TPTP Problem File: SLH0141^1.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 : Multiset_Ordering_NPC/0002_Multiset_Ordering_in_NP/prob_00614_027427__13778874_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1564 ( 644 unt; 291 typ; 0 def)
% Number of atoms : 3598 (1953 equ; 0 cnn)
% Maximal formula atoms : 19 ( 2 avg)
% Number of connectives : 13804 ( 473 ~; 61 |; 390 &;11341 @)
% ( 0 <=>;1539 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 46 ( 45 usr)
% Number of type conns : 711 ( 711 >; 0 *; 0 +; 0 <<)
% Number of symbols : 249 ( 246 usr; 26 con; 0-4 aty)
% Number of variables : 4241 ( 72 ^;3862 !; 307 ?;4241 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:29:48.765
%------------------------------------------------------------------------------
% Could-be-implicit typings (45)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Nat__Onat_J_Mt__Multiset__Omultiset_It__Nat__Onat_J_J_J,type,
set_Pr7086239977703175297et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Int__Oint_J_Mt__Multiset__Omultiset_It__Int__Oint_J_J_J,type,
set_Pr3168025420435445049et_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc254973753779126261st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc5834231552977413017st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr3451248702717554689st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
set_Pr7861072320784411741st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr5578615432719617117st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
set_Pr765067013931698361st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Nat__Onat_J_Mt__Multiset__Omultiset_It__Nat__Onat_J_J,type,
produc7112813594591200289et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Int__Oint_J_Mt__Multiset__Omultiset_It__Int__Oint_J_J,type,
produc7636037365327937753et_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
list_P8198026277950538467nt_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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set_Pr7995236796853374141at_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
multis1201202736280713200et_nat: $tType ).
thf(ty_n_t__List__Olist_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
list_multiset_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
product_prod_nat_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
product_prod_int_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
set_multiset_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
list_int_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $tType ).
thf(ty_n_t__Multiset____Ordering____in____NP__OPropVar,type,
multis3193088007478089820ropVar: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
multiset_int: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (246)
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
inj_on_int_int: ( int > int ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
inj_on720019086181695851st_int: ( list_int > list_int ) > set_list_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
inj_on5670230764983331635et_nat: ( multiset_nat > multiset_nat ) > set_multiset_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Int__Oint_J,type,
minus_4344325018492214997et_int: multiset_int > multiset_int > multiset_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
minus_4897669229644054985et_nat: multis1201202736280713200et_nat > multis1201202736280713200et_nat > multis1201202736280713200et_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Int__Oint_J,type,
plus_p2156642923369911685et_int: multiset_int > multiset_int > multiset_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
plus_p8768199597779566713et_nat: multis1201202736280713200et_nat > multis1201202736280713200et_nat > multis1201202736280713200et_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Int__Oint_J,type,
zero_z3170743180189231877et_int: multiset_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
zero_z9085034013355480569et_nat: multis1201202736280713200et_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups5394990218802192790at_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > ( nat > multiset_nat ) > set_nat > multiset_nat ).
thf(sy_c_Groups__List_Omonoid__add_Osum__list_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
groups2887787882517827221et_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > list_multiset_nat > multiset_nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Int__Oint,type,
groups4559388385066561235st_int: list_int > int ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
groups8053510108761903431et_nat: list_multiset_nat > multiset_nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
sup_su6024340866399070445nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
append_multiset_nat: list_multiset_nat > list_multiset_nat > list_multiset_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
append7030698103840186580nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Obind_001t__Int__Oint_001t__Nat__Onat,type,
bind_int_nat: list_int > ( int > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obutlast_001t__Int__Oint,type,
butlast_int: list_int > list_int ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Odistinct_001t__Int__Oint,type,
distinct_int: list_int > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
distin6294748288989586407et_nat: list_multiset_nat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
distin3744728255968310194nt_int: list_P5707943133018811711nt_int > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
distin7922579275477506902nt_nat: list_P8198026277950538467nt_nat > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
distin6923225563576452346at_nat: list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
enumerate_int: nat > list_int > list_P3521021558325789923at_int ).
thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
gen_length_int: nat > list_int > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olast_001t__Int__Oint,type,
last_int: list_int > int ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
last_P3305686521732843992nt_int: list_P5707943133018811711nt_int > product_prod_int_int ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
last_P6484183829340986144at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).
thf(sy_c_List_Olenlex_001t__Int__Oint,type,
lenlex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001t__Int__Oint,type,
lex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olist_OCons_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
cons_int_nat: ( int > nat ) > list_int_nat > list_int_nat ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
cons_multiset_nat: multiset_nat > list_multiset_nat > list_multiset_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
cons_P3334398858971670639nt_int: product_prod_int_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
cons_P7512249878480867347nt_nat: product_prod_int_nat > list_P8198026277950538467nt_nat > list_P8198026277950538467nt_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
cons_P2335045147070616083at_int: product_prod_nat_int > list_P3521021558325789923at_int > list_P3521021558325789923at_int ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_ONil_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
nil_int_nat: list_int_nat ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
map_int_int: ( int > int ) > list_int > list_int ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Nat__Onat,type,
map_int_nat: ( int > nat ) > list_int > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
map_in7157766398909135175nt_int: ( int > product_prod_int_int ) > list_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
map_nat_int: ( nat > int ) > list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
map_nat_multiset_nat: ( nat > multiset_nat ) > list_nat > list_multiset_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
set_multiset_nat2: list_multiset_nat > set_multiset_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
set_Pr6647972299459129970nt_nat: list_P8198026277950538467nt_nat > set_Pr3448869479623346877nt_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Olist__ex_001t__Int__Oint,type,
list_ex_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__update_001t__Int__Oint,type,
list_update_int: list_int > nat > int > list_int ).
thf(sy_c_List_Olist__update_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
list_u3438943574295160626et_nat: list_multiset_nat > nat > multiset_nat > list_multiset_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel_001t__Int__Oint_001t__Int__Oint,type,
listrel_int_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olistrel_001t__Int__Oint_001t__Nat__Onat,type,
listrel_int_nat: set_Pr3448869479623346877nt_nat > set_Pr5578615432719617117st_nat ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Int__Oint,type,
listrel_nat_int: set_Pr7995236796853374141at_int > set_Pr7861072320784411741st_int ).
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_Omeasures_001t__Int__Oint,type,
measures_int: list_int_nat > set_Pr958786334691620121nt_int ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
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__Multiset__Omultiset_It__Nat__Onat_J,type,
nth_multiset_nat: list_multiset_nat > nat > multiset_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).
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thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
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__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oremove1_001t__Int__Oint,type,
remove1_int: int > list_int > list_int ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
removeAll_int: int > list_int > list_int ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Int__Oint,type,
take_int: nat > list_int > list_int ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Int__Oint,type,
zip_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).
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__Order_Olist__order__extension_001t__Int__Oint,type,
list_l790271996378993376on_int: ( set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ) > ( set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ) > $o ).
thf(sy_c_List__Order_Olist__order__extension_001t__Nat__Onat,type,
list_l792762466888043652on_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ) > $o ).
thf(sy_c_Multiset_Oadd__mset_001t__Int__Oint,type,
add_mset_int: int > multiset_int > multiset_int ).
thf(sy_c_Multiset_Oadd__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
add_ms5124500668711485122et_nat: multiset_nat > multis1201202736280713200et_nat > multis1201202736280713200et_nat ).
thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
add_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
comm_m5787568287065167983et_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > multis1201202736280713200et_nat > multiset_nat ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Int__Oint,type,
comm_m759698451323652583et_int: multiset_int > int ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
comm_m8595621181775931995et_nat: multis1201202736280713200et_nat > multiset_nat ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Nat__Onat,type,
comm_m762188921832702859et_nat: multiset_nat > nat ).
thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
fold_m1829410296857755981et_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > multis1201202736280713200et_nat > multiset_nat ).
thf(sy_c_Multiset_Ofold__mset_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
fold_m2600682269844132093et_nat: ( nat > multiset_nat > multiset_nat ) > multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Int__Oint,type,
linord3045382416894633534et_int: multiset_int > list_int ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
linord3047872887403683810et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset_Omset_001t__Int__Oint,type,
mset_int: list_int > multiset_int ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
mset_P6383711406899277590at_nat: list_P6011104703257516679at_nat > multis2468970476368604999at_nat ).
thf(sy_c_Multiset_Omset__set_001t__Nat__Onat,type,
mset_set_nat: set_nat > multiset_nat ).
thf(sy_c_Multiset_Omult1_001t__Int__Oint,type,
mult1_int: set_Pr958786334691620121nt_int > set_Pr3168025420435445049et_int ).
thf(sy_c_Multiset_Omult1_001t__Nat__Onat,type,
mult1_nat: set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).
thf(sy_c_Multiset_Omult_001t__Int__Oint,type,
mult_int: set_Pr958786334691620121nt_int > set_Pr3168025420435445049et_int ).
thf(sy_c_Multiset_Omult_001t__Nat__Onat,type,
mult_nat: set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Int__Oint,type,
set_mset_int: multiset_int > set_int ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
set_ms4188662328148412963et_nat: multis1201202736280713200et_nat > set_multiset_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Osize__multiset_001t__Nat__Onat,type,
size_multiset_nat: ( nat > nat ) > multiset_nat > nat ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset__Extension2_Ons__mul__ext_001t__Int__Oint,type,
multis8828838126066458039xt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr3168025420435445049et_int ).
thf(sy_c_Multiset__Extension2_Ons__mul__ext_001t__Nat__Onat,type,
multis8831328596575508315xt_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).
thf(sy_c_Multiset__Extension2_Os__mul__ext_001t__Int__Oint,type,
multis4212723674092261717xt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr3168025420435445049et_int ).
thf(sy_c_Multiset__Extension2_Os__mul__ext_001t__Nat__Onat,type,
multis4215214144601311993xt_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).
thf(sy_c_Multiset__Extension__Pair_Omult2__alt_001t__Int__Oint,type,
multis2693779972258166040lt_int: $o > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr3168025420435445049et_int ).
thf(sy_c_Multiset__Extension__Pair_Omult2__alt_001t__Nat__Onat,type,
multis2696270442767216316lt_nat: $o > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).
thf(sy_c_Multiset__Extension__Pair_Omultpw_001t__Int__Oint,type,
multis5149657266184778664pw_int: set_Pr958786334691620121nt_int > set_Pr3168025420435445049et_int ).
thf(sy_c_Multiset__Extension__Pair_Omultpw_001t__Nat__Onat,type,
multis5152147736693828940pw_nat: set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OEpsilon,type,
multis2544335231667181926psilon: nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OGamma,type,
multis387687052011358179_Gamma: nat > nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_Osize__PropVar,type,
multis2955979900537361535ropVar: multis3193088007478089820ropVar > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > 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__Multiset__Omultiset_It__Nat__Onat_J_J,type,
size_s6386657463320973636et_nat: list_multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Order__Pair_OSN__order__pair_001t__Int__Oint,type,
order_7800132744162871582ir_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Order__Pair_OSN__order__pair_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Order__Pair_OSN__order__pair_001t__List__Olist_It__Nat__Onat_J,type,
order_2144575240049343698st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Order__Pair_OSN__order__pair_001t__Nat__Onat,type,
order_7802623214671921858ir_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord_Omax_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord_Omin_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le5777773500796000884et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001t__List__Olist_It__Int__Oint_J,type,
produc8618682346314911123st_int: ( int > int > $o ) > list_int > produc5834231552977413017st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
product_Pair_int_nat: int > nat > product_prod_int_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4542114716404682293st_nat: list_int > list_nat > produc3676724955757786621st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Int__Oint_J,type,
produc7739558402351520821st_int: list_nat > list_int > produc8561936516282095101st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Int__Oint_J_001t__Multiset__Omultiset_It__Int__Oint_J,type,
produc1570911416673723153et_int: multiset_int > multiset_int > produc7636037365327937753et_int ).
thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
produc2735455520514455641et_nat: multiset_nat > multiset_nat > produc7112813594591200289et_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
product_Pair_nat_int: nat > int > product_prod_nat_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Relation_OId_001t__Int__Oint,type,
id_int: set_Pr958786334691620121nt_int ).
thf(sy_c_Relation_OId_001t__Nat__Onat,type,
id_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Relations_Olocally__refl_001t__Nat__Onat,type,
locally_refl_nat: set_Pr1261947904930325089at_nat > multiset_nat > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
the_elem_int: set_int > int ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $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__Multiset__Omultiset_It__Nat__Onat_J,type,
member_multiset_nat: multiset_nat > set_multiset_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member9189046780804443046st_nat: produc3676724955757786621st_nat > set_Pr5578615432719617117st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member4850886304473975718st_int: produc8561936516282095101st_int > set_Pr7861072320784411741st_int > $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__Multiset__Omultiset_It__Int__Oint_J_Mt__Multiset__Omultiset_It__Int__Oint_J_J,type,
member2849730630941545090et_int: produc7636037365327937753et_int > set_Pr3168025420435445049et_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Nat__Onat_J_Mt__Multiset__Omultiset_It__Nat__Onat_J_J,type,
member2326506860204807626et_nat: produc7112813594591200289et_nat > set_Pr7086239977703175297et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $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_v_cns,type,
cns: nat > nat > $o ).
thf(sy_v_cs,type,
cs: nat > nat > $o ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_i__of__j2____,type,
i_of_j2: nat > nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_p____,type,
p: nat ).
thf(sy_v_pos__of____,type,
pos_of: list_nat > nat > nat ).
thf(sy_v_v____,type,
v: multis3193088007478089820ropVar > $o ).
thf(sy_v_xs1____,type,
xs1: list_nat ).
thf(sy_v_xs2____,type,
xs2: list_nat ).
thf(sy_v_ys1____,type,
ys1: list_nat ).
thf(sy_v_ys2____,type,
ys2: list_nat ).
% Relevant facts (1269)
thf(fact_0_p2,axiom,
ord_less_nat @ p @ ( size_size_list_nat @ ys1 ) ).
% p2
thf(fact_1__092_060open_062ys1_A_B_Apos__of_Ays1_Aj_A_061_Aj_092_060close_062,axiom,
( ( nth_nat @ ys1 @ ( pos_of @ ys1 @ j ) )
= j ) ).
% \<open>ys1 ! pos_of ys1 j = j\<close>
thf(fact_2__092_060open_062_092_060exists_062_Bi_O_Ai_A_060_Alength_Ays1_A_092_060and_062_Ays1_A_B_Ai_A_061_Aj_092_060close_062,axiom,
? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ ys1 ) )
& ( ( nth_nat @ ys1 @ X )
= j )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ ys1 ) )
& ( ( nth_nat @ ys1 @ Y )
= j ) )
=> ( Y = X ) ) ) ).
% \<open>\<exists>!i. i < length ys1 \<and> ys1 ! i = j\<close>
thf(fact_3__092_060open_062pos__of_Ays1_Aj_A_060_Alength_Ays1_092_060close_062,axiom,
ord_less_nat @ ( pos_of @ ys1 @ j ) @ ( size_size_list_nat @ ys1 ) ).
% \<open>pos_of ys1 j < length ys1\<close>
thf(fact_4_j__def,axiom,
( j
= ( nth_nat @ ys1 @ p ) ) ).
% j_def
thf(fact_5_j,axiom,
member_nat @ j @ ( set_nat2 @ ys1 ) ).
% j
thf(fact_6_p__ys_I2_J,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ys1 ) )
=> ( ( nth_nat @ ys1 @ ( pos_of @ ys1 @ X2 ) )
= X2 ) ) ).
% p_ys(2)
thf(fact_7_p1,axiom,
ord_less_nat @ p @ ( size_size_list_nat @ xs1 ) ).
% p1
thf(fact_8__092_060open_062distinct_Ays1_092_060close_062,axiom,
distinct_nat @ ys1 ).
% \<open>distinct ys1\<close>
thf(fact_9_p__ys_I1_J,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ys1 ) )
=> ( ord_less_nat @ ( pos_of @ ys1 @ X2 ) @ ( size_size_list_nat @ ys1 ) ) ) ).
% p_ys(1)
thf(fact_10_j__ys1_I1_J,axiom,
! [J: nat] :
( ( member_nat @ J @ ( set_nat2 @ ys1 ) )
=> ( member_nat @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) @ ( set_nat2 @ xs1 ) ) ) ).
% j_ys1(1)
thf(fact_11_xpi,axiom,
( ( nth_nat @ xs1 @ p )
= i ) ).
% xpi
thf(fact_12_p__ys_I3_J,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ys1 ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ ys1 ) )
& ( ( nth_nat @ ys1 @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ ys1 ) )
& ( ( nth_nat @ ys1 @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ).
% p_ys(3)
thf(fact_13_j__ys1_I2_J,axiom,
! [J: nat] :
( ( member_nat @ J @ ( set_nat2 @ ys1 ) )
=> ( cns @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) @ J ) ) ).
% j_ys1(2)
thf(fact_14_pos_I2_J,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( nth_nat @ Xs @ ( pos_of @ Xs @ X2 ) )
= X2 ) ) ) ).
% pos(2)
thf(fact_15__092_060open_062distinct_Axs1_092_060close_062,axiom,
distinct_nat @ xs1 ).
% \<open>distinct xs1\<close>
thf(fact_16_i__xs_I1_J,axiom,
member_nat @ i @ ( set_nat2 @ xs1 ) ).
% i_xs(1)
thf(fact_17_pos_I3_J,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% pos(3)
thf(fact_18_p__xs_I2_J,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ xs1 ) )
=> ( ( nth_nat @ xs1 @ ( pos_of @ xs1 @ X2 ) )
= X2 ) ) ).
% p_xs(2)
thf(fact_19_p__xs_I3_J,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ xs1 ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ xs1 ) )
& ( ( nth_nat @ xs1 @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ xs1 ) )
& ( ( nth_nat @ xs1 @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ).
% p_xs(3)
thf(fact_20_pos_I1_J,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( pos_of @ Xs @ X2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% pos(1)
thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_O_A_092_060lbrakk_062p_A_060_Alength_Axs1_059_Axs1_A_B_Ap_A_061_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [P: nat] :
( ( ord_less_nat @ P @ ( size_size_list_nat @ xs1 ) )
=> ( ( nth_nat @ xs1 @ P )
!= i ) ) ).
% \<open>\<And>thesis. (\<And>p. \<lbrakk>p < length xs1; xs1 ! p = i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_22_p__xs_I1_J,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ xs1 ) )
=> ( ord_less_nat @ ( pos_of @ xs1 @ X2 ) @ ( size_size_list_nat @ xs1 ) ) ) ).
% p_xs(1)
thf(fact_23_i,axiom,
ord_less_nat @ i @ n ).
% i
thf(fact_24_distinct__Ex1,axiom,
! [Xs: list_P8198026277950538467nt_nat,X2: product_prod_int_nat] :
( ( distin7922579275477506902nt_nat @ Xs )
=> ( ( member216504246829706758nt_nat @ X2 @ ( set_Pr6647972299459129970nt_nat @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_s7647898544948552527nt_nat @ Xs ) )
& ( ( nth_Pr8617346907841251940nt_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s7647898544948552527nt_nat @ Xs ) )
& ( ( nth_Pr8617346907841251940nt_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_25_distinct__Ex1,axiom,
! [Xs: list_P5707943133018811711nt_int,X2: product_prod_int_int] :
( ( distin3744728255968310194nt_int @ Xs )
=> ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_s5157815400016825771nt_int @ Xs ) )
& ( ( nth_Pr4439495888332055232nt_int @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s5157815400016825771nt_int @ Xs ) )
& ( ( nth_Pr4439495888332055232nt_int @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_26_distinct__Ex1,axiom,
! [Xs: list_multiset_nat,X2: multiset_nat] :
( ( distin6294748288989586407et_nat @ Xs )
=> ( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_s6386657463320973636et_nat @ Xs ) )
& ( ( nth_multiset_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s6386657463320973636et_nat @ Xs ) )
& ( ( nth_multiset_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_27_distinct__Ex1,axiom,
! [Xs: list_P6011104703257516679at_nat,X2: product_prod_nat_nat] :
( ( distin6923225563576452346at_nat @ Xs )
=> ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_28_distinct__Ex1,axiom,
! [Xs: list_int,X2: int] :
( ( distinct_int @ Xs )
=> ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_29_distinct__Ex1,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_30_distinct__Ex1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_31_distinct__conv__nth,axiom,
( distin3744728255968310194nt_int
= ( ^ [Xs2: list_P5707943133018811711nt_int] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs2 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
=> ( ( I != J2 )
=> ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ I )
!= ( nth_Pr4439495888332055232nt_int @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_32_distinct__conv__nth,axiom,
( distin6294748288989586407et_nat
= ( ^ [Xs2: list_multiset_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s6386657463320973636et_nat @ Xs2 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s6386657463320973636et_nat @ Xs2 ) )
=> ( ( I != J2 )
=> ( ( nth_multiset_nat @ Xs2 @ I )
!= ( nth_multiset_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_33_distinct__conv__nth,axiom,
( distin6923225563576452346at_nat
= ( ^ [Xs2: list_P6011104703257516679at_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( I != J2 )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I )
!= ( nth_Pr7617993195940197384at_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_34_distinct__conv__nth,axiom,
( distinct_int
= ( ^ [Xs2: list_int] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( I != J2 )
=> ( ( nth_int @ Xs2 @ I )
!= ( nth_int @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_35_distinct__conv__nth,axiom,
( distinct_list_nat
= ( ^ [Xs2: list_list_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( I != J2 )
=> ( ( nth_list_nat @ Xs2 @ I )
!= ( nth_list_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_36_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs2: list_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( I != J2 )
=> ( ( nth_nat @ Xs2 @ I )
!= ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_37_nth__eq__iff__index__eq,axiom,
! [Xs: list_P5707943133018811711nt_int,I2: nat,J: nat] :
( ( distin3744728255968310194nt_int @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( ( nth_Pr4439495888332055232nt_int @ Xs @ I2 )
= ( nth_Pr4439495888332055232nt_int @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_38_nth__eq__iff__index__eq,axiom,
! [Xs: list_multiset_nat,I2: nat,J: nat] :
( ( distin6294748288989586407et_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ( nth_multiset_nat @ Xs @ I2 )
= ( nth_multiset_nat @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_39_nth__eq__iff__index__eq,axiom,
! [Xs: list_P6011104703257516679at_nat,I2: nat,J: nat] :
( ( distin6923225563576452346at_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ( nth_Pr7617993195940197384at_nat @ Xs @ I2 )
= ( nth_Pr7617993195940197384at_nat @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_40_nth__eq__iff__index__eq,axiom,
! [Xs: list_int,I2: nat,J: nat] :
( ( distinct_int @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_41_nth__eq__iff__index__eq,axiom,
! [Xs: list_list_nat,I2: nat,J: nat] :
( ( distinct_list_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( nth_list_nat @ Xs @ I2 )
= ( nth_list_nat @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_42_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I2: nat,J: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_43_nth__mem,axiom,
! [N: nat,Xs: list_P8198026277950538467nt_nat] :
( ( ord_less_nat @ N @ ( size_s7647898544948552527nt_nat @ Xs ) )
=> ( member216504246829706758nt_nat @ ( nth_Pr8617346907841251940nt_nat @ Xs @ N ) @ ( set_Pr6647972299459129970nt_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_44_nth__mem,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( member5262025264175285858nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ N ) @ ( set_Pr2470121279949933262nt_int @ Xs ) ) ) ).
% nth_mem
thf(fact_45_nth__mem,axiom,
! [N: nat,Xs: list_multiset_nat] :
( ( ord_less_nat @ N @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( member_multiset_nat @ ( nth_multiset_nat @ Xs @ N ) @ ( set_multiset_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_46_nth__mem,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_47_nth__mem,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_48_nth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_49_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_50_list__ball__nth,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ! [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_51_list__ball__nth,axiom,
! [N: nat,Xs: list_multiset_nat,P2: multiset_nat > $o] :
( ( ord_less_nat @ N @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_multiset_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_52_list__ball__nth,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_53_list__ball__nth,axiom,
! [N: nat,Xs: list_int,P2: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_54_list__ball__nth,axiom,
! [N: nat,Xs: list_list_nat,P2: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_55_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P2: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_56_in__set__conv__nth,axiom,
! [X2: product_prod_int_nat,Xs: list_P8198026277950538467nt_nat] :
( ( member216504246829706758nt_nat @ X2 @ ( set_Pr6647972299459129970nt_nat @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s7647898544948552527nt_nat @ Xs ) )
& ( ( nth_Pr8617346907841251940nt_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_57_in__set__conv__nth,axiom,
! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs ) )
& ( ( nth_Pr4439495888332055232nt_int @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_58_in__set__conv__nth,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s6386657463320973636et_nat @ Xs ) )
& ( ( nth_multiset_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_59_in__set__conv__nth,axiom,
! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_60_in__set__conv__nth,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_61_in__set__conv__nth,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_62_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_63_all__nth__imp__all__set,axiom,
! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o,X2: product_prod_int_int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( P2 @ ( nth_Pr4439495888332055232nt_int @ Xs @ I3 ) ) )
=> ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ( P2 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_64_all__nth__imp__all__set,axiom,
! [Xs: list_multiset_nat,P2: multiset_nat > $o,X2: multiset_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( P2 @ ( nth_multiset_nat @ Xs @ I3 ) ) )
=> ( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
=> ( P2 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_65_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) ) )
=> ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P2 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_66_all__nth__imp__all__set,axiom,
! [Xs: list_int,P2: int > $o,X2: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( P2 @ ( nth_int @ Xs @ I3 ) ) )
=> ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( P2 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_67_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P2: list_nat > $o,X2: list_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P2 @ ( nth_list_nat @ Xs @ I3 ) ) )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P2 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_68_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P2: nat > $o,X2: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_69_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_70_i__xs_I2_J,axiom,
~ ( member_nat @ i @ ( set_nat2 @ xs2 ) ) ).
% i_xs(2)
thf(fact_71_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_72_Skolem__list__nth,axiom,
! [K: nat,P2: nat > nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X4: nat] : ( P2 @ I @ X4 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P2 @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_73_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
= ( ^ [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_74_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_75_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_76_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_77_length__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_78_dist__xs,axiom,
distinct_nat @ ( append_nat @ xs1 @ xs2 ) ).
% dist_xs
thf(fact_79_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_80_j__ys1_I3_J,axiom,
! [J: nat] :
( ( member_nat @ J @ ( set_nat2 @ ys1 ) )
=> ( v @ ( multis387687052011358179_Gamma @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) @ J ) ) ) ).
% j_ys1(3)
thf(fact_81_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83__C2_C_I1_J,axiom,
member_nat @ i @ ( set_nat2 @ ( upt @ zero_zero_nat @ n ) ) ).
% "2"(1)
thf(fact_84_distinct__swap,axiom,
! [I2: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I2 ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_85_set__swap,axiom,
! [I2: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I2 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_86_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P3: nat > $o,Xs2: list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
& ( P3 @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_87_un__xs,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ xs1 ) @ ( set_nat2 @ xs2 ) )
= ( set_ord_lessThan_nat @ n ) ) ).
% un_xs
thf(fact_88_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_89_v,axiom,
v @ ( multis2544335231667181926psilon @ i ) ).
% v
thf(fact_90_inj__on__nth,axiom,
! [Xs: list_nat,I4: set_nat] :
( ( distinct_nat @ Xs )
=> ( ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ord_less_nat @ X @ ( size_size_list_nat @ Xs ) ) )
=> ( inj_on_nat_nat @ ( nth_nat @ Xs ) @ I4 ) ) ) ).
% inj_on_nth
thf(fact_91_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_92_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_93_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_94_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_95_PropVar_Oinject_I1_J,axiom,
! [X11: nat,X12: nat,Y11: nat,Y12: nat] :
( ( ( multis387687052011358179_Gamma @ X11 @ X12 )
= ( multis387687052011358179_Gamma @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% PropVar.inject(1)
thf(fact_96_PropVar_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( multis2544335231667181926psilon @ X22 )
= ( multis2544335231667181926psilon @ Y22 ) )
= ( X22 = Y22 ) ) ).
% PropVar.inject(2)
thf(fact_97_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_98_length__list__update,axiom,
! [Xs: list_nat,I2: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_99_list__update__id,axiom,
! [Xs: list_nat,I2: nat] :
( ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_100_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs: list_nat,X2: nat] :
( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_101_removeAll__id,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_102_removeAll__append,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( removeAll_nat @ X2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( removeAll_nat @ X2 @ Xs ) @ ( removeAll_nat @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_103_list__ex__append,axiom,
! [P2: nat > $o,Xs: list_nat,Ys: list_nat] :
( ( list_ex_nat @ P2 @ ( append_nat @ Xs @ Ys ) )
= ( ( list_ex_nat @ P2 @ Xs )
| ( list_ex_nat @ P2 @ Ys ) ) ) ).
% list_ex_append
thf(fact_104_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_105_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_106_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ I2 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_107_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_108_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_109_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_110_PropVar_Odistinct_I1_J,axiom,
! [X11: nat,X12: nat,X22: nat] :
( ( multis387687052011358179_Gamma @ X11 @ X12 )
!= ( multis2544335231667181926psilon @ X22 ) ) ).
% PropVar.distinct(1)
thf(fact_111_list__update__append1,axiom,
! [I2: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I2 @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_112_distinct__upt,axiom,
! [I2: nat,J: nat] : ( distinct_nat @ ( upt @ I2 @ J ) ) ).
% distinct_upt
thf(fact_113_distinct__removeAll,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_114_list__ex__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
( ( Xs = Ys )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_ex_nat @ F @ Xs )
= ( list_ex_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_115_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_116_nth__list__update,axiom,
! [I2: nat,Xs: list_nat,J: nat,X2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I2 = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ J )
= X2 ) )
& ( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_117_list__update__same__conv,axiom,
! [I2: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I2 @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I2 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_118_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_119_dist__ys,axiom,
distinct_nat @ ( append_nat @ ys1 @ ys2 ) ).
% dist_ys
thf(fact_120_lessThan__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_121_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_122_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_123_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_124_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_125_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_126_lessThan__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_127_xs,axiom,
( ( mset_nat @ ( upt @ zero_zero_nat @ n ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ xs1 ) @ ( mset_nat @ xs2 ) ) ) ).
% xs
thf(fact_128_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_129_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_130_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_131_add__right__cancel,axiom,
! [B: multiset_nat,A: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B @ A )
= ( plus_p6334493942879108393et_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_132_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_133_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_134_add__left__cancel,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ B )
= ( plus_p6334493942879108393et_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_135_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_136_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_137_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_138_sup__left__idem,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) )
= ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_139_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_140_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_141_add__0,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A )
= A ) ).
% add_0
thf(fact_142_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_143_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_144_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_145_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_146_add__cancel__right__right,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( A
= ( plus_p6334493942879108393et_nat @ A @ B ) )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_right
thf(fact_147_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_148_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_149_add__cancel__right__left,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( A
= ( plus_p6334493942879108393et_nat @ B @ A ) )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_left
thf(fact_150_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_151_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_152_add__cancel__left__right,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ B )
= A )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_right
thf(fact_153_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_154_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_155_add__cancel__left__left,axiom,
! [B: multiset_nat,A: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B @ A )
= A )
= ( B = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_left
thf(fact_156_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_157_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_158_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_159_add_Oright__neutral,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% add.right_neutral
thf(fact_160_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_161_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_162_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_163_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_164_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_165_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_166_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_167_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_168_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_169_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_170_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_171_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_172_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_173_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_174_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_175_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_176_ys,axiom,
( ( mset_nat @ ( upt @ zero_zero_nat @ m ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ ys1 ) @ ( mset_nat @ ys2 ) ) ) ).
% ys
thf(fact_177_add__right__imp__eq,axiom,
! [B: multiset_nat,A: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B @ A )
= ( plus_p6334493942879108393et_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_178_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_179_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_180_add__left__imp__eq,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ B )
= ( plus_p6334493942879108393et_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_181_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_182_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_183_add_Oleft__commute,axiom,
! [B: multiset_nat,A: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ B @ ( plus_p6334493942879108393et_nat @ A @ C ) )
= ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_184_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_185_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_186_add_Ocommute,axiom,
( plus_p6334493942879108393et_nat
= ( ^ [A3: multiset_nat,B3: multiset_nat] : ( plus_p6334493942879108393et_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_187_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_188_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_189_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_190_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_191_add_Oassoc,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ C )
= ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_192_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_193_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_194_group__cancel_Oadd2,axiom,
! [B2: multiset_nat,K: multiset_nat,B: multiset_nat,A: multiset_nat] :
( ( B2
= ( plus_p6334493942879108393et_nat @ K @ B ) )
=> ( ( plus_p6334493942879108393et_nat @ A @ B2 )
= ( plus_p6334493942879108393et_nat @ K @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_195_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_196_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_197_group__cancel_Oadd1,axiom,
! [A2: multiset_nat,K: multiset_nat,A: multiset_nat,B: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ K @ A ) )
=> ( ( plus_p6334493942879108393et_nat @ A2 @ B )
= ( plus_p6334493942879108393et_nat @ K @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_198_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_199_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_200_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: multiset_nat,J: multiset_nat,K: multiset_nat,L: multiset_nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_p6334493942879108393et_nat @ I2 @ K )
= ( plus_p6334493942879108393et_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_201_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_202_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_203_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ C )
= ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_204_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_205_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_206_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_207_add_Ocomm__neutral,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% add.comm_neutral
thf(fact_208_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_209_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_210_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_211_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_212_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_213_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_214_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_215_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_216_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_217_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_218_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_219_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_220_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_221_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_222_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_223_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_224_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_225_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_226_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_227_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_228_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_229_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_230_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_231_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_232_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_233_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_234_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_235_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_236_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_237_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_238_sup__left__commute,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_239_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_240_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_241_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_242_sup__assoc,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_243_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_244_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_245_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_246_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_247_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) )
= ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_248_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_249_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N3 )
& ~ ( P2 @ M ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_250_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N3 )
=> ( P2 @ M ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_251_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_252_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_253_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_254_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_255_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_256_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_257_size__neq__size__imp__neq,axiom,
! [X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
( ( ( size_s6253272723116879048ropVar @ X2 )
!= ( size_s6253272723116879048ropVar @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_258_size__neq__size__imp__neq,axiom,
! [X2: char,Y3: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_259_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C3 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_260_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_261_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_262_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_263_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C3 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_264_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P2 @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_265_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P2 @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_266_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_267_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_268_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_269_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_270_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_271_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_272_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_273_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_274_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_275_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_276_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_277_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_278_sup_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_279_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_280_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_281_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( A3
= ( sup_sup_int @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_282_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_283_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_284_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_285_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_286_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_287_sup_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_288_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_289_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_290_sup_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_291_less__supI2,axiom,
! [X2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X2 @ B )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_292_less__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_nat @ X2 @ B )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_293_less__supI2,axiom,
! [X2: int,B: int,A: int] :
( ( ord_less_int @ X2 @ B )
=> ( ord_less_int @ X2 @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI2
thf(fact_294_less__supI1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X2 @ A )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_295_less__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_296_less__supI1,axiom,
! [X2: int,A: int,B: int] :
( ( ord_less_int @ X2 @ A )
=> ( ord_less_int @ X2 @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI1
thf(fact_297_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N3 )
& ~ ( P2 @ M ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_298_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_299_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_300_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_301_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_302_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_303_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_304_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_305_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_306_j__ys2_I3_J,axiom,
! [J: nat] :
( ( member_nat @ J @ ( set_nat2 @ ys2 ) )
=> ( v @ ( multis387687052011358179_Gamma @ ( i_of_j2 @ J ) @ J ) ) ) ).
% j_ys2(3)
thf(fact_307_j__ys2_I1_J,axiom,
! [J: nat] :
( ( member_nat @ J @ ( set_nat2 @ ys2 ) )
=> ( member_nat @ ( i_of_j2 @ J ) @ ( set_nat2 @ xs2 ) ) ) ).
% j_ys2(1)
thf(fact_308_mset__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( mset_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ ( mset_nat @ Ys ) ) ) ).
% mset_append
thf(fact_309_mset__swap,axiom,
! [I2: nat,Ls: list_nat,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ls ) )
=> ( ( mset_nat @ ( list_update_nat @ ( list_update_nat @ Ls @ J @ ( nth_nat @ Ls @ I2 ) ) @ I2 @ ( nth_nat @ Ls @ J ) ) )
= ( mset_nat @ Ls ) ) ) ) ).
% mset_swap
thf(fact_310_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_311_un__ys,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ys1 ) @ ( set_nat2 @ ys2 ) )
= ( set_ord_lessThan_nat @ m ) ) ).
% un_ys
thf(fact_312_set__eq__iff__mset__eq__distinct,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( distinct_nat @ X2 )
=> ( ( distinct_nat @ Y3 )
=> ( ( ( set_nat2 @ X2 )
= ( set_nat2 @ Y3 ) )
= ( ( mset_nat @ X2 )
= ( mset_nat @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_313_add__less__zeroD,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y3 ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_314_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X2: multiset_nat,Y3: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ X2 @ Y3 )
= zero_z7348594199698428585et_nat )
= ( ( X2 = zero_z7348594199698428585et_nat )
& ( Y3 = zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_315_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X2: multiset_nat,Y3: multiset_nat] :
( ( zero_z7348594199698428585et_nat
= ( plus_p6334493942879108393et_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_z7348594199698428585et_nat )
& ( Y3 = zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_316_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_317_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_318_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_319_size__union,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( size_s5917832649809541300et_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
= ( plus_plus_nat @ ( size_s5917832649809541300et_nat @ M3 ) @ ( size_s5917832649809541300et_nat @ N4 ) ) ) ).
% size_union
thf(fact_320_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_321_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_322_union__eq__empty,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ M3 @ N4 )
= zero_z7348594199698428585et_nat )
= ( ( M3 = zero_z7348594199698428585et_nat )
& ( N4 = zero_z7348594199698428585et_nat ) ) ) ).
% union_eq_empty
thf(fact_323_empty__eq__union,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( zero_z7348594199698428585et_nat
= ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
= ( ( M3 = zero_z7348594199698428585et_nat )
& ( N4 = zero_z7348594199698428585et_nat ) ) ) ).
% empty_eq_union
thf(fact_324_size__mset,axiom,
! [Xs: list_nat] :
( ( size_s5917832649809541300et_nat @ ( mset_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% size_mset
thf(fact_325_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_326_nth__upt,axiom,
! [I2: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ K ) ) ) ).
% nth_upt
thf(fact_327_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_328_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_329_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_330_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_331_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_332_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_333_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_334_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_335_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_336_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_337_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_338_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_339_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_340_union__assoc,axiom,
! [M3: multiset_nat,N4: multiset_nat,K3: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) @ K3 )
= ( plus_p6334493942879108393et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ K3 ) ) ) ).
% union_assoc
thf(fact_341_union__lcomm,axiom,
! [M3: multiset_nat,N4: multiset_nat,K3: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ K3 ) )
= ( plus_p6334493942879108393et_nat @ N4 @ ( plus_p6334493942879108393et_nat @ M3 @ K3 ) ) ) ).
% union_lcomm
thf(fact_342_union__commute,axiom,
( plus_p6334493942879108393et_nat
= ( ^ [M4: multiset_nat,N5: multiset_nat] : ( plus_p6334493942879108393et_nat @ N5 @ M4 ) ) ) ).
% union_commute
thf(fact_343_union__left__cancel,axiom,
! [K3: multiset_nat,M3: multiset_nat,N4: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ K3 @ M3 )
= ( plus_p6334493942879108393et_nat @ K3 @ N4 ) )
= ( M3 = N4 ) ) ).
% union_left_cancel
thf(fact_344_union__right__cancel,axiom,
! [M3: multiset_nat,K3: multiset_nat,N4: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ M3 @ K3 )
= ( plus_p6334493942879108393et_nat @ N4 @ K3 ) )
= ( M3 = N4 ) ) ).
% union_right_cancel
thf(fact_345_multi__union__self__other__eq,axiom,
! [A2: multiset_nat,X5: multiset_nat,Y5: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A2 @ X5 )
= ( plus_p6334493942879108393et_nat @ A2 @ Y5 ) )
=> ( X5 = Y5 ) ) ).
% multi_union_self_other_eq
thf(fact_346_union__less__mono,axiom,
! [A2: multiset_nat,C3: multiset_nat,B2: multiset_nat,D2: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ A2 @ C3 )
=> ( ( ord_le5777773500796000884et_nat @ B2 @ D2 )
=> ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ ( plus_p6334493942879108393et_nat @ C3 @ D2 ) ) ) ) ).
% union_less_mono
thf(fact_347_union__le__mono2,axiom,
! [B2: multiset_nat,D2: multiset_nat,C3: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ B2 @ D2 )
=> ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ C3 @ B2 ) @ ( plus_p6334493942879108393et_nat @ C3 @ D2 ) ) ) ).
% union_le_mono2
thf(fact_348_union__le__mono1,axiom,
! [B2: multiset_nat,D2: multiset_nat,C3: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ B2 @ D2 )
=> ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ B2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ D2 @ C3 ) ) ) ).
% union_le_mono1
thf(fact_349_empty__neutral_I1_J,axiom,
! [X2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_350_empty__neutral_I2_J,axiom,
! [X2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ X2 @ zero_z7348594199698428585et_nat )
= X2 ) ).
% empty_neutral(2)
thf(fact_351_ex__mset,axiom,
! [X5: multiset_nat] :
? [Xs3: list_nat] :
( ( mset_nat @ Xs3 )
= X5 ) ).
% ex_mset
thf(fact_352_PropVar_Osize_I7_J,axiom,
! [X11: nat,X12: nat] :
( ( size_s6253272723116879048ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
= zero_zero_nat ) ).
% PropVar.size(7)
thf(fact_353_PropVar_Osize_I8_J,axiom,
! [X22: nat] :
( ( size_s6253272723116879048ropVar @ ( multis2544335231667181926psilon @ X22 ) )
= zero_zero_nat ) ).
% PropVar.size(8)
thf(fact_354_mset__eq__setD,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_355_mset__eq__length,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% mset_eq_length
thf(fact_356_mset__eq__imp__distinct__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
= ( distinct_nat @ Ys ) ) ) ).
% mset_eq_imp_distinct_iff
thf(fact_357_j__ys2_I2_J,axiom,
! [J: nat] :
( ( member_nat @ J @ ( set_nat2 @ ys2 ) )
=> ( cs @ ( i_of_j2 @ J ) @ J ) ) ).
% j_ys2(2)
thf(fact_358_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_359_inj__on__add,axiom,
! [A: multiset_nat,A2: set_multiset_nat] : ( inj_on5670230764983331635et_nat @ ( plus_p6334493942879108393et_nat @ A ) @ A2 ) ).
% inj_on_add
thf(fact_360_inj__on__add,axiom,
! [A: nat,A2: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A ) @ A2 ) ).
% inj_on_add
thf(fact_361_inj__on__add,axiom,
! [A: int,A2: set_int] : ( inj_on_int_int @ ( plus_plus_int @ A ) @ A2 ) ).
% inj_on_add
thf(fact_362_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_363_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_364_verit__sum__simplify,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% verit_sum_simplify
thf(fact_365_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_366_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_367_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_368_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_369_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_370_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_371_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_372_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_373_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P2 @ A5 @ B5 )
= ( P2 @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P2 @ A5 @ B5 )
=> ( P2 @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_374_PropVar_Osize__gen_I2_J,axiom,
! [X22: nat] :
( ( multis2955979900537361535ropVar @ ( multis2544335231667181926psilon @ X22 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(2)
thf(fact_375_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_376_subset__mset_Osum__list__eq__0__iff,axiom,
! [Ns: list_multiset_nat] :
( ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Ns )
= zero_z7348594199698428585et_nat )
= ( ! [X3: multiset_nat] :
( ( member_multiset_nat @ X3 @ ( set_multiset_nat2 @ Ns ) )
=> ( X3 = zero_z7348594199698428585et_nat ) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
thf(fact_377_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_378_PropVar_Osize__gen_I1_J,axiom,
! [X11: nat,X12: nat] :
( ( multis2955979900537361535ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(1)
thf(fact_379_subset__mset_Osum__list__nonneg,axiom,
! [Xs: list_multiset_nat] :
( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
=> ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X ) )
=> ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) ) ) ).
% subset_mset.sum_list_nonneg
thf(fact_380_subset__mset_Osum__list__nonpos,axiom,
! [Xs: list_multiset_nat] :
( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
=> ( subseteq_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
=> ( subseteq_mset_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) @ zero_z7348594199698428585et_nat ) ) ).
% subset_mset.sum_list_nonpos
thf(fact_381_subset__mset_Omember__le__sum__list,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
=> ( subseteq_mset_nat @ X2 @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) ) ) ).
% subset_mset.member_le_sum_list
thf(fact_382_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_multiset_nat] :
( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
=> ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X ) )
=> ( ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs )
= zero_z7348594199698428585et_nat )
= ( ! [X3: multiset_nat] :
( ( member_multiset_nat @ X3 @ ( set_multiset_nat2 @ Xs ) )
=> ( X3 = zero_z7348594199698428585et_nat ) ) ) ) ) ).
% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_383_mset__set__upto__eq__mset__upto,axiom,
! [N: nat] :
( ( mset_set_nat @ ( set_ord_lessThan_nat @ N ) )
= ( mset_nat @ ( upt @ zero_zero_nat @ N ) ) ) ).
% mset_set_upto_eq_mset_upto
thf(fact_384_remove__code_I1_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( remove_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_385_member__remove,axiom,
! [X2: nat,Y3: nat,A2: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y3 @ A2 ) )
= ( ( member_nat @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_386_mset__subset__eq__mono__add__right__cancel,axiom,
! [A2: multiset_nat,C3: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B2 @ C3 ) )
= ( subseteq_mset_nat @ A2 @ B2 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_387_mset__subset__eq__mono__add__left__cancel,axiom,
! [C3: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C3 @ A2 ) @ ( plus_p6334493942879108393et_nat @ C3 @ B2 ) )
= ( subseteq_mset_nat @ A2 @ B2 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_388_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_nat,C: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B @ C ) )
= ( subseteq_mset_nat @ A @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_389_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_nat,A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ ( plus_p6334493942879108393et_nat @ C @ B ) )
= ( subseteq_mset_nat @ A @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_390_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_nat,A: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ B @ A ) @ B )
= ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_391_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ B )
= ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_392_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ A @ B ) )
= ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_393_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ A ) )
= ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_394_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_nat
= ( ^ [A4: multiset_nat,B4: multiset_nat] :
? [C5: multiset_nat] :
( B4
= ( plus_p6334493942879108393et_nat @ A4 @ C5 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_395_mset__subset__eq__add__right,axiom,
! [B2: multiset_nat,A2: multiset_nat] : ( subseteq_mset_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ).
% mset_subset_eq_add_right
thf(fact_396_mset__subset__eq__mono__add,axiom,
! [A2: multiset_nat,B2: multiset_nat,C3: multiset_nat,D2: multiset_nat] :
( ( subseteq_mset_nat @ A2 @ B2 )
=> ( ( subseteq_mset_nat @ C3 @ D2 )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B2 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_397_mset__subset__eq__add__left,axiom,
! [A2: multiset_nat,B2: multiset_nat] : ( subseteq_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ).
% mset_subset_eq_add_left
thf(fact_398_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_nat,C: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B @ C ) )
=> ( subseteq_mset_nat @ A @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_399_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_nat,A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ ( plus_p6334493942879108393et_nat @ C @ B ) )
=> ( subseteq_mset_nat @ A @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_400_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_401_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ ( plus_p6334493942879108393et_nat @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_402_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_nat
= ( ^ [A3: multiset_nat,B3: multiset_nat] :
? [C4: multiset_nat] :
( B3
= ( plus_p6334493942879108393et_nat @ A3 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_403_subset__mset_Oless__eqE,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ~ ! [C2: multiset_nat] :
( B
!= ( plus_p6334493942879108393et_nat @ A @ C2 ) ) ) ).
% subset_mset.less_eqE
thf(fact_404_subset__mset_Oadd__mono,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat,D: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( subseteq_mset_nat @ C @ D )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B @ D ) ) ) ) ).
% subset_mset.add_mono
thf(fact_405_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_nat,C: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat )
=> ( ( subseteq_mset_nat @ C @ B )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_406_subset__mset_Oadd__increasing,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A )
=> ( ( subseteq_mset_nat @ B @ C )
=> ( subseteq_mset_nat @ B @ ( plus_p6334493942879108393et_nat @ A @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_407_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_nat,A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ C @ zero_z7348594199698428585et_nat )
=> ( ( subseteq_mset_nat @ A @ B )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_408_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_nat,B: multiset_nat,A: multiset_nat] :
( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ C )
=> ( ( subseteq_mset_nat @ B @ A )
=> ( subseteq_mset_nat @ B @ ( plus_p6334493942879108393et_nat @ A @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_409_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A )
=> ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B )
=> ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_410_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat )
=> ( ( subseteq_mset_nat @ B @ zero_z7348594199698428585et_nat )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_411_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X2: multiset_nat,Y3: multiset_nat] :
( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X2 )
=> ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ Y3 )
=> ( ( ( plus_p6334493942879108393et_nat @ X2 @ Y3 )
= zero_z7348594199698428585et_nat )
= ( ( X2 = zero_z7348594199698428585et_nat )
& ( Y3 = zero_z7348594199698428585et_nat ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_412_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X2: multiset_nat,Y3: multiset_nat] :
( ( subseteq_mset_nat @ X2 @ zero_z7348594199698428585et_nat )
=> ( ( subseteq_mset_nat @ Y3 @ zero_z7348594199698428585et_nat )
=> ( ( ( plus_p6334493942879108393et_nat @ X2 @ Y3 )
= zero_z7348594199698428585et_nat )
= ( ( X2 = zero_z7348594199698428585et_nat )
& ( Y3 = zero_z7348594199698428585et_nat ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_413_mset__set__set,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( mset_set_nat @ ( set_nat2 @ Xs ) )
= ( mset_nat @ Xs ) ) ) ).
% mset_set_set
thf(fact_414_subset__mset_Oelem__le__sum__list,axiom,
! [K: nat,Ns: list_multiset_nat] :
( ( ord_less_nat @ K @ ( size_s6386657463320973636et_nat @ Ns ) )
=> ( subseteq_mset_nat @ ( nth_multiset_nat @ Ns @ K ) @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Ns ) ) ) ).
% subset_mset.elem_le_sum_list
thf(fact_415_mset__upt,axiom,
! [M2: nat,N: nat] :
( ( mset_nat @ ( upt @ M2 @ N ) )
= ( mset_set_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% mset_upt
thf(fact_416_bij__betw__nth,axiom,
! [Xs: list_nat,A2: set_nat,B2: set_nat] :
( ( distinct_nat @ Xs )
=> ( ( A2
= ( set_ord_lessThan_nat @ ( size_size_list_nat @ Xs ) ) )
=> ( ( B2
= ( set_nat2 @ Xs ) )
=> ( bij_betw_nat_nat @ ( nth_nat @ Xs ) @ A2 @ B2 ) ) ) ) ).
% bij_betw_nth
thf(fact_417_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_418_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_419_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_420_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_421_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_422_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_423_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_424_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_425_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P2 @ M5 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_426_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ( P2 @ M5 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P2 @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_427_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_428_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I: nat,J2: nat] : ( set_nat2 @ ( upt @ I @ J2 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_429_subset__mset_Omin__add__distrib__right,axiom,
! [X2: multiset_nat,Y3: multiset_nat,Z2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ X2 @ ( min_multiset_nat @ subseteq_mset_nat @ Y3 @ Z2 ) )
= ( min_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X2 @ Y3 ) @ ( plus_p6334493942879108393et_nat @ X2 @ Z2 ) ) ) ).
% subset_mset.min_add_distrib_right
thf(fact_430_subset__mset_Omax__add__distrib__right,axiom,
! [X2: multiset_nat,Y3: multiset_nat,Z2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ X2 @ ( max_multiset_nat @ subseteq_mset_nat @ Y3 @ Z2 ) )
= ( max_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X2 @ Y3 ) @ ( plus_p6334493942879108393et_nat @ X2 @ Z2 ) ) ) ).
% subset_mset.max_add_distrib_right
thf(fact_431_subset__mset_Omin__add__distrib__left,axiom,
! [X2: multiset_nat,Y3: multiset_nat,Z2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( min_multiset_nat @ subseteq_mset_nat @ X2 @ Y3 ) @ Z2 )
= ( min_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X2 @ Z2 ) @ ( plus_p6334493942879108393et_nat @ Y3 @ Z2 ) ) ) ).
% subset_mset.min_add_distrib_left
thf(fact_432_subset__mset_Omax__add__distrib__left,axiom,
! [X2: multiset_nat,Y3: multiset_nat,Z2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( max_multiset_nat @ subseteq_mset_nat @ X2 @ Y3 ) @ Z2 )
= ( max_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X2 @ Z2 ) @ ( plus_p6334493942879108393et_nat @ Y3 @ Z2 ) ) ) ).
% subset_mset.max_add_distrib_left
thf(fact_433_subset__mset_Osum__list__update,axiom,
! [K: nat,Xs: list_multiset_nat,X2: multiset_nat] :
( ( ord_less_nat @ K @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ ( list_u3438943574295160626et_nat @ Xs @ K @ X2 ) )
= ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) @ X2 ) @ ( nth_multiset_nat @ Xs @ K ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_434_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_435_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_436_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_437_size__multiset__union,axiom,
! [F: nat > nat,M3: multiset_nat,N4: multiset_nat] :
( ( size_multiset_nat @ F @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
= ( plus_plus_nat @ ( size_multiset_nat @ F @ M3 ) @ ( size_multiset_nat @ F @ N4 ) ) ) ).
% size_multiset_union
thf(fact_438_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_439_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_440_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_441_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_442_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_443_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_444_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_445_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_446_add__diff__cancel__right_H,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_447_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_448_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_449_add__diff__cancel__right,axiom,
! [A: multiset_nat,C: multiset_nat,B: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B @ C ) )
= ( minus_8522176038001411705et_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_450_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_451_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_452_add__diff__cancel__left_H,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_453_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_454_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_455_add__diff__cancel__left,axiom,
! [C: multiset_nat,A: multiset_nat,B: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ ( plus_p6334493942879108393et_nat @ C @ B ) )
= ( minus_8522176038001411705et_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_456_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_457_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_458_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_459_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_460_diff__diff__add__mset,axiom,
! [M3: multiset_nat,N4: multiset_nat,P2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M3 @ N4 ) @ P2 )
= ( minus_8522176038001411705et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ P2 ) ) ) ).
% diff_diff_add_mset
thf(fact_461_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_462_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_463_diff__add__zero,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A @ ( plus_p6334493942879108393et_nat @ A @ B ) )
= zero_z7348594199698428585et_nat ) ).
% diff_add_zero
thf(fact_464_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_465_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_466_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_467_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_468_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_469_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_470_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_471_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_472_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_473_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_474_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_475_subset__mset_Oadd__diff__assoc,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( plus_p6334493942879108393et_nat @ C @ ( minus_8522176038001411705et_nat @ B @ A ) )
= ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ B ) @ A ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_476_subset__mset_Oadd__diff__assoc2,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B @ A ) @ C )
= ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B @ C ) @ A ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_477_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B2: multiset_nat,A2: multiset_nat,C3: multiset_nat] :
( ( subseteq_mset_nat @ B2 @ A2 )
=> ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ A2 @ B2 ) @ C3 )
= ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ B2 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_478_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_479_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_480_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_481_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_482_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_483_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_484_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_485_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_486_diff__diff__eq,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A @ B ) @ C )
= ( minus_8522176038001411705et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_487_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_488_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_489_add__implies__diff,axiom,
! [C: multiset_nat,B: multiset_nat,A: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ C @ B )
= A )
=> ( C
= ( minus_8522176038001411705et_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_490_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_491_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_492_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_493_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_494_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_495_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_496_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_497_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_498_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_499_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_500_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_501_zadd__int__left,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_502_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_503_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_504_inj__on__diff__left,axiom,
! [A: int,A2: set_int] : ( inj_on_int_int @ ( minus_minus_int @ A ) @ A2 ) ).
% inj_on_diff_left
thf(fact_505_diff__union__cancelR,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) @ N4 )
= M3 ) ).
% diff_union_cancelR
thf(fact_506_diff__union__cancelL,axiom,
! [N4: multiset_nat,M3: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ N4 @ M3 ) @ N4 )
= M3 ) ).
% diff_union_cancelL
thf(fact_507_Multiset_Odiff__add,axiom,
! [M3: multiset_nat,N4: multiset_nat,Q: multiset_nat] :
( ( minus_8522176038001411705et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ Q ) )
= ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M3 @ N4 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_508_inj__on__of__nat,axiom,
! [N4: set_nat] : ( inj_on_nat_int @ semiri1314217659103216013at_int @ N4 ) ).
% inj_on_of_nat
thf(fact_509_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_510_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_511_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_512_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_513_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_514_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_515_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_516_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_517_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_518_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_519_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_520_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_521_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_522_subset__mset_Odiff__add,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B @ A ) @ A )
= B ) ) ).
% subset_mset.diff_add
thf(fact_523_subset__mset_Ole__add__diff,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( subseteq_mset_nat @ C @ ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B @ C ) @ A ) ) ) ).
% subset_mset.le_add_diff
thf(fact_524_subset__mset_Ole__diff__conv2,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( subseteq_mset_nat @ C @ ( minus_8522176038001411705et_nat @ B @ A ) )
= ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_525_subset__mset_Odiff__add__assoc,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ B ) @ A )
= ( plus_p6334493942879108393et_nat @ C @ ( minus_8522176038001411705et_nat @ B @ A ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_526_subset__mset_Odiff__add__assoc2,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B @ C ) @ A )
= ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B @ A ) @ C ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_527_subset__mset_Odiff__diff__right,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( minus_8522176038001411705et_nat @ C @ ( minus_8522176038001411705et_nat @ B @ A ) )
= ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_528_subset__mset_Oadd__diff__inverse,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( plus_p6334493942879108393et_nat @ A @ ( minus_8522176038001411705et_nat @ B @ A ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_529_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: multiset_nat,B: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ A @ B )
=> ( ( subseteq_mset_nat @ A @ B )
=> ( ( ( minus_8522176038001411705et_nat @ B @ A )
= C )
= ( B
= ( plus_p6334493942879108393et_nat @ C @ A ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_530_subset__eq__diff__conv,axiom,
! [A2: multiset_nat,C3: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ ( minus_8522176038001411705et_nat @ A2 @ C3 ) @ B2 )
= ( subseteq_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C3 ) ) ) ).
% subset_eq_diff_conv
thf(fact_531_multiset__diff__union__assoc,axiom,
! [C3: multiset_nat,B2: multiset_nat,A2: multiset_nat] :
( ( subseteq_mset_nat @ C3 @ B2 )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ C3 )
= ( plus_p6334493942879108393et_nat @ A2 @ ( minus_8522176038001411705et_nat @ B2 @ C3 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_532_mset__update,axiom,
! [I2: nat,Ls: list_nat,V: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ls ) )
=> ( ( mset_nat @ ( list_update_nat @ Ls @ I2 @ V ) )
= ( add_mset_nat @ V @ ( minus_8522176038001411705et_nat @ ( mset_nat @ Ls ) @ ( add_mset_nat @ ( nth_nat @ Ls @ I2 ) @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% mset_update
thf(fact_533_nth__enumerate__eq,axiom,
! [M2: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_534_sum__list__update,axiom,
! [K: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( groups4561878855575611511st_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ X2 ) @ ( nth_nat @ Xs @ K ) ) ) ) ).
% sum_list_update
thf(fact_535_subset__mset_Osum__nonpos,axiom,
! [A2: set_nat,F: nat > multiset_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( subseteq_mset_nat @ ( F @ X ) @ zero_z7348594199698428585et_nat ) )
=> ( subseteq_mset_nat @ ( groups5394990218802192790at_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A2 ) @ zero_z7348594199698428585et_nat ) ) ).
% subset_mset.sum_nonpos
thf(fact_536_subset__mset_Osum__nonneg,axiom,
! [A2: set_nat,F: nat > multiset_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( F @ X ) ) )
=> ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups5394990218802192790at_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A2 ) ) ) ).
% subset_mset.sum_nonneg
thf(fact_537_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_538_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_539_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_540_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_541_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_542_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_543_union__mset__add__mset__left,axiom,
! [A: nat,A2: multiset_nat,B2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ A @ A2 ) @ B2 )
= ( add_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ) ).
% union_mset_add_mset_left
thf(fact_544_union__mset__add__mset__right,axiom,
! [A2: multiset_nat,A: nat,B2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A2 @ ( add_mset_nat @ A @ B2 ) )
= ( add_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ) ).
% union_mset_add_mset_right
thf(fact_545_lessThan__minus__lessThan,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M2 ) )
= ( set_or4665077453230672383an_nat @ M2 @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_546_length__upt,axiom,
! [I2: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I2 @ J ) )
= ( minus_minus_nat @ J @ I2 ) ) ).
% length_upt
thf(fact_547_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ns ) )
=> ( X3 = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_548_sum__list__append,axiom,
! [Xs: list_multiset_nat,Ys: list_multiset_nat] :
( ( groups8053510108761903431et_nat @ ( append_multiset_nat @ Xs @ Ys ) )
= ( plus_p6334493942879108393et_nat @ ( groups8053510108761903431et_nat @ Xs ) @ ( groups8053510108761903431et_nat @ Ys ) ) ) ).
% sum_list_append
thf(fact_549_sum__list__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( groups4561878855575611511st_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% sum_list_append
thf(fact_550_sum__list__append,axiom,
! [Xs: list_int,Ys: list_int] :
( ( groups4559388385066561235st_int @ ( append_int @ Xs @ Ys ) )
= ( plus_plus_int @ ( groups4559388385066561235st_int @ Xs ) @ ( groups4559388385066561235st_int @ Ys ) ) ) ).
% sum_list_append
thf(fact_551_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_552_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_553_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M6: nat,N3: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_554_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_555_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_556_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_557_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_558_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_559_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_560_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_561_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_562_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C3 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C3 ) @ ( minus_minus_set_nat @ B2 @ C3 ) ) ) ).
% Un_Diff
thf(fact_563_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_564_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_565_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_566_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_567_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_568_add__mset__add__single,axiom,
( add_mset_nat
= ( ^ [A3: nat,A4: multiset_nat] : ( plus_p6334493942879108393et_nat @ A4 @ ( add_mset_nat @ A3 @ zero_z7348594199698428585et_nat ) ) ) ) ).
% add_mset_add_single
thf(fact_569_union__is__single,axiom,
! [M3: multiset_nat,N4: multiset_nat,A: nat] :
( ( ( plus_p6334493942879108393et_nat @ M3 @ N4 )
= ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
= ( ( ( M3
= ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
& ( N4 = zero_z7348594199698428585et_nat ) )
| ( ( M3 = zero_z7348594199698428585et_nat )
& ( N4
= ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% union_is_single
thf(fact_570_single__is__union,axiom,
! [A: nat,M3: multiset_nat,N4: multiset_nat] :
( ( ( add_mset_nat @ A @ zero_z7348594199698428585et_nat )
= ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
= ( ( ( ( add_mset_nat @ A @ zero_z7348594199698428585et_nat )
= M3 )
& ( N4 = zero_z7348594199698428585et_nat ) )
| ( ( M3 = zero_z7348594199698428585et_nat )
& ( ( add_mset_nat @ A @ zero_z7348594199698428585et_nat )
= N4 ) ) ) ) ).
% single_is_union
thf(fact_571_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_572_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_573_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_574_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_575_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_576_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_577_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_578_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_579_mset__remove1,axiom,
! [A: nat,Xs: list_nat] :
( ( mset_nat @ ( remove1_nat @ A @ Xs ) )
= ( minus_8522176038001411705et_nat @ ( mset_nat @ Xs ) @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ).
% mset_remove1
thf(fact_580_nth__zip,axiom,
! [I2: nat,Xs: list_int,Ys: list_int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys ) @ I2 )
= ( product_Pair_int_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Ys @ I2 ) ) ) ) ) ).
% nth_zip
thf(fact_581_nth__zip,axiom,
! [I2: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I2 )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).
% nth_zip
thf(fact_582_listrel1__iff__update,axiom,
! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
= ( ? [Y4: int,N2: nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ Y4 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
& ( Ys
= ( list_update_int @ Xs @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_583_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
= ( ? [Y4: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y4 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_584_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_585_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_586_in__set__remove1,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_587_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_588_append__listrel1I,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).
% append_listrel1I
thf(fact_589_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_590_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_591_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_592_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_593_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_594_notin__set__remove1,axiom,
! [X2: nat,Xs: list_nat,Y3: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat @ X2 @ ( set_nat2 @ ( remove1_nat @ Y3 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_595_remove1__idem,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_596_distinct__remove1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( remove1_nat @ X2 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_597_zip__same,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
= ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_598_zip__same,axiom,
! [A: int,B: int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
= ( ( member_int @ A @ ( set_int2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_599_in__set__zipE,axiom,
! [X2: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
=> ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_600_in__set__zipE,axiom,
! [X2: int,Y3: int,Xs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
=> ~ ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ~ ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_601_set__zip__leftD,axiom,
! [X2: int,Y3: int,Xs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
=> ( member_int @ X2 @ ( set_int2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_602_set__zip__rightD,axiom,
! [X2: int,Y3: int,Xs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
=> ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_603_ex__mset__zip__left,axiom,
! [Xs: list_nat,Ys: list_nat,Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( mset_nat @ Xs4 )
= ( mset_nat @ Xs ) )
=> ? [Ys4: list_nat] :
( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Xs4 ) )
& ( ( mset_P6383711406899277590at_nat @ ( zip_nat_nat @ Xs4 @ Ys4 ) )
= ( mset_P6383711406899277590at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ) ).
% ex_mset_zip_left
thf(fact_604_zip__update,axiom,
! [Xs: list_int,I2: nat,X2: int,Ys: list_int,Y3: int] :
( ( zip_int_int @ ( list_update_int @ Xs @ I2 @ X2 ) @ ( list_update_int @ Ys @ I2 @ Y3 ) )
= ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I2 @ ( product_Pair_int_int @ X2 @ Y3 ) ) ) ).
% zip_update
thf(fact_605_remove1__append,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remove1_nat @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( remove1_nat @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_606_in__set__impl__in__set__zip1,axiom,
! [Xs: list_int,Ys: list_int,X2: int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ~ ! [Y6: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y6 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_607_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 ) )
=> ~ ! [Y6: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y6 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_608_in__set__impl__in__set__zip2,axiom,
! [Xs: list_int,Ys: list_int,Y3: int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( member_int @ Y3 @ ( set_int2 @ Ys ) )
=> ~ ! [X: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_609_in__set__impl__in__set__zip2,axiom,
! [Xs: list_nat,Ys: list_nat,Y3: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
=> ~ ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_610_distinct__remove1__removeAll,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( remove1_nat @ X2 @ Xs )
= ( removeAll_nat @ X2 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_611_enumerate__eq__zip,axiom,
( enumerate_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( zip_nat_nat @ ( upt @ N2 @ ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) ) @ Xs2 ) ) ) ).
% enumerate_eq_zip
thf(fact_612_listrel__iff__nth,axiom,
! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ ( nth_int @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_613_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_614_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M2: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_615_list__order__extension_Oall__ns__imp__ns,axiom,
! [S_list: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int,Ns_list: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int,As: list_int,Bs: list_int,NS: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( list_l790271996378993376on_int @ S_list @ Ns_list )
=> ( ( ( size_size_list_int @ As )
= ( size_size_list_int @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Bs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ As @ I3 ) @ ( nth_int @ Bs @ I3 ) ) @ NS ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).
% list_order_extension.all_ns_imp_ns
thf(fact_616_list__order__extension_Oall__ns__imp__ns,axiom,
! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
=> ( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ NS ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).
% list_order_extension.all_ns_imp_ns
thf(fact_617_ns__mul__ext__intro,axiom,
! [Xs: multiset_int,Xs1: list_int,Xs22: list_int,Ys: multiset_int,Ys1: list_int,Ys22: list_int,NS: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( Xs
= ( plus_p2156642923369911685et_int @ ( mset_int @ Xs1 ) @ ( mset_int @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p2156642923369911685et_int @ ( mset_int @ Ys1 ) @ ( mset_int @ Ys22 ) ) )
=> ( ( ( size_size_list_int @ Xs1 )
= ( size_size_list_int @ Ys1 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys1 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs1 @ I3 ) @ ( nth_int @ Ys1 @ I3 ) ) @ NS ) )
=> ( ! [Y6: int] :
( ( member_int @ Y6 @ ( set_int2 @ Ys22 ) )
=> ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs22 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Y6 ) @ S2 ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ Xs @ Ys ) @ ( multis8828838126066458039xt_int @ NS @ S2 ) ) ) ) ) ) ) ).
% ns_mul_ext_intro
thf(fact_618_ns__mul__ext__intro,axiom,
! [Xs: multiset_nat,Xs1: list_nat,Xs22: list_nat,Ys: multiset_nat,Ys1: list_nat,Ys22: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( Xs
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs1 ) @ ( mset_nat @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys22 ) ) )
=> ( ( ( size_size_list_nat @ Xs1 )
= ( size_size_list_nat @ Ys1 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys1 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I3 ) @ ( nth_nat @ Ys1 @ I3 ) ) @ NS ) )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ ( set_nat2 @ Ys22 ) )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs22 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y6 ) @ S2 ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) ) ) ) ).
% ns_mul_ext_intro
thf(fact_619_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_620_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_621_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_622_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_623_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_624_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_625_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ? [Y4: nat] :
( X3
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_626_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_627_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_628_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_629_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_630_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_631_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_632_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_633_mset__map__split,axiom,
! [F: nat > nat,Xs: list_nat,Ys1: list_nat,Ys22: list_nat] :
( ( ( mset_nat @ ( map_nat_nat @ F @ Xs ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys22 ) ) )
=> ? [Zs1: list_nat,Zs2: list_nat] :
( ( ( mset_nat @ Xs )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
& ( Ys1
= ( map_nat_nat @ F @ Zs1 ) )
& ( Ys22
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% mset_map_split
thf(fact_634_map__update,axiom,
! [F: nat > nat,Xs: list_nat,K: nat,Y3: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y3 ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).
% map_update
thf(fact_635_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_636_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_637_map__inj__on,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_638_inj__on__map__eq__map,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_639_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_640_distinct__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_641_sum__list__map__remove1,axiom,
! [X2: nat,Xs: list_nat,F: nat > multiset_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( groups8053510108761903431et_nat @ ( map_nat_multiset_nat @ F @ Xs ) )
= ( plus_p6334493942879108393et_nat @ ( F @ X2 ) @ ( groups8053510108761903431et_nat @ ( map_nat_multiset_nat @ F @ ( remove1_nat @ X2 @ Xs ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_642_sum__list__map__remove1,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) )
= ( plus_plus_nat @ ( F @ X2 ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ ( remove1_nat @ X2 @ Xs ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_643_sum__list__map__remove1,axiom,
! [X2: nat,Xs: list_nat,F: nat > int] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( groups4559388385066561235st_int @ ( map_nat_int @ F @ Xs ) )
= ( plus_plus_int @ ( F @ X2 ) @ ( groups4559388385066561235st_int @ ( map_nat_int @ F @ ( remove1_nat @ X2 @ Xs ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_644_nth__map__upt,axiom,
! [I2: nat,N: nat,M2: nat,F: nat > nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N @ M2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M2 @ N ) ) @ I2 )
= ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).
% nth_map_upt
thf(fact_645_ns__mul__ext__elim,axiom,
! [Xs: multiset_int,Ys: multiset_int,NS: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ Xs @ Ys ) @ ( multis8828838126066458039xt_int @ NS @ S2 ) )
=> ? [Xs12: list_int,Xs23: list_int] :
( ( Xs
= ( plus_p2156642923369911685et_int @ ( mset_int @ Xs12 ) @ ( mset_int @ Xs23 ) ) )
& ? [Ys12: list_int,Ys23: list_int] :
( ( Ys
= ( plus_p2156642923369911685et_int @ ( mset_int @ Ys12 ) @ ( mset_int @ Ys23 ) ) )
& ( ( size_size_list_int @ Xs12 )
= ( size_size_list_int @ Ys12 ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_int @ Ys12 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs12 @ I5 ) @ ( nth_int @ Ys12 @ I5 ) ) @ NS ) )
& ! [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Ys23 ) )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ ( set_int2 @ Xs23 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).
% ns_mul_ext_elim
thf(fact_646_ns__mul__ext__elim,axiom,
! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) )
=> ? [Xs12: list_nat,Xs23: list_nat] :
( ( Xs
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs23 ) ) )
& ? [Ys12: list_nat,Ys23: list_nat] :
( ( Ys
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys23 ) ) )
& ( ( size_size_list_nat @ Xs12 )
= ( size_size_list_nat @ Ys12 ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Ys12 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs12 @ I5 ) @ ( nth_nat @ Ys12 @ I5 ) ) @ NS ) )
& ! [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Ys23 ) )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Xs23 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).
% ns_mul_ext_elim
thf(fact_647_ns__mul__ext__singleton,axiom,
! [A: int,B: int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ Ns )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( add_mset_int @ A @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ B @ zero_z3170743180189231877et_int ) ) @ ( multis8828838126066458039xt_int @ Ns @ S ) ) ) ).
% ns_mul_ext_singleton
thf(fact_648_all__ns__ns__mul__ext,axiom,
! [As: list_int,Bs: list_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( ( size_size_list_int @ As )
= ( size_size_list_int @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Bs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ As @ I3 ) @ ( nth_int @ Bs @ I3 ) ) @ Ns ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( mset_int @ As ) @ ( mset_int @ Bs ) ) @ ( multis8828838126066458039xt_int @ Ns @ S ) ) ) ) ).
% all_ns_ns_mul_ext
thf(fact_649_all__ns__ns__mul__ext,axiom,
! [As: list_nat,Bs: list_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ Ns ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As ) @ ( mset_nat @ Bs ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ).
% all_ns_ns_mul_ext
thf(fact_650_ns__mul__ext__singleton2,axiom,
! [A: int,B: int,S: set_Pr958786334691620121nt_int,Ns: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( add_mset_int @ A @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ B @ zero_z3170743180189231877et_int ) ) @ ( multis8828838126066458039xt_int @ Ns @ S ) ) ) ).
% ns_mul_ext_singleton2
thf(fact_651_ns__ns__mul__ext__union__compat,axiom,
! [A2: multiset_nat,B2: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,C3: multiset_nat,D2: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ C3 @ D2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B2 @ D2 ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ).
% ns_ns_mul_ext_union_compat
thf(fact_652_ns__mul__extI__old,axiom,
! [A2: multiset_int,Xs: list_int,A22: multiset_int,B2: multiset_int,Ys: list_int,B22: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( A2
= ( plus_p2156642923369911685et_int @ ( mset_int @ Xs ) @ A22 ) )
=> ( ( B2
= ( plus_p2156642923369911685et_int @ ( mset_int @ Ys ) @ B22 ) )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Ys @ I3 ) ) @ Ns ) )
=> ( ! [B5: int] :
( ( member_int @ B5 @ ( set_mset_int @ B22 ) )
=> ? [A6: int] :
( ( member_int @ A6 @ ( set_mset_int @ A22 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B5 ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis8828838126066458039xt_int @ Ns @ S ) ) ) ) ) ) ) ).
% ns_mul_extI_old
thf(fact_653_ns__mul__extI__old,axiom,
! [A2: multiset_nat,Xs: list_nat,A22: multiset_nat,B2: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
=> ( ( B2
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ Ns ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ ( set_mset_nat @ B22 ) )
=> ? [A6: nat] :
( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ) ) ) ).
% ns_mul_extI_old
thf(fact_654_list__order__extension__def,axiom,
( list_l790271996378993376on_int
= ( ^ [S_list2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int,Ns_list2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int] :
( ! [S3: set_Pr958786334691620121nt_int,NS2: set_Pr958786334691620121nt_int] :
( ( order_7800132744162871582ir_int @ S3 @ NS2 )
=> ( order_7190096257394922798st_int @ ( S_list2 @ S3 @ NS2 ) @ ( Ns_list2 @ S3 @ NS2 ) ) )
& ! [S3: set_Pr958786334691620121nt_int,F2: int > int,NS2: set_Pr958786334691620121nt_int,As2: list_int,Bs2: list_int] :
( ! [A3: int,B3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ S3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ S3 ) )
=> ( ! [A3: int,B3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ NS2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ NS2 ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ As2 @ Bs2 ) @ ( S_list2 @ S3 @ NS2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( map_int_int @ F2 @ As2 ) @ ( map_int_int @ F2 @ Bs2 ) ) @ ( S_list2 @ S3 @ NS2 ) ) ) ) )
& ! [S3: set_Pr958786334691620121nt_int,F2: int > int,NS2: set_Pr958786334691620121nt_int,As2: list_int,Bs2: list_int] :
( ! [A3: int,B3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ S3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ S3 ) )
=> ( ! [A3: int,B3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ NS2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ NS2 ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ As2 @ Bs2 ) @ ( Ns_list2 @ S3 @ NS2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( map_int_int @ F2 @ As2 ) @ ( map_int_int @ F2 @ Bs2 ) ) @ ( Ns_list2 @ S3 @ NS2 ) ) ) ) )
& ! [As2: list_int,Bs2: list_int,NS2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ( ( size_size_list_int @ As2 )
= ( size_size_list_int @ Bs2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Bs2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ As2 @ I ) @ ( nth_int @ Bs2 @ I ) ) @ NS2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ As2 @ Bs2 ) @ ( Ns_list2 @ S3 @ NS2 ) ) ) ) ) ) ) ).
% list_order_extension_def
thf(fact_655_list__order__extension__def,axiom,
( list_l792762466888043652on_nat
= ( ^ [S_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
( ! [S3: set_Pr1261947904930325089at_nat,NS2: set_Pr1261947904930325089at_nat] :
( ( order_7802623214671921858ir_nat @ S3 @ NS2 )
=> ( order_2144575240049343698st_nat @ ( S_list2 @ S3 @ NS2 ) @ ( Ns_list2 @ S3 @ NS2 ) ) )
& ! [S3: set_Pr1261947904930325089at_nat,F2: nat > nat,NS2: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
( ! [A3: nat,B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ S3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ NS2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ NS2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( S_list2 @ S3 @ NS2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F2 @ As2 ) @ ( map_nat_nat @ F2 @ Bs2 ) ) @ ( S_list2 @ S3 @ NS2 ) ) ) ) )
& ! [S3: set_Pr1261947904930325089at_nat,F2: nat > nat,NS2: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
( ! [A3: nat,B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ S3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ NS2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ NS2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S3 @ NS2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F2 @ As2 ) @ ( map_nat_nat @ F2 @ Bs2 ) ) @ ( Ns_list2 @ S3 @ NS2 ) ) ) ) )
& ! [As2: list_nat,Bs2: list_nat,NS2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ As2 )
= ( size_size_list_nat @ Bs2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Bs2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As2 @ I ) @ ( nth_nat @ Bs2 @ I ) ) @ NS2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S3 @ NS2 ) ) ) ) ) ) ) ).
% list_order_extension_def
thf(fact_656_list__order__extension_Ointro,axiom,
! [S_list: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int,Ns_list: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int] :
( ! [S4: set_Pr958786334691620121nt_int,NS3: set_Pr958786334691620121nt_int] :
( ( order_7800132744162871582ir_int @ S4 @ NS3 )
=> ( order_7190096257394922798st_int @ ( S_list @ S4 @ NS3 ) @ ( Ns_list @ S4 @ NS3 ) ) )
=> ( ! [S4: set_Pr958786334691620121nt_int,F3: int > int,NS3: set_Pr958786334691620121nt_int,As3: list_int,Bs3: list_int] :
( ! [A6: int,B6: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ S4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ S4 ) )
=> ( ! [A6: int,B6: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ NS3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ NS3 ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ As3 @ Bs3 ) @ ( S_list @ S4 @ NS3 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( map_int_int @ F3 @ As3 ) @ ( map_int_int @ F3 @ Bs3 ) ) @ ( S_list @ S4 @ NS3 ) ) ) ) )
=> ( ! [S4: set_Pr958786334691620121nt_int,F3: int > int,NS3: set_Pr958786334691620121nt_int,As3: list_int,Bs3: list_int] :
( ! [A6: int,B6: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ S4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ S4 ) )
=> ( ! [A6: int,B6: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ NS3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ NS3 ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS3 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( map_int_int @ F3 @ As3 ) @ ( map_int_int @ F3 @ Bs3 ) ) @ ( Ns_list @ S4 @ NS3 ) ) ) ) )
=> ( ! [As3: list_int,Bs3: list_int,NS3: set_Pr958786334691620121nt_int,S4: set_Pr958786334691620121nt_int] :
( ( ( size_size_list_int @ As3 )
= ( size_size_list_int @ Bs3 ) )
=> ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_int @ Bs3 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ As3 @ I5 ) @ ( nth_int @ Bs3 @ I5 ) ) @ NS3 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS3 ) ) ) )
=> ( list_l790271996378993376on_int @ S_list @ Ns_list ) ) ) ) ) ).
% list_order_extension.intro
thf(fact_657_list__order__extension_Ointro,axiom,
! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
( ! [S4: set_Pr1261947904930325089at_nat,NS3: set_Pr1261947904930325089at_nat] :
( ( order_7802623214671921858ir_nat @ S4 @ NS3 )
=> ( order_2144575240049343698st_nat @ ( S_list @ S4 @ NS3 ) @ ( Ns_list @ S4 @ NS3 ) ) )
=> ( ! [S4: set_Pr1261947904930325089at_nat,F3: nat > nat,NS3: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
( ! [A6: nat,B6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ S4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ S4 ) )
=> ( ! [A6: nat,B6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ NS3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ NS3 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( S_list @ S4 @ NS3 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F3 @ As3 ) @ ( map_nat_nat @ F3 @ Bs3 ) ) @ ( S_list @ S4 @ NS3 ) ) ) ) )
=> ( ! [S4: set_Pr1261947904930325089at_nat,F3: nat > nat,NS3: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
( ! [A6: nat,B6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ S4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ S4 ) )
=> ( ! [A6: nat,B6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ NS3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ NS3 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS3 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F3 @ As3 ) @ ( map_nat_nat @ F3 @ Bs3 ) ) @ ( Ns_list @ S4 @ NS3 ) ) ) ) )
=> ( ! [As3: list_nat,Bs3: list_nat,NS3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ As3 )
= ( size_size_list_nat @ Bs3 ) )
=> ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Bs3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As3 @ I5 ) @ ( nth_nat @ Bs3 @ I5 ) ) @ NS3 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS3 ) ) ) )
=> ( list_l792762466888043652on_nat @ S_list @ Ns_list ) ) ) ) ) ).
% list_order_extension.intro
thf(fact_658_set__mset__union,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
= ( sup_sup_set_nat @ ( set_mset_nat @ M3 ) @ ( set_mset_nat @ N4 ) ) ) ).
% set_mset_union
thf(fact_659_set__mset__mset,axiom,
! [Xs: list_nat] :
( ( set_mset_nat @ ( mset_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_mset_mset
thf(fact_660_diff__add__mset__swap,axiom,
! [B: nat,A2: multiset_nat,M3: multiset_nat] :
( ~ ( member_nat @ B @ ( set_mset_nat @ A2 ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ M3 ) @ A2 )
= ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ M3 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_661_single__subset__iff,axiom,
! [A: nat,M3: multiset_nat] :
( ( subseteq_mset_nat @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) @ M3 )
= ( member_nat @ A @ ( set_mset_nat @ M3 ) ) ) ).
% single_subset_iff
thf(fact_662_insert__DiffM,axiom,
! [X2: nat,M3: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ( add_mset_nat @ X2 @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) )
= M3 ) ) ).
% insert_DiffM
thf(fact_663_diff__union__swap2,axiom,
! [Y3: nat,M3: multiset_nat,X2: nat] :
( ( member_nat @ Y3 @ ( set_mset_nat @ M3 ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ X2 @ M3 ) @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) )
= ( add_mset_nat @ X2 @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_swap2
thf(fact_664_in__diffD,axiom,
! [A: nat,M3: multiset_nat,N4: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M3 @ N4 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ M3 ) ) ) ).
% in_diffD
thf(fact_665_union__iff,axiom,
! [A: nat,A2: multiset_nat,B2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) )
= ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
| ( member_nat @ A @ ( set_mset_nat @ B2 ) ) ) ) ).
% union_iff
thf(fact_666_multiset__nonemptyE,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ~ ! [X: nat] :
~ ( member_nat @ X @ ( set_mset_nat @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_667_mset__subset__eqD,axiom,
! [A2: multiset_nat,B2: multiset_nat,X2: nat] :
( ( subseteq_mset_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ ( set_mset_nat @ A2 ) )
=> ( member_nat @ X2 @ ( set_mset_nat @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_668_mset__add,axiom,
! [A: nat,A2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
=> ~ ! [B7: multiset_nat] :
( A2
!= ( add_mset_nat @ A @ B7 ) ) ) ).
% mset_add
thf(fact_669_multi__member__split,axiom,
! [X2: nat,M3: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ? [A7: multiset_nat] :
( M3
= ( add_mset_nat @ X2 @ A7 ) ) ) ).
% multi_member_split
thf(fact_670_insert__noteq__member,axiom,
! [B: nat,B2: multiset_nat,C: nat,C3: multiset_nat] :
( ( ( add_mset_nat @ B @ B2 )
= ( add_mset_nat @ C @ C3 ) )
=> ( ( B != C )
=> ( member_nat @ C @ ( set_mset_nat @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_671_union__single__eq__member,axiom,
! [X2: nat,M3: multiset_nat,N4: multiset_nat] :
( ( ( add_mset_nat @ X2 @ M3 )
= N4 )
=> ( member_nat @ X2 @ ( set_mset_nat @ N4 ) ) ) ).
% union_single_eq_member
thf(fact_672_in__multiset__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
= ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_673_multi__member__last,axiom,
! [X2: nat] : ( member_nat @ X2 @ ( set_mset_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) ).
% multi_member_last
thf(fact_674_mset__set__set__mset__msubset,axiom,
! [A2: multiset_nat] : ( subseteq_mset_nat @ ( mset_set_nat @ ( set_mset_nat @ A2 ) ) @ A2 ) ).
% mset_set_set_mset_msubset
thf(fact_675_multi__subset__induct,axiom,
! [F4: multiset_nat,A2: multiset_nat,P2: multiset_nat > $o] :
( ( subseteq_mset_nat @ F4 @ A2 )
=> ( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [A5: nat,F5: multiset_nat] :
( ( member_nat @ A5 @ ( set_mset_nat @ A2 ) )
=> ( ( P2 @ F5 )
=> ( P2 @ ( add_mset_nat @ A5 @ F5 ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% multi_subset_induct
thf(fact_676_mset__subset__eq__single,axiom,
! [A: nat,B2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ B2 ) )
=> ( subseteq_mset_nat @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_677_multi__member__this,axiom,
! [X2: nat,XS: multiset_nat] : ( member_nat @ X2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ).
% multi_member_this
thf(fact_678_multi__member__skip,axiom,
! [X2: nat,XS: multiset_nat,Y3: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ XS ) )
=> ( member_nat @ X2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_679_diff__single__trivial,axiom,
! [X2: nat,M3: multiset_nat] :
( ~ ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= M3 ) ) ).
% diff_single_trivial
thf(fact_680_diff__single__eq__union,axiom,
! [X2: nat,M3: multiset_nat,N4: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ( ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= N4 )
= ( M3
= ( add_mset_nat @ X2 @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_681_multi__drop__mem__not__eq,axiom,
! [C: nat,B2: multiset_nat] :
( ( member_nat @ C @ ( set_mset_nat @ B2 ) )
=> ( ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_682_add__mset__remove__trivial__If,axiom,
! [A: nat,N4: multiset_nat] :
( ( ( member_nat @ A @ ( set_mset_nat @ N4 ) )
=> ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= N4 ) )
& ( ~ ( member_nat @ A @ ( set_mset_nat @ N4 ) )
=> ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( add_mset_nat @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_683_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_nat,A: nat] :
( ( N4
= ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
= ( member_nat @ A @ ( set_mset_nat @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_684_multiset__add__sub__el__shuffle,axiom,
! [C: nat,B2: multiset_nat,B: nat] :
( ( member_nat @ C @ ( set_mset_nat @ B2 ) )
=> ( ( B != C )
=> ( ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) )
= ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ B2 ) @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_685_more__than__one__mset__mset__diff,axiom,
! [A: nat,M3: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
=> ( ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( set_mset_nat @ M3 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_686_nth__mem__mset,axiom,
! [I2: nat,Ls: list_nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ls ) )
=> ( member_nat @ ( nth_nat @ Ls @ I2 ) @ ( set_mset_nat @ ( mset_nat @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_687_insert__subset__eq__iff,axiom,
! [A: nat,A2: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ ( add_mset_nat @ A @ A2 ) @ B2 )
= ( ( member_nat @ A @ ( set_mset_nat @ B2 ) )
& ( subseteq_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_688_insert__DiffM2,axiom,
! [X2: nat,M3: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= M3 ) ) ).
% insert_DiffM2
thf(fact_689_diff__union__single__conv,axiom,
! [A: nat,J3: multiset_nat,I4: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ J3 ) )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ I4 @ J3 ) @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
= ( plus_p6334493942879108393et_nat @ I4 @ ( minus_8522176038001411705et_nat @ J3 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_690_ns__mul__ext__point,axiom,
! [As: multiset_nat,Bs: multiset_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,B: nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ As @ Bs ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) )
=> ( ( member_nat @ B @ ( set_mset_nat @ Bs ) )
=> ? [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ As ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ B ) @ ( sup_su6327502436637775413at_nat @ NS @ S2 ) ) ) ) ) ).
% ns_mul_ext_point
thf(fact_691_ns__mul__ext__point,axiom,
! [As: multiset_int,Bs: multiset_int,NS: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int,B: int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ As @ Bs ) @ ( multis8828838126066458039xt_int @ NS @ S2 ) )
=> ( ( member_int @ B @ ( set_mset_int @ Bs ) )
=> ? [X: int] :
( ( member_int @ X @ ( set_mset_int @ As ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ B ) @ ( sup_su6024340866399070445nt_int @ NS @ S2 ) ) ) ) ) ).
% ns_mul_ext_point
thf(fact_692_size__Diff1__less,axiom,
! [X2: nat,M3: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M3 ) ) ) ).
% size_Diff1_less
thf(fact_693_size__Diff2__less,axiom,
! [X2: nat,M3: multiset_nat,Y3: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ( member_nat @ Y3 @ ( set_mset_nat @ M3 ) )
=> ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M3 ) ) ) ) ).
% size_Diff2_less
thf(fact_694_s__mul__extI__old,axiom,
! [A2: multiset_int,Xs: list_int,A22: multiset_int,B2: multiset_int,Ys: list_int,B22: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( A2
= ( plus_p2156642923369911685et_int @ ( mset_int @ Xs ) @ A22 ) )
=> ( ( B2
= ( plus_p2156642923369911685et_int @ ( mset_int @ Ys ) @ B22 ) )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Ys @ I3 ) ) @ Ns ) )
=> ( ( A22 != zero_z3170743180189231877et_int )
=> ( ! [B5: int] :
( ( member_int @ B5 @ ( set_mset_int @ B22 ) )
=> ? [A6: int] :
( ( member_int @ A6 @ ( set_mset_int @ A22 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B5 ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis4212723674092261717xt_int @ Ns @ S ) ) ) ) ) ) ) ) ).
% s_mul_extI_old
thf(fact_695_s__mul__extI__old,axiom,
! [A2: multiset_nat,Xs: list_nat,A22: multiset_nat,B2: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
=> ( ( B2
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ Ns ) )
=> ( ( A22 != zero_z7348594199698428585et_nat )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ ( set_mset_nat @ B22 ) )
=> ? [A6: nat] :
( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ) ) ) ) ).
% s_mul_extI_old
thf(fact_696_subset__mset_Osum__mset__0__iff,axiom,
! [M3: multis1201202736280713200et_nat] :
( ( ( comm_m5787568287065167983et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ M3 )
= zero_z7348594199698428585et_nat )
= ( ! [X3: multiset_nat] :
( ( member_multiset_nat @ X3 @ ( set_ms4188662328148412963et_nat @ M3 ) )
=> ( X3 = zero_z7348594199698428585et_nat ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_697_less__add,axiom,
! [N4: multiset_int,A: int,M0: multiset_int,R: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ N4 @ ( add_mset_int @ A @ M0 ) ) @ ( mult1_int @ R ) )
=> ( ? [M7: multiset_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ M7 @ M0 ) @ ( mult1_int @ R ) )
& ( N4
= ( add_mset_int @ A @ M7 ) ) )
| ? [K4: multiset_int] :
( ! [B6: int] :
( ( member_int @ B6 @ ( set_mset_int @ K4 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ A ) @ R ) )
& ( N4
= ( plus_p2156642923369911685et_int @ M0 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_698_less__add,axiom,
! [N4: multiset_nat,A: nat,M0: multiset_nat,R: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ N4 @ ( add_mset_nat @ A @ M0 ) ) @ ( mult1_nat @ R ) )
=> ( ? [M7: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M7 @ M0 ) @ ( mult1_nat @ R ) )
& ( N4
= ( add_mset_nat @ A @ M7 ) ) )
| ? [K4: multiset_nat] :
( ! [B6: nat] :
( ( member_nat @ B6 @ ( set_mset_nat @ K4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ A ) @ R ) )
& ( N4
= ( plus_p6334493942879108393et_nat @ M0 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_699_s__mul__ext__singleton,axiom,
! [A: int,B: int,S: set_Pr958786334691620121nt_int,Ns: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( add_mset_int @ A @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ B @ zero_z3170743180189231877et_int ) ) @ ( multis4212723674092261717xt_int @ Ns @ S ) ) ) ).
% s_mul_ext_singleton
thf(fact_700_s__mul__ext__extend__left,axiom,
! [B2: multiset_nat,C3: multiset_nat,W: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,A2: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ B2 @ C3 ) @ ( multis4215214144601311993xt_nat @ W @ S2 ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ C3 ) @ ( multis4215214144601311993xt_nat @ W @ S2 ) ) ) ).
% s_mul_ext_extend_left
thf(fact_701_mult1__union,axiom,
! [B2: multiset_nat,D2: multiset_nat,R: set_Pr1261947904930325089at_nat,C3: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ B2 @ D2 ) @ ( mult1_nat @ R ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ C3 @ B2 ) @ ( plus_p6334493942879108393et_nat @ C3 @ D2 ) ) @ ( mult1_nat @ R ) ) ) ).
% mult1_union
thf(fact_702_s__ns__mul__ext__union__compat,axiom,
! [A2: multiset_nat,B2: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,C3: multiset_nat,D2: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ C3 @ D2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B2 @ D2 ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ).
% s_ns_mul_ext_union_compat
thf(fact_703_all__s__s__mul__ext,axiom,
! [A2: multiset_nat,B2: multiset_nat,S: set_Pr1261947904930325089at_nat,Ns: set_Pr1261947904930325089at_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ ( set_mset_nat @ B2 ) )
=> ? [A6: nat] :
( ( member_nat @ A6 @ ( set_mset_nat @ A2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_704_all__s__s__mul__ext,axiom,
! [A2: multiset_int,B2: multiset_int,S: set_Pr958786334691620121nt_int,Ns: set_Pr958786334691620121nt_int] :
( ( A2 != zero_z3170743180189231877et_int )
=> ( ! [B5: int] :
( ( member_int @ B5 @ ( set_mset_int @ B2 ) )
=> ? [A6: int] :
( ( member_int @ A6 @ ( set_mset_int @ A2 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B5 ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis4212723674092261717xt_int @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_705_s__mul__ext__point,axiom,
! [As: multiset_nat,Bs: multiset_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,B: nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ As @ Bs ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) )
=> ( ( member_nat @ B @ ( set_mset_nat @ Bs ) )
=> ? [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ As ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ B ) @ ( sup_su6327502436637775413at_nat @ NS @ S2 ) ) ) ) ) ).
% s_mul_ext_point
thf(fact_706_s__mul__ext__point,axiom,
! [As: multiset_int,Bs: multiset_int,NS: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int,B: int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ As @ Bs ) @ ( multis4212723674092261717xt_int @ NS @ S2 ) )
=> ( ( member_int @ B @ ( set_mset_int @ Bs ) )
=> ? [X: int] :
( ( member_int @ X @ ( set_mset_int @ As ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ B ) @ ( sup_su6024340866399070445nt_int @ NS @ S2 ) ) ) ) ) ).
% s_mul_ext_point
thf(fact_707_s__mul__ext__ne__extend__left,axiom,
! [A2: multiset_nat,B2: multiset_nat,C3: multiset_nat,W: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ B2 @ C3 ) @ ( multis8831328596575508315xt_nat @ W @ S2 ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ C3 ) @ ( multis4215214144601311993xt_nat @ W @ S2 ) ) ) ) ).
% s_mul_ext_ne_extend_left
thf(fact_708_ns__s__mul__ext__union__multiset__l,axiom,
! [A2: multiset_int,B2: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,C3: multiset_int,D2: multiset_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis8828838126066458039xt_int @ Ns @ S ) )
=> ( ( C3 != zero_z3170743180189231877et_int )
=> ( ! [D4: int] :
( ( member_int @ D4 @ ( set_mset_int @ D2 ) )
=> ? [C6: int] :
( ( member_int @ C6 @ ( set_mset_int @ C3 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ C6 @ D4 ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( plus_p2156642923369911685et_int @ A2 @ C3 ) @ ( plus_p2156642923369911685et_int @ B2 @ D2 ) ) @ ( multis4212723674092261717xt_int @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_709_ns__s__mul__ext__union__multiset__l,axiom,
! [A2: multiset_nat,B2: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,C3: multiset_nat,D2: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ( ( C3 != zero_z7348594199698428585et_nat )
=> ( ! [D4: nat] :
( ( member_nat @ D4 @ ( set_mset_nat @ D2 ) )
=> ? [C6: nat] :
( ( member_nat @ C6 @ ( set_mset_nat @ C3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C6 @ D4 ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B2 @ D2 ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_710_mult1E,axiom,
! [N4: multiset_int,M3: multiset_int,R: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ N4 @ M3 ) @ ( mult1_int @ R ) )
=> ~ ! [A5: int,M02: multiset_int] :
( ( M3
= ( add_mset_int @ A5 @ M02 ) )
=> ! [K4: multiset_int] :
( ( N4
= ( plus_p2156642923369911685et_int @ M02 @ K4 ) )
=> ~ ! [B6: int] :
( ( member_int @ B6 @ ( set_mset_int @ K4 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ A5 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_711_mult1E,axiom,
! [N4: multiset_nat,M3: multiset_nat,R: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ N4 @ M3 ) @ ( mult1_nat @ R ) )
=> ~ ! [A5: nat,M02: multiset_nat] :
( ( M3
= ( add_mset_nat @ A5 @ M02 ) )
=> ! [K4: multiset_nat] :
( ( N4
= ( plus_p6334493942879108393et_nat @ M02 @ K4 ) )
=> ~ ! [B6: nat] :
( ( member_nat @ B6 @ ( set_mset_nat @ K4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ A5 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_712_mult1I,axiom,
! [M3: multiset_int,A: int,M0: multiset_int,N4: multiset_int,K3: multiset_int,R: set_Pr958786334691620121nt_int] :
( ( M3
= ( add_mset_int @ A @ M0 ) )
=> ( ( N4
= ( plus_p2156642923369911685et_int @ M0 @ K3 ) )
=> ( ! [B5: int] :
( ( member_int @ B5 @ ( set_mset_int @ K3 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B5 @ A ) @ R ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ N4 @ M3 ) @ ( mult1_int @ R ) ) ) ) ) ).
% mult1I
thf(fact_713_mult1I,axiom,
! [M3: multiset_nat,A: nat,M0: multiset_nat,N4: multiset_nat,K3: multiset_nat,R: set_Pr1261947904930325089at_nat] :
( ( M3
= ( add_mset_nat @ A @ M0 ) )
=> ( ( N4
= ( plus_p6334493942879108393et_nat @ M0 @ K3 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ ( set_mset_nat @ K3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A ) @ R ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ N4 @ M3 ) @ ( mult1_nat @ R ) ) ) ) ) ).
% mult1I
thf(fact_714_s__mul__ext__elim,axiom,
! [Xs: multiset_int,Ys: multiset_int,NS: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ Xs @ Ys ) @ ( multis4212723674092261717xt_int @ NS @ S2 ) )
=> ? [Xs12: list_int,Xs23: list_int] :
( ( Xs
= ( plus_p2156642923369911685et_int @ ( mset_int @ Xs12 ) @ ( mset_int @ Xs23 ) ) )
& ? [Ys12: list_int,Ys23: list_int] :
( ( Ys
= ( plus_p2156642923369911685et_int @ ( mset_int @ Ys12 ) @ ( mset_int @ Ys23 ) ) )
& ( ( size_size_list_int @ Xs12 )
= ( size_size_list_int @ Ys12 ) )
& ( Xs23 != nil_int )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_int @ Ys12 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs12 @ I5 ) @ ( nth_int @ Ys12 @ I5 ) ) @ NS ) )
& ! [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Ys23 ) )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ ( set_int2 @ Xs23 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).
% s_mul_ext_elim
thf(fact_715_s__mul__ext__elim,axiom,
! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) )
=> ? [Xs12: list_nat,Xs23: list_nat] :
( ( Xs
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs23 ) ) )
& ? [Ys12: list_nat,Ys23: list_nat] :
( ( Ys
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys23 ) ) )
& ( ( size_size_list_nat @ Xs12 )
= ( size_size_list_nat @ Ys12 ) )
& ( Xs23 != nil_nat )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Ys12 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs12 @ I5 ) @ ( nth_nat @ Ys12 @ I5 ) ) @ NS ) )
& ! [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Ys23 ) )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Xs23 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).
% s_mul_ext_elim
thf(fact_716_s__mul__ext__intro,axiom,
! [Xs: multiset_int,Xs1: list_int,Xs22: list_int,Ys: multiset_int,Ys1: list_int,Ys22: list_int,NS: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( Xs
= ( plus_p2156642923369911685et_int @ ( mset_int @ Xs1 ) @ ( mset_int @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p2156642923369911685et_int @ ( mset_int @ Ys1 ) @ ( mset_int @ Ys22 ) ) )
=> ( ( ( size_size_list_int @ Xs1 )
= ( size_size_list_int @ Ys1 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys1 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs1 @ I3 ) @ ( nth_int @ Ys1 @ I3 ) ) @ NS ) )
=> ( ( Xs22 != nil_int )
=> ( ! [Y6: int] :
( ( member_int @ Y6 @ ( set_int2 @ Ys22 ) )
=> ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs22 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Y6 ) @ S2 ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ Xs @ Ys ) @ ( multis4212723674092261717xt_int @ NS @ S2 ) ) ) ) ) ) ) ) ).
% s_mul_ext_intro
thf(fact_717_s__mul__ext__intro,axiom,
! [Xs: multiset_nat,Xs1: list_nat,Xs22: list_nat,Ys: multiset_nat,Ys1: list_nat,Ys22: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( Xs
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs1 ) @ ( mset_nat @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys22 ) ) )
=> ( ( ( size_size_list_nat @ Xs1 )
= ( size_size_list_nat @ Ys1 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys1 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I3 ) @ ( nth_nat @ Ys1 @ I3 ) ) @ NS ) )
=> ( ( Xs22 != nil_nat )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ ( set_nat2 @ Ys22 ) )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs22 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y6 ) @ S2 ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).
% s_mul_ext_intro
thf(fact_718_size__Diff__singleton__if,axiom,
! [X2: nat,A2: multiset_nat] :
( ( ( member_nat @ X2 @ ( set_mset_nat @ A2 ) )
=> ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) )
= ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X2 @ ( set_mset_nat @ A2 ) )
=> ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) )
= ( size_s5917832649809541300et_nat @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_719_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_720_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_721_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_722_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_723_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_724_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_725_self__append__conv,axiom,
! [Y3: list_nat,Ys: list_nat] :
( ( Y3
= ( append_nat @ Y3 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_726_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_727_self__append__conv2,axiom,
! [Y3: list_nat,Xs: list_nat] :
( ( Y3
= ( append_nat @ Xs @ Y3 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_728_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_729_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_730_zip__eq__Nil__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( zip_nat_nat @ Xs @ Ys )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_731_Nil__eq__zip__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( zip_nat_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_732_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X2: nat] :
( ( ( list_update_nat @ Xs @ K @ X2 )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_733_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_nat @ N @ nil_nat )
= nil_Pr5478986624290739719at_nat ) ).
% enumerate_simps(1)
thf(fact_734_list__ex__simps_I2_J,axiom,
! [P2: nat > $o] :
~ ( list_ex_nat @ P2 @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_735_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_736_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_737_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_738_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_739_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_740_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_741_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_742_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_743_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_744_mset__zero__iff__right,axiom,
! [X2: list_nat] :
( ( zero_z7348594199698428585et_nat
= ( mset_nat @ X2 ) )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff_right
thf(fact_745_mset__zero__iff,axiom,
! [X2: list_nat] :
( ( ( mset_nat @ X2 )
= zero_z7348594199698428585et_nat )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff
thf(fact_746_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_747_sum__list_ONil,axiom,
( ( groups4559388385066561235st_int @ nil_int )
= zero_zero_int ) ).
% sum_list.Nil
thf(fact_748_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_749_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_750_remove1_Osimps_I1_J,axiom,
! [X2: nat] :
( ( remove1_nat @ X2 @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_751_removeAll_Osimps_I1_J,axiom,
! [X2: nat] :
( ( removeAll_nat @ X2 @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_752_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_753_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_754_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_755_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_756_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_757_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_758_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_759_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_760_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_761_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_762_list__update_Osimps_I1_J,axiom,
! [I2: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I2 @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_763_list__update__code_I1_J,axiom,
! [I2: nat,Y3: nat] :
( ( list_update_nat @ nil_nat @ I2 @ Y3 )
= nil_nat ) ).
% list_update_code(1)
thf(fact_764_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_765_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_766_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_767_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_768_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_769_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_770_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_771_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_772_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_773_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_774_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_775_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_776_upt__0,axiom,
! [I2: nat] :
( ( upt @ I2 @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_777_mset_Osimps_I1_J,axiom,
( ( mset_nat @ nil_nat )
= zero_z7348594199698428585et_nat ) ).
% mset.simps(1)
thf(fact_778_not__Nil__listrel1,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).
% not_Nil_listrel1
thf(fact_779_not__listrel1__Nil,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).
% not_listrel1_Nil
thf(fact_780_listrel__Nil2,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil2
thf(fact_781_listrel__Nil1,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil1
thf(fact_782_listrel_ONil,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).
% listrel.Nil
thf(fact_783_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_784_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_785_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs != nil_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_786_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > int,G: nat > int] :
( ( Xs != nil_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_int @ ( groups4559388385066561235st_int @ ( map_nat_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_nat_int @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_787_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_788_size__Diff__singleton,axiom,
! [X2: nat,M3: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) )
= ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ M3 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_789_s__mul__ext__self__extend__left,axiom,
! [A2: multiset_nat,W: set_Pr1261947904930325089at_nat,B2: multiset_nat,S2: set_Pr1261947904930325089at_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ( ( locally_refl_nat @ W @ B2 )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ B2 ) @ ( multis4215214144601311993xt_nat @ W @ S2 ) ) ) ) ).
% s_mul_ext_self_extend_left
thf(fact_790_in__mset__fold__plus__iff,axiom,
! [X2: nat,M3: multiset_nat,NN: multis1201202736280713200et_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ ( fold_m1829410296857755981et_nat @ plus_p6334493942879108393et_nat @ M3 @ NN ) ) )
= ( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
| ? [N5: multiset_nat] :
( ( member_multiset_nat @ N5 @ ( set_ms4188662328148412963et_nat @ NN ) )
& ( member_nat @ X2 @ ( set_mset_nat @ N5 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_791_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_792_nth__Cons__pos,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_793_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_794_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_795_list__ex__simps_I1_J,axiom,
! [P2: nat > $o,X2: nat,Xs: list_nat] :
( ( list_ex_nat @ P2 @ ( cons_nat @ X2 @ Xs ) )
= ( ( P2 @ X2 )
| ( list_ex_nat @ P2 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_796_list__ex__simps_I1_J,axiom,
! [P2: int > $o,X2: int,Xs: list_int] :
( ( list_ex_int @ P2 @ ( cons_int @ X2 @ Xs ) )
= ( ( P2 @ X2 )
| ( list_ex_int @ P2 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_797_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_798_append1__eq__conv,axiom,
! [Xs: list_int,X2: int,Ys: list_int,Y3: int] :
( ( ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y3 @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_799_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_800_nth__Cons__0,axiom,
! [X2: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_801_zip__Cons__Cons,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_802_zip__Cons__Cons,axiom,
! [X2: nat,Xs: list_nat,Y3: int,Ys: list_int] :
( ( zip_nat_int @ ( cons_nat @ X2 @ Xs ) @ ( cons_int @ Y3 @ Ys ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ X2 @ Y3 ) @ ( zip_nat_int @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_803_zip__Cons__Cons,axiom,
! [X2: int,Xs: list_int,Y3: nat,Ys: list_nat] :
( ( zip_int_nat @ ( cons_int @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( cons_P7512249878480867347nt_nat @ ( product_Pair_int_nat @ X2 @ Y3 ) @ ( zip_int_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_804_zip__Cons__Cons,axiom,
! [X2: int,Xs: list_int,Y3: int,Ys: list_int] :
( ( zip_int_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y3 @ Ys ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( zip_int_int @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_805_sum__list_OCons,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ( groups8053510108761903431et_nat @ ( cons_multiset_nat @ X2 @ Xs ) )
= ( plus_p6334493942879108393et_nat @ X2 @ ( groups8053510108761903431et_nat @ Xs ) ) ) ).
% sum_list.Cons
thf(fact_806_sum__list_OCons,axiom,
! [X2: nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat @ ( cons_nat @ X2 @ Xs ) )
= ( plus_plus_nat @ X2 @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% sum_list.Cons
thf(fact_807_sum__list_OCons,axiom,
! [X2: int,Xs: list_int] :
( ( groups4559388385066561235st_int @ ( cons_int @ X2 @ Xs ) )
= ( plus_plus_int @ X2 @ ( groups4559388385066561235st_int @ Xs ) ) ) ).
% sum_list.Cons
thf(fact_808_nth__append__length,axiom,
! [Xs: list_int,X2: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_809_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_810_list__update__length,axiom,
! [Xs: list_int,X2: int,Ys: list_int,Y3: int] :
( ( list_update_int @ ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) ) @ ( size_size_list_int @ Xs ) @ Y3 )
= ( append_int @ Xs @ ( cons_int @ Y3 @ Ys ) ) ) ).
% list_update_length
thf(fact_811_list__update__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y3 )
= ( append_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) ) ).
% list_update_length
thf(fact_812_Cons__listrel1__Cons,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
& ( Xs = Ys ) )
| ( ( X2 = Y3 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_813_Cons__listrel1__Cons,axiom,
! [X2: int,Xs: list_int,Y3: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y3 @ Ys ) ) @ ( listrel1_int @ R ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
& ( Xs = Ys ) )
| ( ( X2 = Y3 )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_814_mset__single__iff__right,axiom,
! [X2: nat,Xs: list_nat] :
( ( ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat )
= ( mset_nat @ Xs ) )
= ( Xs
= ( cons_nat @ X2 @ nil_nat ) ) ) ).
% mset_single_iff_right
thf(fact_815_mset__single__iff__right,axiom,
! [X2: int,Xs: list_int] :
( ( ( add_mset_int @ X2 @ zero_z3170743180189231877et_int )
= ( mset_int @ Xs ) )
= ( Xs
= ( cons_int @ X2 @ nil_int ) ) ) ).
% mset_single_iff_right
thf(fact_816_mset__single__iff,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( mset_nat @ Xs )
= ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= ( Xs
= ( cons_nat @ X2 @ nil_nat ) ) ) ).
% mset_single_iff
thf(fact_817_mset__single__iff,axiom,
! [Xs: list_int,X2: int] :
( ( ( mset_int @ Xs )
= ( add_mset_int @ X2 @ zero_z3170743180189231877et_int ) )
= ( Xs
= ( cons_int @ X2 @ nil_int ) ) ) ).
% mset_single_iff
thf(fact_818_list__nonempty__induct,axiom,
! [Xs: list_nat,P2: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P2 @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_819_list__nonempty__induct,axiom,
! [Xs: list_int,P2: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X: int] : ( P2 @ ( cons_int @ X @ nil_int ) )
=> ( ! [X: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_820_list__induct2_H,axiom,
! [P2: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X @ Xs3 ) @ nil_nat )
=> ( ! [Y6: nat,Ys5: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y6 @ Ys5 ) )
=> ( ! [X: nat,Xs3: list_nat,Y6: nat,Ys5: list_nat] :
( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_821_list__induct2_H,axiom,
! [P2: list_nat > list_int > $o,Xs: list_nat,Ys: list_int] :
( ( P2 @ nil_nat @ nil_int )
=> ( ! [X: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X @ Xs3 ) @ nil_int )
=> ( ! [Y6: int,Ys5: list_int] : ( P2 @ nil_nat @ ( cons_int @ Y6 @ Ys5 ) )
=> ( ! [X: nat,Xs3: list_nat,Y6: int,Ys5: list_int] :
( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_822_list__induct2_H,axiom,
! [P2: list_int > list_nat > $o,Xs: list_int,Ys: list_nat] :
( ( P2 @ nil_int @ nil_nat )
=> ( ! [X: int,Xs3: list_int] : ( P2 @ ( cons_int @ X @ Xs3 ) @ nil_nat )
=> ( ! [Y6: nat,Ys5: list_nat] : ( P2 @ nil_int @ ( cons_nat @ Y6 @ Ys5 ) )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat] :
( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_823_list__induct2_H,axiom,
! [P2: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P2 @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int] : ( P2 @ ( cons_int @ X @ Xs3 ) @ nil_int )
=> ( ! [Y6: int,Ys5: list_int] : ( P2 @ nil_int @ ( cons_int @ Y6 @ Ys5 ) )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int] :
( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_824_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_825_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y4: int,Ys2: list_int] :
( Xs
= ( cons_int @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_826_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y6: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ Y6 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_827_remdups__adj_Ocases,axiom,
! [X2: list_int] :
( ( X2 != nil_int )
=> ( ! [X: int] :
( X2
!= ( cons_int @ X @ nil_int ) )
=> ~ ! [X: int,Y6: int,Xs3: list_int] :
( X2
!= ( cons_int @ X @ ( cons_int @ Y6 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_828_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_829_transpose_Ocases,axiom,
! [X2: list_list_int] :
( ( X2 != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X2
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X: int,Xs3: list_int,Xss: list_list_int] :
( X2
!= ( cons_list_int @ ( cons_int @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_830_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_831_min__list_Ocases,axiom,
! [X2: list_int] :
( ! [X: int,Xs3: list_int] :
( X2
!= ( cons_int @ X @ Xs3 ) )
=> ( X2 = nil_int ) ) ).
% min_list.cases
thf(fact_832_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y3
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_833_list_Oexhaust,axiom,
! [Y3: list_int] :
( ( Y3 != nil_int )
=> ~ ! [X212: int,X223: list_int] :
( Y3
!= ( cons_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_834_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_835_list_OdiscI,axiom,
! [List: list_int,X21: int,X222: list_int] :
( ( List
= ( cons_int @ X21 @ X222 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_836_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_837_list_Odistinct_I1_J,axiom,
! [X21: int,X222: list_int] :
( nil_int
!= ( cons_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_838_sorted__wrt_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P4: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
=> ~ ! [P4: nat > nat > $o,X: nat,Ys5: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_839_sorted__wrt_Ocases,axiom,
! [X2: produc5834231552977413017st_int] :
( ! [P4: int > int > $o] :
( X2
!= ( produc8618682346314911123st_int @ P4 @ nil_int ) )
=> ~ ! [P4: int > int > $o,X: int,Ys5: list_int] :
( X2
!= ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_840_successively_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P4: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
=> ( ! [P4: nat > nat > $o,X: nat] :
( X2
!= ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X @ nil_nat ) ) )
=> ~ ! [P4: nat > nat > $o,X: nat,Y6: nat,Xs3: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X @ ( cons_nat @ Y6 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_841_successively_Ocases,axiom,
! [X2: produc5834231552977413017st_int] :
( ! [P4: int > int > $o] :
( X2
!= ( produc8618682346314911123st_int @ P4 @ nil_int ) )
=> ( ! [P4: int > int > $o,X: int] :
( X2
!= ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X @ nil_int ) ) )
=> ~ ! [P4: int > int > $o,X: int,Y6: int,Xs3: list_int] :
( X2
!= ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X @ ( cons_int @ Y6 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_842_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 ) )
=> ~ ! [X: nat,Xs5: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs5 ) )
=> ! [Y6: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y6 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X @ Y6 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_843_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
( ( ( zip_nat_int @ Xs @ Ys )
= ( cons_P2335045147070616083at_int @ Xy @ Xys ) )
=> ~ ! [X: nat,Xs5: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs5 ) )
=> ! [Y6: int,Ys4: list_int] :
( ( Ys
= ( cons_int @ Y6 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_nat_int @ X @ Y6 ) )
=> ( Xys
!= ( zip_nat_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_844_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
( ( ( zip_int_nat @ Xs @ Ys )
= ( cons_P7512249878480867347nt_nat @ Xy @ Xys ) )
=> ~ ! [X: int,Xs5: list_int] :
( ( Xs
= ( cons_int @ X @ Xs5 ) )
=> ! [Y6: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y6 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_int_nat @ X @ Y6 ) )
=> ( Xys
!= ( zip_int_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_845_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
( ( ( zip_int_int @ Xs @ Ys )
= ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
=> ~ ! [X: int,Xs5: list_int] :
( ( Xs
= ( cons_int @ X @ Xs5 ) )
=> ! [Y6: int,Ys4: list_int] :
( ( Ys
= ( cons_int @ Y6 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_int_int @ X @ Y6 ) )
=> ( Xys
!= ( zip_int_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_846_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_847_distinct__length__2__or__more,axiom,
! [A: int,B: int,Xs: list_int] :
( ( distinct_int @ ( cons_int @ A @ ( cons_int @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_int @ ( cons_int @ A @ Xs ) )
& ( distinct_int @ ( cons_int @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_848_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_849_append__Cons,axiom,
! [X2: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X2 @ Xs ) @ Ys )
= ( cons_int @ X2 @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_850_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_851_Cons__eq__appendI,axiom,
! [X2: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X2 @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_852_set__ConsD,axiom,
! [Y3: nat,X2: nat,Xs: list_nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y3 = X2 )
| ( member_nat @ Y3 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_853_set__ConsD,axiom,
! [Y3: int,X2: int,Xs: list_int] :
( ( member_int @ Y3 @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) )
=> ( ( Y3 = X2 )
| ( member_int @ Y3 @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_854_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_855_list_Oset__cases,axiom,
! [E: int,A: list_int] :
( ( member_int @ E @ ( set_int2 @ A ) )
=> ( ! [Z22: list_int] :
( A
!= ( cons_int @ E @ Z22 ) )
=> ~ ! [Z1: int,Z22: list_int] :
( ( A
= ( cons_int @ Z1 @ Z22 ) )
=> ~ ( member_int @ E @ ( set_int2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_856_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_857_list_Oset__intros_I1_J,axiom,
! [X21: int,X222: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_858_list_Oset__intros_I2_J,axiom,
! [Y3: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_859_list_Oset__intros_I2_J,axiom,
! [Y3: int,X222: list_int,X21: int] :
( ( member_int @ Y3 @ ( set_int2 @ X222 ) )
=> ( member_int @ Y3 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_860_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_861_not__Cons__self2,axiom,
! [X2: int,Xs: list_int] :
( ( cons_int @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_862_inj__on__Cons1,axiom,
! [X2: nat,A2: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X2 ) @ A2 ) ).
% inj_on_Cons1
thf(fact_863_inj__on__Cons1,axiom,
! [X2: int,A2: set_list_int] : ( inj_on720019086181695851st_int @ ( cons_int @ X2 ) @ A2 ) ).
% inj_on_Cons1
thf(fact_864_removeAll_Osimps_I2_J,axiom,
! [X2: nat,Y3: nat,Xs: list_nat] :
( ( ( X2 = Y3 )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y3 @ Xs ) )
= ( removeAll_nat @ X2 @ Xs ) ) )
& ( ( X2 != Y3 )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y3 @ Xs ) )
= ( cons_nat @ Y3 @ ( removeAll_nat @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_865_removeAll_Osimps_I2_J,axiom,
! [X2: int,Y3: int,Xs: list_int] :
( ( ( X2 = Y3 )
=> ( ( removeAll_int @ X2 @ ( cons_int @ Y3 @ Xs ) )
= ( removeAll_int @ X2 @ Xs ) ) )
& ( ( X2 != Y3 )
=> ( ( removeAll_int @ X2 @ ( cons_int @ Y3 @ Xs ) )
= ( cons_int @ Y3 @ ( removeAll_int @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_866_remove1_Osimps_I2_J,axiom,
! [X2: nat,Y3: nat,Xs: list_nat] :
( ( ( X2 = Y3 )
=> ( ( remove1_nat @ X2 @ ( cons_nat @ Y3 @ Xs ) )
= Xs ) )
& ( ( X2 != Y3 )
=> ( ( remove1_nat @ X2 @ ( cons_nat @ Y3 @ Xs ) )
= ( cons_nat @ Y3 @ ( remove1_nat @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_867_remove1_Osimps_I2_J,axiom,
! [X2: int,Y3: int,Xs: list_int] :
( ( ( X2 = Y3 )
=> ( ( remove1_int @ X2 @ ( cons_int @ Y3 @ Xs ) )
= Xs ) )
& ( ( X2 != Y3 )
=> ( ( remove1_int @ X2 @ ( cons_int @ Y3 @ Xs ) )
= ( cons_int @ Y3 @ ( remove1_int @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_868_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_869_list_Osimps_I9_J,axiom,
! [F: nat > int,X21: nat,X222: list_nat] :
( ( map_nat_int @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_nat_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_870_list_Osimps_I9_J,axiom,
! [F: int > nat,X21: int,X222: list_int] :
( ( map_int_nat @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_int_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_871_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X222: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_872_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,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_873_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: int > nat,Ys: list_int] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_int_nat @ F @ Ys ) )
=> ? [Z3: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_int_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_874_Cons__eq__map__D,axiom,
! [X2: int,Xs: list_int,F: nat > int,Ys: list_nat] :
( ( ( cons_int @ X2 @ Xs )
= ( map_nat_int @ F @ Ys ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_875_Cons__eq__map__D,axiom,
! [X2: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( map_int_int @ F @ Ys ) )
=> ? [Z3: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_int_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_876_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y3 )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_877_map__eq__Cons__D,axiom,
! [F: int > nat,Xs: list_int,Y3: nat,Ys: list_nat] :
( ( ( map_int_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ? [Z3: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y3 )
& ( ( map_int_nat @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_878_map__eq__Cons__D,axiom,
! [F: nat > int,Xs: list_nat,Y3: int,Ys: list_int] :
( ( ( map_nat_int @ F @ Xs )
= ( cons_int @ Y3 @ Ys ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y3 )
& ( ( map_nat_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_879_map__eq__Cons__D,axiom,
! [F: int > int,Xs: list_int,Y3: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y3 @ Ys ) )
=> ? [Z3: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y3 )
& ( ( map_int_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_880_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,Zs4: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs4 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs4 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_881_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: int > nat,Ys: list_int] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_int_nat @ F @ Ys ) )
= ( ? [Z4: int,Zs4: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs4 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_int_nat @ F @ Zs4 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_882_Cons__eq__map__conv,axiom,
! [X2: int,Xs: list_int,F: nat > int,Ys: list_nat] :
( ( ( cons_int @ X2 @ Xs )
= ( map_nat_int @ F @ Ys ) )
= ( ? [Z4: nat,Zs4: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs4 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_int @ F @ Zs4 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_883_Cons__eq__map__conv,axiom,
! [X2: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( map_int_int @ F @ Ys ) )
= ( ? [Z4: int,Zs4: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs4 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_int_int @ F @ Zs4 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_884_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
= ( ? [Z4: nat,Zs4: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs4 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_nat_nat @ F @ Zs4 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_885_map__eq__Cons__conv,axiom,
! [F: int > nat,Xs: list_int,Y3: nat,Ys: list_nat] :
( ( ( map_int_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
= ( ? [Z4: int,Zs4: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs4 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_int_nat @ F @ Zs4 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_886_map__eq__Cons__conv,axiom,
! [F: nat > int,Xs: list_nat,Y3: int,Ys: list_int] :
( ( ( map_nat_int @ F @ Xs )
= ( cons_int @ Y3 @ Ys ) )
= ( ? [Z4: nat,Zs4: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs4 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_nat_int @ F @ Zs4 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_887_map__eq__Cons__conv,axiom,
! [F: int > int,Xs: list_int,Y3: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y3 @ Ys ) )
= ( ? [Z4: int,Zs4: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs4 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_int_int @ F @ Zs4 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_888_list__induct4,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_int,P2: list_int > list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int,Z3: int,Zs3: list_int,W2: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( ( size_size_list_int @ Zs3 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_889_list__induct4,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_nat,P2: list_int > list_int > list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_int @ nil_nat )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int,Z3: int,Zs3: list_int,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( ( size_size_list_int @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_890_list__induct4,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_int,P2: list_int > list_int > list_nat > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_nat @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int,Z3: nat,Zs3: list_nat,W2: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_891_list__induct4,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_nat,P2: list_int > list_int > list_nat > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_nat @ nil_nat )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int,Z3: nat,Zs3: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_892_list__induct4,axiom,
! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_int,P2: list_int > list_nat > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P2 @ nil_int @ nil_nat @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat,Z3: int,Zs3: list_int,W2: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( ( size_size_list_int @ Zs3 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_893_list__induct4,axiom,
! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_nat,P2: list_int > list_nat > list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_int @ nil_nat @ nil_int @ nil_nat )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat,Z3: int,Zs3: list_int,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( ( size_size_list_int @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_894_list__induct4,axiom,
! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_int,P2: list_int > list_nat > list_nat > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P2 @ nil_int @ nil_nat @ nil_nat @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat,Z3: nat,Zs3: list_nat,W2: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_895_list__induct4,axiom,
! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_nat,P2: list_int > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_int @ 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 ) )
=> ( ( P2 @ nil_int @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat,Z3: nat,Zs3: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_896_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_int,P2: list_nat > list_int > list_int > list_int > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_int @ nil_int @ nil_int )
=> ( ! [X: nat,Xs3: list_nat,Y6: int,Ys5: list_int,Z3: int,Zs3: list_int,W2: int,Ws2: list_int] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( ( size_size_list_int @ Zs3 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_897_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_nat,P2: list_nat > list_int > list_int > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_int @ nil_int @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y6: int,Ys5: list_int,Z3: int,Zs3: list_int,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( ( size_size_list_int @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_898_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,P2: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int,Z3: int,Zs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_899_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_nat,P2: list_int > list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_nat )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_900_list__induct3,axiom,
! [Xs: list_int,Ys: list_nat,Zs: list_int,P2: list_int > list_nat > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_int @ nil_nat @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat,Z3: int,Zs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_901_list__induct3,axiom,
! [Xs: list_int,Ys: list_nat,Zs: list_nat,P2: list_int > list_nat > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_int @ nil_nat @ nil_nat )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_902_list__induct3,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_int,P2: list_nat > list_int > list_int > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_int @ nil_int )
=> ( ! [X: nat,Xs3: list_nat,Y6: int,Ys5: list_int,Z3: int,Zs3: list_int] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_903_list__induct3,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_nat,P2: list_nat > list_int > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_int @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y6: int,Ys5: list_int,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_904_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_int,P2: list_nat > list_nat > list_int > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_int )
=> ( ! [X: nat,Xs3: list_nat,Y6: nat,Ys5: list_nat,Z3: int,Zs3: list_int] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_int @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_905_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P2: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y6: nat,Ys5: list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( ( size_size_list_nat @ Ys5 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs3 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_906_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P2: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_907_list__induct2,axiom,
! [Xs: list_int,Ys: list_nat,P2: list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P2 @ nil_int @ nil_nat )
=> ( ! [X: int,Xs3: list_int,Y6: nat,Ys5: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_908_list__induct2,axiom,
! [Xs: list_nat,Ys: list_int,P2: list_nat > list_int > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_nat @ nil_int )
=> ( ! [X: nat,Xs3: list_nat,Y6: int,Ys5: list_int] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_909_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P2: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y6: nat,Ys5: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_910_rev__nonempty__induct,axiom,
! [Xs: list_nat,P2: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P2 @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_911_rev__nonempty__induct,axiom,
! [Xs: list_int,P2: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X: int] : ( P2 @ ( cons_int @ X @ nil_int ) )
=> ( ! [X: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_int @ Xs3 @ ( cons_int @ X @ nil_int ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_912_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 ) ) )
| ? [Ys6: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys6 ) )
& ( ( append_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_913_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs: list_int,X2: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( cons_int @ X2 @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs
= ( cons_int @ X2 @ Xs ) ) )
| ? [Ys6: list_int] :
( ( Ys
= ( cons_int @ X2 @ Ys6 ) )
& ( ( append_int @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_914_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 ) )
| ? [Ys6: list_nat] :
( ( ( cons_nat @ X2 @ Ys6 )
= Ys )
& ( Xs
= ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_915_Cons__eq__append__conv,axiom,
! [X2: int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X2 @ Xs )
= Zs ) )
| ? [Ys6: list_int] :
( ( ( cons_int @ X2 @ Ys6 )
= Ys )
& ( Xs
= ( append_int @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_916_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys5: list_nat,Y6: nat] :
( Xs
!= ( append_nat @ Ys5 @ ( cons_nat @ Y6 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_917_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys5: list_int,Y6: int] :
( Xs
!= ( append_int @ Ys5 @ ( cons_int @ Y6 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_918_rev__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ( P2 @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_919_rev__induct,axiom,
! [P2: list_int > $o,Xs: list_int] :
( ( P2 @ nil_int )
=> ( ! [X: int,Xs3: list_int] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_int @ Xs3 @ ( cons_int @ X @ nil_int ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_920_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys5: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_921_split__list,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys5: list_int,Zs3: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_922_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys5: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_923_split__list__last,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys5: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X2 @ Zs3 ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_924_split__list__prop,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys5: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_925_split__list__prop,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys5: list_int,X: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X @ Zs3 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_926_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys5: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_927_split__list__first,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys5: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X2 @ Zs3 ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_928_split__list__propE,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys5: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs3 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_929_split__list__propE,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys5: list_int,X: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X @ Zs3 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_930_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys7: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_931_append__Cons__eq__iff,axiom,
! [X2: int,Xs: list_int,Ys: list_int,Xs4: list_int,Ys7: list_int] :
( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int @ X2 @ ( set_int2 @ Ys ) )
=> ( ( ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) )
= ( append_int @ Xs4 @ ( cons_int @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_932_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs4: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs4 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_933_in__set__conv__decomp,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs4: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs4 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_934_split__list__last__prop,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys5: list_nat,X: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_935_split__list__last__prop,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys5: list_int,X: int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Zs3 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_936_split__list__first__prop,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys5: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Ys5 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_937_split__list__first__prop,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys5: list_int,X: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Ys5 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_938_split__list__last__propE,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys5: list_nat,X: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_939_split__list__last__propE,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys5: list_int,X: int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys5 @ ( cons_int @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Zs3 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_940_split__list__first__propE,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys5: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Ys5 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_941_split__list__first__propE,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys5: list_int,X: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys5 @ ( cons_int @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_int2 @ Ys5 ) )
=> ~ ( P2 @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_942_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs4: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs4 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs4 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_943_in__set__conv__decomp__last,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs4: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs4 ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Zs4 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_944_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs4: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs4 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_945_in__set__conv__decomp__first,axiom,
! [X2: int,Xs: list_int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs4: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs4 ) ) )
& ~ ( member_int @ X2 @ ( set_int2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_946_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_nat,X3: nat,Zs4: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs4 ) ) )
& ( P2 @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Zs4 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_947_split__list__last__prop__iff,axiom,
! [Xs: list_int,P2: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_int,X3: int,Zs4: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs4 ) ) )
& ( P2 @ X3 )
& ! [Y4: int] :
( ( member_int @ Y4 @ ( set_int2 @ Zs4 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_948_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_nat,X3: nat] :
( ? [Zs4: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs4 ) ) )
& ( P2 @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_949_split__list__first__prop__iff,axiom,
! [Xs: list_int,P2: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_int,X3: int] :
( ? [Zs4: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs4 ) ) )
& ( P2 @ X3 )
& ! [Y4: int] :
( ( member_int @ Y4 @ ( set_int2 @ Ys2 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_950_shuffles_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys5: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys5 ) )
=> ( ! [Xs3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
=> ~ ! [X: nat,Xs3: list_nat,Y6: nat,Ys5: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y6 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_951_shuffles_Ocases,axiom,
! [X2: produc1186641810826059865st_int] :
( ! [Ys5: list_int] :
( X2
!= ( produc364263696895485585st_int @ nil_int @ Ys5 ) )
=> ( ! [Xs3: list_int] :
( X2
!= ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
=> ~ ! [X: int,Xs3: list_int,Y6: int,Ys5: list_int] :
( X2
!= ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y6 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_952_subset__eq__mset__impl_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys5: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys5 ) )
=> ~ ! [X: nat,Xs3: list_nat,Ys5: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ Ys5 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_953_subset__eq__mset__impl_Ocases,axiom,
! [X2: produc1186641810826059865st_int] :
( ! [Ys5: list_int] :
( X2
!= ( produc364263696895485585st_int @ nil_int @ Ys5 ) )
=> ~ ! [X: int,Xs3: list_int,Ys5: list_int] :
( X2
!= ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ Ys5 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_954_distinct__singleton,axiom,
! [X2: nat] : ( distinct_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_singleton
thf(fact_955_distinct__singleton,axiom,
! [X2: int] : ( distinct_int @ ( cons_int @ X2 @ nil_int ) ) ).
% distinct_singleton
thf(fact_956_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_957_distinct_Osimps_I2_J,axiom,
! [X2: int,Xs: list_int] :
( ( distinct_int @ ( cons_int @ X2 @ Xs ) )
= ( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( distinct_int @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_958_mset_Osimps_I2_J,axiom,
! [A: nat,X2: list_nat] :
( ( mset_nat @ ( cons_nat @ A @ X2 ) )
= ( add_mset_nat @ A @ ( mset_nat @ X2 ) ) ) ).
% mset.simps(2)
thf(fact_959_mset_Osimps_I2_J,axiom,
! [A: int,X2: list_int] :
( ( mset_int @ ( cons_int @ A @ X2 ) )
= ( add_mset_int @ A @ ( mset_int @ X2 ) ) ) ).
% mset.simps(2)
thf(fact_960_list__update__code_I2_J,axiom,
! [X2: nat,Xs: list_nat,Y3: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
= ( cons_nat @ Y3 @ Xs ) ) ).
% list_update_code(2)
thf(fact_961_list__update__code_I2_J,axiom,
! [X2: int,Xs: list_int,Y3: int] :
( ( list_update_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
= ( cons_int @ Y3 @ Xs ) ) ).
% list_update_code(2)
thf(fact_962_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_963_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_964_listrel1I2,axiom,
! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int,X2: int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ X2 @ Ys ) ) @ ( listrel1_int @ R ) ) ) ).
% listrel1I2
thf(fact_965_zless__add1__eq,axiom,
! [W3: int,Z2: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W3 @ Z2 )
| ( W3 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_966_int__gr__induct,axiom,
! [K: int,I2: int,P2: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_967_int__less__induct,axiom,
! [I2: int,K: int,P2: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_968_union__fold__mset__add__mset,axiom,
( plus_p6334493942879108393et_nat
= ( fold_m2600682269844132093et_nat @ add_mset_nat ) ) ).
% union_fold_mset_add_mset
thf(fact_969_same__length__different,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != Ys )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ? [Pre: list_int,X: int,Xs5: list_int,Y6: int,Ys4: list_int] :
( ( X != Y6 )
& ( Xs
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X @ nil_int ) @ Xs5 ) ) )
& ( Ys
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y6 @ nil_int ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_970_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,X: nat,Xs5: list_nat,Y6: nat,Ys4: list_nat] :
( ( X != Y6 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y6 @ nil_nat ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_971_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs3: list_nat,Ys5: list_nat,Zs3: list_nat,Y6: nat] :
( Ws
= ( append_nat @ Xs3 @ ( append_nat @ ( cons_nat @ Y6 @ nil_nat ) @ ( append_nat @ Ys5 @ ( append_nat @ ( cons_nat @ Y6 @ nil_nat ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_972_not__distinct__decomp,axiom,
! [Ws: list_int] :
( ~ ( distinct_int @ Ws )
=> ? [Xs3: list_int,Ys5: list_int,Zs3: list_int,Y6: int] :
( Ws
= ( append_int @ Xs3 @ ( append_int @ ( cons_int @ Y6 @ nil_int ) @ ( append_int @ Ys5 @ ( append_int @ ( cons_int @ Y6 @ nil_int ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_973_not__distinct__conv__prefix,axiom,
! [As: list_nat] :
( ( ~ ( distinct_nat @ As ) )
= ( ? [Xs2: list_nat,Y4: nat,Ys2: list_nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 )
& ( As
= ( append_nat @ Xs2 @ ( cons_nat @ Y4 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_974_not__distinct__conv__prefix,axiom,
! [As: list_int] :
( ( ~ ( distinct_int @ As ) )
= ( ? [Xs2: list_int,Y4: int,Ys2: list_int] :
( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
& ( distinct_int @ Xs2 )
& ( As
= ( append_int @ Xs2 @ ( cons_int @ Y4 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_975_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X: nat] :
( ( Xs
= ( cons_nat @ X @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
=> ~ ! [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs3 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_976_Cons__listrel1E2,axiom,
! [Xs: list_int,Y3: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y3 @ Ys ) ) @ ( listrel1_int @ R ) )
=> ( ! [X: int] :
( ( Xs
= ( cons_int @ X @ Ys ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R ) )
=> ~ ! [Zs3: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Zs3 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs3 @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_977_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 ) )
=> ( ! [Y6: nat] :
( ( Ys
= ( cons_nat @ Y6 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y6 ) @ R ) )
=> ~ ! [Zs3: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs3 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_978_Cons__listrel1E1,axiom,
! [X2: int,Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ Ys ) @ ( listrel1_int @ R ) )
=> ( ! [Y6: int] :
( ( Ys
= ( cons_int @ Y6 @ Xs ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y6 ) @ R ) )
=> ~ ! [Zs3: list_int] :
( ( Ys
= ( cons_int @ X2 @ Zs3 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs3 ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_979_listrel1I1,axiom,
! [X2: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_980_listrel1I1,axiom,
! [X2: int,Y3: int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y3 @ Xs ) ) @ ( listrel1_int @ R ) ) ) ).
% listrel1I1
thf(fact_981_listrel_OCons,axiom,
! [X2: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_982_listrel_OCons,axiom,
! [X2: nat,Y3: int,R: set_Pr7995236796853374141at_int,Xs: list_nat,Ys: list_int] :
( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X2 @ Y3 ) @ R )
=> ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs @ Ys ) @ ( listrel_nat_int @ R ) )
=> ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ ( cons_nat @ X2 @ Xs ) @ ( cons_int @ Y3 @ Ys ) ) @ ( listrel_nat_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_983_listrel_OCons,axiom,
! [X2: int,Y3: nat,R: set_Pr3448869479623346877nt_nat,Xs: list_int,Ys: list_nat] :
( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X2 @ Y3 ) @ R )
=> ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs @ Ys ) @ ( listrel_int_nat @ R ) )
=> ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ ( cons_int @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_int_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_984_listrel_OCons,axiom,
! [X2: int,Y3: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y3 @ Ys ) ) @ ( listrel_int_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_985_listrel__Cons1,axiom,
! [Y3: nat,Ys: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [Y6: nat,Ys5: list_nat] :
( ( Xs
= ( cons_nat @ Y6 @ Ys5 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Y6 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys5 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_986_listrel__Cons1,axiom,
! [Y3: nat,Ys: list_nat,Xs: list_int,R: set_Pr7995236796853374141at_int] :
( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listrel_nat_int @ R ) )
=> ~ ! [Y6: int,Ys5: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys5 ) )
=> ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ Y3 @ Y6 ) @ R )
=> ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Ys @ Ys5 ) @ ( listrel_nat_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_987_listrel__Cons1,axiom,
! [Y3: int,Ys: list_int,Xs: list_nat,R: set_Pr3448869479623346877nt_nat] :
( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ ( cons_int @ Y3 @ Ys ) @ Xs ) @ ( listrel_int_nat @ R ) )
=> ~ ! [Y6: nat,Ys5: list_nat] :
( ( Xs
= ( cons_nat @ Y6 @ Ys5 ) )
=> ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ Y3 @ Y6 ) @ R )
=> ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Ys @ Ys5 ) @ ( listrel_int_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_988_listrel__Cons1,axiom,
! [Y3: int,Ys: list_int,Xs: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y3 @ Ys ) @ Xs ) @ ( listrel_int_int @ R ) )
=> ~ ! [Y6: int,Ys5: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys5 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Y6 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys5 ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_989_listrel__Cons2,axiom,
! [Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [X: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_990_listrel__Cons2,axiom,
! [Xs: list_int,Y3: nat,Ys: list_nat,R: set_Pr3448869479623346877nt_nat] :
( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_int_nat @ R ) )
=> ~ ! [X: int,Xs3: list_int] :
( ( Xs
= ( cons_int @ X @ Xs3 ) )
=> ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y3 ) @ R )
=> ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs3 @ Ys ) @ ( listrel_int_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_991_listrel__Cons2,axiom,
! [Xs: list_nat,Y3: int,Ys: list_int,R: set_Pr7995236796853374141at_int] :
( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs @ ( cons_int @ Y3 @ Ys ) ) @ ( listrel_nat_int @ R ) )
=> ~ ! [X: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs3 ) )
=> ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y3 ) @ R )
=> ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs3 @ Ys ) @ ( listrel_nat_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_992_listrel__Cons2,axiom,
! [Xs: list_int,Y3: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y3 @ Ys ) ) @ ( listrel_int_int @ R ) )
=> ~ ! [X: int,Xs3: list_int] :
( ( Xs
= ( cons_int @ X @ Xs3 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_993_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_994_remove1__split,axiom,
! [A: int,Xs: list_int,Ys: list_int] :
( ( member_int @ A @ ( set_int2 @ Xs ) )
=> ( ( ( remove1_int @ A @ Xs )
= Ys )
= ( ? [Ls2: list_int,Rs: list_int] :
( ( Xs
= ( append_int @ Ls2 @ ( cons_int @ A @ Rs ) ) )
& ~ ( member_int @ A @ ( set_int2 @ Ls2 ) )
& ( Ys
= ( append_int @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_995_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_996_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_997_nth__Cons_H,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_998_upt__eq__Cons__conv,axiom,
! [I2: nat,J: nat,X2: nat,Xs: list_nat] :
( ( ( upt @ I2 @ J )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ord_less_nat @ I2 @ J )
& ( I2 = X2 )
& ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_999_listrel_Osimps,axiom,
! [A1: list_nat,A23: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A23 ) @ ( listrel_nat_nat @ R ) )
= ( ( ( A1 = nil_nat )
& ( A23 = nil_nat ) )
| ? [X3: nat,Y4: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_1000_listrel_Osimps,axiom,
! [A1: list_nat,A23: list_int,R: set_Pr7995236796853374141at_int] :
( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ A1 @ A23 ) @ ( listrel_nat_int @ R ) )
= ( ( ( A1 = nil_nat )
& ( A23 = nil_int ) )
| ? [X3: nat,Y4: int,Xs2: list_nat,Ys2: list_int] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
& ( A23
= ( cons_int @ Y4 @ Ys2 ) )
& ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X3 @ Y4 ) @ R )
& ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs2 @ Ys2 ) @ ( listrel_nat_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_1001_listrel_Osimps,axiom,
! [A1: list_int,A23: list_nat,R: set_Pr3448869479623346877nt_nat] :
( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ A1 @ A23 ) @ ( listrel_int_nat @ R ) )
= ( ( ( A1 = nil_int )
& ( A23 = nil_nat ) )
| ? [X3: int,Y4: nat,Xs2: list_int,Ys2: list_nat] :
( ( A1
= ( cons_int @ X3 @ Xs2 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys2 ) )
& ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X3 @ Y4 ) @ R )
& ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs2 @ Ys2 ) @ ( listrel_int_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_1002_listrel_Osimps,axiom,
! [A1: list_int,A23: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A23 ) @ ( listrel_int_int @ R ) )
= ( ( ( A1 = nil_int )
& ( A23 = nil_int ) )
| ? [X3: int,Y4: int,Xs2: list_int,Ys2: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs2 ) )
& ( A23
= ( cons_int @ Y4 @ Ys2 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y4 ) @ R )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys2 ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_1003_listrel_Ocases,axiom,
! [A1: list_nat,A23: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A23 ) @ ( listrel_nat_nat @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A23 != nil_nat ) )
=> ~ ! [X: nat,Y6: nat,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs3 ) )
=> ! [Ys5: list_nat] :
( ( A23
= ( cons_nat @ Y6 @ Ys5 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys5 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1004_listrel_Ocases,axiom,
! [A1: list_nat,A23: list_int,R: set_Pr7995236796853374141at_int] :
( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ A1 @ A23 ) @ ( listrel_nat_int @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A23 != nil_int ) )
=> ~ ! [X: nat,Y6: int,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs3 ) )
=> ! [Ys5: list_int] :
( ( A23
= ( cons_int @ Y6 @ Ys5 ) )
=> ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y6 ) @ R )
=> ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs3 @ Ys5 ) @ ( listrel_nat_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1005_listrel_Ocases,axiom,
! [A1: list_int,A23: list_nat,R: set_Pr3448869479623346877nt_nat] :
( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ A1 @ A23 ) @ ( listrel_int_nat @ R ) )
=> ( ( ( A1 = nil_int )
=> ( A23 != nil_nat ) )
=> ~ ! [X: int,Y6: nat,Xs3: list_int] :
( ( A1
= ( cons_int @ X @ Xs3 ) )
=> ! [Ys5: list_nat] :
( ( A23
= ( cons_nat @ Y6 @ Ys5 ) )
=> ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y6 ) @ R )
=> ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs3 @ Ys5 ) @ ( listrel_int_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1006_listrel_Ocases,axiom,
! [A1: list_int,A23: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A23 ) @ ( listrel_int_int @ R ) )
=> ( ( ( A1 = nil_int )
=> ( A23 != nil_int ) )
=> ~ ! [X: int,Y6: int,Xs3: list_int] :
( ( A1
= ( cons_int @ X @ Xs3 ) )
=> ! [Ys5: list_int] :
( ( A23
= ( cons_int @ Y6 @ Ys5 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y6 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys5 ) @ ( listrel_int_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1007_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ~ ! [X: nat,Y6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R )
=> ! [Us3: list_nat,Vs3: list_nat] :
( ( Xs
= ( append_nat @ Us3 @ ( cons_nat @ X @ Vs3 ) ) )
=> ( Ys
!= ( append_nat @ Us3 @ ( cons_nat @ Y6 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1008_listrel1E,axiom,
! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
=> ~ ! [X: int,Y6: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y6 ) @ R )
=> ! [Us3: list_int,Vs3: list_int] :
( ( Xs
= ( append_int @ Us3 @ ( cons_int @ X @ Vs3 ) ) )
=> ( Ys
!= ( append_int @ Us3 @ ( cons_int @ Y6 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1009_listrel1I,axiom,
! [X2: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_nat @ Us @ ( cons_nat @ Y3 @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_1010_listrel1I,axiom,
! [X2: int,Y3: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Us: list_int,Vs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
=> ( ( Xs
= ( append_int @ Us @ ( cons_int @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_int @ Us @ ( cons_int @ Y3 @ Vs ) ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).
% listrel1I
thf(fact_1011_nth__non__equal__first__eq,axiom,
! [X2: nat,Y3: nat,Xs: list_nat,N: nat] :
( ( X2 != Y3 )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= Y3 )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1012_nth__non__equal__first__eq,axiom,
! [X2: int,Y3: int,Xs: list_int,N: nat] :
( ( X2 != Y3 )
=> ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
= Y3 )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1013_ns__mul__ext__union__compat,axiom,
! [A2: multiset_nat,B2: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,C3: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ( ( locally_refl_nat @ Ns @ C3 )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B2 @ C3 ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ).
% ns_mul_ext_union_compat
thf(fact_1014_s__mul__ext__union__compat,axiom,
! [A2: multiset_nat,B2: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,C3: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) )
=> ( ( locally_refl_nat @ Ns @ C3 )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B2 @ C3 ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ).
% s_mul_ext_union_compat
thf(fact_1015_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
& ( X2 = Y3 ) )
| ( ( Xs = Ys )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1016_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_int,X2: int,Ys: list_int,Y3: int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) @ ( append_int @ Ys @ ( cons_int @ Y3 @ nil_int ) ) ) @ ( listrel1_int @ R ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
& ( X2 = Y3 ) )
| ( ( Xs = Ys )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1017_sorted__list__of__multiset__singleton,axiom,
! [X2: nat] :
( ( linord3047872887403683810et_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_list_of_multiset_singleton
thf(fact_1018_sorted__list__of__multiset__singleton,axiom,
! [X2: int] :
( ( linord3045382416894633534et_int @ ( add_mset_int @ X2 @ zero_z3170743180189231877et_int ) )
= ( cons_int @ X2 @ nil_int ) ) ).
% sorted_list_of_multiset_singleton
thf(fact_1019_lr__trans__r,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: multiset_nat,B2: multiset_nat] :
( ( locally_refl_nat @ R2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) )
=> ( locally_refl_nat @ R2 @ B2 ) ) ).
% lr_trans_r
thf(fact_1020_mset__sorted__list__of__multiset,axiom,
! [M3: multiset_nat] :
( ( mset_nat @ ( linord3047872887403683810et_nat @ M3 ) )
= M3 ) ).
% mset_sorted_list_of_multiset
thf(fact_1021_sorted__list__of__multiset__empty,axiom,
( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
= nil_nat ) ).
% sorted_list_of_multiset_empty
thf(fact_1022_set__sorted__list__of__multiset,axiom,
! [M3: multiset_nat] :
( ( set_nat2 @ ( linord3047872887403683810et_nat @ M3 ) )
= ( set_mset_nat @ M3 ) ) ).
% set_sorted_list_of_multiset
thf(fact_1023_lr__trans__l,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: multiset_nat,B2: multiset_nat] :
( ( locally_refl_nat @ R2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) )
=> ( locally_refl_nat @ R2 @ A2 ) ) ).
% lr_trans_l
thf(fact_1024_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1025_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_1026_distinct__n__lists,axiom,
! [Xs: list_nat,N: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_list_nat @ ( n_lists_nat @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_1027_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_1028_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I ) @ Js @ ( upto_aux @ I @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1029_distinct__product__lists,axiom,
! [Xss2: list_list_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xss2 ) )
=> ( distinct_nat @ X ) )
=> ( distinct_list_nat @ ( product_lists_nat @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_1030_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_1031_the__elem__set,axiom,
! [X2: int] :
( ( the_elem_int @ ( set_int2 @ ( cons_int @ X2 @ nil_int ) ) )
= X2 ) ).
% the_elem_set
thf(fact_1032_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_1033_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_1034_Cons__lenlex__iff,axiom,
! [M2: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M2 @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R ) )
= ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
| ( ( ( size_size_list_int @ Ms )
= ( size_size_list_int @ Ns ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1035_Cons__lenlex__iff,axiom,
! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ 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 @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1036_Cons__in__lex,axiom,
! [X2: int,Xs: list_int,Y3: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y3 @ Ys ) ) @ ( lex_int @ R ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
& ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) ) )
| ( ( X2 = Y3 )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1037_Cons__in__lex,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X2 = Y3 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1038_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_1039_Nil__notin__lex,axiom,
! [Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R ) ) ).
% Nil_notin_lex
thf(fact_1040_Nil2__notin__lex,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R ) ) ).
% Nil2_notin_lex
thf(fact_1041_lex__append__leftI,axiom,
! [Ys: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) ) ) ).
% lex_append_leftI
thf(fact_1042_lenlex__irreflexive,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
( ! [X: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lenlex_int @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_1043_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_1044_lex__append__leftD,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_1045_lex__append__leftD,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ! [X: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ R )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Xs @ Zs ) ) @ ( lex_int @ R ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lex_int @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_1046_lex__append__left__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_1047_lex__append__left__iff,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ! [X: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ R )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Xs @ Zs ) ) @ ( lex_int @ R ) )
= ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lex_int @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_1048_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_1049_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_1050_product_Osimps_I2_J,axiom,
! [X2: int,Xs: list_int,Ys: list_int] :
( ( product_int_int @ ( cons_int @ X2 @ Xs ) @ Ys )
= ( append7030698103840186580nt_int @ ( map_in7157766398909135175nt_int @ ( product_Pair_int_int @ X2 ) @ Ys ) @ ( product_int_int @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_1051_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_1052_bind__simps_I2_J,axiom,
! [X2: int,Xs: list_int,F: int > list_nat] :
( ( bind_int_nat @ ( cons_int @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_int_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1053_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_1054_distinct__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( distinct_nat @ Ys )
=> ( distin6923225563576452346at_nat @ ( product_nat_nat @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_1055_s__mul__ext__IdI,axiom,
! [X5: multiset_int,M3: multiset_int,Z5: multiset_int,N4: multiset_int,Y5: multiset_int,R2: set_Pr958786334691620121nt_int] :
( ( X5 != zero_z3170743180189231877et_int )
=> ( ( M3
= ( plus_p2156642923369911685et_int @ X5 @ Z5 ) )
=> ( ( N4
= ( plus_p2156642923369911685et_int @ Y5 @ Z5 ) )
=> ( ! [X: int] :
( ( member_int @ X @ ( set_mset_int @ Y5 ) )
=> ? [Xa3: int] :
( ( member_int @ Xa3 @ ( set_mset_int @ X5 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa3 @ X ) @ R2 ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ M3 @ N4 ) @ ( multis4212723674092261717xt_int @ id_int @ R2 ) ) ) ) ) ) ).
% s_mul_ext_IdI
thf(fact_1056_s__mul__ext__IdI,axiom,
! [X5: multiset_nat,M3: multiset_nat,Z5: multiset_nat,N4: multiset_nat,Y5: multiset_nat,R2: set_Pr1261947904930325089at_nat] :
( ( X5 != zero_z7348594199698428585et_nat )
=> ( ( M3
= ( plus_p6334493942879108393et_nat @ X5 @ Z5 ) )
=> ( ( N4
= ( plus_p6334493942879108393et_nat @ Y5 @ Z5 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ Y5 ) )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_mset_nat @ X5 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X ) @ R2 ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M3 @ N4 ) @ ( multis4215214144601311993xt_nat @ id_nat @ R2 ) ) ) ) ) ) ).
% s_mul_ext_IdI
thf(fact_1057_s__mul__ext__IdE,axiom,
! [M3: multiset_int,N4: multiset_int,R2: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ M3 @ N4 ) @ ( multis4212723674092261717xt_int @ id_int @ R2 ) )
=> ~ ! [X7: multiset_int] :
( ( X7 != zero_z3170743180189231877et_int )
=> ! [Y7: multiset_int,Z6: multiset_int] :
( ( M3
= ( plus_p2156642923369911685et_int @ X7 @ Z6 ) )
=> ( ( N4
= ( plus_p2156642923369911685et_int @ Y7 @ Z6 ) )
=> ~ ! [X6: int] :
( ( member_int @ X6 @ ( set_mset_int @ Y7 ) )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ ( set_mset_int @ X7 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ) ) ).
% s_mul_ext_IdE
thf(fact_1058_s__mul__ext__IdE,axiom,
! [M3: multiset_nat,N4: multiset_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M3 @ N4 ) @ ( multis4215214144601311993xt_nat @ id_nat @ R2 ) )
=> ~ ! [X7: multiset_nat] :
( ( X7 != zero_z7348594199698428585et_nat )
=> ! [Y7: multiset_nat,Z6: multiset_nat] :
( ( M3
= ( plus_p6334493942879108393et_nat @ X7 @ Z6 ) )
=> ( ( N4
= ( plus_p6334493942879108393et_nat @ Y7 @ Z6 ) )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ ( set_mset_nat @ Y7 ) )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_mset_nat @ X7 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ) ) ).
% s_mul_ext_IdE
thf(fact_1059_ns__mul__ext__IdE,axiom,
! [M3: multiset_int,N4: multiset_int,R2: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ M3 @ N4 ) @ ( multis8828838126066458039xt_int @ id_int @ R2 ) )
=> ~ ! [X7: multiset_int,Y7: multiset_int,Z6: multiset_int] :
( ( M3
= ( plus_p2156642923369911685et_int @ X7 @ Z6 ) )
=> ( ( N4
= ( plus_p2156642923369911685et_int @ Y7 @ Z6 ) )
=> ~ ! [X6: int] :
( ( member_int @ X6 @ ( set_mset_int @ Y7 ) )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ ( set_mset_int @ X7 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ) ).
% ns_mul_ext_IdE
thf(fact_1060_ns__mul__ext__IdE,axiom,
! [M3: multiset_nat,N4: multiset_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M3 @ N4 ) @ ( multis8831328596575508315xt_nat @ id_nat @ R2 ) )
=> ~ ! [X7: multiset_nat,Y7: multiset_nat,Z6: multiset_nat] :
( ( M3
= ( plus_p6334493942879108393et_nat @ X7 @ Z6 ) )
=> ( ( N4
= ( plus_p6334493942879108393et_nat @ Y7 @ Z6 ) )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ ( set_mset_nat @ Y7 ) )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_mset_nat @ X7 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ) ).
% ns_mul_ext_IdE
thf(fact_1061_ns__mul__ext__IdI,axiom,
! [M3: multiset_int,X5: multiset_int,Z5: multiset_int,N4: multiset_int,Y5: multiset_int,R2: set_Pr958786334691620121nt_int] :
( ( M3
= ( plus_p2156642923369911685et_int @ X5 @ Z5 ) )
=> ( ( N4
= ( plus_p2156642923369911685et_int @ Y5 @ Z5 ) )
=> ( ! [X: int] :
( ( member_int @ X @ ( set_mset_int @ Y5 ) )
=> ? [Xa3: int] :
( ( member_int @ Xa3 @ ( set_mset_int @ X5 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa3 @ X ) @ R2 ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ M3 @ N4 ) @ ( multis8828838126066458039xt_int @ id_int @ R2 ) ) ) ) ) ).
% ns_mul_ext_IdI
thf(fact_1062_ns__mul__ext__IdI,axiom,
! [M3: multiset_nat,X5: multiset_nat,Z5: multiset_nat,N4: multiset_nat,Y5: multiset_nat,R2: set_Pr1261947904930325089at_nat] :
( ( M3
= ( plus_p6334493942879108393et_nat @ X5 @ Z5 ) )
=> ( ( N4
= ( plus_p6334493942879108393et_nat @ Y5 @ Z5 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ Y5 ) )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_mset_nat @ X5 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X ) @ R2 ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M3 @ N4 ) @ ( multis8831328596575508315xt_nat @ id_nat @ R2 ) ) ) ) ) ).
% ns_mul_ext_IdI
thf(fact_1063_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1064_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_1065_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1066_s__mul__extI,axiom,
! [A2: multiset_int,A12: multiset_int,A22: multiset_int,B2: multiset_int,B1: multiset_int,B22: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( A2
= ( plus_p2156642923369911685et_int @ A12 @ A22 ) )
=> ( ( B2
= ( plus_p2156642923369911685et_int @ B1 @ B22 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A12 @ B1 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ( A22 != zero_z3170743180189231877et_int )
=> ( ! [B5: int] :
( ( member_int @ B5 @ ( set_mset_int @ B22 ) )
=> ? [A6: int] :
( ( member_int @ A6 @ ( set_mset_int @ A22 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B5 ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis4212723674092261717xt_int @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_1067_s__mul__extI,axiom,
! [A2: multiset_nat,A12: multiset_nat,A22: multiset_nat,B2: multiset_nat,B1: multiset_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ A12 @ A22 ) )
=> ( ( B2
= ( plus_p6334493942879108393et_nat @ B1 @ B22 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A12 @ B1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ( A22 != zero_z7348594199698428585et_nat )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ ( set_mset_nat @ B22 ) )
=> ? [A6: nat] :
( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_1068_s__mul__extE,axiom,
! [A2: multiset_int,B2: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis4212723674092261717xt_int @ Ns @ S ) )
=> ~ ! [A13: multiset_int,A24: multiset_int] :
( ( A2
= ( plus_p2156642923369911685et_int @ A13 @ A24 ) )
=> ! [B12: multiset_int,B23: multiset_int] :
( ( B2
= ( plus_p2156642923369911685et_int @ B12 @ B23 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A13 @ B12 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ( A24 != zero_z3170743180189231877et_int )
=> ~ ! [B6: int] :
( ( member_int @ B6 @ ( set_mset_int @ B23 ) )
=> ? [A5: int] :
( ( member_int @ A5 @ ( set_mset_int @ A24 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A5 @ B6 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_1069_s__mul__extE,axiom,
! [A2: multiset_nat,B2: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) )
=> ~ ! [A13: multiset_nat,A24: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ A13 @ A24 ) )
=> ! [B12: multiset_nat,B23: multiset_nat] :
( ( B2
= ( plus_p6334493942879108393et_nat @ B12 @ B23 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A13 @ B12 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ( A24 != zero_z7348594199698428585et_nat )
=> ~ ! [B6: nat] :
( ( member_nat @ B6 @ ( set_mset_nat @ B23 ) )
=> ? [A5: nat] :
( ( member_nat @ A5 @ ( set_mset_nat @ A24 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B6 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_1070_ns__mul__extE,axiom,
! [A2: multiset_int,B2: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis8828838126066458039xt_int @ Ns @ S ) )
=> ~ ! [A13: multiset_int,A24: multiset_int] :
( ( A2
= ( plus_p2156642923369911685et_int @ A13 @ A24 ) )
=> ! [B12: multiset_int,B23: multiset_int] :
( ( B2
= ( plus_p2156642923369911685et_int @ B12 @ B23 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A13 @ B12 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ~ ! [B6: int] :
( ( member_int @ B6 @ ( set_mset_int @ B23 ) )
=> ? [A5: int] :
( ( member_int @ A5 @ ( set_mset_int @ A24 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A5 @ B6 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_1071_ns__mul__extE,axiom,
! [A2: multiset_nat,B2: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ~ ! [A13: multiset_nat,A24: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ A13 @ A24 ) )
=> ! [B12: multiset_nat,B23: multiset_nat] :
( ( B2
= ( plus_p6334493942879108393et_nat @ B12 @ B23 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A13 @ B12 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [B6: nat] :
( ( member_nat @ B6 @ ( set_mset_nat @ B23 ) )
=> ? [A5: nat] :
( ( member_nat @ A5 @ ( set_mset_nat @ A24 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B6 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_1072_ns__mul__extI,axiom,
! [A2: multiset_int,A12: multiset_int,A22: multiset_int,B2: multiset_int,B1: multiset_int,B22: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( A2
= ( plus_p2156642923369911685et_int @ A12 @ A22 ) )
=> ( ( B2
= ( plus_p2156642923369911685et_int @ B1 @ B22 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A12 @ B1 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ! [B5: int] :
( ( member_int @ B5 @ ( set_mset_int @ B22 ) )
=> ? [A6: int] :
( ( member_int @ A6 @ ( set_mset_int @ A22 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B5 ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A2 @ B2 ) @ ( multis8828838126066458039xt_int @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_1073_ns__mul__extI,axiom,
! [A2: multiset_nat,A12: multiset_nat,A22: multiset_nat,B2: multiset_nat,B1: multiset_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ A12 @ A22 ) )
=> ( ( B2
= ( plus_p6334493942879108393et_nat @ B1 @ B22 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A12 @ B1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ ( set_mset_nat @ B22 ) )
=> ? [A6: nat] :
( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B2 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_1074_multpw__listI,axiom,
! [Xs: list_int,Ys: list_int,X5: multiset_int,Y5: multiset_int,Ns: set_Pr958786334691620121nt_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( X5
= ( mset_int @ Xs ) )
=> ( ( Y5
= ( mset_int @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Ys @ I3 ) ) @ Ns ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis5149657266184778664pw_int @ Ns ) ) ) ) ) ) ).
% multpw_listI
thf(fact_1075_multpw__listI,axiom,
! [Xs: list_nat,Ys: list_nat,X5: multiset_nat,Y5: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( X5
= ( mset_nat @ Xs ) )
=> ( ( Y5
= ( mset_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ Ns ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis5152147736693828940pw_nat @ Ns ) ) ) ) ) ) ).
% multpw_listI
thf(fact_1076_multpw__listE,axiom,
! [X5: multiset_int,Y5: multiset_int,Ns: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ~ ! [Xs3: list_int,Ys5: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( X5
= ( mset_int @ Xs3 ) )
=> ( ( Y5
= ( mset_int @ Ys5 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_int @ Ys5 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs3 @ I5 ) @ ( nth_int @ Ys5 @ I5 ) ) @ Ns ) ) ) ) ) ) ).
% multpw_listE
thf(fact_1077_multpw__listE,axiom,
! [X5: multiset_nat,Y5: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [Xs3: list_nat,Ys5: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys5 ) )
=> ( ( X5
= ( mset_nat @ Xs3 ) )
=> ( ( Y5
= ( mset_nat @ Ys5 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Ys5 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I5 ) @ ( nth_nat @ Ys5 @ I5 ) ) @ Ns ) ) ) ) ) ) ).
% multpw_listE
thf(fact_1078_multpw__add,axiom,
! [X1: multiset_nat,Y1: multiset_nat,Ns: set_Pr1261947904930325089at_nat,X23: multiset_nat,Y23: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y23 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X1 @ X23 ) @ ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) ) @ ( multis5152147736693828940pw_nat @ Ns ) ) ) ) ).
% multpw_add
thf(fact_1079_multpw__splitL,axiom,
! [X5: multiset_nat,Y1: multiset_nat,Y23: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [X13: multiset_nat,X24: multiset_nat] :
( ( X5
= ( plus_p6334493942879108393et_nat @ X13 @ X24 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X13 @ Y1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X24 @ Y23 ) @ ( multis5152147736693828940pw_nat @ Ns ) ) ) ) ) ).
% multpw_splitL
thf(fact_1080_multpw__splitR,axiom,
! [X1: multiset_nat,X23: multiset_nat,Y5: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X1 @ X23 ) @ Y5 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [Y13: multiset_nat,Y24: multiset_nat] :
( ( Y5
= ( plus_p6334493942879108393et_nat @ Y13 @ Y24 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y13 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y24 ) @ ( multis5152147736693828940pw_nat @ Ns ) ) ) ) ) ).
% multpw_splitR
thf(fact_1081_multpw_Ocases,axiom,
! [A1: multiset_int,A23: multiset_int,Ns: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A1 @ A23 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ( ( A1 = zero_z3170743180189231877et_int )
=> ( A23 != zero_z3170743180189231877et_int ) )
=> ~ ! [X: int,Y6: int,X7: multiset_int] :
( ( A1
= ( add_mset_int @ X @ X7 ) )
=> ! [Y7: multiset_int] :
( ( A23
= ( add_mset_int @ Y6 @ Y7 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y6 ) @ Ns )
=> ~ ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X7 @ Y7 ) @ ( multis5149657266184778664pw_int @ Ns ) ) ) ) ) ) ) ).
% multpw.cases
thf(fact_1082_multpw_Osimps,axiom,
! [A1: multiset_int,A23: multiset_int,Ns: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ A1 @ A23 ) @ ( multis5149657266184778664pw_int @ Ns ) )
= ( ( ( A1 = zero_z3170743180189231877et_int )
& ( A23 = zero_z3170743180189231877et_int ) )
| ? [X3: int,Y4: int,X4: multiset_int,Y8: multiset_int] :
( ( A1
= ( add_mset_int @ X3 @ X4 ) )
& ( A23
= ( add_mset_int @ Y4 @ Y8 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y4 ) @ Ns )
& ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X4 @ Y8 ) @ ( multis5149657266184778664pw_int @ Ns ) ) ) ) ) ).
% multpw.simps
thf(fact_1083_multpw__single,axiom,
! [X2: int,Y3: int,Ns: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ Ns )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( add_mset_int @ X2 @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ Y3 @ zero_z3170743180189231877et_int ) ) @ ( multis5149657266184778664pw_int @ Ns ) ) ) ).
% multpw_single
thf(fact_1084_mult2__altE,axiom,
! [X5: multiset_int,Y5: multiset_int,B: $o,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis2693779972258166040lt_int @ B @ Ns @ S ) )
=> ~ ! [X13: multiset_int,X24: multiset_int] :
( ( X5
= ( plus_p2156642923369911685et_int @ X13 @ X24 ) )
=> ! [Y13: multiset_int,Y24: multiset_int] :
( ( Y5
= ( plus_p2156642923369911685et_int @ Y13 @ Y24 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X13 @ Y13 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ( B
| ( Y24 != zero_z3170743180189231877et_int ) )
=> ~ ! [X6: int] :
( ( member_int @ X6 @ ( set_mset_int @ X24 ) )
=> ? [Y6: int] :
( ( member_int @ Y6 @ ( set_mset_int @ Y24 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Y6 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_1085_mult2__altE,axiom,
! [X5: multiset_nat,Y5: multiset_nat,B: $o,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis2696270442767216316lt_nat @ B @ Ns @ S ) )
=> ~ ! [X13: multiset_nat,X24: multiset_nat] :
( ( X5
= ( plus_p6334493942879108393et_nat @ X13 @ X24 ) )
=> ! [Y13: multiset_nat,Y24: multiset_nat] :
( ( Y5
= ( plus_p6334493942879108393et_nat @ Y13 @ Y24 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X13 @ Y13 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ( B
| ( Y24 != zero_z7348594199698428585et_nat ) )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ ( set_mset_nat @ X24 ) )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ ( set_mset_nat @ Y24 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y6 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_1086_mult2__altI,axiom,
! [X5: multiset_int,X1: multiset_int,X23: multiset_int,Y5: multiset_int,Y1: multiset_int,Y23: multiset_int,Ns: set_Pr958786334691620121nt_int,B: $o,S: set_Pr958786334691620121nt_int] :
( ( X5
= ( plus_p2156642923369911685et_int @ X1 @ X23 ) )
=> ( ( Y5
= ( plus_p2156642923369911685et_int @ Y1 @ Y23 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X1 @ Y1 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ( B
| ( Y23 != zero_z3170743180189231877et_int ) )
=> ( ! [X: int] :
( ( member_int @ X @ ( set_mset_int @ X23 ) )
=> ? [Y: int] :
( ( member_int @ Y @ ( set_mset_int @ Y23 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis2693779972258166040lt_int @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_1087_mult2__altI,axiom,
! [X5: multiset_nat,X1: multiset_nat,X23: multiset_nat,Y5: multiset_nat,Y1: multiset_nat,Y23: multiset_nat,Ns: set_Pr1261947904930325089at_nat,B: $o,S: set_Pr1261947904930325089at_nat] :
( ( X5
= ( plus_p6334493942879108393et_nat @ X1 @ X23 ) )
=> ( ( Y5
= ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ( B
| ( Y23 != zero_z7348594199698428585et_nat ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ X23 ) )
=> ? [Y: nat] :
( ( member_nat @ Y @ ( set_mset_nat @ Y23 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis2696270442767216316lt_nat @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_1088_mult2__alt__add,axiom,
! [X1: multiset_nat,Y1: multiset_nat,B13: $o,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X23: multiset_nat,Y23: multiset_nat,B24: $o] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis2696270442767216316lt_nat @ B13 @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y23 ) @ ( multis2696270442767216316lt_nat @ B24 @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X1 @ X23 ) @ ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) )
@ ( multis2696270442767216316lt_nat
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ).
% mult2_alt_add
thf(fact_1089_mult2__alt__s__s__add,axiom,
! [X1: multiset_nat,Y1: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X23: multiset_nat,Y23: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y23 ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X1 @ X23 ) @ ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_s_s_add
thf(fact_1090_mult2__alt__ns__s__add,axiom,
! [X1: multiset_nat,Y1: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X23: multiset_nat,Y23: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis2696270442767216316lt_nat @ $true @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y23 ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X1 @ X23 ) @ ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_ns_s_add
thf(fact_1091_mult2__alt__s__ns__add,axiom,
! [X1: multiset_nat,Y1: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X23: multiset_nat,Y23: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y23 ) @ ( multis2696270442767216316lt_nat @ $true @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X1 @ X23 ) @ ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_s_ns_add
thf(fact_1092_mult2__alt__ns__ns__add,axiom,
! [X1: multiset_nat,Y1: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X23: multiset_nat,Y23: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis2696270442767216316lt_nat @ $true @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y23 ) @ ( multis2696270442767216316lt_nat @ $true @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X1 @ X23 ) @ ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) ) @ ( multis2696270442767216316lt_nat @ $true @ Ns @ S ) ) ) ) ).
% mult2_alt_ns_ns_add
thf(fact_1093_mult2__alt__s__single,axiom,
! [A: int,B: int,S: set_Pr958786334691620121nt_int,Ns: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( add_mset_int @ A @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ B @ zero_z3170743180189231877et_int ) ) @ ( multis2693779972258166040lt_int @ $false @ Ns @ S ) ) ) ).
% mult2_alt_s_single
thf(fact_1094_mult2__alt__nsI,axiom,
! [X5: multiset_int,X1: multiset_int,X23: multiset_int,Y5: multiset_int,Y1: multiset_int,Y23: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( X5
= ( plus_p2156642923369911685et_int @ X1 @ X23 ) )
=> ( ( Y5
= ( plus_p2156642923369911685et_int @ Y1 @ Y23 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X1 @ Y1 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ! [X: int] :
( ( member_int @ X @ ( set_mset_int @ X23 ) )
=> ? [Y: int] :
( ( member_int @ Y @ ( set_mset_int @ Y23 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis2693779972258166040lt_int @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_1095_mult2__alt__nsI,axiom,
! [X5: multiset_nat,X1: multiset_nat,X23: multiset_nat,Y5: multiset_nat,Y1: multiset_nat,Y23: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( X5
= ( plus_p6334493942879108393et_nat @ X1 @ X23 ) )
=> ( ( Y5
= ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ X23 ) )
=> ? [Y: nat] :
( ( member_nat @ Y @ ( set_mset_nat @ Y23 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis2696270442767216316lt_nat @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_1096_mult2__alt__nsE,axiom,
! [X5: multiset_int,Y5: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis2693779972258166040lt_int @ $true @ Ns @ S ) )
=> ~ ! [X13: multiset_int,X24: multiset_int] :
( ( X5
= ( plus_p2156642923369911685et_int @ X13 @ X24 ) )
=> ! [Y13: multiset_int,Y24: multiset_int] :
( ( Y5
= ( plus_p2156642923369911685et_int @ Y13 @ Y24 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X13 @ Y13 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ~ ! [X6: int] :
( ( member_int @ X6 @ ( set_mset_int @ X24 ) )
=> ? [Y6: int] :
( ( member_int @ Y6 @ ( set_mset_int @ Y24 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Y6 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_1097_mult2__alt__nsE,axiom,
! [X5: multiset_nat,Y5: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis2696270442767216316lt_nat @ $true @ Ns @ S ) )
=> ~ ! [X13: multiset_nat,X24: multiset_nat] :
( ( X5
= ( plus_p6334493942879108393et_nat @ X13 @ X24 ) )
=> ! [Y13: multiset_nat,Y24: multiset_nat] :
( ( Y5
= ( plus_p6334493942879108393et_nat @ Y13 @ Y24 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X13 @ Y13 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ ( set_mset_nat @ X24 ) )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ ( set_mset_nat @ Y24 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y6 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_1098_mult2__alt__sI,axiom,
! [X5: multiset_int,X1: multiset_int,X23: multiset_int,Y5: multiset_int,Y1: multiset_int,Y23: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( X5
= ( plus_p2156642923369911685et_int @ X1 @ X23 ) )
=> ( ( Y5
= ( plus_p2156642923369911685et_int @ Y1 @ Y23 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X1 @ Y1 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ( Y23 != zero_z3170743180189231877et_int )
=> ( ! [X: int] :
( ( member_int @ X @ ( set_mset_int @ X23 ) )
=> ? [Y: int] :
( ( member_int @ Y @ ( set_mset_int @ Y23 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis2693779972258166040lt_int @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_1099_mult2__alt__sI,axiom,
! [X5: multiset_nat,X1: multiset_nat,X23: multiset_nat,Y5: multiset_nat,Y1: multiset_nat,Y23: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( X5
= ( plus_p6334493942879108393et_nat @ X1 @ X23 ) )
=> ( ( Y5
= ( plus_p6334493942879108393et_nat @ Y1 @ Y23 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X1 @ Y1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ( Y23 != zero_z7348594199698428585et_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ X23 ) )
=> ? [Y: nat] :
( ( member_nat @ Y @ ( set_mset_nat @ Y23 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_1100_mult2__alt__sE,axiom,
! [X5: multiset_int,Y5: multiset_int,Ns: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X5 @ Y5 ) @ ( multis2693779972258166040lt_int @ $false @ Ns @ S ) )
=> ~ ! [X13: multiset_int,X24: multiset_int] :
( ( X5
= ( plus_p2156642923369911685et_int @ X13 @ X24 ) )
=> ! [Y13: multiset_int,Y24: multiset_int] :
( ( Y5
= ( plus_p2156642923369911685et_int @ Y13 @ Y24 ) )
=> ( ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ X13 @ Y13 ) @ ( multis5149657266184778664pw_int @ Ns ) )
=> ( ( Y24 != zero_z3170743180189231877et_int )
=> ~ ! [X6: int] :
( ( member_int @ X6 @ ( set_mset_int @ X24 ) )
=> ? [Y6: int] :
( ( member_int @ Y6 @ ( set_mset_int @ Y24 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Y6 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_1101_mult2__alt__sE,axiom,
! [X5: multiset_nat,Y5: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X5 @ Y5 ) @ ( multis2696270442767216316lt_nat @ $false @ Ns @ S ) )
=> ~ ! [X13: multiset_nat,X24: multiset_nat] :
( ( X5
= ( plus_p6334493942879108393et_nat @ X13 @ X24 ) )
=> ! [Y13: multiset_nat,Y24: multiset_nat] :
( ( Y5
= ( plus_p6334493942879108393et_nat @ Y13 @ Y24 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X13 @ Y13 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ( Y24 != zero_z7348594199698428585et_nat )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ ( set_mset_nat @ X24 ) )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ ( set_mset_nat @ Y24 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y6 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_1102_bezw__0,axiom,
! [X2: nat] :
( ( bezw @ X2 @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1103_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_1104_last__appendR,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_1105_last__appendL,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_1106_last__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( last_nat @ ( upt @ I2 @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1107_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_1108_last__snoc,axiom,
! [Xs: list_int,X2: int] :
( ( last_int @ ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) )
= X2 ) ).
% last_snoc
thf(fact_1109_last__zip,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != nil_int )
=> ( ( Ys != nil_int )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( last_P3305686521732843992nt_int @ ( zip_int_int @ Xs @ Ys ) )
= ( product_Pair_int_int @ ( last_int @ Xs ) @ ( last_int @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1110_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_1111_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_1112_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_1113_last__ConsR,axiom,
! [Xs: list_int,X2: int] :
( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X2 @ Xs ) )
= ( last_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_1114_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_1115_last__ConsL,axiom,
! [Xs: list_int,X2: int] :
( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_1116_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_1117_last_Osimps,axiom,
! [Xs: list_int,X2: int] :
( ( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X2 @ Xs ) )
= ( last_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_1118_last__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_1119_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs5: list_nat,Ys4: list_nat] :
( ( Xs
= ( append_nat @ Xs5 @ Ss ) )
& ( Ys
= ( append_nat @ Ys4 @ Ss ) )
& ( ( Xs5 = nil_nat )
| ( Ys4 = nil_nat )
| ( ( last_nat @ Xs5 )
!= ( last_nat @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_1120_last__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_1121_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_1122_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_1123_append__butlast__last__id,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( append_int @ ( butlast_int @ Xs ) @ ( cons_int @ ( last_int @ Xs ) @ nil_int ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_1124_one__step__implies__mult,axiom,
! [J3: multiset_int,K3: multiset_int,R: set_Pr958786334691620121nt_int,I4: multiset_int] :
( ( J3 != zero_z3170743180189231877et_int )
=> ( ! [X: int] :
( ( member_int @ X @ ( set_mset_int @ K3 ) )
=> ? [Xa3: int] :
( ( member_int @ Xa3 @ ( set_mset_int @ J3 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Xa3 ) @ R ) ) )
=> ( member2849730630941545090et_int @ ( produc1570911416673723153et_int @ ( plus_p2156642923369911685et_int @ I4 @ K3 ) @ ( plus_p2156642923369911685et_int @ I4 @ J3 ) ) @ ( mult_int @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_1125_one__step__implies__mult,axiom,
! [J3: multiset_nat,K3: multiset_nat,R: set_Pr1261947904930325089at_nat,I4: multiset_nat] :
( ( J3 != zero_z7348594199698428585et_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ K3 ) )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_mset_nat @ J3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Xa3 ) @ R ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ I4 @ K3 ) @ ( plus_p6334493942879108393et_nat @ I4 @ J3 ) ) @ ( mult_nat @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_1126_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1127_butlast__snoc,axiom,
! [Xs: list_int,X2: int] :
( ( butlast_int @ ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1128_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_1129_map__butlast,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_1130_butlast__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1131_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_1132_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_1133_butlast_Osimps_I2_J,axiom,
! [Xs: list_int,X2: int] :
( ( ( Xs = nil_int )
=> ( ( butlast_int @ ( cons_int @ X2 @ Xs ) )
= nil_int ) )
& ( ( Xs != nil_int )
=> ( ( butlast_int @ ( cons_int @ X2 @ Xs ) )
= ( cons_int @ X2 @ ( butlast_int @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1134_in__set__butlast__appendI,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1135_in__set__butlastD,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1136_distinct__butlast,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( butlast_nat @ Xs ) ) ) ).
% distinct_butlast
thf(fact_1137_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_1138_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_1139_snoc__eq__iff__butlast,axiom,
! [Xs: list_int,X2: int,Ys: list_int] :
( ( ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) )
= Ys )
= ( ( Ys != nil_int )
& ( ( butlast_int @ Ys )
= Xs )
& ( ( last_int @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1140_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_1141_lex__take__index,axiom,
! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys ) )
=> ( ( ( take_int @ I3 @ Xs )
= ( take_int @ I3 @ Ys ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_1142_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_1143_in__measures_I2_J,axiom,
! [X2: int,Y3: int,F: int > nat,Fs: list_int_nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
| ( ( ( F @ X2 )
= ( F @ Y3 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measures_int @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_1144_in__measures_I1_J,axiom,
! [X2: int,Y3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measures_int @ nil_int_nat ) ) ).
% in_measures(1)
thf(fact_1145_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs2: list_nat] : nil_nat ) ) ).
% take0
thf(fact_1146_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_1147_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_1148_nth__take,axiom,
! [I2: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I2 )
= ( nth_nat @ Xs @ I2 ) ) ) ).
% nth_take
thf(fact_1149_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_1150_distinct__take,axiom,
! [Xs: list_nat,I2: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( take_nat @ I2 @ Xs ) ) ) ).
% distinct_take
thf(fact_1151_in__set__takeD,axiom,
! [X2: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1152_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_1153_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_1154_take__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_1155_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_1156_measures__less,axiom,
! [F: int > nat,X2: int,Y3: int,Fs: list_int_nat] :
( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_1157_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_1158_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_1159_take__Cons_H,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( take_int @ N @ ( cons_int @ X2 @ Xs ) )
= nil_int ) )
& ( ( N != zero_zero_nat )
=> ( ( take_int @ N @ ( cons_int @ X2 @ Xs ) )
= ( cons_int @ X2 @ ( take_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1160_take__Suc__conv__app__nth,axiom,
! [I2: nat,Xs: list_int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( take_int @ ( suc @ I2 ) @ Xs )
= ( append_int @ ( take_int @ I2 @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I2 ) @ nil_int ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1161_take__Suc__conv__app__nth,axiom,
! [I2: nat,Xs: list_nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I2 ) @ Xs )
= ( append_nat @ ( take_nat @ I2 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I2 ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1162_sum__mset_Oremove,axiom,
! [X2: multiset_nat,A2: multis1201202736280713200et_nat] :
( ( member_multiset_nat @ X2 @ ( set_ms4188662328148412963et_nat @ A2 ) )
=> ( ( comm_m8595621181775931995et_nat @ A2 )
= ( plus_p6334493942879108393et_nat @ X2 @ ( comm_m8595621181775931995et_nat @ ( minus_4897669229644054985et_nat @ A2 @ ( add_ms5124500668711485122et_nat @ X2 @ zero_z9085034013355480569et_nat ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_1163_sum__mset_Oremove,axiom,
! [X2: nat,A2: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ A2 ) )
=> ( ( comm_m762188921832702859et_nat @ A2 )
= ( plus_plus_nat @ X2 @ ( comm_m762188921832702859et_nat @ ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_1164_sum__mset_Oremove,axiom,
! [X2: int,A2: multiset_int] :
( ( member_int @ X2 @ ( set_mset_int @ A2 ) )
=> ( ( comm_m759698451323652583et_int @ A2 )
= ( plus_plus_int @ X2 @ ( comm_m759698451323652583et_int @ ( minus_4344325018492214997et_int @ A2 @ ( add_mset_int @ X2 @ zero_z3170743180189231877et_int ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_1165_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1166_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1167_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1168_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1169_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1170_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1171_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1172_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1173_in__Union__mset__iff,axiom,
! [X2: nat,MM: multis1201202736280713200et_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ ( comm_m8595621181775931995et_nat @ MM ) ) )
= ( ? [M4: multiset_nat] :
( ( member_multiset_nat @ M4 @ ( set_ms4188662328148412963et_nat @ MM ) )
& ( member_nat @ X2 @ ( set_mset_nat @ M4 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_1174_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1175_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1176_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_1177_nth__Cons__Suc,axiom,
! [X2: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1178_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1179_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_1180_take__Suc__Cons,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( take_int @ ( suc @ N ) @ ( cons_int @ X2 @ Xs ) )
= ( cons_int @ X2 @ ( take_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1181_sum__mset_Oempty,axiom,
( ( comm_m762188921832702859et_nat @ zero_z7348594199698428585et_nat )
= zero_zero_nat ) ).
% sum_mset.empty
thf(fact_1182_sum__mset_Oempty,axiom,
( ( comm_m759698451323652583et_int @ zero_z3170743180189231877et_int )
= zero_zero_int ) ).
% sum_mset.empty
thf(fact_1183_sum__mset__0__iff,axiom,
! [M3: multiset_nat] :
( ( ( comm_m762188921832702859et_nat @ M3 )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ M3 ) )
=> ( X3 = zero_zero_nat ) ) ) ) ).
% sum_mset_0_iff
thf(fact_1184_sum__mset_Ounion,axiom,
! [M3: multis1201202736280713200et_nat,N4: multis1201202736280713200et_nat] :
( ( comm_m8595621181775931995et_nat @ ( plus_p8768199597779566713et_nat @ M3 @ N4 ) )
= ( plus_p6334493942879108393et_nat @ ( comm_m8595621181775931995et_nat @ M3 ) @ ( comm_m8595621181775931995et_nat @ N4 ) ) ) ).
% sum_mset.union
thf(fact_1185_sum__mset_Ounion,axiom,
! [M3: multiset_int,N4: multiset_int] :
( ( comm_m759698451323652583et_int @ ( plus_p2156642923369911685et_int @ M3 @ N4 ) )
= ( plus_plus_int @ ( comm_m759698451323652583et_int @ M3 ) @ ( comm_m759698451323652583et_int @ N4 ) ) ) ).
% sum_mset.union
thf(fact_1186_sum__mset_Ounion,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( comm_m762188921832702859et_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
= ( plus_plus_nat @ ( comm_m762188921832702859et_nat @ M3 ) @ ( comm_m762188921832702859et_nat @ N4 ) ) ) ).
% sum_mset.union
thf(fact_1187_sum__mset_Oadd__mset,axiom,
! [X2: multiset_nat,N4: multis1201202736280713200et_nat] :
( ( comm_m8595621181775931995et_nat @ ( add_ms5124500668711485122et_nat @ X2 @ N4 ) )
= ( plus_p6334493942879108393et_nat @ X2 @ ( comm_m8595621181775931995et_nat @ N4 ) ) ) ).
% sum_mset.add_mset
thf(fact_1188_sum__mset_Oadd__mset,axiom,
! [X2: nat,N4: multiset_nat] :
( ( comm_m762188921832702859et_nat @ ( add_mset_nat @ X2 @ N4 ) )
= ( plus_plus_nat @ X2 @ ( comm_m762188921832702859et_nat @ N4 ) ) ) ).
% sum_mset.add_mset
thf(fact_1189_sum__mset_Oadd__mset,axiom,
! [X2: int,N4: multiset_int] :
( ( comm_m759698451323652583et_int @ ( add_mset_int @ X2 @ N4 ) )
= ( plus_plus_int @ X2 @ ( comm_m759698451323652583et_int @ N4 ) ) ) ).
% sum_mset.add_mset
thf(fact_1190_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1191_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1192_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1193_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_1194_enumerate__simps_I2_J,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( enumerate_int @ N @ ( cons_int @ X2 @ Xs ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X2 ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1195_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1196_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y4: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y4 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1197_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1198_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y4: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y4 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1199_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1200_list__update__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,I2: nat,Y3: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ I2 ) @ Y3 )
= ( cons_nat @ X2 @ ( list_update_nat @ Xs @ I2 @ Y3 ) ) ) ).
% list_update_code(3)
thf(fact_1201_list__update__code_I3_J,axiom,
! [X2: int,Xs: list_int,I2: nat,Y3: int] :
( ( list_update_int @ ( cons_int @ X2 @ Xs ) @ ( suc @ I2 ) @ Y3 )
= ( cons_int @ X2 @ ( list_update_int @ Xs @ I2 @ Y3 ) ) ) ).
% list_update_code(3)
thf(fact_1202_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
= ( upt @ M2 @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1203_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_1204_gen__length__code_I2_J,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X2 @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1205_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1206_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I2: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1207_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1208_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_1209_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1210_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1211_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1212_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1213_strict__inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1214_less__Suc__induct,axiom,
! [I2: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J4: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J4 )
=> ( ( ord_less_nat @ J4 @ K2 )
=> ( ( P2 @ I3 @ J4 )
=> ( ( P2 @ J4 @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1215_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1216_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1217_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1218_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M8: nat] :
( ( M2
= ( suc @ M8 ) )
& ( ord_less_nat @ N @ M8 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1219_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1220_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1221_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1222_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1223_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1224_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1225_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1226_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I2 @ J4 )
=> ( K
!= ( suc @ J4 ) ) ) ) ).
% Suc_lessE
thf(fact_1227_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1228_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I2 @ J4 )
=> ( K
!= ( suc @ J4 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1229_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_1230_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1231_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1232_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1233_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1234_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X: nat] : ( P2 @ X @ zero_zero_nat )
=> ( ! [Y6: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y6 ) )
=> ( ! [X: nat,Y6: nat] :
( ( P2 @ X @ Y6 )
=> ( P2 @ ( suc @ X ) @ ( suc @ Y6 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1235_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1236_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1237_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1238_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1239_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1240_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1241_sum__mset__sum__list,axiom,
! [Xs: list_nat] :
( ( comm_m762188921832702859et_nat @ ( mset_nat @ Xs ) )
= ( groups4561878855575611511st_nat @ Xs ) ) ).
% sum_mset_sum_list
thf(fact_1242_sum__mset_Oneutral,axiom,
! [A2: multiset_nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ A2 ) )
=> ( X = zero_zero_nat ) )
=> ( ( comm_m762188921832702859et_nat @ A2 )
= zero_zero_nat ) ) ).
% sum_mset.neutral
thf(fact_1243_sum__mset_Oneutral,axiom,
! [A2: multiset_int] :
( ! [X: int] :
( ( member_int @ X @ ( set_mset_int @ A2 ) )
=> ( X = zero_zero_int ) )
=> ( ( comm_m759698451323652583et_int @ A2 )
= zero_zero_int ) ) ).
% sum_mset.neutral
thf(fact_1244_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1245_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1246_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1247_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1248_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1249_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1250_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1251_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1252_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
? [K5: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M5 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1253_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1254_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1255_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1256_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1257_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1258_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1259_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1260_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1261_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1262_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1263_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1264_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1265_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1266_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1267_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1268_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X2: list_int,Y3: list_int] :
( ( if_list_int @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X2: list_int,Y3: list_int] :
( ( if_list_int @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( pos_of @ ys1 @ j )
= p ) ).
%------------------------------------------------------------------------------