiom,
! [X2: produc8196726482729697190st_nat] :
~ ( is_non6231852249821756618st_nat @ ( some_P617598486967069697st_nat @ X2 ) ) ).
% is_none_code(2)
thf(fact_505_is__none__code_I2_J,axiom,
! [X2: produc6173137527213542198list_o] :
~ ( is_non8031834422447889170list_o @ ( some_P3766925538565183643list_o @ X2 ) ) ).
% is_none_code(2)
thf(fact_506_is__none__code_I2_J,axiom,
! [X2: nat] :
~ ( is_none_nat @ ( some_nat @ X2 ) ) ).
% is_none_code(2)
thf(fact_507_is__none__code_I2_J,axiom,
! [X2: list_o] :
~ ( is_none_list_o @ ( some_list_o @ X2 ) ) ).
% is_none_code(2)
thf(fact_508_is__none__code_I1_J,axiom,
is_none_list_o @ none_list_o ).
% is_none_code(1)
thf(fact_509_append1__eq__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,X2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y2: product_prod_nat_nat] :
( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
= ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y2 @ nil_Pr5478986624290739719at_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_510_append1__eq__conv,axiom,
! [Xs: list_option_list_o,X2: option_list_o,Ys: list_option_list_o,Y2: option_list_o] :
( ( ( append_option_list_o @ Xs @ ( cons_option_list_o @ X2 @ nil_option_list_o ) )
= ( append_option_list_o @ Ys @ ( cons_option_list_o @ Y2 @ nil_option_list_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_511_append1__eq__conv,axiom,
! [Xs: list_real,X2: real,Ys: list_real,Y2: real] :
( ( ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) )
= ( append_real @ Ys @ ( cons_real @ Y2 @ nil_real ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_512_append1__eq__conv,axiom,
! [Xs: list_list_o,X2: list_o,Ys: list_list_o,Y2: list_o] :
( ( ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) )
= ( append_list_o @ Ys @ ( cons_list_o @ Y2 @ nil_list_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_513_append1__eq__conv,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys: list_list_nat,Y2: list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) )
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_514_append1__eq__conv,axiom,
! [Xs: list_o8358219866161907755_o_nat,X2: option_list_o > nat,Ys: list_o8358219866161907755_o_nat,Y2: option_list_o > nat] :
( ( ( append2548158351732429248_o_nat @ Xs @ ( cons_o6089670980329342427_o_nat @ X2 @ nil_op686717097995950635_o_nat ) )
= ( append2548158351732429248_o_nat @ Ys @ ( cons_o6089670980329342427_o_nat @ Y2 @ nil_op686717097995950635_o_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_515_append1__eq__conv,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y2: $o] :
( ( ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) )
= ( append_o @ Ys @ ( cons_o @ Y2 @ nil_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_516_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_517_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_o] :
( ( bind_nat_o @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_o @ ( F @ X2 ) @ ( bind_nat_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_518_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_519_bind__simps_I2_J,axiom,
! [X2: real,Xs: list_real,F: real > list_o] :
( ( bind_real_o @ ( cons_real @ X2 @ Xs ) @ F )
= ( append_o @ ( F @ X2 ) @ ( bind_real_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_520_bind__simps_I2_J,axiom,
! [X2: real,Xs: list_real,F: real > list_nat] :
( ( bind_real_nat @ ( cons_real @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_real_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_521_bind__simps_I2_J,axiom,
! [X2: $o,Xs: list_o,F: $o > list_o] :
( ( bind_o_o @ ( cons_o @ X2 @ Xs ) @ F )
= ( append_o @ ( F @ X2 ) @ ( bind_o_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_522_bind__simps_I2_J,axiom,
! [X2: $o,Xs: list_o,F: $o > list_nat] :
( ( bind_o_nat @ ( cons_o @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_o_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_523_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_list_o] :
( ( bind_nat_list_o @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_list_o @ ( F @ X2 ) @ ( bind_nat_list_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_524_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_list_nat] :
( ( bind_nat_list_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_list_nat @ ( F @ X2 ) @ ( bind_nat_list_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_525_bind__simps_I2_J,axiom,
! [X2: real,Xs: list_real,F: real > list_list_o] :
( ( bind_real_list_o @ ( cons_real @ X2 @ Xs ) @ F )
= ( append_list_o @ ( F @ X2 ) @ ( bind_real_list_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_526_bind__simps_I2_J,axiom,
! [X2: real,Xs: list_real,F: real > list_list_nat] :
( ( bind_real_list_nat @ ( cons_real @ X2 @ Xs ) @ F )
= ( append_list_nat @ ( F @ X2 ) @ ( bind_real_list_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_527_not__Cons__self2,axiom,
! [X2: real,Xs: list_real] :
( ( cons_real @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_528_not__Cons__self2,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( cons_list_o @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_529_not__Cons__self2,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( cons_list_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_530_not__Cons__self2,axiom,
! [X2: $o,Xs: list_o] :
( ( cons_o @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_531_not__Cons__self2,axiom,
! [X2: option_list_o > nat,Xs: list_o8358219866161907755_o_nat] :
( ( cons_o6089670980329342427_o_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_532_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_533_transpose_Ocases,axiom,
! [X2: list_list_real] :
( ( X2 != nil_list_real )
=> ( ! [Xss: list_list_real] :
( X2
!= ( cons_list_real @ nil_real @ Xss ) )
=> ~ ! [X3: real,Xs3: list_real,Xss: list_list_real] :
( X2
!= ( cons_list_real @ ( cons_real @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_534_transpose_Ocases,axiom,
! [X2: list_list_list_o] :
( ( X2 != nil_list_list_o )
=> ( ! [Xss: list_list_list_o] :
( X2
!= ( cons_list_list_o @ nil_list_o @ Xss ) )
=> ~ ! [X3: list_o,Xs3: list_list_o,Xss: list_list_list_o] :
( X2
!= ( cons_list_list_o @ ( cons_list_o @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_535_transpose_Ocases,axiom,
! [X2: list_list_list_nat] :
( ( X2 != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X3: list_nat,Xs3: list_list_nat,Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_536_transpose_Ocases,axiom,
! [X2: list_l1581440573117054897_o_nat] :
( ( X2 != nil_li35917879731135281_o_nat )
=> ( ! [Xss: list_l1581440573117054897_o_nat] :
( X2
!= ( cons_l8100113683866236641_o_nat @ nil_op686717097995950635_o_nat @ Xss ) )
=> ~ ! [X3: option_list_o > nat,Xs3: list_o8358219866161907755_o_nat,Xss: list_l1581440573117054897_o_nat] :
( X2
!= ( cons_l8100113683866236641_o_nat @ ( cons_o6089670980329342427_o_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_537_transpose_Ocases,axiom,
! [X2: list_list_o] :
( ( X2 != nil_list_o )
=> ( ! [Xss: list_list_o] :
( X2
!= ( cons_list_o @ nil_o @ Xss ) )
=> ~ ! [X3: $o,Xs3: list_o,Xss: list_list_o] :
( X2
!= ( cons_list_o @ ( cons_o @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_538_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_539_sorted__wrt_Ocases,axiom,
! [X2: produc2087297845004381081t_real] :
( ! [P5: real > real > $o] :
( X2
!= ( produc8266819007309214227t_real @ P5 @ nil_real ) )
=> ~ ! [P5: real > real > $o,X3: real,Ys3: list_real] :
( X2
!= ( produc8266819007309214227t_real @ P5 @ ( cons_real @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_540_sorted__wrt_Ocases,axiom,
! [X2: produc867208691148388573list_o] :
( ! [P5: list_o > list_o > $o] :
( X2
!= ( produc5708246048495597389list_o @ P5 @ nil_list_o ) )
=> ~ ! [P5: list_o > list_o > $o,X3: list_o,Ys3: list_list_o] :
( X2
!= ( produc5708246048495597389list_o @ P5 @ ( cons_list_o @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_541_sorted__wrt_Ocases,axiom,
! [X2: produc2068713617857708901st_nat] :
( ! [P5: list_nat > list_nat > $o] :
( X2
!= ( produc6344517630863869919st_nat @ P5 @ nil_list_nat ) )
=> ~ ! [P5: list_nat > list_nat > $o,X3: list_nat,Ys3: list_list_nat] :
( X2
!= ( produc6344517630863869919st_nat @ P5 @ ( cons_list_nat @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_542_sorted__wrt_Ocases,axiom,
! [X2: produc8642409424279824599list_o] :
( ! [P5: $o > $o > $o] :
( X2
!= ( produc8744836578217649351list_o @ P5 @ nil_o ) )
=> ~ ! [P5: $o > $o > $o,X3: $o,Ys3: list_o] :
( X2
!= ( produc8744836578217649351list_o @ P5 @ ( cons_o @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_543_sorted__wrt_Ocases,axiom,
! [X2: produc7714731344432443428_o_nat] :
( ! [P5: ( option_list_o > nat ) > ( option_list_o > nat ) > $o] :
( X2
!= ( produc1027695606731132692_o_nat @ P5 @ nil_op686717097995950635_o_nat ) )
=> ~ ! [P5: ( option_list_o > nat ) > ( option_list_o > nat ) > $o,X3: option_list_o > nat,Ys3: list_o8358219866161907755_o_nat] :
( X2
!= ( produc1027695606731132692_o_nat @ P5 @ ( cons_o6089670980329342427_o_nat @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_544_sorted__wrt_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P5: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
=> ~ ! [P5: nat > nat > $o,X3: nat,Ys3: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_545_successively_Ocases,axiom,
! [X2: produc2087297845004381081t_real] :
( ! [P5: real > real > $o] :
( X2
!= ( produc8266819007309214227t_real @ P5 @ nil_real ) )
=> ( ! [P5: real > real > $o,X3: real] :
( X2
!= ( produc8266819007309214227t_real @ P5 @ ( cons_real @ X3 @ nil_real ) ) )
=> ~ ! [P5: real > real > $o,X3: real,Y3: real,Xs3: list_real] :
( X2
!= ( produc8266819007309214227t_real @ P5 @ ( cons_real @ X3 @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_546_successively_Ocases,axiom,
! [X2: produc867208691148388573list_o] :
( ! [P5: list_o > list_o > $o] :
( X2
!= ( produc5708246048495597389list_o @ P5 @ nil_list_o ) )
=> ( ! [P5: list_o > list_o > $o,X3: list_o] :
( X2
!= ( produc5708246048495597389list_o @ P5 @ ( cons_list_o @ X3 @ nil_list_o ) ) )
=> ~ ! [P5: list_o > list_o > $o,X3: list_o,Y3: list_o,Xs3: list_list_o] :
( X2
!= ( produc5708246048495597389list_o @ P5 @ ( cons_list_o @ X3 @ ( cons_list_o @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_547_successively_Ocases,axiom,
! [X2: produc2068713617857708901st_nat] :
( ! [P5: list_nat > list_nat > $o] :
( X2
!= ( produc6344517630863869919st_nat @ P5 @ nil_list_nat ) )
=> ( ! [P5: list_nat > list_nat > $o,X3: list_nat] :
( X2
!= ( produc6344517630863869919st_nat @ P5 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) )
=> ~ ! [P5: list_nat > list_nat > $o,X3: list_nat,Y3: list_nat,Xs3: list_list_nat] :
( X2
!= ( produc6344517630863869919st_nat @ P5 @ ( cons_list_nat @ X3 @ ( cons_list_nat @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_548_successively_Ocases,axiom,
! [X2: produc8642409424279824599list_o] :
( ! [P5: $o > $o > $o] :
( X2
!= ( produc8744836578217649351list_o @ P5 @ nil_o ) )
=> ( ! [P5: $o > $o > $o,X3: $o] :
( X2
!= ( produc8744836578217649351list_o @ P5 @ ( cons_o @ X3 @ nil_o ) ) )
=> ~ ! [P5: $o > $o > $o,X3: $o,Y3: $o,Xs3: list_o] :
( X2
!= ( produc8744836578217649351list_o @ P5 @ ( cons_o @ X3 @ ( cons_o @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_549_successively_Ocases,axiom,
! [X2: produc7714731344432443428_o_nat] :
( ! [P5: ( option_list_o > nat ) > ( option_list_o > nat ) > $o] :
( X2
!= ( produc1027695606731132692_o_nat @ P5 @ nil_op686717097995950635_o_nat ) )
=> ( ! [P5: ( option_list_o > nat ) > ( option_list_o > nat ) > $o,X3: option_list_o > nat] :
( X2
!= ( produc1027695606731132692_o_nat @ P5 @ ( cons_o6089670980329342427_o_nat @ X3 @ nil_op686717097995950635_o_nat ) ) )
=> ~ ! [P5: ( option_list_o > nat ) > ( option_list_o > nat ) > $o,X3: option_list_o > nat,Y3: option_list_o > nat,Xs3: list_o8358219866161907755_o_nat] :
( X2
!= ( produc1027695606731132692_o_nat @ P5 @ ( cons_o6089670980329342427_o_nat @ X3 @ ( cons_o6089670980329342427_o_nat @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_550_successively_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P5: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
=> ( ! [P5: nat > nat > $o,X3: nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [P5: nat > nat > $o,X3: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_551_map__tailrec__rev_Ocases,axiom,
! [X2: produc1616951275169580055st_nat] :
( ! [F2: nat > nat,Bs: list_nat] :
( X2
!= ( produc4626581765195395529st_nat @ F2 @ ( produc2694037385005941721st_nat @ nil_nat @ Bs ) ) )
=> ~ ! [F2: nat > nat,A2: nat,As: list_nat,Bs: list_nat] :
( X2
!= ( produc4626581765195395529st_nat @ F2 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A2 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_552_longest__common__prefix_Ocases,axiom,
! [X2: produc478978216448986841t_real] :
( ! [X3: real,Xs3: list_real,Y3: real,Ys3: list_real] :
( X2
!= ( produc1408950526243324945t_real @ ( cons_real @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_real] :
( X2
!= ( produc1408950526243324945t_real @ nil_real @ Uv2 ) )
=> ~ ! [Uu2: list_real] :
( X2
!= ( produc1408950526243324945t_real @ Uu2 @ nil_real ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_553_longest__common__prefix_Ocases,axiom,
! [X2: produc4690905340047322919list_o] :
( ! [X3: list_o,Xs3: list_list_o,Y3: list_o,Ys3: list_list_o] :
( X2
!= ( produc2957346356364703511list_o @ ( cons_list_o @ X3 @ Xs3 ) @ ( cons_list_o @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_list_o] :
( X2
!= ( produc2957346356364703511list_o @ nil_list_o @ Uv2 ) )
=> ~ ! [Uu2: list_list_o] :
( X2
!= ( produc2957346356364703511list_o @ Uu2 @ nil_list_o ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_554_longest__common__prefix_Ocases,axiom,
! [X2: produc4326814125627636033st_nat] :
( ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys3: list_list_nat] :
( X2
!= ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_list_nat] :
( X2
!= ( produc7129799990162260089st_nat @ nil_list_nat @ Uv2 ) )
=> ~ ! [Uu2: list_list_nat] :
( X2
!= ( produc7129799990162260089st_nat @ Uu2 @ nil_list_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_555_longest__common__prefix_Ocases,axiom,
! [X2: produc7102631898165422375list_o] :
( ! [X3: $o,Xs3: list_o,Y3: $o,Ys3: list_o] :
( X2
!= ( produc8435520187683070743list_o @ ( cons_o @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_o] :
( X2
!= ( produc8435520187683070743list_o @ nil_o @ Uv2 ) )
=> ~ ! [Uu2: list_o] :
( X2
!= ( produc8435520187683070743list_o @ Uu2 @ nil_o ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_556_longest__common__prefix_Ocases,axiom,
! [X2: produc5022493669877164263_o_nat] :
( ! [X3: option_list_o > nat,Xs3: list_o8358219866161907755_o_nat,Y3: option_list_o > nat,Ys3: list_o8358219866161907755_o_nat] :
( X2
!= ( produc4668812882758624855_o_nat @ ( cons_o6089670980329342427_o_nat @ X3 @ Xs3 ) @ ( cons_o6089670980329342427_o_nat @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_o8358219866161907755_o_nat] :
( X2
!= ( produc4668812882758624855_o_nat @ nil_op686717097995950635_o_nat @ Uv2 ) )
=> ~ ! [Uu2: list_o8358219866161907755_o_nat] :
( X2
!= ( produc4668812882758624855_o_nat @ Uu2 @ nil_op686717097995950635_o_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_557_longest__common__prefix_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Uv2 ) )
=> ~ ! [Uu2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Uu2 @ nil_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_558_shuffles_Ocases,axiom,
! [X2: produc478978216448986841t_real] :
( ! [Ys3: list_real] :
( X2
!= ( produc1408950526243324945t_real @ nil_real @ Ys3 ) )
=> ( ! [Xs3: list_real] :
( X2
!= ( produc1408950526243324945t_real @ Xs3 @ nil_real ) )
=> ~ ! [X3: real,Xs3: list_real,Y3: real,Ys3: list_real] :
( X2
!= ( produc1408950526243324945t_real @ ( cons_real @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_559_shuffles_Ocases,axiom,
! [X2: produc4690905340047322919list_o] :
( ! [Ys3: list_list_o] :
( X2
!= ( produc2957346356364703511list_o @ nil_list_o @ Ys3 ) )
=> ( ! [Xs3: list_list_o] :
( X2
!= ( produc2957346356364703511list_o @ Xs3 @ nil_list_o ) )
=> ~ ! [X3: list_o,Xs3: list_list_o,Y3: list_o,Ys3: list_list_o] :
( X2
!= ( produc2957346356364703511list_o @ ( cons_list_o @ X3 @ Xs3 ) @ ( cons_list_o @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_560_shuffles_Ocases,axiom,
! [X2: produc4326814125627636033st_nat] :
( ! [Ys3: list_list_nat] :
( X2
!= ( produc7129799990162260089st_nat @ nil_list_nat @ Ys3 ) )
=> ( ! [Xs3: list_list_nat] :
( X2
!= ( produc7129799990162260089st_nat @ Xs3 @ nil_list_nat ) )
=> ~ ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys3: list_list_nat] :
( X2
!= ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_561_shuffles_Ocases,axiom,
! [X2: produc7102631898165422375list_o] :
( ! [Ys3: list_o] :
( X2
!= ( produc8435520187683070743list_o @ nil_o @ Ys3 ) )
=> ( ! [Xs3: list_o] :
( X2
!= ( produc8435520187683070743list_o @ Xs3 @ nil_o ) )
=> ~ ! [X3: $o,Xs3: list_o,Y3: $o,Ys3: list_o] :
( X2
!= ( produc8435520187683070743list_o @ ( cons_o @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_562_shuffles_Ocases,axiom,
! [X2: produc5022493669877164263_o_nat] :
( ! [Ys3: list_o8358219866161907755_o_nat] :
( X2
!= ( produc4668812882758624855_o_nat @ nil_op686717097995950635_o_nat @ Ys3 ) )
=> ( ! [Xs3: list_o8358219866161907755_o_nat] :
( X2
!= ( produc4668812882758624855_o_nat @ Xs3 @ nil_op686717097995950635_o_nat ) )
=> ~ ! [X3: option_list_o > nat,Xs3: list_o8358219866161907755_o_nat,Y3: option_list_o > nat,Ys3: list_o8358219866161907755_o_nat] :
( X2
!= ( produc4668812882758624855_o_nat @ ( cons_o6089670980329342427_o_nat @ X3 @ Xs3 ) @ ( cons_o6089670980329342427_o_nat @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_563_shuffles_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ( ! [Xs3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
=> ~ ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_564_splice_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ~ ! [X3: nat,Xs3: list_nat,Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_565_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_566_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys3 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_567_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_568_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_569_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_570_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_571_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_572_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_573_append__Cons,axiom,
! [X2: $o,Xs: list_o,Ys: list_o] :
( ( append_o @ ( cons_o @ X2 @ Xs ) @ Ys )
= ( cons_o @ X2 @ ( append_o @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_574_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_575_Cons__eq__appendI,axiom,
! [X2: $o,Xs1: list_o,Ys: list_o,Xs: list_o,Zs: list_o] :
( ( ( cons_o @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_o @ Xs1 @ Zs ) )
=> ( ( cons_o @ X2 @ Xs )
= ( append_o @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_576_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_577_is__none__simps_I2_J,axiom,
! [X2: list_o] :
~ ( is_none_list_o @ ( some_list_o @ X2 ) ) ).
% is_none_simps(2)
thf(fact_578_is__none__simps_I1_J,axiom,
is_none_list_o @ none_list_o ).
% is_none_simps(1)
thf(fact_579_Option_Ois__none__def,axiom,
( is_none_list_o
= ( ^ [X: option_list_o] : ( X = none_list_o ) ) ) ).
% Option.is_none_def
thf(fact_580_rev__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X3: $o] : ( P @ ( cons_o @ X3 @ nil_o ) )
=> ( ! [X3: $o,Xs3: list_o] :
( ( Xs3 != nil_o )
=> ( ( P @ Xs3 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X3 @ nil_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_581_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_582_append__eq__Cons__conv,axiom,
! [Ys: list_o,Zs: list_o,X2: $o,Xs: list_o] :
( ( ( append_o @ Ys @ Zs )
= ( cons_o @ X2 @ Xs ) )
= ( ( ( Ys = nil_o )
& ( Zs
= ( cons_o @ X2 @ Xs ) ) )
| ? [Ys4: list_o] :
( ( Ys
= ( cons_o @ X2 @ Ys4 ) )
& ( ( append_o @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_583_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_584_Cons__eq__append__conv,axiom,
! [X2: $o,Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( cons_o @ X2 @ Xs )
= ( append_o @ Ys @ Zs ) )
= ( ( ( Ys = nil_o )
& ( ( cons_o @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_o] :
( ( ( cons_o @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_o @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_585_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_586_rev__exhaust,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ~ ! [Ys3: list_o,Y3: $o] :
( Xs
!= ( append_o @ Ys3 @ ( cons_o @ Y3 @ nil_o ) ) ) ) ).
% rev_exhaust
thf(fact_587_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys3: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_588_rev__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ( P @ nil_o )
=> ( ! [X3: $o,Xs3: list_o] :
( ( P @ Xs3 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X3 @ nil_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_589_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_590_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( ( X3 != Y3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( ( X3 = Y3 )
=> ( ~ ( prefix_nat @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_591_not__prefix__induct,axiom,
! [Ps: list_o,Ls: list_o,P: list_o > list_o > $o] :
( ~ ( prefix_o @ Ps @ Ls )
=> ( ! [X3: $o,Xs3: list_o] : ( P @ ( cons_o @ X3 @ Xs3 ) @ nil_o )
=> ( ! [X3: $o,Xs3: list_o,Y3: $o,Ys3: list_o] :
( ( X3 != Y3 )
=> ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys3 ) ) )
=> ( ! [X3: $o,Xs3: list_o,Y3: $o,Ys3: list_o] :
( ( X3 = Y3 )
=> ( ~ ( prefix_o @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_592_not__prefix__cases,axiom,
! [Ps: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A2: nat,As: list_nat] :
( ( Ps
= ( cons_nat @ A2 @ As ) )
=> ! [X3: nat,Xs3: list_nat] :
( ( Ls
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( X3 = A2 )
=> ( prefix_nat @ As @ Xs3 ) ) ) )
=> ~ ! [A2: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A2 @ As ) )
=> ! [X3: nat] :
( ? [Xs3: list_nat] :
( Ls
= ( cons_nat @ X3 @ Xs3 ) )
=> ( X3 = A2 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_593_not__prefix__cases,axiom,
! [Ps: list_o,Ls: list_o] :
( ~ ( prefix_o @ Ps @ Ls )
=> ( ( ( Ps != nil_o )
=> ( Ls != nil_o ) )
=> ( ! [A2: $o,As: list_o] :
( ( Ps
= ( cons_o @ A2 @ As ) )
=> ! [X3: $o,Xs3: list_o] :
( ( Ls
= ( cons_o @ X3 @ Xs3 ) )
=> ( ( X3 = A2 )
=> ( prefix_o @ As @ Xs3 ) ) ) )
=> ~ ! [A2: $o] :
( ? [As: list_o] :
( Ps
= ( cons_o @ A2 @ As ) )
=> ! [X3: $o] :
( ? [Xs3: list_o] :
( Ls
= ( cons_o @ X3 @ Xs3 ) )
=> ( X3 = A2 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_594_prefix__Cons,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) )
= ( ( Xs = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_595_prefix__Cons,axiom,
! [Xs: list_o,Y2: $o,Ys: list_o] :
( ( prefix_o @ Xs @ ( cons_o @ Y2 @ Ys ) )
= ( ( Xs = nil_o )
| ? [Zs3: list_o] :
( ( Xs
= ( cons_o @ Y2 @ Zs3 ) )
& ( prefix_o @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_596_prefix__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_597_prefix__code_I2_J,axiom,
! [X2: $o,Xs: list_o] :
~ ( prefix_o @ ( cons_o @ X2 @ Xs ) @ nil_o ) ).
% prefix_code(2)
thf(fact_598_accp_Ocases,axiom,
! [R: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,A: produc4882884732533091879list_o] :
( ( accp_P4987914878654715440list_o @ R @ A )
=> ! [Y5: produc4882884732533091879list_o] :
( ( R @ Y5 @ A )
=> ( accp_P4987914878654715440list_o @ R @ Y5 ) ) ) ).
% accp.cases
thf(fact_599_accp_Osimps,axiom,
( accp_P4987914878654715440list_o
= ( ^ [R3: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,A3: produc4882884732533091879list_o] :
? [X: produc4882884732533091879list_o] :
( ( A3 = X )
& ! [Y: produc4882884732533091879list_o] :
( ( R3 @ Y @ X )
=> ( accp_P4987914878654715440list_o @ R3 @ Y ) ) ) ) ) ).
% accp.simps
thf(fact_600_maps__simps_I1_J,axiom,
! [F: nat > list_o,X2: nat,Xs: list_nat] :
( ( maps_nat_o @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_o @ ( F @ X2 ) @ ( maps_nat_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_601_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_602_prefixes__snoc,axiom,
! [Xs: list_o,X2: $o] :
( ( prefixes_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= ( append_list_o @ ( prefixes_o @ Xs ) @ ( cons_list_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ nil_list_o ) ) ) ).
% prefixes_snoc
thf(fact_603_prefixes__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_604_butlast__snoc,axiom,
! [Xs: list_o,X2: $o] :
( ( butlast_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_605_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_606_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y2: $o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ ( append_o @ Ys @ ( cons_o @ Y2 @ nil_o ) ) ) @ ( listrel1_o @ R ) )
= ( ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R ) )
& ( X2 = Y2 ) )
| ( ( Xs = Ys )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_607_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
& ( X2 = Y2 ) )
| ( ( Xs = Ys )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_608_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_option_list_o,X2: option_list_o,Ys: list_option_list_o,Y2: option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( append_option_list_o @ Xs @ ( cons_option_list_o @ X2 @ nil_option_list_o ) ) @ ( append_option_list_o @ Ys @ ( cons_option_list_o @ Y2 @ nil_option_list_o ) ) ) @ ( listre3104272764119459907list_o @ R ) )
= ( ( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( listre3104272764119459907list_o @ R ) )
& ( X2 = Y2 ) )
| ( ( Xs = Ys )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_609_last__snoc,axiom,
! [Xs: list_o,X2: $o] :
( ( last_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= X2 ) ).
% last_snoc
thf(fact_610_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_611_rotate1_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( rotate1_o @ ( cons_o @ X2 @ Xs ) )
= ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) ) ).
% rotate1.simps(2)
thf(fact_612_rotate1_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_613_splice_Opinduct,axiom,
! [A0: list_nat,A1: list_nat,P: list_nat > list_nat > $o] :
( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ A0 @ A1 ) )
=> ( ! [Ys3: list_nat] :
( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ( P @ nil_nat @ Ys3 ) )
=> ( ! [X3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ Ys3 ) )
=> ( ( P @ Ys3 @ Xs3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_614_prefix__snocD,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ Ys )
=> ( strict_prefix_nat @ Xs @ Ys ) ) ).
% prefix_snocD
thf(fact_615_prefix__snocD,axiom,
! [Xs: list_o,X2: $o,Ys: list_o] :
( ( prefix_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ Ys )
=> ( strict_prefix_o @ Xs @ Ys ) ) ).
% prefix_snocD
thf(fact_616_strict__prefix__simps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( strict_prefix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( strict_prefix_nat @ Xs @ Ys ) ) ) ).
% strict_prefix_simps(3)
thf(fact_617_strict__prefix__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat] : ( strict_prefix_nat @ nil_nat @ ( cons_nat @ X2 @ Xs ) ) ).
% strict_prefix_simps(2)
thf(fact_618_last__appendL,axiom,
! [Ys: list_o,Xs: list_o] :
( ( Ys = nil_o )
=> ( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( last_o @ Xs ) ) ) ).
% last_appendL
thf(fact_619_last__appendR,axiom,
! [Ys: list_o,Xs: list_o] :
( ( Ys != nil_o )
=> ( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( last_o @ Ys ) ) ) ).
% last_appendR
thf(fact_620_Cons__listrel1__Cons,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
& ( Xs = Ys ) )
| ( ( X2 = Y2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_621_Cons__listrel1__Cons,axiom,
! [X2: option_list_o,Xs: list_option_list_o,Y2: option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ X2 @ Xs ) @ ( cons_option_list_o @ Y2 @ Ys ) ) @ ( listre3104272764119459907list_o @ R ) )
= ( ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ R )
& ( Xs = Ys ) )
| ( ( X2 = Y2 )
& ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( listre3104272764119459907list_o @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_622_append__butlast__last__id,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( ( append_o @ ( butlast_o @ Xs ) @ ( cons_o @ ( last_o @ Xs ) @ nil_o ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_623_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_624_strict__prefix__def,axiom,
( strict_prefix_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
( ( prefix_o @ Xs2 @ Ys2 )
& ( Xs2 != Ys2 ) ) ) ) ).
% strict_prefix_def
thf(fact_625_strict__prefixI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ( Xs != Ys )
=> ( strict_prefix_o @ Xs @ Ys ) ) ) ).
% strict_prefixI
thf(fact_626_strict__prefixE,axiom,
! [Xs: list_o,Ys: list_o] :
( ( strict_prefix_o @ Xs @ Ys )
=> ~ ( ( prefix_o @ Xs @ Ys )
=> ( Xs = Ys ) ) ) ).
% strict_prefixE
thf(fact_627_prefix__order_Odual__order_Onot__eq__order__implies__strict,axiom,
! [A: list_o,B: list_o] :
( ( A != B )
=> ( ( prefix_o @ B @ A )
=> ( strict_prefix_o @ B @ A ) ) ) ).
% prefix_order.dual_order.not_eq_order_implies_strict
thf(fact_628_prefix__order_Odual__order_Ostrict__implies__order,axiom,
! [B: list_o,A: list_o] :
( ( strict_prefix_o @ B @ A )
=> ( prefix_o @ B @ A ) ) ).
% prefix_order.dual_order.strict_implies_order
thf(fact_629_prefix__order_Onot__eq__order__implies__strict,axiom,
! [A: list_o,B: list_o] :
( ( A != B )
=> ( ( prefix_o @ A @ B )
=> ( strict_prefix_o @ A @ B ) ) ) ).
% prefix_order.not_eq_order_implies_strict
thf(fact_630_prefix__order_Odual__order_Ostrict__iff__order,axiom,
( strict_prefix_o
= ( ^ [B3: list_o,A3: list_o] :
( ( prefix_o @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% prefix_order.dual_order.strict_iff_order
thf(fact_631_prefix__order_Odual__order_Oorder__iff__strict,axiom,
( prefix_o
= ( ^ [B3: list_o,A3: list_o] :
( ( strict_prefix_o @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% prefix_order.dual_order.order_iff_strict
thf(fact_632_prefix__order_Odual__order_Ostrict__iff__not,axiom,
( strict_prefix_o
= ( ^ [B3: list_o,A3: list_o] :
( ( prefix_o @ B3 @ A3 )
& ~ ( prefix_o @ A3 @ B3 ) ) ) ) ).
% prefix_order.dual_order.strict_iff_not
thf(fact_633_prefix__order_Odual__order_Ostrict__trans2,axiom,
! [B: list_o,A: list_o,C: list_o] :
( ( strict_prefix_o @ B @ A )
=> ( ( prefix_o @ C @ B )
=> ( strict_prefix_o @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans2
thf(fact_634_prefix__order_Odual__order_Ostrict__trans1,axiom,
! [B: list_o,A: list_o,C: list_o] :
( ( prefix_o @ B @ A )
=> ( ( strict_prefix_o @ C @ B )
=> ( strict_prefix_o @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans1
thf(fact_635_prefix__order_Ostrict__implies__order,axiom,
! [A: list_o,B: list_o] :
( ( strict_prefix_o @ A @ B )
=> ( prefix_o @ A @ B ) ) ).
% prefix_order.strict_implies_order
thf(fact_636_prefix__order_Ole__imp__less__or__eq,axiom,
! [X2: list_o,Y2: list_o] :
( ( prefix_o @ X2 @ Y2 )
=> ( ( strict_prefix_o @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% prefix_order.le_imp_less_or_eq
thf(fact_637_prefix__order_Ostrict__iff__order,axiom,
( strict_prefix_o
= ( ^ [A3: list_o,B3: list_o] :
( ( prefix_o @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% prefix_order.strict_iff_order
thf(fact_638_prefix__order_Oorder__iff__strict,axiom,
( prefix_o
= ( ^ [A3: list_o,B3: list_o] :
( ( strict_prefix_o @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% prefix_order.order_iff_strict
thf(fact_639_prefix__order_Ostrict__iff__not,axiom,
( strict_prefix_o
= ( ^ [A3: list_o,B3: list_o] :
( ( prefix_o @ A3 @ B3 )
& ~ ( prefix_o @ B3 @ A3 ) ) ) ) ).
% prefix_order.strict_iff_not
thf(fact_640_prefix__order_Oless__le__not__le,axiom,
( strict_prefix_o
= ( ^ [X: list_o,Y: list_o] :
( ( prefix_o @ X @ Y )
& ~ ( prefix_o @ Y @ X ) ) ) ) ).
% prefix_order.less_le_not_le
thf(fact_641_prefix__order_Ostrict__trans2,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( strict_prefix_o @ A @ B )
=> ( ( prefix_o @ B @ C )
=> ( strict_prefix_o @ A @ C ) ) ) ).
% prefix_order.strict_trans2
thf(fact_642_prefix__order_Ostrict__trans1,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( prefix_o @ A @ B )
=> ( ( strict_prefix_o @ B @ C )
=> ( strict_prefix_o @ A @ C ) ) ) ).
% prefix_order.strict_trans1
thf(fact_643_prefix__order_Oless__le__trans,axiom,
! [X2: list_o,Y2: list_o,Z: list_o] :
( ( strict_prefix_o @ X2 @ Y2 )
=> ( ( prefix_o @ Y2 @ Z )
=> ( strict_prefix_o @ X2 @ Z ) ) ) ).
% prefix_order.less_le_trans
thf(fact_644_prefix__order_Ole__less__trans,axiom,
! [X2: list_o,Y2: list_o,Z: list_o] :
( ( prefix_o @ X2 @ Y2 )
=> ( ( strict_prefix_o @ Y2 @ Z )
=> ( strict_prefix_o @ X2 @ Z ) ) ) ).
% prefix_order.le_less_trans
thf(fact_645_prefix__order_Oantisym__conv2,axiom,
! [X2: list_o,Y2: list_o] :
( ( prefix_o @ X2 @ Y2 )
=> ( ( ~ ( strict_prefix_o @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% prefix_order.antisym_conv2
thf(fact_646_prefix__order_Oantisym__conv1,axiom,
! [X2: list_o,Y2: list_o] :
( ~ ( strict_prefix_o @ X2 @ Y2 )
=> ( ( prefix_o @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% prefix_order.antisym_conv1
thf(fact_647_prefix__order_Ole__neq__trans,axiom,
! [A: list_o,B: list_o] :
( ( prefix_o @ A @ B )
=> ( ( A != B )
=> ( strict_prefix_o @ A @ B ) ) ) ).
% prefix_order.le_neq_trans
thf(fact_648_prefix__order_Oless__imp__le,axiom,
! [X2: list_o,Y2: list_o] :
( ( strict_prefix_o @ X2 @ Y2 )
=> ( prefix_o @ X2 @ Y2 ) ) ).
% prefix_order.less_imp_le
thf(fact_649_prefix__order_Onless__le,axiom,
! [A: list_o,B: list_o] :
( ( ~ ( strict_prefix_o @ A @ B ) )
= ( ~ ( prefix_o @ A @ B )
| ( A = B ) ) ) ).
% prefix_order.nless_le
thf(fact_650_prefix__order_Oless__le,axiom,
( strict_prefix_o
= ( ^ [X: list_o,Y: list_o] :
( ( prefix_o @ X @ Y )
& ( X != Y ) ) ) ) ).
% prefix_order.less_le
thf(fact_651_prefix__order_Ole__less,axiom,
( prefix_o
= ( ^ [X: list_o,Y: list_o] :
( ( strict_prefix_o @ X @ Y )
| ( X = Y ) ) ) ) ).
% prefix_order.le_less
thf(fact_652_prefix__order_OleD,axiom,
! [Y2: list_o,X2: list_o] :
( ( prefix_o @ Y2 @ X2 )
=> ~ ( strict_prefix_o @ X2 @ Y2 ) ) ).
% prefix_order.leD
thf(fact_653_listrel1I2,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X2: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ X2 @ Ys ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I2
thf(fact_654_append__listrel1I,axiom,
! [Xs: list_o,Ys: list_o,R: set_Product_prod_o_o,Us: list_o,Vs: list_o] :
( ( ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Us @ Vs ) @ ( listrel1_o @ R ) ) ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ Us ) @ ( append_o @ Ys @ Vs ) ) @ ( listrel1_o @ R ) ) ) ).
% append_listrel1I
thf(fact_655_prefixeq__butlast,axiom,
! [Xs: list_o] : ( prefix_o @ ( butlast_o @ Xs ) @ Xs ) ).
% prefixeq_butlast
thf(fact_656_snoc__eq__iff__butlast,axiom,
! [Xs: list_o,X2: $o,Ys: list_o] :
( ( ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) )
= Ys )
= ( ( Ys != nil_o )
& ( ( butlast_o @ Ys )
= Xs )
& ( ( last_o @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_657_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs )
& ( ( last_nat @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_658_strict__prefixI_H,axiom,
! [Ys: list_o,Xs: list_o,Z: $o,Zs: list_o] :
( ( Ys
= ( append_o @ Xs @ ( cons_o @ Z @ Zs ) ) )
=> ( strict_prefix_o @ Xs @ Ys ) ) ).
% strict_prefixI'
thf(fact_659_strict__prefixI_H,axiom,
! [Ys: list_nat,Xs: list_nat,Z: nat,Zs: list_nat] :
( ( Ys
= ( append_nat @ Xs @ ( cons_nat @ Z @ Zs ) ) )
=> ( strict_prefix_nat @ Xs @ Ys ) ) ).
% strict_prefixI'
thf(fact_660_strict__prefixE_H,axiom,
! [Xs: list_o,Ys: list_o] :
( ( strict_prefix_o @ Xs @ Ys )
=> ~ ! [Z3: $o,Zs2: list_o] :
( Ys
!= ( append_o @ Xs @ ( cons_o @ Z3 @ Zs2 ) ) ) ) ).
% strict_prefixE'
thf(fact_661_strict__prefixE_H,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( strict_prefix_nat @ Xs @ Ys )
=> ~ ! [Z3: nat,Zs2: list_nat] :
( Ys
!= ( append_nat @ Xs @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) ).
% strict_prefixE'
thf(fact_662_listrel1I1,axiom,
! [X2: nat,Y2: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_663_listrel1I1,axiom,
! [X2: option_list_o,Y2: option_list_o,R: set_Pr7497825696620840711list_o,Xs: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ R )
=> ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ X2 @ Xs ) @ ( cons_option_list_o @ Y2 @ Xs ) ) @ ( listre3104272764119459907list_o @ R ) ) ) ).
% listrel1I1
thf(fact_664_Cons__listrel1E1,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ! [Y3: nat] :
( ( Ys
= ( cons_nat @ Y3 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_665_Cons__listrel1E1,axiom,
! [X2: option_list_o,Xs: list_option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ X2 @ Xs ) @ Ys ) @ ( listre3104272764119459907list_o @ R ) )
=> ( ! [Y3: option_list_o] :
( ( Ys
= ( cons_option_list_o @ Y3 @ Xs ) )
=> ~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_option_list_o] :
( ( Ys
= ( cons_option_list_o @ X2 @ Zs2 ) )
=> ~ ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Zs2 ) @ ( listre3104272764119459907list_o @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_666_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X3: nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_667_Cons__listrel1E2,axiom,
! [Xs: list_option_list_o,Y2: option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ ( cons_option_list_o @ Y2 @ Ys ) ) @ ( listre3104272764119459907list_o @ R ) )
=> ( ! [X3: option_list_o] :
( ( Xs
= ( cons_option_list_o @ X3 @ Ys ) )
=> ~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_option_list_o] :
( ( Xs
= ( cons_option_list_o @ Y2 @ Zs2 ) )
=> ~ ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Zs2 @ Ys ) @ ( listre3104272764119459907list_o @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_668_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_669_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_670_last_Osimps,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_671_last__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( ( ( Ys = nil_o )
=> ( last_o @ Xs ) )
& ( ( Ys != nil_o )
=> ( last_o @ Ys ) ) ) ) ).
% last_append
thf(fact_672_longest__common__suffix,axiom,
! [Xs: list_o,Ys: list_o] :
? [Ss: list_o,Xs4: list_o,Ys5: list_o] :
( ( Xs
= ( append_o @ Xs4 @ Ss ) )
& ( Ys
= ( append_o @ Ys5 @ Ss ) )
& ( ( Xs4 = nil_o )
| ( Ys5 = nil_o )
| ( ( last_o @ Xs4 )
!= ( last_o @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_673_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_674_butlast__append,axiom,
! [Ys: list_o,Xs: list_o] :
( ( ( Ys = nil_o )
=> ( ( butlast_o @ ( append_o @ Xs @ Ys ) )
= ( butlast_o @ Xs ) ) )
& ( ( Ys != nil_o )
=> ( ( butlast_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ Xs @ ( butlast_o @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_675_listrel1I,axiom,
! [X2: $o,Y2: $o,R: set_Product_prod_o_o,Xs: list_o,Us: list_o,Vs: list_o,Ys: list_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R )
=> ( ( Xs
= ( append_o @ Us @ ( cons_o @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_o @ Us @ ( cons_o @ Y2 @ Vs ) ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R ) ) ) ) ) ).
% listrel1I
thf(fact_676_listrel1I,axiom,
! [X2: nat,Y2: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_nat @ Us @ ( cons_nat @ Y2 @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_677_listrel1I,axiom,
! [X2: option_list_o,Y2: option_list_o,R: set_Pr7497825696620840711list_o,Xs: list_option_list_o,Us: list_option_list_o,Vs: list_option_list_o,Ys: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ R )
=> ( ( Xs
= ( append_option_list_o @ Us @ ( cons_option_list_o @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_option_list_o @ Us @ ( cons_option_list_o @ Y2 @ Vs ) ) )
=> ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( listre3104272764119459907list_o @ R ) ) ) ) ) ).
% listrel1I
thf(fact_678_listrel1E,axiom,
! [Xs: list_o,Ys: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R ) )
=> ~ ! [X3: $o,Y3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R )
=> ! [Us3: list_o,Vs2: list_o] :
( ( Xs
= ( append_o @ Us3 @ ( cons_o @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_o @ Us3 @ ( cons_o @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_679_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ! [Us3: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us3 @ ( cons_nat @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_nat @ Us3 @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_680_listrel1E,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( listre3104272764119459907list_o @ R ) )
=> ~ ! [X3: option_list_o,Y3: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ R )
=> ! [Us3: list_option_list_o,Vs2: list_option_list_o] :
( ( Xs
= ( append_option_list_o @ Us3 @ ( cons_option_list_o @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_option_list_o @ Us3 @ ( cons_option_list_o @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_681_prefixes__eq__snoc,axiom,
! [Ys: list_o,Xs: list_list_o,X2: list_o] :
( ( ( prefixes_o @ Ys )
= ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) )
= ( ( ( ( Ys = nil_o )
& ( Xs = nil_list_o ) )
| ? [Z4: $o,Zs3: list_o] :
( ( Ys
= ( append_o @ Zs3 @ ( cons_o @ Z4 @ nil_o ) ) )
& ( Xs
= ( prefixes_o @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_682_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z4 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_683_splice_Opelims,axiom,
! [X2: list_nat,Xa: list_nat,Y2: list_nat] :
( ( ( splice_nat @ X2 @ Xa )
= Y2 )
=> ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_nat )
=> ( ( Y2 = Xa )
=> ~ ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa ) ) ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( Y2
= ( cons_nat @ X3 @ ( splice_nat @ Xa @ Xs3 ) ) )
=> ~ ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_684_shuffles_Opinduct,axiom,
! [A0: list_nat,A1: list_nat,P: list_nat > list_nat > $o] :
( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ A0 @ A1 ) )
=> ( ! [Ys3: list_nat] :
( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ( P @ nil_nat @ Ys3 ) )
=> ( ! [Xs3: list_nat] :
( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
=> ( P @ Xs3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( ( P @ Xs3 @ ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons_nat @ X3 @ Xs3 ) @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_685_splice_Opsimps_I2_J,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) )
=> ( ( splice_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( splice_nat @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_686_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( Xs
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_687_concat__eq__append__conv,axiom,
! [Xss2: list_list_o,Ys: list_o,Zs: list_o] :
( ( ( concat_o @ Xss2 )
= ( append_o @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_o )
=> ( ( Ys = nil_o )
& ( Zs = nil_o ) ) )
& ( ( Xss2 != nil_list_o )
=> ? [Xss1: list_list_o,Xs2: list_o,Xs5: list_o,Xss22: list_list_o] :
( ( Xss2
= ( append_list_o @ Xss1 @ ( cons_list_o @ ( append_o @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_o @ ( concat_o @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_o @ Xs5 @ ( concat_o @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_688_concat__append,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( concat_o @ ( append_list_o @ Xs @ Ys ) )
= ( append_o @ ( concat_o @ Xs ) @ ( concat_o @ Ys ) ) ) ).
% concat_append
thf(fact_689_splice_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( splice_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( splice_nat @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_690_concat_Osimps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( concat_o @ ( cons_list_o @ X2 @ Xs ) )
= ( append_o @ X2 @ ( concat_o @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_691_splice_Oelims,axiom,
! [X2: list_nat,Xa: list_nat,Y2: list_nat] :
( ( ( splice_nat @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_nat )
=> ( Y2 != Xa ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs3 ) )
=> ( Y2
!= ( cons_nat @ X3 @ ( splice_nat @ Xa @ Xs3 ) ) ) ) ) ) ).
% splice.elims
thf(fact_692_concat__eq__appendD,axiom,
! [Xss2: list_list_o,Ys: list_o,Zs: list_o] :
( ( ( concat_o @ Xss2 )
= ( append_o @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_o )
=> ? [Xss12: list_list_o,Xs3: list_o,Xs4: list_o,Xss23: list_list_o] :
( ( Xss2
= ( append_list_o @ Xss12 @ ( cons_list_o @ ( append_o @ Xs3 @ Xs4 ) @ Xss23 ) ) )
& ( Ys
= ( append_o @ ( concat_o @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_o @ Xs4 @ ( concat_o @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_693_suffixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_694_map__tailrec__rev_Opelims,axiom,
! [X2: nat > nat,Xa: list_nat,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X2 @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_nat )
=> ( ( Y2 = Xb )
=> ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X2 @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
=> ~ ! [A2: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A2 @ As ) )
=> ( ( Y2
= ( map_ta7164188454487880599at_nat @ X2 @ As @ ( cons_nat @ ( X2 @ A2 ) @ Xb ) ) )
=> ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X2 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A2 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_695_lexord__append__left__rightI,axiom,
! [A: $o,B: $o,R: set_Product_prod_o_o,U: list_o,X2: list_o,Y2: list_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ U @ ( cons_o @ A @ X2 ) ) @ ( append_o @ U @ ( cons_o @ B @ Y2 ) ) ) @ ( lexord_o @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_696_lexord__append__left__rightI,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X2: list_nat,Y2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X2 ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y2 ) ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_697_lexord__append__left__rightI,axiom,
! [A: option_list_o,B: option_list_o,R: set_Pr7497825696620840711list_o,U: list_option_list_o,X2: list_option_list_o,Y2: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A @ B ) @ R )
=> ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( append_option_list_o @ U @ ( cons_option_list_o @ A @ X2 ) ) @ ( append_option_list_o @ U @ ( cons_option_list_o @ B @ Y2 ) ) ) @ ( lexord_option_list_o @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_698_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 ) )
| ? [X: nat,Y: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
& ( A22
= ( cons_nat @ Y @ Ys2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_699_listrel_Osimps,axiom,
! [A1: list_option_list_o,A22: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ A1 @ A22 ) @ ( listre2892261910510432881list_o @ R ) )
= ( ( ( A1 = nil_option_list_o )
& ( A22 = nil_option_list_o ) )
| ? [X: option_list_o,Y: option_list_o,Xs2: list_option_list_o,Ys2: list_option_list_o] :
( ( A1
= ( cons_option_list_o @ X @ Xs2 ) )
& ( A22
= ( cons_option_list_o @ Y @ Ys2 ) )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ R )
& ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs2 @ Ys2 ) @ ( listre2892261910510432881list_o @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_700_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_P6834414599653733731al_nat,R: set_Pr836851361167288120al_nat] :
( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ A1 @ A22 ) @ ( listre1017866332378306906al_nat @ R ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_Pr1917482104270272867al_nat ) )
| ? [X: nat,Y: produc3741383161447143261al_nat,Xs2: list_nat,Ys2: list_P6834414599653733731al_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
& ( A22
= ( cons_P500833500243608851al_nat @ Y @ Ys2 ) )
& ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X @ Y ) @ R )
& ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs2 @ Ys2 ) @ ( listre1017866332378306906al_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_701_listrel_Osimps,axiom,
! [A1: list_real,A22: list_nat,R: set_Pr3510011417693777981al_nat] :
( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ A1 @ A22 ) @ ( listrel_real_nat @ R ) )
= ( ( ( A1 = nil_real )
& ( A22 = nil_nat ) )
| ? [X: real,Y: nat,Xs2: list_real,Ys2: list_nat] :
( ( A1
= ( cons_real @ X @ Xs2 ) )
& ( A22
= ( cons_nat @ Y @ Ys2 ) )
& ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y ) @ R )
& ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs2 @ Ys2 ) @ ( listrel_real_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_702_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 ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_703_listrel_Ocases,axiom,
! [A1: list_option_list_o,A22: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ A1 @ A22 ) @ ( listre2892261910510432881list_o @ R ) )
=> ( ( ( A1 = nil_option_list_o )
=> ( A22 != nil_option_list_o ) )
=> ~ ! [X3: option_list_o,Y3: option_list_o,Xs3: list_option_list_o] :
( ( A1
= ( cons_option_list_o @ X3 @ Xs3 ) )
=> ! [Ys3: list_option_list_o] :
( ( A22
= ( cons_option_list_o @ Y3 @ Ys3 ) )
=> ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ R )
=> ~ ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs3 @ Ys3 ) @ ( listre2892261910510432881list_o @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_704_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_P6834414599653733731al_nat,R: set_Pr836851361167288120al_nat] :
( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ A1 @ A22 ) @ ( listre1017866332378306906al_nat @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_Pr1917482104270272867al_nat ) )
=> ~ ! [X3: nat,Y3: produc3741383161447143261al_nat,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Ys3: list_P6834414599653733731al_nat] :
( ( A22
= ( cons_P500833500243608851al_nat @ Y3 @ Ys3 ) )
=> ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X3 @ Y3 ) @ R )
=> ~ ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs3 @ Ys3 ) @ ( listre1017866332378306906al_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_705_listrel_Ocases,axiom,
! [A1: list_real,A22: list_nat,R: set_Pr3510011417693777981al_nat] :
( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ A1 @ A22 ) @ ( listrel_real_nat @ R ) )
=> ( ( ( A1 = nil_real )
=> ( A22 != nil_nat ) )
=> ~ ! [X3: real,Y3: nat,Xs3: list_real] :
( ( A1
= ( cons_real @ X3 @ Xs3 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y3 ) @ R )
=> ~ ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs3 @ Ys3 ) @ ( listrel_real_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_706_lexord__cons__cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y2 ) ) @ ( lexord_nat @ R ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
| ( ( A = B )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y2 ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_707_lexord__cons__cons,axiom,
! [A: option_list_o,X2: list_option_list_o,B: option_list_o,Y2: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ A @ X2 ) @ ( cons_option_list_o @ B @ Y2 ) ) @ ( lexord_option_list_o @ R ) )
= ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A @ B ) @ R )
| ( ( A = B )
& ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ X2 @ Y2 ) @ ( lexord_option_list_o @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_708_lexord__Nil__left,axiom,
! [Y2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y2 ) @ ( lexord_nat @ R ) )
= ( ? [A3: nat,X: list_nat] :
( Y2
= ( cons_nat @ A3 @ X ) ) ) ) ).
% lexord_Nil_left
thf(fact_709_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A: nat,As2: list_nat,Bs2: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A @ As2 ) @ Bs2 )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_710_lexord__linear,axiom,
! [R: set_Pr7497825696620840711list_o,X2: list_option_list_o,Y2: list_option_list_o] :
( ! [A2: option_list_o,B2: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A2 @ B2 ) @ R )
| ( A2 = B2 )
| ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ B2 @ A2 ) @ R ) )
=> ( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ X2 @ Y2 ) @ ( lexord_option_list_o @ R ) )
| ( X2 = Y2 )
| ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Y2 @ X2 ) @ ( lexord_option_list_o @ R ) ) ) ) ).
% lexord_linear
thf(fact_711_lexord__irreflexive,axiom,
! [R: set_Pr7497825696620840711list_o,Xs: list_option_list_o] :
( ! [X3: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ X3 ) @ R )
=> ~ ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Xs ) @ ( lexord_option_list_o @ R ) ) ) ).
% lexord_irreflexive
thf(fact_712_lexord__append__leftI,axiom,
! [U: list_o,V2: list_o,R: set_Product_prod_o_o,X2: list_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ U @ V2 ) @ ( lexord_o @ R ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ X2 @ U ) @ ( append_o @ X2 @ V2 ) ) @ ( lexord_o @ R ) ) ) ).
% lexord_append_leftI
thf(fact_713_listrel_OCons,axiom,
! [X2: nat,Y2: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_714_listrel_OCons,axiom,
! [X2: option_list_o,Y2: option_list_o,R: set_Pr7497825696620840711list_o,Xs: list_option_list_o,Ys: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ R )
=> ( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( listre2892261910510432881list_o @ R ) )
=> ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ X2 @ Xs ) @ ( cons_option_list_o @ Y2 @ Ys ) ) @ ( listre2892261910510432881list_o @ R ) ) ) ) ).
% listrel.Cons
thf(fact_715_listrel_OCons,axiom,
! [X2: nat,Y2: produc3741383161447143261al_nat,R: set_Pr836851361167288120al_nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y2 ) @ R )
=> ( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs @ Ys ) @ ( listre1017866332378306906al_nat @ R ) )
=> ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_P500833500243608851al_nat @ Y2 @ Ys ) ) @ ( listre1017866332378306906al_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_716_listrel_OCons,axiom,
! [X2: real,Y2: nat,R: set_Pr3510011417693777981al_nat,Xs: list_real,Ys: list_nat] :
( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y2 ) @ R )
=> ( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs @ Ys ) @ ( listrel_real_nat @ R ) )
=> ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ ( cons_real @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_real_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_717_listrel__Cons1,axiom,
! [Y2: nat,Ys: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y2 @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Y3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_718_listrel__Cons1,axiom,
! [Y2: option_list_o,Ys: list_option_list_o,Xs: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ Y2 @ Ys ) @ Xs ) @ ( listre2892261910510432881list_o @ R ) )
=> ~ ! [Y3: option_list_o,Ys3: list_option_list_o] :
( ( Xs
= ( cons_option_list_o @ Y3 @ Ys3 ) )
=> ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ Y2 @ Y3 ) @ R )
=> ~ ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Ys @ Ys3 ) @ ( listre2892261910510432881list_o @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_719_listrel__Cons1,axiom,
! [Y2: nat,Ys: list_nat,Xs: list_P6834414599653733731al_nat,R: set_Pr836851361167288120al_nat] :
( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ ( cons_nat @ Y2 @ Ys ) @ Xs ) @ ( listre1017866332378306906al_nat @ R ) )
=> ~ ! [Y3: produc3741383161447143261al_nat,Ys3: list_P6834414599653733731al_nat] :
( ( Xs
= ( cons_P500833500243608851al_nat @ Y3 @ Ys3 ) )
=> ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ Y2 @ Y3 ) @ R )
=> ~ ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Ys @ Ys3 ) @ ( listre1017866332378306906al_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_720_listrel__Cons1,axiom,
! [Y2: real,Ys: list_real,Xs: list_nat,R: set_Pr3510011417693777981al_nat] :
( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ ( cons_real @ Y2 @ Ys ) @ Xs ) @ ( listrel_real_nat @ R ) )
=> ~ ! [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ Y2 @ Y3 ) @ R )
=> ~ ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Ys @ Ys3 ) @ ( listrel_real_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_721_listrel__Cons2,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_722_listrel__Cons2,axiom,
! [Xs: list_option_list_o,Y2: option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ ( cons_option_list_o @ Y2 @ Ys ) ) @ ( listre2892261910510432881list_o @ R ) )
=> ~ ! [X3: option_list_o,Xs3: list_option_list_o] :
( ( Xs
= ( cons_option_list_o @ X3 @ Xs3 ) )
=> ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y2 ) @ R )
=> ~ ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs3 @ Ys ) @ ( listre2892261910510432881list_o @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_723_listrel__Cons2,axiom,
! [Xs: list_nat,Y2: produc3741383161447143261al_nat,Ys: list_P6834414599653733731al_nat,R: set_Pr836851361167288120al_nat] :
( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs @ ( cons_P500833500243608851al_nat @ Y2 @ Ys ) ) @ ( listre1017866332378306906al_nat @ R ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X3 @ Y2 ) @ R )
=> ~ ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs3 @ Ys ) @ ( listre1017866332378306906al_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_724_listrel__Cons2,axiom,
! [Xs: list_real,Y2: nat,Ys: list_nat,R: set_Pr3510011417693777981al_nat] :
( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_real_nat @ R ) )
=> ~ ! [X3: real,Xs3: list_real] :
( ( Xs
= ( cons_real @ X3 @ Xs3 ) )
=> ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y2 ) @ R )
=> ~ ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs3 @ Ys ) @ ( listrel_real_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_725_lexord__append__leftD,axiom,
! [X2: list_o,U: list_o,V2: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ X2 @ U ) @ ( append_o @ X2 @ V2 ) ) @ ( lexord_o @ R ) )
=> ( ! [A2: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ U @ V2 ) @ ( lexord_o @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_726_lexord__append__leftD,axiom,
! [X2: list_option_list_o,U: list_option_list_o,V2: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( append_option_list_o @ X2 @ U ) @ ( append_option_list_o @ X2 @ V2 ) ) @ ( lexord_option_list_o @ R ) )
=> ( ! [A2: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A2 @ A2 ) @ R )
=> ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ U @ V2 ) @ ( lexord_option_list_o @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_727_lexord__append__rightI,axiom,
! [Y2: list_o,X2: list_o,R: set_Product_prod_o_o] :
( ? [B5: $o,Z5: list_o] :
( Y2
= ( cons_o @ B5 @ Z5 ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ X2 @ ( append_o @ X2 @ Y2 ) ) @ ( lexord_o @ R ) ) ) ).
% lexord_append_rightI
thf(fact_728_lexord__append__rightI,axiom,
! [Y2: list_nat,X2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ? [B5: nat,Z5: list_nat] :
( Y2
= ( cons_nat @ B5 @ Z5 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ ( append_nat @ X2 @ Y2 ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_rightI
thf(fact_729_map__tailrec__rev_Oelims,axiom,
! [X2: nat > nat,Xa: list_nat,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != Xb ) )
=> ~ ! [A2: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A2 @ As ) )
=> ( Y2
!= ( map_ta7164188454487880599at_nat @ X2 @ As @ ( cons_nat @ ( X2 @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_730_sublists_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_731_longest__common__prefix_Opelims,axiom,
! [X2: list_nat,Xa: list_nat,Y2: list_nat] :
( ( ( longes266370323089874118ix_nat @ X2 @ Xa )
= Y2 )
=> ( ( accp_P8037286306265792042st_nat @ longes4198368034990555989el_nat @ ( produc2694037385005941721st_nat @ X2 @ Xa ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Y3: nat,Ys3: list_nat] :
( ( Xa
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( ( ( X3 = Y3 )
=> ( Y2
= ( cons_nat @ X3 @ ( longes266370323089874118ix_nat @ Xs3 @ Ys3 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2 = nil_nat ) ) )
=> ~ ( accp_P8037286306265792042st_nat @ longes4198368034990555989el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) ) ) )
=> ( ( ( X2 = nil_nat )
=> ( ( Y2 = nil_nat )
=> ~ ( accp_P8037286306265792042st_nat @ longes4198368034990555989el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa ) ) ) )
=> ~ ( ( Xa = nil_nat )
=> ( ( Y2 = nil_nat )
=> ~ ( accp_P8037286306265792042st_nat @ longes4198368034990555989el_nat @ ( produc2694037385005941721st_nat @ X2 @ nil_nat ) ) ) ) ) ) ) ) ).
% longest_common_prefix.pelims
thf(fact_732_lexord__same__pref__iff,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Xs @ Zs ) ) @ ( lexord_o @ R ) )
= ( ? [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ X ) @ R ) )
| ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys @ Zs ) @ ( lexord_o @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_733_lexord__same__pref__iff,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( append_option_list_o @ Xs @ Ys ) @ ( append_option_list_o @ Xs @ Zs ) ) @ ( lexord_option_list_o @ R ) )
= ( ? [X: option_list_o] :
( ( member_option_list_o @ X @ ( set_option_list_o2 @ Xs ) )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ X ) @ R ) )
| ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Ys @ Zs ) @ ( lexord_option_list_o @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_734_lex__append__left__iff,axiom,
! [R: set_Product_prod_o_o,Xs: list_o,Ys: list_o,Zs: list_o] :
( ! [X3: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Xs @ Zs ) ) @ ( lex_o @ R ) )
= ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys @ Zs ) @ ( lex_o @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_735_lex__append__left__iff,axiom,
! [R: set_Pr7497825696620840711list_o,Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ! [X3: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ X3 ) @ R )
=> ( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( append_option_list_o @ Xs @ Ys ) @ ( append_option_list_o @ Xs @ Zs ) ) @ ( lex_option_list_o @ R ) )
= ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Ys @ Zs ) @ ( lex_option_list_o @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_736_lex__append__leftD,axiom,
! [R: set_Product_prod_o_o,Xs: list_o,Ys: list_o,Zs: list_o] :
( ! [X3: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R )
=> ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Xs @ Zs ) ) @ ( lex_o @ R ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys @ Zs ) @ ( lex_o @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_737_lex__append__leftD,axiom,
! [R: set_Pr7497825696620840711list_o,Xs: list_option_list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ! [X3: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ X3 ) @ R )
=> ( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( append_option_list_o @ Xs @ Ys ) @ ( append_option_list_o @ Xs @ Zs ) ) @ ( lex_option_list_o @ R ) )
=> ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Ys @ Zs ) @ ( lex_option_list_o @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_738_map__append,axiom,
! [F: $o > $o,Xs: list_o,Ys: list_o] :
( ( map_o_o @ F @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( map_o_o @ F @ Xs ) @ ( map_o_o @ F @ Ys ) ) ) ).
% map_append
thf(fact_739_in__set__prefixes,axiom,
! [Xs: list_o,Ys: list_o] :
( ( member_list_o @ Xs @ ( set_list_o2 @ ( prefixes_o @ Ys ) ) )
= ( prefix_o @ Xs @ Ys ) ) ).
% in_set_prefixes
thf(fact_740_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_741_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_742_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_743_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_744_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_745_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X222: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_746_append__eq__map__conv,axiom,
! [Ys: list_o,Zs: list_o,F: $o > $o,Xs: list_o] :
( ( ( append_o @ Ys @ Zs )
= ( map_o_o @ F @ Xs ) )
= ( ? [Us2: list_o,Vs3: list_o] :
( ( Xs
= ( append_o @ Us2 @ Vs3 ) )
& ( Ys
= ( map_o_o @ F @ Us2 ) )
& ( Zs
= ( map_o_o @ F @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_747_map__eq__append__conv,axiom,
! [F: $o > $o,Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( map_o_o @ F @ Xs )
= ( append_o @ Ys @ Zs ) )
= ( ? [Us2: list_o,Vs3: list_o] :
( ( Xs
= ( append_o @ Us2 @ Vs3 ) )
& ( Ys
= ( map_o_o @ F @ Us2 ) )
& ( Zs
= ( map_o_o @ F @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_748_map__mono__prefix,axiom,
! [Xs: list_o,Ys: list_o,F: $o > $o] :
( ( prefix_o @ Xs @ Ys )
=> ( prefix_o @ ( map_o_o @ F @ Xs ) @ ( map_o_o @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_749_prefix__map__rightE,axiom,
! [Xs: list_o,F: $o > $o,Ys: list_o] :
( ( prefix_o @ Xs @ ( map_o_o @ F @ Ys ) )
=> ? [Xs4: list_o] :
( ( prefix_o @ Xs4 @ Ys )
& ( Xs
= ( map_o_o @ F @ Xs4 ) ) ) ) ).
% prefix_map_rightE
thf(fact_750_set__ConsD,axiom,
! [Y2: nat,X2: nat,Xs: list_nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_nat @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_751_list_Oset__cases,axiom,
! [E2: nat,A: list_nat] :
( ( member_nat @ E2 @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_752_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_753_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_754_split__list,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ? [Ys3: list_o,Zs2: list_o] :
( Xs
= ( append_o @ Ys3 @ ( cons_o @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_755_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_756_split__list__last,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ? [Ys3: list_o,Zs2: list_o] :
( ( Xs
= ( append_o @ Ys3 @ ( cons_o @ X2 @ Zs2 ) ) )
& ~ ( member_o @ X2 @ ( set_o2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_757_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_758_split__list__prop,axiom,
! [Xs: list_o,P: $o > $o] :
( ? [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_o,X3: $o] :
( ? [Zs2: list_o] :
( Xs
= ( append_o @ Ys3 @ ( cons_o @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_759_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_760_split__list__first,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ? [Ys3: list_o,Zs2: list_o] :
( ( Xs
= ( append_o @ Ys3 @ ( cons_o @ X2 @ Zs2 ) ) )
& ~ ( member_o @ X2 @ ( set_o2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_761_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_762_split__list__propE,axiom,
! [Xs: list_o,P: $o > $o] :
( ? [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_o,X3: $o] :
( ? [Zs2: list_o] :
( Xs
= ( append_o @ Ys3 @ ( cons_o @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_763_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_764_append__Cons__eq__iff,axiom,
! [X2: $o,Xs: list_o,Ys: list_o,Xs6: list_o,Ys6: list_o] :
( ~ ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( ~ ( member_o @ X2 @ ( set_o2 @ Ys ) )
=> ( ( ( append_o @ Xs @ ( cons_o @ X2 @ Ys ) )
= ( append_o @ Xs6 @ ( cons_o @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_765_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs6: list_nat,Ys6: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs6 @ ( cons_nat @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_766_in__set__conv__decomp,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
= ( ? [Ys2: list_o,Zs3: list_o] :
( Xs
= ( append_o @ Ys2 @ ( cons_o @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_767_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_768_split__list__last__prop,axiom,
! [Xs: list_o,P: $o > $o] :
( ? [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_o,X3: $o,Zs2: list_o] :
( ( Xs
= ( append_o @ Ys3 @ ( cons_o @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ ( set_o2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_769_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_770_split__list__first__prop,axiom,
! [Xs: list_o,P: $o > $o] :
( ? [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_o,X3: $o] :
( ? [Zs2: list_o] :
( Xs
= ( append_o @ Ys3 @ ( cons_o @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ ( set_o2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_771_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_772_split__list__last__propE,axiom,
! [Xs: list_o,P: $o > $o] :
( ? [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_o,X3: $o,Zs2: list_o] :
( ( Xs
= ( append_o @ Ys3 @ ( cons_o @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: $o] :
( ( member_o @ Xa2 @ ( set_o2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_773_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_774_split__list__first__propE,axiom,
! [Xs: list_o,P: $o > $o] :
( ? [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_o,X3: $o] :
( ? [Zs2: list_o] :
( Xs
= ( append_o @ Ys3 @ ( cons_o @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: $o] :
( ( member_o @ Xa2 @ ( set_o2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_775_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_776_in__set__conv__decomp__last,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
= ( ? [Ys2: list_o,Zs3: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ X2 @ Zs3 ) ) )
& ~ ( member_o @ X2 @ ( set_o2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_777_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_778_in__set__conv__decomp__first,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
= ( ? [Ys2: list_o,Zs3: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ X2 @ Zs3 ) ) )
& ~ ( member_o @ X2 @ ( set_o2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_779_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_780_split__list__last__prop__iff,axiom,
! [Xs: list_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys2: list_o,X: $o,Zs3: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y: $o] :
( ( member_o @ Y @ ( set_o2 @ Zs3 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_781_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys2: list_nat,X: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_782_split__list__first__prop__iff,axiom,
! [Xs: list_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys2: list_o,X: $o] :
( ? [Zs3: list_o] :
( Xs
= ( append_o @ Ys2 @ ( cons_o @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y: $o] :
( ( member_o @ Y @ ( set_o2 @ Ys2 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_783_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys2: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_784_longest__common__prefix__prefix1,axiom,
! [Xs: list_o,Ys: list_o] : ( prefix_o @ ( longes218265413398645026efix_o @ Xs @ Ys ) @ Xs ) ).
% longest_common_prefix_prefix1
thf(fact_785_longest__common__prefix__prefix2,axiom,
! [Xs: list_o,Ys: list_o] : ( prefix_o @ ( longes218265413398645026efix_o @ Xs @ Ys ) @ Ys ) ).
% longest_common_prefix_prefix2
thf(fact_786_longest__common__prefix__max__prefix,axiom,
! [Ps: list_o,Xs: list_o,Ys: list_o] :
( ( prefix_o @ Ps @ Xs )
=> ( ( prefix_o @ Ps @ Ys )
=> ( prefix_o @ Ps @ ( longes218265413398645026efix_o @ Xs @ Ys ) ) ) ) ).
% longest_common_prefix_max_prefix
thf(fact_787_in__set__butlast__appendI,axiom,
! [X2: $o,Xs: list_o,Ys: list_o] :
( ( ( member_o @ X2 @ ( set_o2 @ ( butlast_o @ Xs ) ) )
| ( member_o @ X2 @ ( set_o2 @ ( butlast_o @ Ys ) ) ) )
=> ( member_o @ X2 @ ( set_o2 @ ( butlast_o @ ( append_o @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_788_Cons__in__subseqsD,axiom,
! [Y2: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y2 @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_789_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_790_lex__append__leftI,axiom,
! [Ys: list_o,Zs: list_o,R: set_Product_prod_o_o,Xs: list_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys @ Zs ) @ ( lex_o @ R ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Xs @ Zs ) ) @ ( lex_o @ R ) ) ) ).
% lex_append_leftI
thf(fact_791_lenlex__irreflexive,axiom,
! [R: set_Pr7497825696620840711list_o,Xs: list_option_list_o] :
( ! [X3: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ X3 ) @ R )
=> ~ ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Xs ) @ ( lenlex_option_list_o @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_792_lexord__partial__trans,axiom,
! [Xs: list_option_list_o,R: set_Pr7497825696620840711list_o,Ys: list_option_list_o,Zs: list_option_list_o] :
( ! [X3: option_list_o,Y3: option_list_o,Z3: option_list_o] :
( ( member_option_list_o @ X3 @ ( set_option_list_o2 @ Xs ) )
=> ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ R )
=> ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ Y3 @ Z3 ) @ R )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Z3 ) @ R ) ) ) )
=> ( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( lexord_option_list_o @ R ) )
=> ( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Ys @ Zs ) @ ( lexord_option_list_o @ R ) )
=> ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Zs ) @ ( lexord_option_list_o @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_793_longest__common__prefix_Oelims,axiom,
! [X2: list_nat,Xa: list_nat,Y2: list_nat] :
( ( ( longes266370323089874118ix_nat @ X2 @ Xa )
= Y2 )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Y3: nat,Ys3: list_nat] :
( ( Xa
= ( cons_nat @ Y3 @ Ys3 ) )
=> ~ ( ( ( X3 = Y3 )
=> ( Y2
= ( cons_nat @ X3 @ ( longes266370323089874118ix_nat @ Xs3 @ Ys3 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2 = nil_nat ) ) ) ) )
=> ( ( ( X2 = nil_nat )
=> ( Y2 != nil_nat ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y2 != nil_nat ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_794_longest__common__prefix_Osimps_I1_J,axiom,
! [X2: nat,Y2: nat,Xs: list_nat,Ys: list_nat] :
( ( ( X2 = Y2 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( cons_nat @ X2 @ ( longes266370323089874118ix_nat @ Xs @ Ys ) ) ) )
& ( ( X2 != Y2 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= nil_nat ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_795_prefixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_796_extract__Some__iff,axiom,
! [P: $o > $o,Xs: list_o,Ys: list_o,Y2: $o,Zs: list_o] :
( ( ( extract_o @ P @ Xs )
= ( some_P3766925538565183643list_o @ ( produc6388330606509188902list_o @ Ys @ ( produc7263596898809104029list_o @ Y2 @ Zs ) ) ) )
= ( ( Xs
= ( append_o @ Ys @ ( cons_o @ Y2 @ Zs ) ) )
& ( P @ Y2 )
& ~ ? [X: $o] :
( ( member_o @ X @ ( set_o2 @ Ys ) )
& ( P @ X ) ) ) ) ).
% extract_Some_iff
thf(fact_797_extract__Some__iff,axiom,
! [P: nat > $o,Xs: list_nat,Ys: list_nat,Y2: nat,Zs: list_nat] :
( ( ( extract_nat @ P @ Xs )
= ( some_P617598486967069697st_nat @ ( produc7518127839388293336st_nat @ Ys @ ( produc8282810413953273033st_nat @ Y2 @ Zs ) ) ) )
= ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ Zs ) ) )
& ( P @ Y2 )
& ~ ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
& ( P @ X ) ) ) ) ).
% extract_Some_iff
thf(fact_798_extract__SomeE,axiom,
! [P: $o > $o,Xs: list_o,Ys: list_o,Y2: $o,Zs: list_o] :
( ( ( extract_o @ P @ Xs )
= ( some_P3766925538565183643list_o @ ( produc6388330606509188902list_o @ Ys @ ( produc7263596898809104029list_o @ Y2 @ Zs ) ) ) )
=> ( ( Xs
= ( append_o @ Ys @ ( cons_o @ Y2 @ Zs ) ) )
& ( P @ Y2 )
& ~ ? [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Ys ) )
& ( P @ X5 ) ) ) ) ).
% extract_SomeE
thf(fact_799_extract__SomeE,axiom,
! [P: nat > $o,Xs: list_nat,Ys: list_nat,Y2: nat,Zs: list_nat] :
( ( ( extract_nat @ P @ Xs )
= ( some_P617598486967069697st_nat @ ( produc7518127839388293336st_nat @ Ys @ ( produc8282810413953273033st_nat @ Y2 @ Zs ) ) ) )
=> ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ Zs ) ) )
& ( P @ Y2 )
& ~ ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
& ( P @ X5 ) ) ) ) ).
% extract_SomeE
thf(fact_800_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_801_Cons__in__lex,axiom,
! [X2: option_list_o,Xs: list_option_list_o,Y2: option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ X2 @ Xs ) @ ( cons_option_list_o @ Y2 @ Ys ) ) @ ( lex_option_list_o @ R ) )
= ( ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ R )
& ( ( size_s5120752914790010966list_o @ Xs )
= ( size_s5120752914790010966list_o @ Ys ) ) )
| ( ( X2 = Y2 )
& ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( lex_option_list_o @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_802_Cons__in__lex,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X2 = Y2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_803_append__eq__append__conv,axiom,
! [Xs: list_o,Ys: list_o,Us: list_o,Vs: list_o] :
( ( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
| ( ( size_size_list_o @ Us )
= ( size_size_list_o @ Vs ) ) )
=> ( ( ( append_o @ Xs @ Us )
= ( append_o @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_804_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_805_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_806_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_807_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_808_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_809_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_810_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ 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 @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_811_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_812_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_813_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_814_listrel__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_815_same__length__different,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != Ys )
=> ( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ? [Pre: list_o,X3: $o,Xs4: list_o,Y3: $o,Ys5: list_o] :
( ( X3 != Y3 )
& ( Xs
= ( append_o @ Pre @ ( append_o @ ( cons_o @ X3 @ nil_o ) @ Xs4 ) ) )
& ( Ys
= ( append_o @ Pre @ ( append_o @ ( cons_o @ Y3 @ nil_o ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_816_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs4: list_nat,Y3: nat,Ys5: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs4 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_817_lexord__sufE,axiom,
! [Xs: list_o,Zs: list_o,Ys: list_o,Qs: list_o,R: set_Product_prod_o_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ Zs ) @ ( append_o @ Ys @ Qs ) ) @ ( lexord_o @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_size_list_o @ Qs ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( lexord_o @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_818_lexord__sufE,axiom,
! [Xs: list_nat,Zs: list_nat,Ys: list_nat,Qs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Zs ) @ ( append_nat @ Ys @ Qs ) ) @ ( lexord_nat @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Qs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_819_lex__append__rightI,axiom,
! [Xs: list_o,Ys: list_o,R: set_Product_prod_o_o,Vs: list_o,Us: list_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( lex_o @ R ) )
=> ( ( ( size_size_list_o @ Vs )
= ( size_size_list_o @ Us ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs @ Us ) @ ( append_o @ Ys @ Vs ) ) @ ( lex_o @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_820_lex__append__rightI,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_821_lenlex__append1,axiom,
! [Us: list_o,Xs: list_o,R2: set_Product_prod_o_o,Vs: list_o,Ys: list_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Us @ Xs ) @ ( lenlex_o @ R2 ) )
=> ( ( ( size_size_list_o @ Vs )
= ( size_size_list_o @ Ys ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Us @ Vs ) @ ( append_o @ Xs @ Ys ) ) @ ( lenlex_o @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_822_lenlex__append1,axiom,
! [Us: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R2 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_823_lexord__lex,axiom,
! [X2: list_nat,Y2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y2 ) @ ( lex_nat @ R ) )
= ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y2 ) @ ( lexord_nat @ R ) )
& ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) ) ) ) ).
% lexord_lex
thf(fact_824_Cons__lenlex__iff,axiom,
! [M: option_list_o,Ms: list_option_list_o,N: option_list_o,Ns: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ ( cons_option_list_o @ M @ Ms ) @ ( cons_option_list_o @ N @ Ns ) ) @ ( lenlex_option_list_o @ R ) )
= ( ( ord_less_nat @ ( size_s5120752914790010966list_o @ Ms ) @ ( size_s5120752914790010966list_o @ Ns ) )
| ( ( ( size_s5120752914790010966list_o @ Ms )
= ( size_s5120752914790010966list_o @ Ns ) )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ M @ N ) @ R ) )
| ( ( M = N )
& ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Ms @ Ns ) @ ( lenlex_option_list_o @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_825_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_826_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_827_lexord__sufI,axiom,
! [U: list_o,W2: list_o,R: set_Product_prod_o_o,V2: list_o,Z: list_o] :
( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ U @ W2 ) @ ( lexord_o @ R ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_o @ W2 ) @ ( size_size_list_o @ U ) )
=> ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ U @ V2 ) @ ( append_o @ W2 @ Z ) ) @ ( lexord_o @ R ) ) ) ) ).
% lexord_sufI
thf(fact_828_lexord__sufI,axiom,
! [U: list_nat,W2: list_nat,R: set_Pr1261947904930325089at_nat,V2: list_nat,Z: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W2 ) @ ( lexord_nat @ R ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ W2 ) @ ( size_size_list_nat @ U ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V2 ) @ ( append_nat @ W2 @ Z ) ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_sufI
thf(fact_829_lexn__length,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,N: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexn_nat @ R @ N ) )
=> ( ( ( size_size_list_nat @ Xs )
= N )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_830_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_831_rev__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( rev_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( rev_o @ Ys ) @ ( rev_o @ Xs ) ) ) ).
% rev_append
thf(fact_832_length__concat__rev,axiom,
! [Xs: list_list_nat] :
( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
= ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).
% length_concat_rev
thf(fact_833_rev__singleton__conv,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ X2 @ nil_nat ) )
= ( Xs
= ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_834_singleton__rev__conv,axiom,
! [X2: nat,Xs: list_nat] :
( ( ( cons_nat @ X2 @ nil_nat )
= ( rev_nat @ Xs ) )
= ( ( cons_nat @ X2 @ nil_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_835_rev__eq__Cons__iff,axiom,
! [Xs: list_o,Y2: $o,Ys: list_o] :
( ( ( rev_o @ Xs )
= ( cons_o @ Y2 @ Ys ) )
= ( Xs
= ( append_o @ ( rev_o @ Ys ) @ ( cons_o @ Y2 @ nil_o ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_836_rev__eq__Cons__iff,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( Xs
= ( append_nat @ ( rev_nat @ Ys ) @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_837_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys7: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys7 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys7 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_838_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_839_prefix__length__less,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( strict_prefix_nat @ Xs @ Ys )
=> ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% prefix_length_less
thf(fact_840_prefix__length__le,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% prefix_length_le
thf(fact_841_prefix__length__le,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% prefix_length_le
thf(fact_842_prefix__length__prefix,axiom,
! [Ps: list_o,Xs: list_o,Qs: list_o] :
( ( prefix_o @ Ps @ Xs )
=> ( ( prefix_o @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_o @ Ps ) @ ( size_size_list_o @ Qs ) )
=> ( prefix_o @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_843_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs )
=> ( ( prefix_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_844_rev_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( rev_o @ ( cons_o @ X2 @ Xs ) )
= ( append_o @ ( rev_o @ Xs ) @ ( cons_o @ X2 @ nil_o ) ) ) ).
% rev.simps(2)
thf(fact_845_rev_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( rev_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ ( rev_nat @ Xs ) @ ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rev.simps(2)
thf(fact_846_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_847_in__measure,axiom,
! [X2: option_list_o,Y2: option_list_o,F: option_list_o > nat] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( measur8478699559534842796list_o @ F ) )
= ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ).
% in_measure
thf(fact_848_in__measures_I2_J,axiom,
! [X2: option_list_o,Y2: option_list_o,F: option_list_o > nat,Fs: list_o8358219866161907755_o_nat] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( measur3775467863237363684list_o @ ( cons_o6089670980329342427_o_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) )
| ( ( ( F @ X2 )
= ( F @ Y2 ) )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( measur3775467863237363684list_o @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_849_append__one__prefix,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) )
=> ( prefix_o @ ( append_o @ Xs @ ( cons_o @ ( nth_o @ Ys @ ( size_size_list_o @ Xs ) ) @ nil_o ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_850_append__one__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( prefix_nat @ ( append_nat @ Xs @ ( cons_nat @ ( nth_nat @ Ys @ ( size_size_list_nat @ Xs ) ) @ nil_nat ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_851_in__measures_I1_J,axiom,
! [X2: option_list_o,Y2: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( measur3775467863237363684list_o @ nil_op686717097995950635_o_nat ) ) ).
% in_measures(1)
thf(fact_852_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_853_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_854_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_855_nth__append__length,axiom,
! [Xs: list_o,X2: $o,Ys: list_o] :
( ( nth_o @ ( append_o @ Xs @ ( cons_o @ X2 @ Ys ) ) @ ( size_size_list_o @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_856_nth__append__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_857_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_858_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_859_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_860_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X6: nat] : ( P @ I @ X6 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_861_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_862_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_863_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_864_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_865_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_866_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_867_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_868_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_869_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_870_measures__less,axiom,
! [F: option_list_o > nat,X2: option_list_o,Y2: option_list_o,Fs: list_o8358219866161907755_o_nat] :
( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( measur3775467863237363684list_o @ ( cons_o6089670980329342427_o_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_871_measures__lesseq,axiom,
! [F: option_list_o > nat,X2: option_list_o,Y2: option_list_o,Fs: list_o8358219866161907755_o_nat] :
( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) )
=> ( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( measur3775467863237363684list_o @ Fs ) )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( measur3775467863237363684list_o @ ( cons_o6089670980329342427_o_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_872_listrel__iff__nth,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( listre2892261910510432881list_o @ R ) )
= ( ( ( size_s5120752914790010966list_o @ Xs )
= ( size_s5120752914790010966list_o @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s5120752914790010966list_o @ Xs ) )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ ( nth_option_list_o @ Xs @ N2 ) @ ( nth_option_list_o @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_873_listrel__iff__nth,axiom,
! [Xs: list_real,Ys: list_nat,R: set_Pr3510011417693777981al_nat] :
( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs @ Ys ) @ ( listrel_real_nat @ R ) )
= ( ( ( size_size_list_real @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
=> ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ ( nth_real @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_874_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_P6834414599653733731al_nat,R: set_Pr836851361167288120al_nat] :
( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs @ Ys ) @ ( listre1017866332378306906al_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s1877336372972134351al_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_Pr3792769368220640996al_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_875_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_876_find__Some__iff,axiom,
! [P: list_o > $o,Xs: list_list_o,X2: list_o] :
( ( ( find_list_o @ P @ Xs )
= ( some_list_o @ X2 ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s2710708370519433104list_o @ Xs ) )
& ( P @ ( nth_list_o @ Xs @ I ) )
& ( X2
= ( nth_list_o @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_list_o @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_877_find__Some__iff,axiom,
! [P: nat > $o,Xs: list_nat,X2: nat] :
( ( ( find_nat @ P @ Xs )
= ( some_nat @ X2 ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I ) )
& ( X2
= ( nth_nat @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_878_find__Some__iff2,axiom,
! [X2: list_o,P: list_o > $o,Xs: list_list_o] :
( ( ( some_list_o @ X2 )
= ( find_list_o @ P @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s2710708370519433104list_o @ Xs ) )
& ( P @ ( nth_list_o @ Xs @ I ) )
& ( X2
= ( nth_list_o @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_list_o @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_879_find__Some__iff2,axiom,
! [X2: nat,P: nat > $o,Xs: list_nat] :
( ( ( some_nat @ X2 )
= ( find_nat @ P @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I ) )
& ( X2
= ( nth_nat @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_880_listrel1__iff__update,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( listre3104272764119459907list_o @ R ) )
= ( ? [Y: option_list_o,N2: nat] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ ( nth_option_list_o @ Xs @ N2 ) @ Y ) @ R )
& ( ord_less_nat @ N2 @ ( size_s5120752914790010966list_o @ Xs ) )
& ( Ys
= ( list_u6420874592553512018list_o @ Xs @ N2 @ Y ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_881_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
= ( ? [Y: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N2 @ Y ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_882_lex__take__index,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ Xs @ Ys ) @ ( lex_option_list_o @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5120752914790010966list_o @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s5120752914790010966list_o @ Ys ) )
=> ( ( ( take_option_list_o @ I2 @ Xs )
= ( take_option_list_o @ I2 @ Ys ) )
=> ~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ ( nth_option_list_o @ Xs @ I2 ) @ ( nth_option_list_o @ Ys @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_883_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_884_length__list__update,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_885_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_nat,X2: nat] :
( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_886_list__update__id,axiom,
! [Xs: list_nat,I3: nat] :
( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_887_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_888_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_889_list__update__beyond,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
=> ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_890_nth__take,axiom,
! [I3: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I3 )
= ( nth_nat @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_891_list__update__length,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y2: $o] :
( ( list_update_o @ ( append_o @ Xs @ ( cons_o @ X2 @ Ys ) ) @ ( size_size_list_o @ Xs ) @ Y2 )
= ( append_o @ Xs @ ( cons_o @ Y2 @ Ys ) ) ) ).
% list_update_length
thf(fact_892_list__update__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y2 )
= ( append_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) ) ).
% list_update_length
thf(fact_893_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_894_set__swap,axiom,
! [I3: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_895_take__is__prefix,axiom,
! [N: nat,Xs: list_o] : ( prefix_o @ ( take_o @ N @ Xs ) @ Xs ) ).
% take_is_prefix
thf(fact_896_list__update__code_I2_J,axiom,
! [X2: nat,Xs: list_nat,Y2: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y2 )
= ( cons_nat @ Y2 @ Xs ) ) ).
% list_update_code(2)
thf(fact_897_take__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_898_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_899_list__update__append1,axiom,
! [I3: nat,Xs: list_o,Ys: list_o,X2: $o] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ ( append_o @ Xs @ Ys ) @ I3 @ X2 )
= ( append_o @ ( list_update_o @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_900_list__update__append1,axiom,
! [I3: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I3 @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_901_list__update__same__conv,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_902_nth__list__update,axiom,
! [I3: nat,Xs: list_nat,J2: nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_903_find_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( some_nat @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( find_nat @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_904_find_Osimps_I2_J,axiom,
! [P: list_o > $o,X2: list_o,Xs: list_list_o] :
( ( ( P @ X2 )
=> ( ( find_list_o @ P @ ( cons_list_o @ X2 @ Xs ) )
= ( some_list_o @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( find_list_o @ P @ ( cons_list_o @ X2 @ Xs ) )
= ( find_list_o @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_905_find_Osimps_I1_J,axiom,
! [Uu: list_o > $o] :
( ( find_list_o @ Uu @ nil_list_o )
= none_list_o ) ).
% find.simps(1)
thf(fact_906_find__None__iff,axiom,
! [P: list_o > $o,Xs: list_list_o] :
( ( ( find_list_o @ P @ Xs )
= none_list_o )
= ( ~ ? [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% find_None_iff
thf(fact_907_find__None__iff2,axiom,
! [P: list_o > $o,Xs: list_list_o] :
( ( none_list_o
= ( find_list_o @ P @ Xs ) )
= ( ~ ? [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% find_None_iff2
thf(fact_908_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_909_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_o] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( ( take_o @ ( suc @ I3 ) @ Xs )
= ( append_o @ ( take_o @ I3 @ Xs ) @ ( cons_o @ ( nth_o @ Xs @ I3 ) @ nil_o ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_910_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I3 ) @ Xs )
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_911_lexord__take__index__conv,axiom,
! [X2: list_option_list_o,Y2: list_option_list_o,R: set_Pr7497825696620840711list_o] :
( ( member2844550799419677264list_o @ ( produc3395777322662382615list_o @ X2 @ Y2 ) @ ( lexord_option_list_o @ R ) )
= ( ( ( ord_less_nat @ ( size_s5120752914790010966list_o @ X2 ) @ ( size_s5120752914790010966list_o @ Y2 ) )
& ( ( take_option_list_o @ ( size_s5120752914790010966list_o @ X2 ) @ Y2 )
= X2 ) )
| ? [I: nat] :
( ( ord_less_nat @ I @ ( ord_min_nat @ ( size_s5120752914790010966list_o @ X2 ) @ ( size_s5120752914790010966list_o @ Y2 ) ) )
& ( ( take_option_list_o @ I @ X2 )
= ( take_option_list_o @ I @ Y2 ) )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ ( nth_option_list_o @ X2 @ I ) @ ( nth_option_list_o @ Y2 @ I ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_912_lexord__take__index__conv,axiom,
! [X2: list_nat,Y2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y2 ) @ ( lexord_nat @ R ) )
= ( ( ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ ( size_size_list_nat @ Y2 ) )
& ( ( take_nat @ ( size_size_list_nat @ X2 ) @ Y2 )
= X2 ) )
| ? [I: nat] :
( ( ord_less_nat @ I @ ( ord_min_nat @ ( size_size_list_nat @ X2 ) @ ( size_size_list_nat @ Y2 ) ) )
& ( ( take_nat @ I @ X2 )
= ( take_nat @ I @ Y2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X2 @ I ) @ ( nth_nat @ Y2 @ I ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_913_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_914_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_915_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_916_take__Suc__Cons,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_917_length__take,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( take_nat @ N @ Xs ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_take
thf(fact_918_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_919_sorted__wrt__take,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_920_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_921_accp__subset__induct,axiom,
! [D2: produc4882884732533091879list_o > $o,R2: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,X2: produc4882884732533091879list_o,P: produc4882884732533091879list_o > $o] :
( ( ord_le6604716143944071414st_o_o @ D2 @ ( accp_P4987914878654715440list_o @ R2 ) )
=> ( ! [X3: produc4882884732533091879list_o,Z3: produc4882884732533091879list_o] :
( ( D2 @ X3 )
=> ( ( R2 @ Z3 @ X3 )
=> ( D2 @ Z3 ) ) )
=> ( ( D2 @ X2 )
=> ( ! [X3: produc4882884732533091879list_o] :
( ( D2 @ X3 )
=> ( ! [Z5: produc4882884732533091879list_o] :
( ( R2 @ Z5 @ X3 )
=> ( P @ Z5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_922_accp__subset,axiom,
! [R1: produc4882884732533091879list_o > produc4882884732533091879list_o > $o,R22: produc4882884732533091879list_o > produc4882884732533091879list_o > $o] :
( ( ord_le727495567946737358st_o_o @ R1 @ R22 )
=> ( ord_le6604716143944071414st_o_o @ ( accp_P4987914878654715440list_o @ R22 ) @ ( accp_P4987914878654715440list_o @ R1 ) ) ) ).
% accp_subset
thf(fact_923_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_924_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_925_sorted2,axiom,
! [X2: nat,Y2: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Zs ) ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y2 @ Zs ) ) ) ) ).
% sorted2
thf(fact_926_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_927_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_928_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_929_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ ( suc @ I ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_930_sorted__antimono__prefix,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ( sorted_wrt_o @ ord_less_eq_o @ Ys )
=> ( sorted_wrt_o @ ord_less_eq_o @ Xs ) ) ) ).
% sorted_antimono_prefix
thf(fact_931_sorted__antimono__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ) ).
% sorted_antimono_prefix
thf(fact_932_sorted__wrt1,axiom,
! [P: nat > nat > $o,X2: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_933_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I3: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I3 @ ( nth_nat @ Ns @ I3 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_934_sorted__wrt__append,axiom,
! [P: $o > $o > $o,Xs: list_o,Ys: list_o] :
( ( sorted_wrt_o @ P @ ( append_o @ Xs @ Ys ) )
= ( ( sorted_wrt_o @ P @ Xs )
& ( sorted_wrt_o @ P @ Ys )
& ! [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ! [Y: $o] :
( ( member_o @ Y @ ( set_o2 @ Ys ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_935_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_936_sorted__take,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_take
thf(fact_937_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_938_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_939_prefix__order_Olift__Suc__antimono__le,axiom,
! [F: nat > list_o,N: nat,N3: nat] :
( ! [N4: nat] : ( prefix_o @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_o @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_940_prefix__order_Olift__Suc__mono__le,axiom,
! [F: nat > list_o,N: nat,N3: nat] :
( ! [N4: nat] : ( prefix_o @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_o @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_941_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_942_set__mono__prefix,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ ( set_o2 @ Ys ) ) ) ).
% set_mono_prefix
thf(fact_943_list__update__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,I3: nat,Y2: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ I3 ) @ Y2 )
= ( cons_nat @ X2 @ ( list_update_nat @ Xs @ I3 @ Y2 ) ) ) ).
% list_update_code(3)
thf(fact_944_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I ) ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_945_sorted1,axiom,
! [X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted1
thf(fact_946_sorted__simps_I2_J,axiom,
! [X2: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X2 @ X ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_947_strict__sorted__simps_I2_J,axiom,
! [X2: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X2 @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ord_less_nat @ X2 @ X ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).
% strict_sorted_simps(2)
thf(fact_948_sorted__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( sorted_wrt_o @ ord_less_eq_o @ ( append_o @ Xs @ Ys ) )
= ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
& ( sorted_wrt_o @ ord_less_eq_o @ Ys )
& ! [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ! [Y: $o] :
( ( member_o @ Y @ ( set_o2 @ Ys ) )
=> ( ord_less_eq_o @ X @ Y ) ) ) ) ) ).
% sorted_append
thf(fact_949_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% sorted_append
thf(fact_950_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_951_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs2: list_nat] :
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P3 @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_952_sorted__butlast,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( butlast_nat @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_953_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_954_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_955_length__Suc__conv__rev,axiom,
! [Xs: list_o,N: nat] :
( ( ( size_size_list_o @ Xs )
= ( suc @ N ) )
= ( ? [Y: $o,Ys2: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ Y @ nil_o ) ) )
& ( ( size_size_list_o @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_956_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_957_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_958_sorted__nth__mono,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_959_length__append__singleton,axiom,
! [Xs: list_o,X2: $o] :
( ( size_size_list_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= ( suc @ ( size_size_list_o @ Xs ) ) ) ).
% length_append_singleton
thf(fact_960_length__append__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_961_option_Osize__gen_I1_J,axiom,
! [X2: list_o > nat] :
( ( size_option_list_o @ X2 @ none_list_o )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_962_option_Osize_I4_J,axiom,
! [X22: list_o] :
( ( size_s9086679804311813456list_o @ ( some_list_o @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_963_less__list__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ X2 @ Xs ) ) ).
% less_list_code(2)
thf(fact_964_Nil__less__Cons,axiom,
! [A: nat,X2: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X2 ) ) ).
% Nil_less_Cons
thf(fact_965_less__eq__list__code_I1_J,axiom,
! [X2: nat,Xs: list_nat] :
~ ( ord_less_eq_list_nat @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% less_eq_list_code(1)
thf(fact_966_option_Osize_I3_J,axiom,
( ( size_s9086679804311813456list_o @ none_list_o )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_967_length__transpose__sorted,axiom,
! [Xs: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs ) ) )
=> ( ( ( Xs = nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= zero_zero_nat ) )
& ( ( Xs != nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= ( size_size_list_nat @ ( nth_list_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_968_subrelI,axiom,
! [R: set_Pr7497825696620840711list_o,S: set_Pr7497825696620840711list_o] :
( ! [X3: option_list_o,Y3: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ R )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ S ) )
=> ( ord_le7655427003059986087list_o @ R @ S ) ) ).
% subrelI
thf(fact_969_subrelI,axiom,
! [R: set_Pr836851361167288120al_nat,S: set_Pr836851361167288120al_nat] :
( ! [X3: nat,Y3: produc3741383161447143261al_nat] :
( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X3 @ Y3 ) @ R )
=> ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le4407014575791007640al_nat @ R @ S ) ) ).
% subrelI
thf(fact_970_subrelI,axiom,
! [R: set_Pr3510011417693777981al_nat,S: set_Pr3510011417693777981al_nat] :
( ! [X3: real,Y3: nat] :
( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y3 ) @ R )
=> ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le8992644243367634909al_nat @ R @ S ) ) ).
% subrelI
thf(fact_971_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_o,A: $o] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ Xs @ I3 @ A )
= ( append_o @ ( take_o @ I3 @ Xs ) @ ( cons_o @ A @ ( drop_o @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_972_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I3 @ A )
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_973_drop__Suc__Cons,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X2 @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_974_append__take__drop__id,axiom,
! [N: nat,Xs: list_o] :
( ( append_o @ ( take_o @ N @ Xs ) @ ( drop_o @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_975_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_976_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_977_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_978_last__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_drop
thf(fact_979_sorted__wrt__drop,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_980_sorted__drop,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_drop
thf(fact_981_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y2 ) ) ).
% nth_via_drop
thf(fact_982_append__eq__conv__conj,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( append_o @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_o @ ( size_size_list_o @ Xs ) @ Zs ) )
& ( Ys
= ( drop_o @ ( size_size_list_o @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_983_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_984_append__eq__append__conv__if,axiom,
! [Xs_1: list_o,Xs_2: list_o,Ys_1: list_o,Ys_2: list_o] :
( ( ( append_o @ Xs_1 @ Xs_2 )
= ( append_o @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs_1 ) @ ( size_size_list_o @ Ys_1 ) )
=> ( ( Xs_1
= ( take_o @ ( size_size_list_o @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_o @ ( drop_o @ ( size_size_list_o @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_o @ Xs_1 ) @ ( size_size_list_o @ Ys_1 ) )
=> ( ( ( take_o @ ( size_size_list_o @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_o @ ( drop_o @ ( size_size_list_o @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_985_append__eq__append__conv__if,axiom,
! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
( ( ( append_nat @ Xs_1 @ Xs_2 )
= ( append_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_986_sorted__transpose,axiom,
! [Xs: list_list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( transpose_nat @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_987_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) )
= ( drop_nat @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_988_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_o] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( Xs
= ( append_o @ ( take_o @ I3 @ Xs ) @ ( cons_o @ ( nth_o @ Xs @ I3 ) @ ( drop_o @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_989_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_990_take__hd__drop,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( append_o @ ( take_o @ N @ Xs ) @ ( cons_o @ ( hd_o @ ( drop_o @ N @ Xs ) ) @ nil_o ) )
= ( take_o @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_991_take__hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_992_SuccI,axiom,
! [Kl: list_o,K: $o,Kl2: set_list_o] :
( ( member_list_o @ ( append_o @ Kl @ ( cons_o @ K @ nil_o ) ) @ Kl2 )
=> ( member_o @ K @ ( bNF_Greatest_Succ_o @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_993_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_994_SuccD,axiom,
! [K: $o,Kl2: set_list_o,Kl: list_o] :
( ( member_o @ K @ ( bNF_Greatest_Succ_o @ Kl2 @ Kl ) )
=> ( member_list_o @ ( append_o @ Kl @ ( cons_o @ K @ nil_o ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_995_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_996_hd__append2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != nil_o )
=> ( ( hd_o @ ( append_o @ Xs @ Ys ) )
= ( hd_o @ Xs ) ) ) ).
% hd_append2
thf(fact_997_list_Osel_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_998_longest__common__prefix,axiom,
! [Xs: list_o,Ys: list_o] :
? [Ps2: list_o,Xs4: list_o,Ys5: list_o] :
( ( Xs
= ( append_o @ Ps2 @ Xs4 ) )
& ( Ys
= ( append_o @ Ps2 @ Ys5 ) )
& ( ( Xs4 = nil_o )
| ( Ys5 = nil_o )
| ( ( hd_o @ Xs4 )
!= ( hd_o @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_999_hd__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( hd_o @ ( append_o @ Xs @ Ys ) )
= ( ( ( Xs = nil_o )
=> ( hd_o @ Ys ) )
& ( ( Xs != nil_o )
=> ( hd_o @ Xs ) ) ) ) ).
% hd_append
thf(fact_1000_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1001_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_1002_power__down_Ocases,axiom,
! [X2: produc9155354469731034754al_nat] :
( ! [P6: nat,X3: real] :
( X2
!= ( produc4490577844368043572al_nat @ P6 @ ( produc3181502643871035669al_nat @ X3 @ zero_zero_nat ) ) )
=> ~ ! [P6: nat,X3: real,N4: nat] :
( X2
!= ( produc4490577844368043572al_nat @ P6 @ ( produc3181502643871035669al_nat @ X3 @ ( suc @ N4 ) ) ) ) ) ).
% power_down.cases
thf(fact_1003_enumerate__simps_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X2 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1004_enumerate__simps_I2_J,axiom,
! [N: nat,X2: produc3741383161447143261al_nat,Xs: list_P6834414599653733731al_nat] :
( ( enumer7722876367396059202al_nat @ N @ ( cons_P500833500243608851al_nat @ X2 @ Xs ) )
= ( cons_P1152050239662505292al_nat @ ( produc4490577844368043572al_nat @ N @ X2 ) @ ( enumer7722876367396059202al_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1005_distinct__adj__append__iff,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_adj_o @ ( append_o @ Xs @ Ys ) )
= ( ( distinct_adj_o @ Xs )
& ( distinct_adj_o @ Ys )
& ( ( Xs = nil_o )
| ( Ys = nil_o )
| ( ( last_o @ Xs )
!= ( hd_o @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1006_distinct__adj__Cons__Cons,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( ( X2 != Y2 )
& ( distinct_adj_nat @ ( cons_nat @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_1007_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_1008_distinct__adj__ConsD,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_1009_distinct__adj__appendD2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_adj_o @ ( append_o @ Xs @ Ys ) )
=> ( distinct_adj_o @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_1010_distinct__adj__appendD1,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_adj_o @ ( append_o @ Xs @ Ys ) )
=> ( distinct_adj_o @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_1011_distinct__adj__singleton,axiom,
! [X2: nat] : ( distinct_adj_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_adj_singleton
thf(fact_1012_distinct__adj__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( X2
!= ( hd_nat @ Xs ) )
& ( distinct_adj_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_1013_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_P6834414599653733731al_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_s1877336372972134351al_nat @ Xs ) )
=> ( ( nth_Pr7063540399719084603al_nat @ ( enumer7722876367396059202al_nat @ N @ Xs ) @ M )
= ( produc4490577844368043572al_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_Pr3792769368220640996al_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1014_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1015_rotate1__hd__tl,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( ( rotate1_o @ Xs )
= ( append_o @ ( tl_o @ Xs ) @ ( cons_o @ ( hd_o @ Xs ) @ nil_o ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1016_rotate1__hd__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( rotate1_nat @ Xs )
= ( append_nat @ ( tl_nat @ Xs ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1017_nth__drop,axiom,
! [N: nat,Xs: list_nat,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I3 )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_1018_length__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( size_size_list_o @ ( append_o @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_append
thf(fact_1019_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_1020_tl__append2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != nil_o )
=> ( ( tl_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( tl_o @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_1021_length__splice,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_splice
thf(fact_1022_nth__append__length__plus,axiom,
! [Xs: list_o,Ys: list_o,N: nat] :
( ( nth_o @ ( append_o @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_o @ Xs ) @ N ) )
= ( nth_o @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1023_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1024_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_1025_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1026_list_Osel_I3_J,axiom,
! [X21: nat,X222: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_1027_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X: nat] :
( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_1028_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X: nat] :
( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_1029_tl__append__if,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( Xs = nil_o )
=> ( ( tl_o @ ( append_o @ Xs @ Ys ) )
= ( tl_o @ Ys ) ) )
& ( ( Xs != nil_o )
=> ( ( tl_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( tl_o @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_1030_sorted__tl,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( tl_nat @ Xs ) ) ) ).
% sorted_tl
thf(fact_1031_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1032_take__add,axiom,
! [I3: nat,J2: nat,Xs: list_o] :
( ( take_o @ ( plus_plus_nat @ I3 @ J2 ) @ Xs )
= ( append_o @ ( take_o @ I3 @ Xs ) @ ( take_o @ J2 @ ( drop_o @ I3 @ Xs ) ) ) ) ).
% take_add
thf(fact_1033_enumerate__append__eq,axiom,
! [N: nat,Xs: list_o,Ys: list_o] :
( ( enumerate_o @ N @ ( append_o @ Xs @ Ys ) )
= ( append1535412585758129762_nat_o @ ( enumerate_o @ N @ Xs ) @ ( enumerate_o @ ( plus_plus_nat @ N @ ( size_size_list_o @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_1034_enumerate__append__eq,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_1035_nth__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_1036_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1037_option_Osize__gen_I2_J,axiom,
! [X2: list_o > nat,X22: list_o] :
( ( size_option_list_o @ X2 @ ( some_list_o @ X22 ) )
= ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1038_take__Suc,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ ( suc @ N ) @ Xs )
= ( cons_nat @ ( hd_nat @ Xs ) @ ( take_nat @ N @ ( tl_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_1039_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs2: list_nat] : ( if_nat @ ( Xs2 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs2 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1040_product_Osimps_I2_J,axiom,
! [X2: option_list_o,Xs: list_option_list_o,Ys: list_option_list_o] :
( ( produc2445308772338649029list_o @ ( cons_option_list_o @ X2 @ Xs ) @ Ys )
= ( append9085468213580610754list_o @ ( map_op4472081130091516167list_o @ ( produc5745850523778858007list_o @ X2 ) @ Ys ) @ ( produc2445308772338649029list_o @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_1041_product_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( produc7250786521479175942al_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( append3750557741271749927al_nat @ ( map_Pr6623428534301005723al_nat @ ( produc4490577844368043572al_nat @ X2 ) @ Ys ) @ ( produc7250786521479175942al_nat @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_1042_product_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real,Ys: list_nat] :
( ( product_real_nat @ ( cons_real @ X2 @ Xs ) @ Ys )
= ( append2703634050814417656al_nat @ ( map_na1309146153819616583al_nat @ ( produc3181502643871035669al_nat @ X2 ) @ Ys ) @ ( product_real_nat @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_1043_nth__zip,axiom,
! [I3: nat,Xs: list_option_list_o,Ys: list_option_list_o] :
( ( ord_less_nat @ I3 @ ( size_s5120752914790010966list_o @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s5120752914790010966list_o @ Ys ) )
=> ( ( nth_Pr5324855644858119342list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) @ I3 )
= ( produc5745850523778858007list_o @ ( nth_option_list_o @ Xs @ I3 ) @ ( nth_option_list_o @ Ys @ I3 ) ) ) ) ) ).
% nth_zip
thf(fact_1044_nth__zip,axiom,
! [I3: nat,Xs: list_real,Ys: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr3792769368220640996al_nat @ ( zip_real_nat @ Xs @ Ys ) @ I3 )
= ( produc3181502643871035669al_nat @ ( nth_real @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ).
% nth_zip
thf(fact_1045_nth__zip,axiom,
! [I3: nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s1877336372972134351al_nat @ Ys ) )
=> ( ( nth_Pr7063540399719084603al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) @ I3 )
= ( produc4490577844368043572al_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_Pr3792769368220640996al_nat @ Ys @ I3 ) ) ) ) ) ).
% nth_zip
thf(fact_1046_nth__zip,axiom,
! [I3: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I3 )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ).
% nth_zip
thf(fact_1047_size__list__append,axiom,
! [F: $o > nat,Xs: list_o,Ys: list_o] :
( ( size_list_o @ F @ ( append_o @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_o @ F @ Xs ) @ ( size_list_o @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_1048_zip__Cons__Cons,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_1049_zip__Cons__Cons,axiom,
! [X2: option_list_o,Xs: list_option_list_o,Y2: option_list_o,Ys: list_option_list_o] :
( ( zip_op2665896746229601079list_o @ ( cons_option_list_o @ X2 @ Xs ) @ ( cons_option_list_o @ Y2 @ Ys ) )
= ( cons_P8615093581336785373list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_1050_zip__Cons__Cons,axiom,
! [X2: nat,Xs: list_nat,Y2: produc3741383161447143261al_nat,Ys: list_P6834414599653733731al_nat] :
( ( zip_na1967359233052233492al_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_P500833500243608851al_nat @ Y2 @ Ys ) )
= ( cons_P1152050239662505292al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y2 ) @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_1051_zip__Cons__Cons,axiom,
! [X2: real,Xs: list_real,Y2: nat,Ys: list_nat] :
( ( zip_real_nat @ ( cons_real @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( cons_P500833500243608851al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y2 ) @ ( zip_real_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_1052_zip__append,axiom,
! [Xs: list_o,Us: list_o,Ys: list_o,Vs: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Us ) )
=> ( ( zip_o_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Us @ Vs ) )
= ( append2614242729457001410od_o_o @ ( zip_o_o @ Xs @ Us ) @ ( zip_o_o @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1053_zip__append,axiom,
! [Xs: list_o,Us: list_nat,Ys: list_o,Vs: list_nat] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_o_nat @ ( append_o @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append7250250098557412540_o_nat @ ( zip_o_nat @ Xs @ Us ) @ ( zip_o_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1054_zip__append,axiom,
! [Xs: list_nat,Us: list_o,Ys: list_nat,Vs: list_o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_o @ Us ) )
=> ( ( zip_nat_o @ ( append_nat @ Xs @ Ys ) @ ( append_o @ Us @ Vs ) )
= ( append1535412585758129762_nat_o @ ( zip_nat_o @ Xs @ Us ) @ ( zip_nat_o @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1055_zip__append,axiom,
! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1056_length__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_zip
thf(fact_1057_zip__same,axiom,
! [A: option_list_o,B: option_list_o,Xs: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A @ B ) @ ( set_Pr797121458932016060list_o @ ( zip_op2665896746229601079list_o @ Xs @ Xs ) ) )
= ( ( member_option_list_o @ A @ ( set_option_list_o2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1058_in__set__zipE,axiom,
! [X2: option_list_o,Y2: option_list_o,Xs: list_option_list_o,Ys: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( set_Pr797121458932016060list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) ) )
=> ~ ( ( member_option_list_o @ X2 @ ( set_option_list_o2 @ Xs ) )
=> ~ ( member_option_list_o @ Y2 @ ( set_option_list_o2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1059_in__set__zipE,axiom,
! [X2: nat,Y2: produc3741383161447143261al_nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y2 ) @ ( set_Pr7354016594238973869al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) ) )
=> ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member5805532792777349510al_nat @ Y2 @ ( set_Pr3174298344852596722al_nat @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1060_in__set__zipE,axiom,
! [X2: real,Y2: nat,Xs: list_real,Ys: list_nat] :
( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y2 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
=> ~ ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1061_set__zip__leftD,axiom,
! [X2: option_list_o,Y2: option_list_o,Xs: list_option_list_o,Ys: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( set_Pr797121458932016060list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) ) )
=> ( member_option_list_o @ X2 @ ( set_option_list_o2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1062_set__zip__leftD,axiom,
! [X2: nat,Y2: produc3741383161447143261al_nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y2 ) @ ( set_Pr7354016594238973869al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1063_set__zip__leftD,axiom,
! [X2: real,Y2: nat,Xs: list_real,Ys: list_nat] :
( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y2 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
=> ( member_real @ X2 @ ( set_real2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1064_set__zip__rightD,axiom,
! [X2: option_list_o,Y2: option_list_o,Xs: list_option_list_o,Ys: list_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( set_Pr797121458932016060list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) ) )
=> ( member_option_list_o @ Y2 @ ( set_option_list_o2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_1065_set__zip__rightD,axiom,
! [X2: nat,Y2: produc3741383161447143261al_nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y2 ) @ ( set_Pr7354016594238973869al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) ) )
=> ( member5805532792777349510al_nat @ Y2 @ ( set_Pr3174298344852596722al_nat @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_1066_set__zip__rightD,axiom,
! [X2: real,Y2: nat,Xs: list_real,Ys: list_nat] :
( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y2 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
=> ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_1067_zip__update,axiom,
! [Xs: list_option_list_o,I3: nat,X2: option_list_o,Ys: list_option_list_o,Y2: option_list_o] :
( ( zip_op2665896746229601079list_o @ ( list_u6420874592553512018list_o @ Xs @ I3 @ X2 ) @ ( list_u6420874592553512018list_o @ Ys @ I3 @ Y2 ) )
= ( list_u2015267059982357653list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) @ I3 @ ( produc5745850523778858007list_o @ X2 @ Y2 ) ) ) ).
% zip_update
thf(fact_1068_zip__update,axiom,
! [Xs: list_nat,I3: nat,X2: nat,Ys: list_P6834414599653733731al_nat,Y2: produc3741383161447143261al_nat] :
( ( zip_na1967359233052233492al_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ ( list_u873804745024221131al_nat @ Ys @ I3 @ Y2 ) )
= ( list_u1500079835982728340al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) @ I3 @ ( produc4490577844368043572al_nat @ X2 @ Y2 ) ) ) ).
% zip_update
thf(fact_1069_zip__update,axiom,
! [Xs: list_real,I3: nat,X2: real,Ys: list_nat,Y2: nat] :
( ( zip_real_nat @ ( list_update_real @ Xs @ I3 @ X2 ) @ ( list_update_nat @ Ys @ I3 @ Y2 ) )
= ( list_u873804745024221131al_nat @ ( zip_real_nat @ Xs @ Ys ) @ I3 @ ( produc3181502643871035669al_nat @ X2 @ Y2 ) ) ) ).
% zip_update
thf(fact_1070_zip__rev,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( zip_nat_nat @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys ) )
= ( rev_Pr6102188148953555047at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_1071_in__set__impl__in__set__zip2,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Y2: option_list_o] :
( ( ( size_s5120752914790010966list_o @ Xs )
= ( size_s5120752914790010966list_o @ Ys ) )
=> ( ( member_option_list_o @ Y2 @ ( set_option_list_o2 @ Ys ) )
=> ~ ! [X3: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y2 ) @ ( set_Pr797121458932016060list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1072_in__set__impl__in__set__zip2,axiom,
! [Xs: list_real,Ys: list_nat,Y2: nat] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
=> ~ ! [X3: real] :
~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y2 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1073_in__set__impl__in__set__zip2,axiom,
! [Xs: list_nat,Ys: list_P6834414599653733731al_nat,Y2: produc3741383161447143261al_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s1877336372972134351al_nat @ Ys ) )
=> ( ( member5805532792777349510al_nat @ Y2 @ ( set_Pr3174298344852596722al_nat @ Ys ) )
=> ~ ! [X3: nat] :
~ ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X3 @ Y2 ) @ ( set_Pr7354016594238973869al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1074_in__set__impl__in__set__zip2,axiom,
! [Xs: list_nat,Ys: list_nat,Y2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
=> ~ ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1075_in__set__impl__in__set__zip1,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,X2: option_list_o] :
( ( ( size_s5120752914790010966list_o @ Xs )
= ( size_s5120752914790010966list_o @ Ys ) )
=> ( ( member_option_list_o @ X2 @ ( set_option_list_o2 @ Xs ) )
=> ~ ! [Y3: option_list_o] :
~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y3 ) @ ( set_Pr797121458932016060list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1076_in__set__impl__in__set__zip1,axiom,
! [Xs: list_real,Ys: list_nat,X2: real] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ~ ! [Y3: nat] :
~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1077_in__set__impl__in__set__zip1,axiom,
! [Xs: list_nat,Ys: list_P6834414599653733731al_nat,X2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s1877336372972134351al_nat @ Ys ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ! [Y3: produc3741383161447143261al_nat] :
~ ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y3 ) @ ( set_Pr7354016594238973869al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1078_in__set__impl__in__set__zip1,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ! [Y3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1079_hd__zip,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( Xs != nil_option_list_o )
=> ( ( Ys != nil_option_list_o )
=> ( ( hd_Pro1830912869442821906list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) )
= ( produc5745850523778858007list_o @ ( hd_option_list_o @ Xs ) @ ( hd_option_list_o @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1080_hd__zip,axiom,
! [Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_Pr1917482104270272867al_nat )
=> ( ( hd_Pro3543069110806503127al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) )
= ( produc4490577844368043572al_nat @ ( hd_nat @ Xs ) @ ( hd_Pro1205043226352004424al_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1081_hd__zip,axiom,
! [Xs: list_real,Ys: list_nat] :
( ( Xs != nil_real )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro1205043226352004424al_nat @ ( zip_real_nat @ Xs @ Ys ) )
= ( produc3181502643871035669al_nat @ ( hd_real @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1082_zip__obtain__same__length,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_P6011104703257516679at_nat > $o] :
( ! [Zs2: list_nat,Ws2: list_nat,N4: nat] :
( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( N4
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( Zs2
= ( take_nat @ N4 @ Xs ) )
=> ( ( Ws2
= ( take_nat @ N4 @ Ys ) )
=> ( P @ ( zip_nat_nat @ Zs2 @ Ws2 ) ) ) ) ) )
=> ( P @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_1083_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
( ( ( zip_nat_nat @ Xs @ Ys )
= ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
=> ~ ! [X3: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs4 ) )
=> ! [Y3: nat,Ys5: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1084_zip__eq__ConsE,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o,Xy: produc4882884732533091879list_o,Xys: list_P1348966709184090669list_o] :
( ( ( zip_op2665896746229601079list_o @ Xs @ Ys )
= ( cons_P8615093581336785373list_o @ Xy @ Xys ) )
=> ~ ! [X3: option_list_o,Xs4: list_option_list_o] :
( ( Xs
= ( cons_option_list_o @ X3 @ Xs4 ) )
=> ! [Y3: option_list_o,Ys5: list_option_list_o] :
( ( Ys
= ( cons_option_list_o @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( produc5745850523778858007list_o @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_op2665896746229601079list_o @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1085_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys: list_P6834414599653733731al_nat,Xy: produc9155354469731034754al_nat,Xys: list_P189569681052899986al_nat] :
( ( ( zip_na1967359233052233492al_nat @ Xs @ Ys )
= ( cons_P1152050239662505292al_nat @ Xy @ Xys ) )
=> ~ ! [X3: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs4 ) )
=> ! [Y3: produc3741383161447143261al_nat,Ys5: list_P6834414599653733731al_nat] :
( ( Ys
= ( cons_P500833500243608851al_nat @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( produc4490577844368043572al_nat @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_na1967359233052233492al_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1086_zip__eq__ConsE,axiom,
! [Xs: list_real,Ys: list_nat,Xy: produc3741383161447143261al_nat,Xys: list_P6834414599653733731al_nat] :
( ( ( zip_real_nat @ Xs @ Ys )
= ( cons_P500833500243608851al_nat @ Xy @ Xys ) )
=> ~ ! [X3: real,Xs4: list_real] :
( ( Xs
= ( cons_real @ X3 @ Xs4 ) )
=> ! [Y3: nat,Ys5: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( produc3181502643871035669al_nat @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_real_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1087_last__zip,axiom,
! [Xs: list_option_list_o,Ys: list_option_list_o] :
( ( Xs != nil_option_list_o )
=> ( ( Ys != nil_option_list_o )
=> ( ( ( size_s5120752914790010966list_o @ Xs )
= ( size_s5120752914790010966list_o @ Ys ) )
=> ( ( last_P7106378149766555590list_o @ ( zip_op2665896746229601079list_o @ Xs @ Ys ) )
= ( produc5745850523778858007list_o @ ( last_option_list_o @ Xs ) @ ( last_option_list_o @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1088_last__zip,axiom,
! [Xs: list_real,Ys: list_nat] :
( ( Xs != nil_real )
=> ( ( Ys != nil_nat )
=> ( ( ( size_size_list_real @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P800818296573919740al_nat @ ( zip_real_nat @ Xs @ Ys ) )
= ( produc3181502643871035669al_nat @ ( last_real @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1089_last__zip,axiom,
! [Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_Pr1917482104270272867al_nat )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_s1877336372972134351al_nat @ Ys ) )
=> ( ( last_P1311776657978906915al_nat @ ( zip_na1967359233052233492al_nat @ Xs @ Ys ) )
= ( produc4490577844368043572al_nat @ ( last_nat @ Xs ) @ ( last_P800818296573919740al_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1090_last__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( product_Pair_nat_nat @ ( last_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1091_length__prefixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_1092_length__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_1093_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_1094_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1095_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_1096_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_1097_gen__length__code_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1098_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_1099_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1100_last__list__update,axiom,
! [Xs: list_nat,K: nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1101_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_1102_take__append,axiom,
! [N: nat,Xs: list_o,Ys: list_o] :
( ( take_o @ N @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( take_o @ N @ Xs ) @ ( take_o @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1103_take__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1104_drop__append,axiom,
! [N: nat,Xs: list_o,Ys: list_o] :
( ( drop_o @ N @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( drop_o @ N @ Xs ) @ ( drop_o @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1105_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1106_length__tl,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1107_length__butlast,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1108_nth__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1109_nth__append,axiom,
! [Xs: list_o,Ys: list_o,N: nat] :
( ( nth_o @ ( append_o @ Xs @ Ys ) @ N )
= ( ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( nth_o @ Xs @ N ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( nth_o @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1110_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1111_drop__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1112_list__update__append,axiom,
! [N: nat,Xs: list_o,Ys: list_o,X2: $o] :
( ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ ( append_o @ Xs @ Ys ) @ N @ X2 )
= ( append_o @ ( list_update_o @ Xs @ N @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ ( append_o @ Xs @ Ys ) @ N @ X2 )
= ( append_o @ Xs @ ( list_update_o @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_1113_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_1114_take__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_1115_rev__take,axiom,
! [I3: nat,Xs: list_nat] :
( ( rev_nat @ ( take_nat @ I3 @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I3 ) @ ( rev_nat @ Xs ) ) ) ).
% rev_take
thf(fact_1116_rev__drop,axiom,
! [I3: nat,Xs: list_nat] :
( ( rev_nat @ ( drop_nat @ I3 @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I3 ) @ ( rev_nat @ Xs ) ) ) ).
% rev_drop
thf(fact_1117_drop__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_1118_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs2: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_conv_take
thf(fact_1119_butlast__list__update,axiom,
! [K: nat,Xs: list_nat,X2: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_1120_nth__non__equal__first__eq,axiom,
! [X2: nat,Y2: nat,Xs: list_nat,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= Y2 )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1121_take__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1122_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_1123_last__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ Xs )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1124_rev__update,axiom,
! [K: nat,Xs: list_nat,Y2: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y2 ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y2 ) ) ) ).
% rev_update
thf(fact_1125_butlast__take,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1126_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A0: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,A1: nat,A22: nat,A32: produc3741383161447143261al_nat,P: ( nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat ) > nat > nat > produc3741383161447143261al_nat > $o] :
( ( accp_P1387443933282790731al_nat @ set_fo6615314570929945908al_nat @ ( produc5211167162158505542al_nat @ A0 @ ( produc2879132834507146987al_nat @ A1 @ ( produc4490577844368043572al_nat @ A22 @ A32 ) ) ) )
=> ( ! [F2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,A2: nat,B2: nat,Acc: produc3741383161447143261al_nat] :
( ( accp_P1387443933282790731al_nat @ set_fo6615314570929945908al_nat @ ( produc5211167162158505542al_nat @ F2 @ ( produc2879132834507146987al_nat @ A2 @ ( produc4490577844368043572al_nat @ B2 @ Acc ) ) ) )
=> ( ( ~ ( ord_less_nat @ B2 @ A2 )
=> ( P @ F2 @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F2 @ A2 @ Acc ) ) )
=> ( P @ F2 @ A2 @ B2 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_1127_drop__Cons__numeral,axiom,
! [V2: num,X2: nat,Xs: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_1128_nth__Cons__numeral,axiom,
! [X2: nat,Xs: list_nat,V2: num] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( numeral_numeral_nat @ V2 ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1129_take__Cons__numeral,axiom,
! [V2: num,X2: nat,Xs: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1130_fold__atLeastAtMost__nat_Ocases,axiom,
! [X2: produc1019696013813152916al_nat] :
~ ! [F2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,A2: nat,B2: nat,Acc: produc3741383161447143261al_nat] :
( X2
!= ( produc5211167162158505542al_nat @ F2 @ ( produc2879132834507146987al_nat @ A2 @ ( produc4490577844368043572al_nat @ B2 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1131_fold__atLeastAtMost__nat_Opsimps,axiom,
! [F: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,A: nat,B: nat,Acc2: produc3741383161447143261al_nat] :
( ( accp_P1387443933282790731al_nat @ set_fo6615314570929945908al_nat @ ( produc5211167162158505542al_nat @ F @ ( produc2879132834507146987al_nat @ A @ ( produc4490577844368043572al_nat @ B @ Acc2 ) ) ) )
=> ( ( ( ord_less_nat @ B @ A )
=> ( ( set_fo5235541981465435595al_nat @ F @ A @ B @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less_nat @ B @ A )
=> ( ( set_fo5235541981465435595al_nat @ F @ A @ B @ Acc2 )
= ( set_fo5235541981465435595al_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_1132_fold__atLeastAtMost__nat_Opelims,axiom,
! [X2: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,Xa: nat,Xb: nat,Xc: produc3741383161447143261al_nat,Y2: produc3741383161447143261al_nat] :
( ( ( set_fo5235541981465435595al_nat @ X2 @ Xa @ Xb @ Xc )
= Y2 )
=> ( ( accp_P1387443933282790731al_nat @ set_fo6615314570929945908al_nat @ ( produc5211167162158505542al_nat @ X2 @ ( produc2879132834507146987al_nat @ Xa @ ( produc4490577844368043572al_nat @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa )
=> ( Y2
= ( set_fo5235541981465435595al_nat @ X2 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) )
=> ~ ( accp_P1387443933282790731al_nat @ set_fo6615314570929945908al_nat @ ( produc5211167162158505542al_nat @ X2 @ ( produc2879132834507146987al_nat @ Xa @ ( produc4490577844368043572al_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_1133_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1134_card__set__suffixes,axiom,
! [Xs: list_nat] :
( ( finite_card_list_nat @ ( set_list_nat2 @ ( suffixes_nat @ Xs ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% card_set_suffixes
thf(fact_1135_card__set__prefixes,axiom,
! [Xs: list_nat] :
( ( finite_card_list_nat @ ( set_list_nat2 @ ( prefixes_nat @ Xs ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% card_set_prefixes
thf(fact_1136_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_1137_product__nth,axiom,
! [N: nat,Xs: list_option_list_o,Ys: list_option_list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s5120752914790010966list_o @ Xs ) @ ( size_s5120752914790010966list_o @ Ys ) ) )
=> ( ( nth_Pr5324855644858119342list_o @ ( produc2445308772338649029list_o @ Xs @ Ys ) @ N )
= ( produc5745850523778858007list_o @ ( nth_option_list_o @ Xs @ ( divide_divide_nat @ N @ ( size_s5120752914790010966list_o @ Ys ) ) ) @ ( nth_option_list_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_s5120752914790010966list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1138_product__nth,axiom,
! [N: nat,Xs: list_real,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_real @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr3792769368220640996al_nat @ ( product_real_nat @ Xs @ Ys ) @ N )
= ( produc3181502643871035669al_nat @ ( nth_real @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1139_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s1877336372972134351al_nat @ Ys ) ) )
=> ( ( nth_Pr7063540399719084603al_nat @ ( produc7250786521479175942al_nat @ Xs @ Ys ) @ N )
= ( produc4490577844368043572al_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s1877336372972134351al_nat @ Ys ) ) ) @ ( nth_Pr3792769368220640996al_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s1877336372972134351al_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1140_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1141_rotate__drop__take,axiom,
( rotate_o
= ( ^ [N2: nat,Xs2: list_o] : ( append_o @ ( drop_o @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Xs2 ) ) @ Xs2 ) @ ( take_o @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_1142_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) @ ( take_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_1143_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_1144_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1145_rotate__id,axiom,
! [N: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1146_rotate__append,axiom,
! [L: list_o,Q2: list_o] :
( ( rotate_o @ ( size_size_list_o @ L ) @ ( append_o @ L @ Q2 ) )
= ( append_o @ Q2 @ L ) ) ).
% rotate_append
thf(fact_1147_rotate__append,axiom,
! [L: list_nat,Q2: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q2 ) )
= ( append_nat @ Q2 @ L ) ) ).
% rotate_append
thf(fact_1148_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ).
% rotate_conv_mod
thf(fact_1149_rotate__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1150_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1151_hd__rotate__conv__nth,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( rotate_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_1152_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1153_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A4 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A4 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1154_length__replicate,axiom,
! [N: nat,X2: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_1155_nth__replicate,axiom,
! [I3: nat,N: nat,X2: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I3 )
= X2 ) ) ).
% nth_replicate
thf(fact_1156_zip__replicate,axiom,
! [I3: nat,X2: option_list_o,J2: nat,Y2: option_list_o] :
( ( zip_op2665896746229601079list_o @ ( replic6237453613039860388list_o @ I3 @ X2 ) @ ( replic6237453613039860388list_o @ J2 @ Y2 ) )
= ( replic6149313115625166311list_o @ ( ord_min_nat @ I3 @ J2 ) @ ( produc5745850523778858007list_o @ X2 @ Y2 ) ) ) ).
% zip_replicate
thf(fact_1157_zip__replicate,axiom,
! [I3: nat,X2: nat,J2: nat,Y2: produc3741383161447143261al_nat] :
( ( zip_na1967359233052233492al_nat @ ( replicate_nat @ I3 @ X2 ) @ ( replic1728136009727115037al_nat @ J2 @ Y2 ) )
= ( replic2733456754749247170al_nat @ ( ord_min_nat @ I3 @ J2 ) @ ( produc4490577844368043572al_nat @ X2 @ Y2 ) ) ) ).
% zip_replicate
thf(fact_1158_zip__replicate,axiom,
! [I3: nat,X2: real,J2: nat,Y2: nat] :
( ( zip_real_nat @ ( replicate_real @ I3 @ X2 ) @ ( replicate_nat @ J2 @ Y2 ) )
= ( replic1728136009727115037al_nat @ ( ord_min_nat @ I3 @ J2 ) @ ( produc3181502643871035669al_nat @ X2 @ Y2 ) ) ) ).
% zip_replicate
thf(fact_1159_append__replicate__commute,axiom,
! [N: nat,X2: $o,K: nat] :
( ( append_o @ ( replicate_o @ N @ X2 ) @ ( replicate_o @ K @ X2 ) )
= ( append_o @ ( replicate_o @ K @ X2 ) @ ( replicate_o @ N @ X2 ) ) ) ).
% append_replicate_commute
thf(fact_1160_replicate__add,axiom,
! [N: nat,M: nat,X2: $o] :
( ( replicate_o @ ( plus_plus_nat @ N @ M ) @ X2 )
= ( append_o @ ( replicate_o @ N @ X2 ) @ ( replicate_o @ M @ X2 ) ) ) ).
% replicate_add
thf(fact_1161_prefix__order_Ofinite__has__minimal2,axiom,
! [A4: set_list_o,A: list_o] :
( ( finite_finite_list_o @ A4 )
=> ( ( member_list_o @ A @ A4 )
=> ? [X3: list_o] :
( ( member_list_o @ X3 @ A4 )
& ( prefix_o @ X3 @ A )
& ! [Xa2: list_o] :
( ( member_list_o @ Xa2 @ A4 )
=> ( ( prefix_o @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% prefix_order.finite_has_minimal2
thf(fact_1162_prefix__order_Ofinite__has__maximal2,axiom,
! [A4: set_list_o,A: list_o] :
( ( finite_finite_list_o @ A4 )
=> ( ( member_list_o @ A @ A4 )
=> ? [X3: list_o] :
( ( member_list_o @ X3 @ A4 )
& ( prefix_o @ A @ X3 )
& ! [Xa2: list_o] :
( ( member_list_o @ Xa2 @ A4 )
=> ( ( prefix_o @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% prefix_order.finite_has_maximal2
thf(fact_1163_sorted__replicate,axiom,
! [N: nat,X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_1164_comm__append__are__replicate,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= ( append_o @ Ys @ Xs ) )
=> ? [M2: nat,N4: nat,Zs2: list_o] :
( ( ( concat_o @ ( replicate_list_o @ M2 @ Zs2 ) )
= Xs )
& ( ( concat_o @ ( replicate_list_o @ N4 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_1165_replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( replicate_nat @ ( suc @ N ) @ X2 )
= ( cons_nat @ X2 @ ( replicate_nat @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_1166_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X2: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X2 ) )
=> ( Xs
= ( replicate_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_1167_replicate__length__same,axiom,
! [Xs: list_nat,X2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = X2 ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_1168_finite__maxlen,axiom,
! [M3: set_list_nat] :
( ( finite8100373058378681591st_nat @ M3 )
=> ? [N4: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M3 )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_1169_replicate__app__Cons__same,axiom,
! [N: nat,X2: $o,Xs: list_o] :
( ( append_o @ ( replicate_o @ N @ X2 ) @ ( cons_o @ X2 @ Xs ) )
= ( cons_o @ X2 @ ( append_o @ ( replicate_o @ N @ X2 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_1170_replicate__app__Cons__same,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( append_nat @ ( replicate_nat @ N @ X2 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_1171_replicate__append__same,axiom,
! [I3: nat,X2: $o] :
( ( append_o @ ( replicate_o @ I3 @ X2 ) @ ( cons_o @ X2 @ nil_o ) )
= ( cons_o @ X2 @ ( replicate_o @ I3 @ X2 ) ) ) ).
% replicate_append_same
thf(fact_1172_replicate__append__same,axiom,
! [I3: nat,X2: nat] :
( ( append_nat @ ( replicate_nat @ I3 @ X2 ) @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ X2 @ ( replicate_nat @ I3 @ X2 ) ) ) ).
% replicate_append_same
thf(fact_1173_zip__replicate1,axiom,
! [N: nat,X2: option_list_o,Ys: list_option_list_o] :
( ( zip_op2665896746229601079list_o @ ( replic6237453613039860388list_o @ N @ X2 ) @ Ys )
= ( map_op4472081130091516167list_o @ ( produc5745850523778858007list_o @ X2 ) @ ( take_option_list_o @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_1174_zip__replicate1,axiom,
! [N: nat,X2: nat,Ys: list_P6834414599653733731al_nat] :
( ( zip_na1967359233052233492al_nat @ ( replicate_nat @ N @ X2 ) @ Ys )
= ( map_Pr6623428534301005723al_nat @ ( produc4490577844368043572al_nat @ X2 ) @ ( take_P5981256949253159243al_nat @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_1175_zip__replicate1,axiom,
! [N: nat,X2: real,Ys: list_nat] :
( ( zip_real_nat @ ( replicate_real @ N @ X2 ) @ Ys )
= ( map_na1309146153819616583al_nat @ ( produc3181502643871035669al_nat @ X2 ) @ ( take_nat @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_1176_Cons__replicate__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat,Y2: nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( replicate_nat @ N @ Y2 ) )
= ( ( X2 = Y2 )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1177_comm__append__is__replicate,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != nil_o )
=> ( ( Ys != nil_o )
=> ( ( ( append_o @ Xs @ Ys )
= ( append_o @ Ys @ Xs ) )
=> ? [N4: nat,Zs2: list_o] :
( ( ord_less_nat @ one_one_nat @ N4 )
& ( ( concat_o @ ( replicate_list_o @ N4 @ Zs2 ) )
= ( append_o @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_1178_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o,S2: set_nat,F: nat > nat,A4: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq2 @ Less @ S2 @ F )
=> ( ( ord_less_eq_set_nat @ A4 @ S2 )
=> ( ( finite_finite_nat @ A4 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F @ L2 ) )
=> ( ( ( set_nat2 @ L2 )
= A4 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A4 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1179_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A4: set_nat,L: list_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A4 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A4 ) ) )
= ( ( linord2614967742042102400et_nat @ A4 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1180_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( linord2614967742042102400et_nat @ A4 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1181_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A4 ) )
= A4 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1182_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A4: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A4 ) )
= ( finite_card_nat @ A4 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_1183_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A4: set_nat,B6: set_nat] :
( ( ( linord2614967742042102400et_nat @ A4 )
= ( linord2614967742042102400et_nat @ B6 ) )
=> ( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B6 )
=> ( A4 = B6 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1184_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A4: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_1185_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A4: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_1186_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o,S2: set_nat,F: nat > nat,A4: set_nat,L: list_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq2 @ Less @ S2 @ F )
=> ( ( ord_less_eq_set_nat @ A4 @ S2 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F @ L ) )
& ( ( set_nat2 @ L )
= A4 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A4 ) ) )
= ( ( sorted5905597674102116260at_nat @ Less_eq2 @ F @ A4 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1187_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_1188_remdups__adj__Cons__alt,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_1189_distinct__swap,axiom,
! [I3: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_1190_distinct__singleton,axiom,
! [X2: nat] : ( distinct_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_singleton
thf(fact_1191_distinct__length__2__or__more,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ A @ ( cons_nat @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_nat @ ( cons_nat @ A @ Xs ) )
& ( distinct_nat @ ( cons_nat @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1192_remdups__adj_Osimps_I3_J,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( ( X2 = Y2 )
=> ( ( remdups_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) )
& ( ( X2 != Y2 )
=> ( ( remdups_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( cons_nat @ X2 @ ( remdups_adj_nat @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_1193_distinct_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1194_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_1195_remdups__adj_Oelims,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( remdups_adj_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_nat )
=> ( Y2 != nil_nat ) )
=> ( ! [X3: nat] :
( ( X2
= ( cons_nat @ X3 @ nil_nat ) )
=> ( Y2
!= ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ~ ( ( ( X3 = Y3 )
=> ( Y2
= ( remdups_adj_nat @ ( cons_nat @ X3 @ Xs3 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2
= ( cons_nat @ X3 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_1196_remdups__adj_Osimps_I2_J,axiom,
! [X2: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_1197_prefix__remdups__adj,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( prefix_o @ ( remdups_adj_o @ Xs ) @ ( remdups_adj_o @ Ys ) ) ) ).
% prefix_remdups_adj
thf(fact_1198_sorted__remdups__adj,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_adj_nat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_1199_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A4: set_nat] : ( distinct_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_1200_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_1201_not__distinct__decomp,axiom,
! [Ws: list_o] :
( ~ ( distinct_o @ Ws )
=> ? [Xs3: list_o,Ys3: list_o,Zs2: list_o,Y3: $o] :
( Ws
= ( append_o @ Xs3 @ ( append_o @ ( cons_o @ Y3 @ nil_o ) @ ( append_o @ Ys3 @ ( append_o @ ( cons_o @ Y3 @ nil_o ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1202_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs3: list_nat,Ys3: list_nat,Zs2: list_nat,Y3: nat] :
( Ws
= ( append_nat @ Xs3 @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ ( append_nat @ Ys3 @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1203_not__distinct__conv__prefix,axiom,
! [As2: list_o] :
( ( ~ ( distinct_o @ As2 ) )
= ( ? [Xs2: list_o,Y: $o,Ys2: list_o] :
( ( member_o @ Y @ ( set_o2 @ Xs2 ) )
& ( distinct_o @ Xs2 )
& ( As2
= ( append_o @ Xs2 @ ( cons_o @ Y @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1204_not__distinct__conv__prefix,axiom,
! [As2: list_nat] :
( ( ~ ( distinct_nat @ As2 ) )
= ( ? [Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 )
& ( As2
= ( append_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1205_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1206_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_1207_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_1208_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs2: list_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( I != J )
=> ( ( nth_nat @ Xs2 @ I )
!= ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1209_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Xs @ J2 ) )
= ( I3 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1210_remdups__adj__append__two,axiom,
! [Xs: list_o,X2: $o,Y2: $o] :
( ( remdups_adj_o @ ( append_o @ Xs @ ( cons_o @ X2 @ ( cons_o @ Y2 @ nil_o ) ) ) )
= ( append_o @ ( remdups_adj_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) ) @ ( if_list_o @ ( X2 = Y2 ) @ nil_o @ ( cons_o @ Y2 @ nil_o ) ) ) ) ).
% remdups_adj_append_two
thf(fact_1211_remdups__adj__append__two,axiom,
! [Xs: list_nat,X2: nat,Y2: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) @ ( if_list_nat @ ( X2 = Y2 ) @ nil_nat @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_1212_finite__sorted__distinct__unique,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [X3: list_nat] :
( ( ( set_nat2 @ X3 )
= A4 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X3 )
& ( distinct_nat @ X3 )
& ! [Y5: list_nat] :
( ( ( ( set_nat2 @ Y5 )
= A4 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
& ( distinct_nat @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1213_distinct__Ex1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1214_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_1215_remdups__adj__adjacent,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I3 )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_1216_remdups__adj__replicate,axiom,
! [N: nat,X2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X2 ) )
= ( cons_nat @ X2 @ nil_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_1217_remdups__adj__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X2 @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 ) ) ) ).
% remdups_adj_singleton
thf(fact_1218_remdups__adj__append,axiom,
! [Xs_1: list_o,X2: $o,Xs_2: list_o] :
( ( remdups_adj_o @ ( append_o @ Xs_1 @ ( cons_o @ X2 @ Xs_2 ) ) )
= ( append_o @ ( remdups_adj_o @ ( append_o @ Xs_1 @ ( cons_o @ X2 @ nil_o ) ) ) @ ( tl_o @ ( remdups_adj_o @ ( cons_o @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_1219_remdups__adj__append,axiom,
! [Xs_1: list_nat,X2: nat,Xs_2: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X2 @ Xs_2 ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X2 @ nil_nat ) ) ) @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_1220_remdups__adj__append_H,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( Xs = nil_o )
| ( Ys = nil_o )
| ( ( last_o @ Xs )
!= ( hd_o @ Ys ) ) )
=> ( ( remdups_adj_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( remdups_adj_o @ Xs ) @ ( remdups_adj_o @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_1221_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_1222_remdups__adj_Opelims,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( remdups_adj_nat @ X2 )
= Y2 )
=> ( ( accp_list_nat @ remdups_adj_rel_nat @ X2 )
=> ( ( ( X2 = nil_nat )
=> ( ( Y2 = nil_nat )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ nil_nat ) ) )
=> ( ! [X3: nat] :
( ( X2
= ( cons_nat @ X3 @ nil_nat ) )
=> ( ( Y2
= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ( ( ( ( X3 = Y3 )
=> ( Y2
= ( remdups_adj_nat @ ( cons_nat @ X3 @ Xs3 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2
= ( cons_nat @ X3 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_1223_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J2: nat,I3: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ ( suc @ I3 ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I3 @ J2 ) ) @ N )
= ( suc @ ( plus_plus_nat @ I3 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_1224_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
= nil_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_1225_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( linord2614967742042102400et_nat @ A4 )
= nil_nat )
= ( A4 = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1226_sorted__list__of__set__greaterThanLessThan,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I3 @ J2 ) )
= ( cons_nat @ ( suc @ I3 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I3 ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_1227_prefix__order_Ofinite__has__minimal,axiom,
! [A4: set_list_o] :
( ( finite_finite_list_o @ A4 )
=> ( ( A4 != bot_bot_set_list_o )
=> ? [X3: list_o] :
( ( member_list_o @ X3 @ A4 )
& ! [Xa2: list_o] :
( ( member_list_o @ Xa2 @ A4 )
=> ( ( prefix_o @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% prefix_order.finite_has_minimal
thf(fact_1228_prefix__order_Ofinite__has__maximal,axiom,
! [A4: set_list_o] :
( ( finite_finite_list_o @ A4 )
=> ( ( A4 != bot_bot_set_list_o )
=> ? [X3: list_o] :
( ( member_list_o @ X3 @ A4 )
& ! [Xa2: list_o] :
( ( member_list_o @ Xa2 @ A4 )
=> ( ( prefix_o @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% prefix_order.finite_has_maximal
thf(fact_1229_Longest__common__prefix__ex,axiom,
! [L3: set_list_o] :
( ( L3 != bot_bot_set_list_o )
=> ? [Ps2: list_o] :
( ! [X5: list_o] :
( ( member_list_o @ X5 @ L3 )
=> ( prefix_o @ Ps2 @ X5 ) )
& ! [Qs2: list_o] :
( ! [X3: list_o] :
( ( member_list_o @ X3 @ L3 )
=> ( prefix_o @ Qs2 @ X3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Qs2 ) @ ( size_size_list_o @ Ps2 ) ) ) ) ) ).
% Longest_common_prefix_ex
thf(fact_1230_Longest__common__prefix__ex,axiom,
! [L3: set_list_nat] :
( ( L3 != bot_bot_set_list_nat )
=> ? [Ps2: list_nat] :
( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ L3 )
=> ( prefix_nat @ Ps2 @ X5 ) )
& ! [Qs2: list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ L3 )
=> ( prefix_nat @ Qs2 @ X3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs2 ) @ ( size_size_list_nat @ Ps2 ) ) ) ) ) ).
% Longest_common_prefix_ex
thf(fact_1231_Longest__common__prefix__unique,axiom,
! [L3: set_list_o] :
( ( L3 != bot_bot_set_list_o )
=> ? [X3: list_o] :
( ! [Xa2: list_o] :
( ( member_list_o @ Xa2 @ L3 )
=> ( prefix_o @ X3 @ Xa2 ) )
& ! [Qs2: list_o] :
( ! [Xa3: list_o] :
( ( member_list_o @ Xa3 @ L3 )
=> ( prefix_o @ Qs2 @ Xa3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Qs2 ) @ ( size_size_list_o @ X3 ) ) )
& ! [Y5: list_o] :
( ( ! [Xa3: list_o] :
( ( member_list_o @ Xa3 @ L3 )
=> ( prefix_o @ Y5 @ Xa3 ) )
& ! [Qs3: list_o] :
( ! [Xa2: list_o] :
( ( member_list_o @ Xa2 @ L3 )
=> ( prefix_o @ Qs3 @ Xa2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Qs3 ) @ ( size_size_list_o @ Y5 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% Longest_common_prefix_unique
thf(fact_1232_Longest__common__prefix__unique,axiom,
! [L3: set_list_nat] :
( ( L3 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ! [Xa2: list_nat] :
( ( member_list_nat @ Xa2 @ L3 )
=> ( prefix_nat @ X3 @ Xa2 ) )
& ! [Qs2: list_nat] :
( ! [Xa3: list_nat] :
( ( member_list_nat @ Xa3 @ L3 )
=> ( prefix_nat @ Qs2 @ Xa3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs2 ) @ ( size_size_list_nat @ X3 ) ) )
& ! [Y5: list_nat] :
( ( ! [Xa3: list_nat] :
( ( member_list_nat @ Xa3 @ L3 )
=> ( prefix_nat @ Y5 @ Xa3 ) )
& ! [Qs3: list_nat] :
( ! [Xa2: list_nat] :
( ( member_list_nat @ Xa2 @ L3 )
=> ( prefix_nat @ Qs3 @ Xa2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs3 ) @ ( size_size_list_nat @ Y5 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% Longest_common_prefix_unique
thf(fact_1233_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J2: nat,I3: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ I3 ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I3 @ J2 ) ) @ N )
= ( suc @ ( plus_plus_nat @ I3 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_1234_distinct__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_o @ ( append_o @ Xs @ Ys ) )
= ( ( distinct_o @ Xs )
& ( distinct_o @ Ys )
& ( ( inf_inf_set_o @ ( set_o2 @ Xs ) @ ( set_o2 @ Ys ) )
= bot_bot_set_o ) ) ) ).
% distinct_append
thf(fact_1235_sorted__list__of__set__greaterThanAtMost,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ ( suc @ I3 ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I3 @ J2 ) )
= ( cons_nat @ ( suc @ I3 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I3 ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_1236_Longest__common__prefix__longest,axiom,
! [L3: set_list_o,Ps: list_o] :
( ( L3 != bot_bot_set_list_o )
=> ( ! [X3: list_o] :
( ( member_list_o @ X3 @ L3 )
=> ( prefix_o @ Ps @ X3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Ps ) @ ( size_size_list_o @ ( longes2916941589570675010efix_o @ L3 ) ) ) ) ) ).
% Longest_common_prefix_longest
thf(fact_1237_Longest__common__prefix__longest,axiom,
! [L3: set_list_nat,Ps: list_nat] :
( ( L3 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ L3 )
=> ( prefix_nat @ Ps @ X3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ ( longes514542611558403238ix_nat @ L3 ) ) ) ) ) ).
% Longest_common_prefix_longest
thf(fact_1238_removeAll__append,axiom,
! [X2: $o,Xs: list_o,Ys: list_o] :
( ( removeAll_o @ X2 @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( removeAll_o @ X2 @ Xs ) @ ( removeAll_o @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_1239_removeAll_Osimps_I2_J,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( ( X2 = Y2 )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
= ( removeAll_nat @ X2 @ Xs ) ) )
& ( ( X2 != Y2 )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
= ( cons_nat @ Y2 @ ( removeAll_nat @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1240_length__removeAll__less__eq,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_1241_Longest__common__prefix__prefix,axiom,
! [Xs: list_o,L3: set_list_o] :
( ( member_list_o @ Xs @ L3 )
=> ( prefix_o @ ( longes2916941589570675010efix_o @ L3 ) @ Xs ) ) ).
% Longest_common_prefix_prefix
thf(fact_1242_Longest__common__prefix__eq__Nil,axiom,
! [X2: nat,Ys: list_nat,L3: set_list_nat,Y2: nat,Zs: list_nat] :
( ( member_list_nat @ ( cons_nat @ X2 @ Ys ) @ L3 )
=> ( ( member_list_nat @ ( cons_nat @ Y2 @ Zs ) @ L3 )
=> ( ( X2 != Y2 )
=> ( ( longes514542611558403238ix_nat @ L3 )
= nil_nat ) ) ) ) ).
% Longest_common_prefix_eq_Nil
thf(fact_1243_Longest__common__prefix__max__prefix,axiom,
! [L3: set_list_o,Ps: list_o] :
( ( L3 != bot_bot_set_list_o )
=> ( ! [X3: list_o] :
( ( member_list_o @ X3 @ L3 )
=> ( prefix_o @ Ps @ X3 ) )
=> ( prefix_o @ Ps @ ( longes2916941589570675010efix_o @ L3 ) ) ) ) ).
% Longest_common_prefix_max_prefix
thf(fact_1244_length__removeAll__less,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1245_Longest__common__prefix__eq,axiom,
! [L3: set_list_o,Ps: list_o] :
( ( L3 != bot_bot_set_list_o )
=> ( ! [X3: list_o] :
( ( member_list_o @ X3 @ L3 )
=> ( prefix_o @ Ps @ X3 ) )
=> ( ! [Qs3: list_o] :
( ! [X5: list_o] :
( ( member_list_o @ X5 @ L3 )
=> ( prefix_o @ Qs3 @ X5 ) )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Qs3 ) @ ( size_size_list_o @ Ps ) ) )
=> ( ( longes2916941589570675010efix_o @ L3 )
= Ps ) ) ) ) ).
% Longest_common_prefix_eq
thf(fact_1246_Longest__common__prefix__eq,axiom,
! [L3: set_list_nat,Ps: list_nat] :
( ( L3 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ L3 )
=> ( prefix_nat @ Ps @ X3 ) )
=> ( ! [Qs3: list_nat] :
( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ L3 )
=> ( prefix_nat @ Qs3 @ X5 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs3 ) @ ( size_size_list_nat @ Ps ) ) )
=> ( ( longes514542611558403238ix_nat @ L3 )
= Ps ) ) ) ) ).
% Longest_common_prefix_eq
thf(fact_1247_max__ext_Omax__extI,axiom,
! [X7: set_option_list_o,Y6: set_option_list_o,R2: set_Pr7497825696620840711list_o] :
( ( finite7007496012504252301list_o @ X7 )
=> ( ( finite7007496012504252301list_o @ Y6 )
=> ( ( Y6 != bot_bo3275880064859030064list_o )
=> ( ! [X3: option_list_o] :
( ( member_option_list_o @ X3 @ X7 )
=> ? [Xa2: option_list_o] :
( ( member_option_list_o @ Xa2 @ Y6 )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Xa2 ) @ R2 ) ) )
=> ( member6168996299790611792list_o @ ( produc1805441192919750679list_o @ X7 @ Y6 ) @ ( max_ex20081290693658143list_o @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_1248_max__ext_Ocases,axiom,
! [A1: set_option_list_o,A22: set_option_list_o,R2: set_Pr7497825696620840711list_o] :
( ( member6168996299790611792list_o @ ( produc1805441192919750679list_o @ A1 @ A22 ) @ ( max_ex20081290693658143list_o @ R2 ) )
=> ~ ( ( finite7007496012504252301list_o @ A1 )
=> ( ( finite7007496012504252301list_o @ A22 )
=> ( ( A22 != bot_bo3275880064859030064list_o )
=> ~ ! [X5: option_list_o] :
( ( member_option_list_o @ X5 @ A1 )
=> ? [Xa3: option_list_o] :
( ( member_option_list_o @ Xa3 @ A22 )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_1249_max__ext_Osimps,axiom,
! [A1: set_option_list_o,A22: set_option_list_o,R2: set_Pr7497825696620840711list_o] :
( ( member6168996299790611792list_o @ ( produc1805441192919750679list_o @ A1 @ A22 ) @ ( max_ex20081290693658143list_o @ R2 ) )
= ( ( finite7007496012504252301list_o @ A1 )
& ( finite7007496012504252301list_o @ A22 )
& ( A22 != bot_bo3275880064859030064list_o )
& ! [X: option_list_o] :
( ( member_option_list_o @ X @ A1 )
=> ? [Y: option_list_o] :
( ( member_option_list_o @ Y @ A22 )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_1250_list_Osimps_I15_J,axiom,
! [X21: nat,X222: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X222 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_1251_set__update__distinct,axiom,
! [Xs: list_nat,N: nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) )
= ( insert_nat2 @ X2 @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1252_distinct__list__update,axiom,
! [Xs: list_nat,A: nat,I3: nat] :
( ( distinct_nat @ Xs )
=> ( ~ ( member_nat @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ I3 ) @ bot_bot_set_nat ) ) )
=> ( distinct_nat @ ( list_update_nat @ Xs @ I3 @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1253_remove1_Osimps_I2_J,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( ( X2 = Y2 )
=> ( ( remove1_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
= Xs ) )
& ( ( X2 != Y2 )
=> ( ( remove1_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
= ( cons_nat @ Y2 @ ( remove1_nat @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_1254_sorted__remove1,axiom,
! [Xs: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).
% sorted_remove1
thf(fact_1255_remove1__append,axiom,
! [X2: $o,Xs: list_o,Ys: list_o] :
( ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( ( remove1_o @ X2 @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( remove1_o @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( ( remove1_o @ X2 @ ( append_o @ Xs @ Ys ) )
= ( append_o @ Xs @ ( remove1_o @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1256_remove1__split,axiom,
! [A: $o,Xs: list_o,Ys: list_o] :
( ( member_o @ A @ ( set_o2 @ Xs ) )
=> ( ( ( remove1_o @ A @ Xs )
= Ys )
= ( ? [Ls2: list_o,Rs: list_o] :
( ( Xs
= ( append_o @ Ls2 @ ( cons_o @ A @ Rs ) ) )
& ~ ( member_o @ A @ ( set_o2 @ Ls2 ) )
& ( Ys
= ( append_o @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1257_remove1__split,axiom,
! [A: nat,Xs: list_nat,Ys: list_nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs ) )
=> ( ( ( remove1_nat @ A @ Xs )
= Ys )
= ( ? [Ls2: list_nat,Rs: list_nat] :
( ( Xs
= ( append_nat @ Ls2 @ ( cons_nat @ A @ Rs ) ) )
& ~ ( member_nat @ A @ ( set_nat2 @ Ls2 ) )
& ( Ys
= ( append_nat @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1258_length__remove1,axiom,
! [X2: nat,Xs: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X2 @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1259_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X2 @ ( linord2614967742042102400et_nat @ A4 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1260_sorted__list__of__set__nonempty,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A4 )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_1261_length__shuffles,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( size_size_list_nat @ Zs )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_1262_Cons__in__shuffles__leftI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z: nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ ( cons_nat @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_1263_Cons__in__shuffles__rightI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z: nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_1264_shufflesE,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_nat ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_nat ) )
=> ( ! [X3: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs4 ) )
=> ! [Z3: nat,Zs4: list_nat] :
( ( Zs
= ( cons_nat @ Z3 @ Zs4 ) )
=> ( ( X3 = Z3 )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y3: nat,Ys5: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys5 ) )
=> ! [Z3: nat,Zs4: list_nat] :
( ( Zs
= ( cons_nat @ Z3 @ Zs4 ) )
=> ( ( Y3 = Z3 )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_1265_Cons__in__shuffles__iff,axiom,
! [Z: nat,Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ Xs @ Ys ) )
= ( ( ( Xs != nil_nat )
& ( ( hd_nat @ Xs )
= Z )
& ( member_list_nat @ Zs @ ( shuffles_nat @ ( tl_nat @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_nat )
& ( ( hd_nat @ Ys )
= Z )
& ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ ( tl_nat @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_1266_Min_Oinfinite,axiom,
! [A4: set_list_o] :
( ~ ( finite_finite_list_o @ A4 )
=> ( ( lattic7994203477452000287list_o @ A4 )
= ( the_list_o @ none_list_o ) ) ) ).
% Min.infinite
thf(fact_1267_Cons__in__lists__iff,axiom,
! [X2: nat,Xs: list_nat,A4: set_nat] :
( ( member_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( lists_nat @ A4 ) )
= ( ( member_nat @ X2 @ A4 )
& ( member_list_nat @ Xs @ ( lists_nat @ A4 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1268_append__in__lists__conv,axiom,
! [Xs: list_o,Ys: list_o,A4: set_o] :
( ( member_list_o @ ( append_o @ Xs @ Ys ) @ ( lists_o @ A4 ) )
= ( ( member_list_o @ Xs @ ( lists_o @ A4 ) )
& ( member_list_o @ Ys @ ( lists_o @ A4 ) ) ) ) ).
% append_in_lists_conv
thf(fact_1269_lists_Osimps,axiom,
! [A: list_nat,A4: set_nat] :
( ( member_list_nat @ A @ ( lists_nat @ A4 ) )
= ( ( A = nil_nat )
| ? [A3: nat,L4: list_nat] :
( ( A
= ( cons_nat @ A3 @ L4 ) )
& ( member_nat @ A3 @ A4 )
& ( member_list_nat @ L4 @ ( lists_nat @ A4 ) ) ) ) ) ).
% lists.simps
thf(fact_1270_lists_Ocases,axiom,
! [A: list_nat,A4: set_nat] :
( ( member_list_nat @ A @ ( lists_nat @ A4 ) )
=> ( ( A != nil_nat )
=> ~ ! [A2: nat,L2: list_nat] :
( ( A
= ( cons_nat @ A2 @ L2 ) )
=> ( ( member_nat @ A2 @ A4 )
=> ~ ( member_list_nat @ L2 @ ( lists_nat @ A4 ) ) ) ) ) ) ).
% lists.cases
thf(fact_1271_lists_OCons,axiom,
! [A: nat,A4: set_nat,L: list_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member_list_nat @ L @ ( lists_nat @ A4 ) )
=> ( member_list_nat @ ( cons_nat @ A @ L ) @ ( lists_nat @ A4 ) ) ) ) ).
% lists.Cons
thf(fact_1272_listsE,axiom,
! [X2: nat,L: list_nat,A4: set_nat] :
( ( member_list_nat @ ( cons_nat @ X2 @ L ) @ ( lists_nat @ A4 ) )
=> ~ ( ( member_nat @ X2 @ A4 )
=> ~ ( member_list_nat @ L @ ( lists_nat @ A4 ) ) ) ) ).
% listsE
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X2: list_o,Y2: list_o] :
( ( if_list_o @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X2: list_o,Y2: list_o] :
( ( if_list_o @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
prefix454693708527911765comp_o @ x @ y ).
%------------------------------------------------------------------------------