TPTP Problem File: SLH0037^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Multiset_Ordering_NPC/0002_Multiset_Ordering_in_NP/prob_00624_027931__13792480_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1480 ( 644 unt; 209 typ; 0 def)
% Number of atoms : 3204 (1687 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 10294 ( 262 ~; 39 |; 259 &;8560 @)
% ( 0 <=>;1174 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 677 ( 677 >; 0 *; 0 +; 0 <<)
% Number of symbols : 193 ( 190 usr; 18 con; 0-3 aty)
% Number of variables : 3340 ( 111 ^;3066 !; 163 ?;3340 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:29:49.118
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__List__Olist_It__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J_J,type,
list_m1471284410698777088et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J_J,type,
set_mu4665810586158949542et_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
multis1201202736280713200et_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__List__Olist_It__Nat__Onat_J_J,type,
multiset_list_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__Set__Oset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
set_multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
multiset_set_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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Multiset____Ordering____in____NP__OPropVar,type,
multis3193088007478089820ropVar: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (190)
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
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__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__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on1816901372521670873et_nat: ( list_nat > set_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__Multiset__Omultiset_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on1845515579845918819at_nat: ( multiset_nat > nat ) > set_multiset_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
inj_on5984871485562810851et_nat: ( nat > multiset_nat ) > 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_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on5467128325884351833st_nat: ( set_nat > list_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
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__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__List__Olist_It__Nat__Onat_J_J,type,
plus_p6257111884068938041st_nat: multiset_list_nat > multiset_list_nat > multiset_list_nat ).
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__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
plus_p8712254050562127327et_nat: multiset_set_nat > multiset_set_nat > multiset_set_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__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__List_Omonoid__add_Osum__list_001t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
groups8131247675274367973et_nat: ( multis1201202736280713200et_nat > multis1201202736280713200et_nat > multis1201202736280713200et_nat ) > multis1201202736280713200et_nat > list_m1471284410698777088et_nat > multis1201202736280713200et_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__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__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
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__List__Olist_It__Nat__Onat_J_J,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
sup_su2450674742475619714et_nat: set_multiset_nat > set_multiset_nat > set_multiset_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_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__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__Set__Oset_It__Nat__Onat_J,type,
append_set_nat: list_set_nat > list_set_nat > list_set_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__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
butlast_multiset_nat: list_multiset_nat > list_multiset_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Set__Oset_It__Nat__Onat_J,type,
butlast_set_nat: list_set_nat > list_set_nat ).
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__Set__Oset_It__Nat__Onat_J,type,
distinct_set_nat: list_set_nat > $o ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
gen_length_list_nat: nat > list_list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
gen_le4137597950846853325et_nat: nat > list_multiset_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_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__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
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_Olist__all_001t__List__Olist_It__Nat__Onat_J,type,
list_all_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
list_a6678426458882917996et_nat: ( multiset_nat > $o ) > list_multiset_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Nat__Onat,type,
list_all_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
map_li41320761485832518et_nat: ( list_nat > multiset_nat ) > list_list_nat > list_multiset_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_list_nat_set_nat: ( list_nat > set_nat ) > list_list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_mu6289883152598356806st_nat: ( multiset_nat > list_nat ) > list_multiset_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
map_mu3115708687246746246et_nat: ( multiset_nat > multiset_nat ) > list_multiset_nat > list_multiset_nat ).
thf(sy_c_List_Olist_Omap_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Nat__Onat,type,
map_multiset_nat_nat: ( multiset_nat > nat ) > list_multiset_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
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_Omap_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
map_nat_set_nat: ( nat > set_nat ) > list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_set_nat_list_nat: ( set_nat > list_nat ) > list_set_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
map_set_nat_nat: ( set_nat > nat ) > list_set_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_set_nat_set_nat: ( set_nat > set_nat ) > list_set_nat > list_set_nat ).
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__Multiset__Omultiset_It__Nat__Onat_J_J,type,
set_mu5931229344184163355et_nat: list_m1471284410698777088et_nat > set_mu4665810586158949542et_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__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Nat__Onat_J,type,
list_ex_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
list_ex_multiset_nat: ( multiset_nat > $o ) > list_multiset_nat > $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__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
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__Set__Oset_It__Nat__Onat_J,type,
list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_ta5245179185533627382et_nat: ( list_nat > set_nat ) > list_list_nat > list_set_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
map_ta4907023928961249024et_nat: ( nat > multiset_nat ) > list_nat > list_multiset_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
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__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__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_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__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Nat__Onat_J,type,
removeAll_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
remove7538432002526776084et_nat: multiset_nat > list_multiset_nat > list_multiset_nat ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Ounion_001t__List__Olist_It__Nat__Onat_J,type,
union_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Ounion_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
union_multiset_nat: list_multiset_nat > list_multiset_nat > list_multiset_nat ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Set__Oset_It__Nat__Onat_J,type,
union_set_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
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__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_Oimage__mset_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
image_1736967878604623021et_nat: ( nat > multiset_nat ) > multiset_nat > multis1201202736280713200et_nat ).
thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001t__Nat__Onat,type,
image_mset_nat_nat: ( nat > nat ) > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Ois__empty_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
is_emp1171713086576547631et_nat: multis1201202736280713200et_nat > $o ).
thf(sy_c_Multiset_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: multiset_nat > $o ).
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__List__Olist_It__Nat__Onat_J,type,
mset_list_nat: list_list_nat > multiset_list_nat ).
thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
mset_multiset_nat: list_multiset_nat > multis1201202736280713200et_nat ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Nat__Onat_J,type,
mset_set_nat: list_set_nat > multiset_set_nat ).
thf(sy_c_Multiset_Omset__set_001t__List__Olist_It__Nat__Onat_J,type,
mset_set_list_nat: set_list_nat > multiset_list_nat ).
thf(sy_c_Multiset_Omset__set_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
mset_s7006675203091291577et_nat: set_multiset_nat > multis1201202736280713200et_nat ).
thf(sy_c_Multiset_Omset__set_001t__Nat__Onat,type,
mset_set_nat2: set_nat > multiset_nat ).
thf(sy_c_Multiset_Omset__set_001t__Set__Oset_It__Nat__Onat_J,type,
mset_set_set_nat: set_set_nat > multiset_set_nat ).
thf(sy_c_Multiset_Oreplicate__mset_001t__Nat__Onat,type,
replicate_mset_nat: nat > nat > multiset_nat ).
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_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
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_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,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__List__Olist_It__Nat__Onat_J_J,type,
size_s2759588557502552900st_nat: multiset_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
size_s359445857611097220et_nat: multis1201202736280713200et_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
size_s5917832649809541300et_nat: multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset____Ordering____in____NP__OPropVar,type,
size_s6253272723116879048ropVar: multis3193088007478089820ropVar > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
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__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
ord_le4494706898095093418et_nat: set_multiset_nat > set_multiset_nat > $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_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le6602235886369790592et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
collect_multiset_nat: ( multiset_nat > $o ) > set_multiset_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
remove_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oremove_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
remove_multiset_nat: multiset_nat > set_multiset_nat > set_multiset_nat ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
set_or2559576294515156797et_nat: multiset_nat > set_multiset_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
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__Multiset__Omultiset_It__Nat__Onat_J_J,type,
member4301349206009146759et_nat: multis1201202736280713200et_nat > set_mu4665810586158949542et_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__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_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_k____,type,
k: 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 (1265)
thf(fact_0__092_060open_062ys1_A_B_Apos__of_Ays1_Ak_A_061_Ak_092_060close_062,axiom,
( ( nth_nat @ ys1 @ ( pos_of @ ys1 @ k ) )
= k ) ).
% \<open>ys1 ! pos_of ys1 k = k\<close>
thf(fact_1__092_060open_062_092_060exists_062_Bi_O_Ai_A_060_Alength_Ays1_A_092_060and_062_Ays1_A_B_Ai_A_061_Ak_092_060close_062,axiom,
? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ ys1 ) )
& ( ( nth_nat @ ys1 @ X )
= k )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ ys1 ) )
& ( ( nth_nat @ ys1 @ Y )
= k ) )
=> ( Y = X ) ) ) ).
% \<open>\<exists>!i. i < length ys1 \<and> ys1 ! i = k\<close>
thf(fact_2__092_060open_062pos__of_Ays1_Ak_A_060_Alength_Ays1_092_060close_062,axiom,
ord_less_nat @ ( pos_of @ ys1 @ k ) @ ( size_size_list_nat @ ys1 ) ).
% \<open>pos_of ys1 k < length ys1\<close>
thf(fact_3_id,axiom,
( ( pos_of @ ys1 @ k )
= p ) ).
% id
thf(fact_4__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_5_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_6_k,axiom,
member_nat @ k @ ( set_nat2 @ ys1 ) ).
% k
thf(fact_7__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_8__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_9_pp,axiom,
( ( pos_of @ ys1 @ j )
= p ) ).
% pp
thf(fact_10__092_060open_062distinct_Ays1_092_060close_062,axiom,
distinct_nat @ ys1 ).
% \<open>distinct ys1\<close>
thf(fact_11_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_12_j__def,axiom,
( j
= ( nth_nat @ ys1 @ p ) ) ).
% j_def
thf(fact_13_p2,axiom,
ord_less_nat @ p @ ( size_size_list_nat @ ys1 ) ).
% p2
thf(fact_14_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_15_j,axiom,
member_nat @ j @ ( set_nat2 @ ys1 ) ).
% j
thf(fact_16_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_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_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_19_p1,axiom,
ord_less_nat @ p @ ( size_size_list_nat @ xs1 ) ).
% p1
thf(fact_20_distinct__Ex1,axiom,
! [Xs: list_set_nat,X2: set_nat] :
( ( distinct_set_nat @ Xs )
=> ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_s3254054031482475050et_nat @ Xs ) )
& ( ( nth_set_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s3254054031482475050et_nat @ Xs ) )
& ( ( nth_set_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_21_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_22_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_23_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_24_distinct__conv__nth,axiom,
( distinct_set_nat
= ( ^ [Xs2: list_set_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( I != J )
=> ( ( nth_set_nat @ Xs2 @ I )
!= ( nth_set_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_25_distinct__conv__nth,axiom,
( distin6294748288989586407et_nat
= ( ^ [Xs2: list_multiset_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s6386657463320973636et_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s6386657463320973636et_nat @ Xs2 ) )
=> ( ( I != J )
=> ( ( nth_multiset_nat @ Xs2 @ I )
!= ( nth_multiset_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_26_distinct__conv__nth,axiom,
( distinct_list_nat
= ( ^ [Xs2: list_list_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( I != J )
=> ( ( nth_list_nat @ Xs2 @ I )
!= ( nth_list_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_27_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs2: list_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( I != J )
=> ( ( nth_nat @ Xs2 @ I )
!= ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_28_nth__eq__iff__index__eq,axiom,
! [Xs: list_set_nat,I2: nat,J2: nat] :
( ( distinct_set_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( ( ( nth_set_nat @ Xs @ I2 )
= ( nth_set_nat @ Xs @ J2 ) )
= ( I2 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_29_nth__eq__iff__index__eq,axiom,
! [Xs: list_multiset_nat,I2: nat,J2: nat] :
( ( distin6294748288989586407et_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ( nth_multiset_nat @ Xs @ I2 )
= ( nth_multiset_nat @ Xs @ J2 ) )
= ( I2 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_30_nth__eq__iff__index__eq,axiom,
! [Xs: list_list_nat,I2: nat,J2: nat] :
( ( distinct_list_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( nth_list_nat @ Xs @ I2 )
= ( nth_list_nat @ Xs @ J2 ) )
= ( I2 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_31_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I2: nat,J2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Xs @ J2 ) )
= ( I2 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_32_all__set__conv__all__nth,axiom,
! [Xs: list_multiset_nat,P: multiset_nat > $o] :
( ( ! [X3: multiset_nat] :
( ( member_multiset_nat @ X3 @ ( set_multiset_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( P @ ( nth_multiset_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_33_all__set__conv__all__nth,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_34_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_35_all__nth__imp__all__set,axiom,
! [Xs: list_multiset_nat,P: multiset_nat > $o,X2: multiset_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( P @ ( nth_multiset_nat @ Xs @ I3 ) ) )
=> ( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_36_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X2: list_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I3 ) ) )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_37_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_38_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_39_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_40_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_41_list__ball__nth,axiom,
! [N: nat,Xs: list_multiset_nat,P: multiset_nat > $o] :
( ( ord_less_nat @ N @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_multiset_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_42_list__ball__nth,axiom,
! [N: nat,Xs: list_list_nat,P: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_43_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_44_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_45_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_46_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_47_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_multiset_nat,Z: list_multiset_nat] : ( Y2 = Z ) )
= ( ^ [Xs2: list_multiset_nat,Ys: list_multiset_nat] :
( ( ( size_s6386657463320973636et_nat @ Xs2 )
= ( size_s6386657463320973636et_nat @ Ys ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s6386657463320973636et_nat @ Xs2 ) )
=> ( ( nth_multiset_nat @ Xs2 @ I )
= ( nth_multiset_nat @ Ys @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_48_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_list_nat,Z: list_list_nat] : ( Y2 = Z ) )
= ( ^ [Xs2: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ Xs2 @ I )
= ( nth_list_nat @ Ys @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_49_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
= ( ^ [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_50_Skolem__list__nth,axiom,
! [K: nat,P: nat > multiset_nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X4: multiset_nat] : ( P @ I @ X4 ) ) )
= ( ? [Xs2: list_multiset_nat] :
( ( ( size_s6386657463320973636et_nat @ Xs2 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_multiset_nat @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_51_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X4: list_nat] : ( P @ I @ X4 ) ) )
= ( ? [Xs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_list_nat @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_52_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X4: nat] : ( P @ I @ X4 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_53_nth__equalityI,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( ( size_s6386657463320973636et_nat @ Xs )
= ( size_s6386657463320973636et_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( nth_multiset_nat @ Xs @ I3 )
= ( nth_multiset_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_54_nth__equalityI,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I3 )
= ( nth_list_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_55_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_56__092_060open_062distinct_Axs1_092_060close_062,axiom,
distinct_nat @ xs1 ).
% \<open>distinct xs1\<close>
thf(fact_57_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_58_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_59_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_60_j__ys1_I1_J,axiom,
! [J2: nat] :
( ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
=> ( member_nat @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) @ ( set_nat2 @ xs1 ) ) ) ).
% j_ys1(1)
thf(fact_61_neq__if__length__neq,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( ( size_s6386657463320973636et_nat @ Xs )
!= ( size_s6386657463320973636et_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_62_neq__if__length__neq,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
!= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_63_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_64_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_multiset_nat] :
( ( size_s6386657463320973636et_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_65_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_66_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_67_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_68_length__induct,axiom,
! [P: list_multiset_nat > $o,Xs: list_multiset_nat] :
( ! [Xs3: list_multiset_nat] :
( ! [Ys3: list_multiset_nat] :
( ( ord_less_nat @ ( size_s6386657463320973636et_nat @ Ys3 ) @ ( size_s6386657463320973636et_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_69_length__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ! [Xs3: list_list_nat] :
( ! [Ys3: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys3 ) @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_70_length__induct,axiom,
! [P: 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 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_71_j__ys1_I2_J,axiom,
! [J2: nat] :
( ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
=> ( cns @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) @ J2 ) ) ).
% j_ys1(2)
thf(fact_72_xpi,axiom,
( ( nth_nat @ xs1 @ p )
= i ) ).
% xpi
thf(fact_73_ik,axiom,
( i
= ( nth_nat @ xs1 @ ( pos_of @ ys1 @ k ) ) ) ).
% ik
thf(fact_74__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,
~ ! [P2: nat] :
( ( ord_less_nat @ P2 @ ( size_size_list_nat @ xs1 ) )
=> ( ( nth_nat @ xs1 @ P2 )
!= i ) ) ).
% \<open>\<And>thesis. (\<And>p. \<lbrakk>p < length xs1; xs1 ! p = i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_75_distinct__union,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( distinct_list_nat @ ( union_list_nat @ Xs @ Ys2 ) )
= ( distinct_list_nat @ Ys2 ) ) ).
% distinct_union
thf(fact_76_distinct__union,axiom,
! [Xs: list_set_nat,Ys2: list_set_nat] :
( ( distinct_set_nat @ ( union_set_nat @ Xs @ Ys2 ) )
= ( distinct_set_nat @ Ys2 ) ) ).
% distinct_union
thf(fact_77_distinct__union,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys2 ) )
= ( distinct_nat @ Ys2 ) ) ).
% distinct_union
thf(fact_78_mem__Collect__eq,axiom,
! [A: multiset_nat,P: multiset_nat > $o] :
( ( member_multiset_nat @ A @ ( collect_multiset_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_multiset_nat] :
( ( collect_multiset_nat
@ ^ [X3: multiset_nat] : ( member_multiset_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_85_j__ys1_I3_J,axiom,
! [J2: nat] :
( ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
=> ( v @ ( multis387687052011358179_Gamma @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) @ J2 ) ) ) ).
% j_ys1(3)
thf(fact_86_jm,axiom,
ord_less_nat @ j @ m ).
% jm
thf(fact_87_distinct__swap,axiom,
! [I2: nat,Xs: list_set_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( ( distinct_set_nat @ ( list_update_set_nat @ ( list_update_set_nat @ Xs @ I2 @ ( nth_set_nat @ Xs @ J2 ) ) @ J2 @ ( nth_set_nat @ Xs @ I2 ) ) )
= ( distinct_set_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_88_distinct__swap,axiom,
! [I2: nat,Xs: list_multiset_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( distin6294748288989586407et_nat @ ( list_u3438943574295160626et_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ ( nth_multiset_nat @ Xs @ J2 ) ) @ J2 @ ( nth_multiset_nat @ Xs @ I2 ) ) )
= ( distin6294748288989586407et_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_89_distinct__swap,axiom,
! [I2: nat,Xs: list_list_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( distinct_list_nat @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I2 @ ( nth_list_nat @ Xs @ J2 ) ) @ J2 @ ( nth_list_nat @ Xs @ I2 ) ) )
= ( distinct_list_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_90_distinct__swap,axiom,
! [I2: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I2 ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_91_set__swap,axiom,
! [I2: nat,Xs: list_multiset_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( set_multiset_nat2 @ ( list_u3438943574295160626et_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ ( nth_multiset_nat @ Xs @ J2 ) ) @ J2 @ ( nth_multiset_nat @ Xs @ I2 ) ) )
= ( set_multiset_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_92_set__swap,axiom,
! [I2: nat,Xs: list_list_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( set_list_nat2 @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I2 @ ( nth_list_nat @ Xs @ J2 ) ) @ J2 @ ( nth_list_nat @ Xs @ I2 ) ) )
= ( set_list_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_93_set__swap,axiom,
! [I2: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I2 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_94_i__xs_I1_J,axiom,
member_nat @ i @ ( set_nat2 @ xs1 ) ).
% i_xs(1)
thf(fact_95_list__ex__length,axiom,
( list_ex_multiset_nat
= ( ^ [P3: multiset_nat > $o,Xs2: list_multiset_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s6386657463320973636et_nat @ Xs2 ) )
& ( P3 @ ( nth_multiset_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_96_list__ex__length,axiom,
( list_ex_list_nat
= ( ^ [P3: list_nat > $o,Xs2: list_list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( P3 @ ( nth_list_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_97_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_98_vk,axiom,
v @ ( multis387687052011358179_Gamma @ i @ k ) ).
% vk
thf(fact_99_list__update__overwrite,axiom,
! [Xs: list_nat,I2: nat,X2: nat,Y3: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ I2 @ Y3 )
= ( list_update_nat @ Xs @ I2 @ Y3 ) ) ).
% list_update_overwrite
thf(fact_100_list__update__overwrite,axiom,
! [Xs: list_multiset_nat,I2: nat,X2: multiset_nat,Y3: multiset_nat] :
( ( list_u3438943574295160626et_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) @ I2 @ Y3 )
= ( list_u3438943574295160626et_nat @ Xs @ I2 @ Y3 ) ) ).
% list_update_overwrite
thf(fact_101_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_102_v,axiom,
v @ ( multis387687052011358179_Gamma @ i @ j ) ).
% v
thf(fact_103_length__list__update,axiom,
! [Xs: list_multiset_nat,I2: nat,X2: multiset_nat] :
( ( size_s6386657463320973636et_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) )
= ( size_s6386657463320973636et_nat @ Xs ) ) ).
% length_list_update
thf(fact_104_length__list__update,axiom,
! [Xs: list_list_nat,I2: nat,X2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( list_update_list_nat @ Xs @ I2 @ X2 ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_list_update
thf(fact_105_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_106_nth__list__update__neq,axiom,
! [I2: nat,J2: nat,Xs: list_multiset_nat,X2: multiset_nat] :
( ( I2 != J2 )
=> ( ( nth_multiset_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) @ J2 )
= ( nth_multiset_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_107_nth__list__update__neq,axiom,
! [I2: nat,J2: nat,Xs: list_nat,X2: nat] :
( ( I2 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_108_list__update__id,axiom,
! [Xs: list_multiset_nat,I2: nat] :
( ( list_u3438943574295160626et_nat @ Xs @ I2 @ ( nth_multiset_nat @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_109_list__update__id,axiom,
! [Xs: list_nat,I2: nat] :
( ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_110_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_multiset_nat,X2: multiset_nat] :
( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( nth_multiset_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) @ I2 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_111_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I2 @ X2 ) @ I2 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_112_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_113_i,axiom,
ord_less_nat @ i @ n ).
% i
thf(fact_114_list__update__swap,axiom,
! [I2: nat,I4: nat,Xs: list_nat,X2: nat,X5: nat] :
( ( I2 != I4 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ I4 @ X5 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ X5 ) @ I2 @ X2 ) ) ) ).
% list_update_swap
thf(fact_115_list__update__swap,axiom,
! [I2: nat,I4: nat,Xs: list_multiset_nat,X2: multiset_nat,X5: multiset_nat] :
( ( I2 != I4 )
=> ( ( list_u3438943574295160626et_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) @ I4 @ X5 )
= ( list_u3438943574295160626et_nat @ ( list_u3438943574295160626et_nat @ Xs @ I4 @ X5 ) @ I2 @ X2 ) ) ) ).
% list_update_swap
thf(fact_116_list__ex__cong,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat,F: multiset_nat > $o,G: multiset_nat > $o] :
( ( Xs = Ys2 )
=> ( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Ys2 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_ex_multiset_nat @ F @ Xs )
= ( list_ex_multiset_nat @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_117_list__ex__cong,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,F: list_nat > $o,G: list_nat > $o] :
( ( Xs = Ys2 )
=> ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_ex_list_nat @ F @ Xs )
= ( list_ex_list_nat @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_118_list__ex__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > $o,G: nat > $o] :
( ( Xs = Ys2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_ex_nat @ F @ Xs )
= ( list_ex_nat @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_119_set__update__memI,axiom,
! [N: nat,Xs: list_multiset_nat,X2: multiset_nat] :
( ( ord_less_nat @ N @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ ( list_u3438943574295160626et_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_120_set__update__memI,axiom,
! [N: nat,Xs: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_121_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_122_list__update__same__conv,axiom,
! [I2: nat,Xs: list_multiset_nat,X2: multiset_nat] :
( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 )
= Xs )
= ( ( nth_multiset_nat @ Xs @ I2 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_123_list__update__same__conv,axiom,
! [I2: nat,Xs: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( list_update_list_nat @ Xs @ I2 @ X2 )
= Xs )
= ( ( nth_list_nat @ Xs @ I2 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_124_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_125_nth__list__update,axiom,
! [I2: nat,Xs: list_multiset_nat,J2: nat,X2: multiset_nat] :
( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( ( I2 = J2 )
=> ( ( nth_multiset_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) @ J2 )
= X2 ) )
& ( ( I2 != J2 )
=> ( ( nth_multiset_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) @ J2 )
= ( nth_multiset_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_126_nth__list__update,axiom,
! [I2: nat,Xs: list_list_nat,J2: nat,X2: list_nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( I2 = J2 )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I2 @ X2 ) @ J2 )
= X2 ) )
& ( ( I2 != J2 )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I2 @ X2 ) @ J2 )
= ( nth_list_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_127_nth__list__update,axiom,
! [I2: nat,Xs: list_nat,J2: nat,X2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I2 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ J2 )
= X2 ) )
& ( ( I2 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_128_i__xs_I2_J,axiom,
~ ( member_nat @ i @ ( set_nat2 @ xs2 ) ) ).
% i_xs(2)
thf(fact_129_length__removeAll__less,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s6386657463320973636et_nat @ ( remove7538432002526776084et_nat @ X2 @ Xs ) ) @ ( size_s6386657463320973636et_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_130_length__removeAll__less,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( removeAll_list_nat @ X2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_131_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_132_inj__on__nth,axiom,
! [Xs: list_set_nat,I5: set_nat] :
( ( distinct_set_nat @ Xs )
=> ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ord_less_nat @ X @ ( size_s3254054031482475050et_nat @ Xs ) ) )
=> ( inj_on_nat_set_nat @ ( nth_set_nat @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_133_inj__on__nth,axiom,
! [Xs: list_multiset_nat,I5: set_nat] :
( ( distin6294748288989586407et_nat @ Xs )
=> ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ord_less_nat @ X @ ( size_s6386657463320973636et_nat @ Xs ) ) )
=> ( inj_on5984871485562810851et_nat @ ( nth_multiset_nat @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_134_inj__on__nth,axiom,
! [Xs: list_list_nat,I5: set_nat] :
( ( distinct_list_nat @ Xs )
=> ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ord_less_nat @ X @ ( size_s3023201423986296836st_nat @ Xs ) ) )
=> ( inj_on_nat_list_nat @ ( nth_list_nat @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_135_inj__on__nth,axiom,
! [Xs: list_nat,I5: set_nat] :
( ( distinct_nat @ Xs )
=> ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ord_less_nat @ X @ ( size_size_list_nat @ Xs ) ) )
=> ( inj_on_nat_nat @ ( nth_nat @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_136_nth__butlast,axiom,
! [N: nat,Xs: list_multiset_nat] :
( ( ord_less_nat @ N @ ( size_s6386657463320973636et_nat @ ( butlast_multiset_nat @ Xs ) ) )
=> ( ( nth_multiset_nat @ ( butlast_multiset_nat @ Xs ) @ N )
= ( nth_multiset_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_137_nth__butlast,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ ( butlast_list_nat @ Xs ) ) )
=> ( ( nth_list_nat @ ( butlast_list_nat @ Xs ) @ N )
= ( nth_list_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_138_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_139_list__all__length,axiom,
( list_a6678426458882917996et_nat
= ( ^ [P3: multiset_nat > $o,Xs2: list_multiset_nat] :
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s6386657463320973636et_nat @ Xs2 ) )
=> ( P3 @ ( nth_multiset_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_all_length
thf(fact_140_list__all__length,axiom,
( list_all_list_nat
= ( ^ [P3: list_nat > $o,Xs2: list_list_nat] :
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P3 @ ( nth_list_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_all_length
thf(fact_141_list__all__length,axiom,
( list_all_nat
= ( ^ [P3: nat > $o,Xs2: list_nat] :
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P3 @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_all_length
thf(fact_142_set__union,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( set_multiset_nat2 @ ( union_multiset_nat @ Xs @ Ys2 ) )
= ( sup_su2450674742475619714et_nat @ ( set_multiset_nat2 @ Xs ) @ ( set_multiset_nat2 @ Ys2 ) ) ) ).
% set_union
thf(fact_143_set__union,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( set_list_nat2 @ ( union_list_nat @ Xs @ Ys2 ) )
= ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys2 ) ) ) ).
% set_union
thf(fact_144_set__union,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys2 ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys2 ) ) ) ).
% set_union
thf(fact_145_length__pos__if__in__set,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6386657463320973636et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_146_length__pos__if__in__set,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_147_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_148_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > multiset_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_multiset_nat @ ( map_nat_multiset_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_149_nth__map,axiom,
! [N: nat,Xs: list_multiset_nat,F: multiset_nat > nat] :
( ( ord_less_nat @ N @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( nth_nat @ ( map_multiset_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_multiset_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_150_nth__map,axiom,
! [N: nat,Xs: list_multiset_nat,F: multiset_nat > multiset_nat] :
( ( ord_less_nat @ N @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( nth_multiset_nat @ ( map_mu3115708687246746246et_nat @ F @ Xs ) @ N )
= ( F @ ( nth_multiset_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_151_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_nat @ ( map_list_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_152_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > multiset_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_multiset_nat @ ( map_li41320761485832518et_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_153_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > set_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_set_nat @ ( map_list_nat_set_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_154_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_155_j__ys2_I3_J,axiom,
! [J2: nat] :
( ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
=> ( v @ ( multis387687052011358179_Gamma @ ( i_of_j2 @ J2 ) @ J2 ) ) ) ).
% j_ys2(3)
thf(fact_156_size__neq__size__imp__neq,axiom,
! [X2: multiset_nat,Y3: multiset_nat] :
( ( ( size_s5917832649809541300et_nat @ X2 )
!= ( size_s5917832649809541300et_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_157_size__neq__size__imp__neq,axiom,
! [X2: list_multiset_nat,Y3: list_multiset_nat] :
( ( ( size_s6386657463320973636et_nat @ X2 )
!= ( size_s6386657463320973636et_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_158_size__neq__size__imp__neq,axiom,
! [X2: list_list_nat,Y3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X2 )
!= ( size_s3023201423986296836st_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_159_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_160_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_161_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_162_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_163_j__ys2_I1_J,axiom,
! [J2: nat] :
( ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
=> ( member_nat @ ( i_of_j2 @ J2 ) @ ( set_nat2 @ xs2 ) ) ) ).
% j_ys2(1)
thf(fact_164_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_165_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_166_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_167_map__eq__conv,axiom,
! [F: nat > multiset_nat,Xs: list_nat,G: nat > multiset_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_nat_multiset_nat @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_168_map__eq__conv,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat,G: list_nat > set_nat] :
( ( ( map_list_nat_set_nat @ F @ Xs )
= ( map_list_nat_set_nat @ G @ Xs ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_169_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_170_length__map,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat] :
( ( size_s3254054031482475050et_nat @ ( map_list_nat_set_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_171_length__map,axiom,
! [F: multiset_nat > nat,Xs: list_multiset_nat] :
( ( size_size_list_nat @ ( map_multiset_nat_nat @ F @ Xs ) )
= ( size_s6386657463320973636et_nat @ Xs ) ) ).
% length_map
thf(fact_172_length__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_173_length__map,axiom,
! [F: nat > multiset_nat,Xs: list_nat] :
( ( size_s6386657463320973636et_nat @ ( map_nat_multiset_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_174_length__map,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat] :
( ( size_s6386657463320973636et_nat @ ( map_mu3115708687246746246et_nat @ F @ Xs ) )
= ( size_s6386657463320973636et_nat @ Xs ) ) ).
% length_map
thf(fact_175_length__map,axiom,
! [F: list_nat > multiset_nat,Xs: list_list_nat] :
( ( size_s6386657463320973636et_nat @ ( map_li41320761485832518et_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_176_length__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_177_length__map,axiom,
! [F: multiset_nat > list_nat,Xs: list_multiset_nat] :
( ( size_s3023201423986296836st_nat @ ( map_mu6289883152598356806st_nat @ F @ Xs ) )
= ( size_s6386657463320973636et_nat @ Xs ) ) ).
% length_map
thf(fact_178_length__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_179_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_180_removeAll__id,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ~ ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
=> ( ( remove7538432002526776084et_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_181_removeAll__id,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( removeAll_list_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_182_removeAll__id,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_183__C2_C_I1_J,axiom,
member_nat @ i @ ( set_nat2 @ ( upt @ zero_zero_nat @ n ) ) ).
% "2"(1)
thf(fact_184_map__butlast,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat] :
( ( map_list_nat_set_nat @ F @ ( butlast_list_nat @ Xs ) )
= ( butlast_set_nat @ ( map_list_nat_set_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_185_map__butlast,axiom,
! [F: nat > multiset_nat,Xs: list_nat] :
( ( map_nat_multiset_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_multiset_nat @ ( map_nat_multiset_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_186_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_187_inj__on__map__eq__map,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( inj_on5670230764983331635et_nat @ F @ ( sup_su2450674742475619714et_nat @ ( set_multiset_nat2 @ Xs ) @ ( set_multiset_nat2 @ Ys2 ) ) )
=> ( ( ( map_mu3115708687246746246et_nat @ F @ Xs )
= ( map_mu3115708687246746246et_nat @ F @ Ys2 ) )
= ( Xs = Ys2 ) ) ) ).
% inj_on_map_eq_map
thf(fact_188_inj__on__map__eq__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys2 ) ) )
=> ( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ F @ Ys2 ) )
= ( Xs = Ys2 ) ) ) ).
% inj_on_map_eq_map
thf(fact_189_inj__on__map__eq__map,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( inj_on1816901372521670873et_nat @ F @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys2 ) ) )
=> ( ( ( map_list_nat_set_nat @ F @ Xs )
= ( map_list_nat_set_nat @ F @ Ys2 ) )
= ( Xs = Ys2 ) ) ) ).
% inj_on_map_eq_map
thf(fact_190_inj__on__map__eq__map,axiom,
! [F: nat > multiset_nat,Xs: list_nat,Ys2: list_nat] :
( ( inj_on5984871485562810851et_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys2 ) ) )
=> ( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_nat_multiset_nat @ F @ Ys2 ) )
= ( Xs = Ys2 ) ) ) ).
% inj_on_map_eq_map
thf(fact_191_inj__on__map__eq__map,axiom,
! [F: nat > nat,Xs: list_nat,Ys2: list_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys2 ) ) )
=> ( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
= ( Xs = Ys2 ) ) ) ).
% inj_on_map_eq_map
thf(fact_192_map__inj__on,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( ( map_mu3115708687246746246et_nat @ F @ Xs )
= ( map_mu3115708687246746246et_nat @ F @ Ys2 ) )
=> ( ( inj_on5670230764983331635et_nat @ F @ ( sup_su2450674742475619714et_nat @ ( set_multiset_nat2 @ Xs ) @ ( set_multiset_nat2 @ Ys2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% map_inj_on
thf(fact_193_map__inj__on,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ F @ Ys2 ) )
=> ( ( inj_on3049792774292151987st_nat @ F @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% map_inj_on
thf(fact_194_map__inj__on,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( map_list_nat_set_nat @ F @ Xs )
= ( map_list_nat_set_nat @ F @ Ys2 ) )
=> ( ( inj_on1816901372521670873et_nat @ F @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% map_inj_on
thf(fact_195_map__inj__on,axiom,
! [F: nat > multiset_nat,Xs: list_nat,Ys2: list_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_nat_multiset_nat @ F @ Ys2 ) )
=> ( ( inj_on5984871485562810851et_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% map_inj_on
thf(fact_196_map__inj__on,axiom,
! [F: nat > nat,Xs: list_nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
=> ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% map_inj_on
thf(fact_197_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_198_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_199_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_200_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_201_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_202_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_203_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_204_PropVar_Osize_I7_J,axiom,
! [X11: nat,X12: nat] :
( ( size_s6253272723116879048ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
= zero_zero_nat ) ).
% PropVar.size(7)
thf(fact_205_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_206_distinct__map,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( distinct_nat @ ( map_set_nat_nat @ F @ Xs ) )
= ( ( distinct_set_nat @ Xs )
& ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_207_distinct__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( distinct_list_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on_nat_list_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_208_distinct__map,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( distinct_set_nat @ ( map_nat_set_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on_nat_set_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_209_distinct__map,axiom,
! [F: nat > multiset_nat,Xs: list_nat] :
( ( distin6294748288989586407et_nat @ ( map_nat_multiset_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on5984871485562810851et_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_210_distinct__map,axiom,
! [F: multiset_nat > nat,Xs: list_multiset_nat] :
( ( distinct_nat @ ( map_multiset_nat_nat @ F @ Xs ) )
= ( ( distin6294748288989586407et_nat @ Xs )
& ( inj_on1845515579845918819at_nat @ F @ ( set_multiset_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_211_distinct__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( distinct_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( ( distinct_list_nat @ Xs )
& ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_212_distinct__map,axiom,
! [F: set_nat > list_nat,Xs: list_set_nat] :
( ( distinct_list_nat @ ( map_set_nat_list_nat @ F @ Xs ) )
= ( ( distinct_set_nat @ Xs )
& ( inj_on5467128325884351833st_nat @ F @ ( set_set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_213_distinct__map,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( distinct_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) )
= ( ( distinct_set_nat @ Xs )
& ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_214_distinct__map,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat] :
( ( distin6294748288989586407et_nat @ ( map_mu3115708687246746246et_nat @ F @ Xs ) )
= ( ( distin6294748288989586407et_nat @ Xs )
& ( inj_on5670230764983331635et_nat @ F @ ( set_multiset_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_215_ex__map__conv,axiom,
! [Ys2: list_set_nat,F: list_nat > set_nat] :
( ( ? [Xs2: list_list_nat] :
( Ys2
= ( map_list_nat_set_nat @ F @ Xs2 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Ys2 ) )
=> ? [Y4: list_nat] :
( X3
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_216_ex__map__conv,axiom,
! [Ys2: list_multiset_nat,F: nat > multiset_nat] :
( ( ? [Xs2: list_nat] :
( Ys2
= ( map_nat_multiset_nat @ F @ Xs2 ) ) )
= ( ! [X3: multiset_nat] :
( ( member_multiset_nat @ X3 @ ( set_multiset_nat2 @ Ys2 ) )
=> ? [Y4: nat] :
( X3
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_217_ex__map__conv,axiom,
! [Ys2: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys2
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys2 ) )
=> ? [Y4: nat] :
( X3
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_218_map__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > multiset_nat,G: nat > multiset_nat] :
( ( Xs = Ys2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_nat_multiset_nat @ G @ Ys2 ) ) ) ) ).
% map_cong
thf(fact_219_map__cong,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,F: list_nat > set_nat,G: list_nat > set_nat] :
( ( Xs = Ys2 )
=> ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_list_nat_set_nat @ F @ Xs )
= ( map_list_nat_set_nat @ G @ Ys2 ) ) ) ) ).
% map_cong
thf(fact_220_map__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) ) ) ) ).
% map_cong
thf(fact_221_map__idI,axiom,
! [Xs: list_multiset_nat,F: multiset_nat > multiset_nat] :
( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_mu3115708687246746246et_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_222_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_223_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_224_map__ext,axiom,
! [Xs: list_nat,F: nat > multiset_nat,G: nat > multiset_nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_nat_multiset_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_225_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > set_nat,G: list_nat > set_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_list_nat_set_nat @ F @ Xs )
= ( map_list_nat_set_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_226_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_227_list_Omap__ident__strong,axiom,
! [T: list_multiset_nat,F: multiset_nat > multiset_nat] :
( ! [Z2: multiset_nat] :
( ( member_multiset_nat @ Z2 @ ( set_multiset_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_mu3115708687246746246et_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_228_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_229_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_230_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > multiset_nat,Fa: nat > multiset_nat] :
( ! [Z2: nat,Za: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_nat_multiset_nat @ F @ X2 )
= ( map_nat_multiset_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_231_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > set_nat,Fa: list_nat > set_nat] :
( ! [Z2: list_nat,Za: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_list_nat_set_nat @ F @ X2 )
= ( map_list_nat_set_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_232_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z2: nat,Za: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_233_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > multiset_nat,G: nat > multiset_nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_nat_multiset_nat @ F @ X2 )
= ( map_nat_multiset_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_234_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > set_nat,G: list_nat > set_nat] :
( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_list_nat_set_nat @ F @ X2 )
= ( map_list_nat_set_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_235_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_236_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > multiset_nat,G: nat > multiset_nat] :
( ( X2 = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_nat_multiset_nat @ F @ X2 )
= ( map_nat_multiset_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_237_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > set_nat,G: list_nat > set_nat] :
( ( X2 = Ya )
=> ( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_list_nat_set_nat @ F @ X2 )
= ( map_list_nat_set_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_238_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_239_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_240_map__eq__imp__length__eq,axiom,
! [F: nat > multiset_nat,Xs: list_nat,G: nat > multiset_nat,Ys2: list_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_nat_multiset_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_241_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: multiset_nat > nat,Ys2: list_multiset_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_multiset_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s6386657463320973636et_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_242_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys2: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_243_map__eq__imp__length__eq,axiom,
! [F: multiset_nat > nat,Xs: list_multiset_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_multiset_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_s6386657463320973636et_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_244_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_245_map__eq__imp__length__eq,axiom,
! [F: nat > multiset_nat,Xs: list_nat,G: multiset_nat > multiset_nat,Ys2: list_multiset_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_mu3115708687246746246et_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s6386657463320973636et_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_246_map__eq__imp__length__eq,axiom,
! [F: nat > set_nat,Xs: list_nat,G: list_nat > set_nat,Ys2: list_list_nat] :
( ( ( map_nat_set_nat @ F @ Xs )
= ( map_list_nat_set_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_247_map__eq__imp__length__eq,axiom,
! [F: nat > multiset_nat,Xs: list_nat,G: list_nat > multiset_nat,Ys2: list_list_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_li41320761485832518et_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_248_map__eq__imp__length__eq,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat,G: nat > multiset_nat,Ys2: list_nat] :
( ( ( map_mu3115708687246746246et_nat @ F @ Xs )
= ( map_nat_multiset_nat @ G @ Ys2 ) )
=> ( ( size_s6386657463320973636et_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_249_map__update,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat,K: nat,Y3: list_nat] :
( ( map_list_nat_set_nat @ F @ ( list_update_list_nat @ Xs @ K @ Y3 ) )
= ( list_update_set_nat @ ( map_list_nat_set_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).
% map_update
thf(fact_250_map__update,axiom,
! [F: nat > multiset_nat,Xs: list_nat,K: nat,Y3: nat] :
( ( map_nat_multiset_nat @ F @ ( list_update_nat @ Xs @ K @ Y3 ) )
= ( list_u3438943574295160626et_nat @ ( map_nat_multiset_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).
% map_update
thf(fact_251_map__update,axiom,
! [F: multiset_nat > nat,Xs: list_multiset_nat,K: nat,Y3: multiset_nat] :
( ( map_multiset_nat_nat @ F @ ( list_u3438943574295160626et_nat @ Xs @ K @ Y3 ) )
= ( list_update_nat @ ( map_multiset_nat_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).
% map_update
thf(fact_252_map__update,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat,K: nat,Y3: multiset_nat] :
( ( map_mu3115708687246746246et_nat @ F @ ( list_u3438943574295160626et_nat @ Xs @ K @ Y3 ) )
= ( list_u3438943574295160626et_nat @ ( map_mu3115708687246746246et_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).
% map_update
thf(fact_253_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_254_list__all__cong,axiom,
! [X2: list_multiset_nat,Ya: list_multiset_nat,P: multiset_nat > $o,Pa: multiset_nat > $o] :
( ( X2 = Ya )
=> ( ! [Z2: multiset_nat] :
( ( member_multiset_nat @ Z2 @ ( set_multiset_nat2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_a6678426458882917996et_nat @ P @ X2 )
= ( list_a6678426458882917996et_nat @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_255_list__all__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,P: list_nat > $o,Pa: list_nat > $o] :
( ( X2 = Ya )
=> ( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_list_nat @ P @ X2 )
= ( list_all_list_nat @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_256_list__all__cong,axiom,
! [X2: list_nat,Ya: list_nat,P: nat > $o,Pa: nat > $o] :
( ( X2 = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_nat @ P @ X2 )
= ( list_all_nat @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_257_list_Opred__mono__strong,axiom,
! [P: multiset_nat > $o,X2: list_multiset_nat,Pa: multiset_nat > $o] :
( ( list_a6678426458882917996et_nat @ P @ X2 )
=> ( ! [Z2: multiset_nat] :
( ( member_multiset_nat @ Z2 @ ( set_multiset_nat2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_a6678426458882917996et_nat @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_258_list_Opred__mono__strong,axiom,
! [P: list_nat > $o,X2: list_list_nat,Pa: list_nat > $o] :
( ( list_all_list_nat @ P @ X2 )
=> ( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_list_nat @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_259_list_Opred__mono__strong,axiom,
! [P: nat > $o,X2: list_nat,Pa: nat > $o] :
( ( list_all_nat @ P @ X2 )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_nat @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_260_in__set__butlastD,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ ( butlast_multiset_nat @ Xs ) ) )
=> ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_261_in__set__butlastD,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Xs ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_262_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_263_distinct__butlast,axiom,
! [Xs: list_list_nat] :
( ( distinct_list_nat @ Xs )
=> ( distinct_list_nat @ ( butlast_list_nat @ Xs ) ) ) ).
% distinct_butlast
thf(fact_264_distinct__butlast,axiom,
! [Xs: list_set_nat] :
( ( distinct_set_nat @ Xs )
=> ( distinct_set_nat @ ( butlast_set_nat @ Xs ) ) ) ).
% distinct_butlast
thf(fact_265_distinct__butlast,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( butlast_nat @ Xs ) ) ) ).
% distinct_butlast
thf(fact_266_distinct__removeAll,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( distinct_list_nat @ Xs )
=> ( distinct_list_nat @ ( removeAll_list_nat @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_267_distinct__removeAll,axiom,
! [Xs: list_set_nat,X2: set_nat] :
( ( distinct_set_nat @ Xs )
=> ( distinct_set_nat @ ( removeAll_set_nat @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_268_distinct__removeAll,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_269_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_270_map__equality__iff,axiom,
! [F: nat > multiset_nat,Xs: list_nat,G: nat > multiset_nat,Ys2: list_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_nat_multiset_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_271_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: multiset_nat > nat,Ys2: list_multiset_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_multiset_nat_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s6386657463320973636et_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s6386657463320973636et_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_multiset_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_272_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys2: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_list_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_273_map__equality__iff,axiom,
! [F: multiset_nat > nat,Xs: list_multiset_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_multiset_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_s6386657463320973636et_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_multiset_nat @ Xs @ I ) )
= ( G @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_274_map__equality__iff,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I ) )
= ( G @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_275_map__equality__iff,axiom,
! [F: nat > multiset_nat,Xs: list_nat,G: multiset_nat > multiset_nat,Ys2: list_multiset_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_mu3115708687246746246et_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s6386657463320973636et_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s6386657463320973636et_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_multiset_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_276_map__equality__iff,axiom,
! [F: nat > set_nat,Xs: list_nat,G: list_nat > set_nat,Ys2: list_list_nat] :
( ( ( map_nat_set_nat @ F @ Xs )
= ( map_list_nat_set_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_list_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_277_map__equality__iff,axiom,
! [F: nat > multiset_nat,Xs: list_nat,G: list_nat > multiset_nat,Ys2: list_list_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( map_li41320761485832518et_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I ) )
= ( G @ ( nth_list_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_278_map__equality__iff,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat,G: nat > multiset_nat,Ys2: list_nat] :
( ( ( map_mu3115708687246746246et_nat @ F @ Xs )
= ( map_nat_multiset_nat @ G @ Ys2 ) )
= ( ( ( size_s6386657463320973636et_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_multiset_nat @ Xs @ I ) )
= ( G @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_279_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_280_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_281_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_282_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_283_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_284_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_285_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_286_j__ys2_I2_J,axiom,
! [J2: nat] :
( ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
=> ( cs @ ( i_of_j2 @ J2 ) @ J2 ) ) ).
% j_ys2(2)
thf(fact_287_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_288_un__ys,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ys1 ) @ ( set_nat2 @ ys2 ) )
= ( set_ord_lessThan_nat @ m ) ) ).
% un_ys
thf(fact_289_un__xs,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ xs1 ) @ ( set_nat2 @ xs2 ) )
= ( set_ord_lessThan_nat @ n ) ) ).
% un_xs
thf(fact_290_dist__ys,axiom,
distinct_nat @ ( append_nat @ ys1 @ ys2 ) ).
% dist_ys
thf(fact_291_dist__xs,axiom,
distinct_nat @ ( append_nat @ xs1 @ xs2 ) ).
% dist_xs
thf(fact_292_Un__iff,axiom,
! [C: multiset_nat,A2: set_multiset_nat,B: set_multiset_nat] :
( ( member_multiset_nat @ C @ ( sup_su2450674742475619714et_nat @ A2 @ B ) )
= ( ( member_multiset_nat @ C @ A2 )
| ( member_multiset_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_293_Un__iff,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) )
= ( ( member_list_nat @ C @ A2 )
| ( member_list_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_294_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_295_UnCI,axiom,
! [C: multiset_nat,B: set_multiset_nat,A2: set_multiset_nat] :
( ( ~ ( member_multiset_nat @ C @ B )
=> ( member_multiset_nat @ C @ A2 ) )
=> ( member_multiset_nat @ C @ ( sup_su2450674742475619714et_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_296_UnCI,axiom,
! [C: list_nat,B: set_list_nat,A2: set_list_nat] :
( ( ~ ( member_list_nat @ C @ B )
=> ( member_list_nat @ C @ A2 ) )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_297_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_298_sup_Oright__idem,axiom,
! [A: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ B2 )
= ( sup_sup_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_299_sup_Oright__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_300_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_301_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_302_sup__idem,axiom,
! [X2: nat] :
( ( sup_sup_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_303_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_304_sup_Oleft__idem,axiom,
! [A: nat,B2: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B2 ) )
= ( sup_sup_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_305_sup_Oleft__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_306_sup__left__idem,axiom,
! [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) )
= ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_307_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_308_same__append__eq,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat,Zs: list_multiset_nat] :
( ( ( append_multiset_nat @ Xs @ Ys2 )
= ( append_multiset_nat @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_309_same__append__eq,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_310_append__same__eq,axiom,
! [Ys2: list_multiset_nat,Xs: list_multiset_nat,Zs: list_multiset_nat] :
( ( ( append_multiset_nat @ Ys2 @ Xs )
= ( append_multiset_nat @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_311_append__same__eq,axiom,
! [Ys2: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys2 @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_312_append__assoc,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat,Zs: list_multiset_nat] :
( ( append_multiset_nat @ ( append_multiset_nat @ Xs @ Ys2 ) @ Zs )
= ( append_multiset_nat @ Xs @ ( append_multiset_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_313_append__assoc,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys2 ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_314_append_Oassoc,axiom,
! [A: list_multiset_nat,B2: list_multiset_nat,C: list_multiset_nat] :
( ( append_multiset_nat @ ( append_multiset_nat @ A @ B2 ) @ C )
= ( append_multiset_nat @ A @ ( append_multiset_nat @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_315_append_Oassoc,axiom,
! [A: list_nat,B2: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B2 ) @ C )
= ( append_nat @ A @ ( append_nat @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_316_append__eq__append__conv,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat,Us: list_multiset_nat,Vs: list_multiset_nat] :
( ( ( ( size_s6386657463320973636et_nat @ Xs )
= ( size_s6386657463320973636et_nat @ Ys2 ) )
| ( ( size_s6386657463320973636et_nat @ Us )
= ( size_s6386657463320973636et_nat @ Vs ) ) )
=> ( ( ( append_multiset_nat @ Xs @ Us )
= ( append_multiset_nat @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_317_append__eq__append__conv,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Us: list_list_nat,Vs: list_list_nat] :
( ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
| ( ( size_s3023201423986296836st_nat @ Us )
= ( size_s3023201423986296836st_nat @ Vs ) ) )
=> ( ( ( append_list_nat @ Xs @ Us )
= ( append_list_nat @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_318_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_319_map__append,axiom,
! [F: multiset_nat > nat,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( map_multiset_nat_nat @ F @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( map_multiset_nat_nat @ F @ Xs ) @ ( map_multiset_nat_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_320_map__append,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( map_mu3115708687246746246et_nat @ F @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( append_multiset_nat @ ( map_mu3115708687246746246et_nat @ F @ Xs ) @ ( map_mu3115708687246746246et_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_321_map__append,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( map_list_nat_set_nat @ F @ ( append_list_nat @ Xs @ Ys2 ) )
= ( append_set_nat @ ( map_list_nat_set_nat @ F @ Xs ) @ ( map_list_nat_set_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_322_map__append,axiom,
! [F: nat > multiset_nat,Xs: list_nat,Ys2: list_nat] :
( ( map_nat_multiset_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
= ( append_multiset_nat @ ( map_nat_multiset_nat @ F @ Xs ) @ ( map_nat_multiset_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_323_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys2: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_324_list__all__append,axiom,
! [P: multiset_nat > $o,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( list_a6678426458882917996et_nat @ P @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( ( list_a6678426458882917996et_nat @ P @ Xs )
& ( list_a6678426458882917996et_nat @ P @ Ys2 ) ) ) ).
% list_all_append
thf(fact_325_list__all__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( list_all_nat @ P @ ( append_nat @ Xs @ Ys2 ) )
= ( ( list_all_nat @ P @ Xs )
& ( list_all_nat @ P @ Ys2 ) ) ) ).
% list_all_append
thf(fact_326_removeAll__append,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( remove7538432002526776084et_nat @ X2 @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( append_multiset_nat @ ( remove7538432002526776084et_nat @ X2 @ Xs ) @ ( remove7538432002526776084et_nat @ X2 @ Ys2 ) ) ) ).
% removeAll_append
thf(fact_327_removeAll__append,axiom,
! [X2: nat,Xs: list_nat,Ys2: list_nat] :
( ( removeAll_nat @ X2 @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( removeAll_nat @ X2 @ Xs ) @ ( removeAll_nat @ X2 @ Ys2 ) ) ) ).
% removeAll_append
thf(fact_328_list__ex__append,axiom,
! [P: multiset_nat > $o,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( list_ex_multiset_nat @ P @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( ( list_ex_multiset_nat @ P @ Xs )
| ( list_ex_multiset_nat @ P @ Ys2 ) ) ) ).
% list_ex_append
thf(fact_329_list__ex__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( list_ex_nat @ P @ ( append_nat @ Xs @ Ys2 ) )
= ( ( list_ex_nat @ P @ Xs )
| ( list_ex_nat @ P @ Ys2 ) ) ) ).
% list_ex_append
thf(fact_330_set__append,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( set_multiset_nat2 @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( sup_su2450674742475619714et_nat @ ( set_multiset_nat2 @ Xs ) @ ( set_multiset_nat2 @ Ys2 ) ) ) ).
% set_append
thf(fact_331_set__append,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( set_list_nat2 @ ( append_list_nat @ Xs @ Ys2 ) )
= ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys2 ) ) ) ).
% set_append
thf(fact_332_set__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys2 ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys2 ) ) ) ).
% set_append
thf(fact_333_append__eq__append__conv2,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat,Zs: list_multiset_nat,Ts: list_multiset_nat] :
( ( ( append_multiset_nat @ Xs @ Ys2 )
= ( append_multiset_nat @ Zs @ Ts ) )
= ( ? [Us2: list_multiset_nat] :
( ( ( Xs
= ( append_multiset_nat @ Zs @ Us2 ) )
& ( ( append_multiset_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_multiset_nat @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_multiset_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_334_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_335_append__eq__appendI,axiom,
! [Xs: list_multiset_nat,Xs1: list_multiset_nat,Zs: list_multiset_nat,Ys2: list_multiset_nat,Us: list_multiset_nat] :
( ( ( append_multiset_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_multiset_nat @ Xs1 @ Us ) )
=> ( ( append_multiset_nat @ Xs @ Ys2 )
= ( append_multiset_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_336_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys2: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_337_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_338_map__eq__append__conv,axiom,
! [F: multiset_nat > nat,Xs: list_multiset_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( map_multiset_nat_nat @ F @ Xs )
= ( append_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_multiset_nat,Vs2: list_multiset_nat] :
( ( Xs
= ( append_multiset_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_multiset_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_multiset_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_339_map__eq__append__conv,axiom,
! [F: multiset_nat > multiset_nat,Xs: list_multiset_nat,Ys2: list_multiset_nat,Zs: list_multiset_nat] :
( ( ( map_mu3115708687246746246et_nat @ F @ Xs )
= ( append_multiset_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_multiset_nat,Vs2: list_multiset_nat] :
( ( Xs
= ( append_multiset_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_mu3115708687246746246et_nat @ F @ Us2 ) )
& ( Zs
= ( map_mu3115708687246746246et_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_340_map__eq__append__conv,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat,Ys2: list_set_nat,Zs: list_set_nat] :
( ( ( map_list_nat_set_nat @ F @ Xs )
= ( append_set_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_list_nat_set_nat @ F @ Us2 ) )
& ( Zs
= ( map_list_nat_set_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_341_map__eq__append__conv,axiom,
! [F: nat > multiset_nat,Xs: list_nat,Ys2: list_multiset_nat,Zs: list_multiset_nat] :
( ( ( map_nat_multiset_nat @ F @ Xs )
= ( append_multiset_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_multiset_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_multiset_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_342_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_343_append__eq__map__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,F: multiset_nat > nat,Xs: list_multiset_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( map_multiset_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_multiset_nat,Vs2: list_multiset_nat] :
( ( Xs
= ( append_multiset_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_multiset_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_multiset_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_344_append__eq__map__conv,axiom,
! [Ys2: list_multiset_nat,Zs: list_multiset_nat,F: multiset_nat > multiset_nat,Xs: list_multiset_nat] :
( ( ( append_multiset_nat @ Ys2 @ Zs )
= ( map_mu3115708687246746246et_nat @ F @ Xs ) )
= ( ? [Us2: list_multiset_nat,Vs2: list_multiset_nat] :
( ( Xs
= ( append_multiset_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_mu3115708687246746246et_nat @ F @ Us2 ) )
& ( Zs
= ( map_mu3115708687246746246et_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_345_append__eq__map__conv,axiom,
! [Ys2: list_set_nat,Zs: list_set_nat,F: list_nat > set_nat,Xs: list_list_nat] :
( ( ( append_set_nat @ Ys2 @ Zs )
= ( map_list_nat_set_nat @ F @ Xs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_list_nat_set_nat @ F @ Us2 ) )
& ( Zs
= ( map_list_nat_set_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_346_append__eq__map__conv,axiom,
! [Ys2: list_multiset_nat,Zs: list_multiset_nat,F: nat > multiset_nat,Xs: list_nat] :
( ( ( append_multiset_nat @ Ys2 @ Zs )
= ( map_nat_multiset_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_multiset_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_multiset_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_347_append__eq__map__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_348_distinct__upt,axiom,
! [I2: nat,J2: nat] : ( distinct_nat @ ( upt @ I2 @ J2 ) ) ).
% distinct_upt
thf(fact_349_in__set__butlast__appendI,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ ( butlast_multiset_nat @ Xs ) ) )
| ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ ( butlast_multiset_nat @ Ys2 ) ) ) )
=> ( member_multiset_nat @ X2 @ ( set_multiset_nat2 @ ( butlast_multiset_nat @ ( append_multiset_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_350_in__set__butlast__appendI,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Xs ) ) )
| ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Ys2 ) ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_351_in__set__butlast__appendI,axiom,
! [X2: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Ys2 ) ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_352_list__update__append1,axiom,
! [I2: nat,Xs: list_multiset_nat,Ys2: list_multiset_nat,X2: multiset_nat] :
( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( list_u3438943574295160626et_nat @ ( append_multiset_nat @ Xs @ Ys2 ) @ I2 @ X2 )
= ( append_multiset_nat @ ( list_u3438943574295160626et_nat @ Xs @ I2 @ X2 ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_353_list__update__append1,axiom,
! [I2: nat,Xs: list_list_nat,Ys2: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( list_update_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ I2 @ X2 )
= ( append_list_nat @ ( list_update_list_nat @ Xs @ I2 @ X2 ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_354_list__update__append1,axiom,
! [I2: nat,Xs: list_nat,Ys2: list_nat,X2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ I2 @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ I2 @ X2 ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_355_zero__reorient,axiom,
! [X2: multiset_nat] :
( ( zero_z7348594199698428585et_nat = X2 )
= ( X2 = zero_z7348594199698428585et_nat ) ) ).
% zero_reorient
thf(fact_356_zero__reorient,axiom,
! [X2: multis1201202736280713200et_nat] :
( ( zero_z9085034013355480569et_nat = X2 )
= ( X2 = zero_z9085034013355480569et_nat ) ) ).
% zero_reorient
thf(fact_357_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_358_inf__sup__aci_I8_J,axiom,
! [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) )
= ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_359_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_360_inf__sup__aci_I7_J,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z3 ) )
= ( sup_sup_nat @ Y3 @ ( sup_sup_nat @ X2 @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_361_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z3 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_362_inf__sup__aci_I6_J,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z3 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_363_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z3 )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_364_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_365_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_366_sup_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ C )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_367_sup_Oassoc,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_368_sup__assoc,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z3 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z3 ) ) ) ).
% sup_assoc
thf(fact_369_sup__assoc,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z3 )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z3 ) ) ) ).
% sup_assoc
thf(fact_370_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_371_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_372_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_373_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_374_sup_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A @ C ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_375_sup_Oleft__commute,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_376_sup__left__commute,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z3 ) )
= ( sup_sup_nat @ Y3 @ ( sup_sup_nat @ X2 @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_377_sup__left__commute,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z3 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_378_UnE,axiom,
! [C: multiset_nat,A2: set_multiset_nat,B: set_multiset_nat] :
( ( member_multiset_nat @ C @ ( sup_su2450674742475619714et_nat @ A2 @ B ) )
=> ( ~ ( member_multiset_nat @ C @ A2 )
=> ( member_multiset_nat @ C @ B ) ) ) ).
% UnE
thf(fact_379_UnE,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) )
=> ( ~ ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B ) ) ) ).
% UnE
thf(fact_380_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_381_UnI1,axiom,
! [C: multiset_nat,A2: set_multiset_nat,B: set_multiset_nat] :
( ( member_multiset_nat @ C @ A2 )
=> ( member_multiset_nat @ C @ ( sup_su2450674742475619714et_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_382_UnI1,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_383_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_384_UnI2,axiom,
! [C: multiset_nat,B: set_multiset_nat,A2: set_multiset_nat] :
( ( member_multiset_nat @ C @ B )
=> ( member_multiset_nat @ C @ ( sup_su2450674742475619714et_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_385_UnI2,axiom,
! [C: list_nat,B: set_list_nat,A2: set_list_nat] :
( ( member_list_nat @ C @ B )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_386_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_387_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_388_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_389_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_390_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_391_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_392_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_393_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_394_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_395_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_396_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_397_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_398_less__supI1,axiom,
! [X2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ X2 @ A )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_399_less__supI1,axiom,
! [X2: nat,A: nat,B2: nat] :
( ( ord_less_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_400_less__supI2,axiom,
! [X2: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X2 @ B2 )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_401_less__supI2,axiom,
! [X2: nat,B2: nat,A: nat] :
( ( ord_less_nat @ X2 @ B2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_402_sup_Oabsorb3,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_403_sup_Oabsorb3,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_404_sup_Oabsorb4,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_405_sup_Oabsorb4,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_406_sup_Ostrict__boundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B2 @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_407_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_408_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_409_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_410_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_411_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_412_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_413_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_414_lessThan__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_415_lessThan__iff,axiom,
! [I2: multiset_nat,K: multiset_nat] :
( ( member_multiset_nat @ I2 @ ( set_or2559576294515156797et_nat @ K ) )
= ( ord_le5777773500796000884et_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_416_lessThan__iff,axiom,
! [I2: num,K: num] :
( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_417_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_418_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_419_ys,axiom,
( ( mset_nat @ ( upt @ zero_zero_nat @ m ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ ys1 ) @ ( mset_nat @ ys2 ) ) ) ).
% ys
thf(fact_420_xs,axiom,
( ( mset_nat @ ( upt @ zero_zero_nat @ n ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ xs1 ) @ ( mset_nat @ xs2 ) ) ) ).
% xs
thf(fact_421_size__empty,axiom,
( ( size_s5917832649809541300et_nat @ zero_z7348594199698428585et_nat )
= zero_zero_nat ) ).
% size_empty
thf(fact_422_size__empty,axiom,
( ( size_s359445857611097220et_nat @ zero_z9085034013355480569et_nat )
= zero_zero_nat ) ).
% size_empty
thf(fact_423_size__eq__0__iff__empty,axiom,
! [M3: multiset_nat] :
( ( ( size_s5917832649809541300et_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_z7348594199698428585et_nat ) ) ).
% size_eq_0_iff_empty
thf(fact_424_size__eq__0__iff__empty,axiom,
! [M3: multis1201202736280713200et_nat] :
( ( ( size_s359445857611097220et_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_z9085034013355480569et_nat ) ) ).
% size_eq_0_iff_empty
thf(fact_425_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_426_lessThan__strict__subset__iff,axiom,
! [M: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_427_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_428_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [I3: nat,J3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( member_nat @ J3 @ A2 )
=> ( ( ord_less_nat @ I3 @ J3 )
=> ( ( F @ I3 )
!= ( F @ J3 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_429_add__left__cancel,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ B2 )
= ( plus_p6334493942879108393et_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_430_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_431_add__right__cancel,axiom,
! [B2: multiset_nat,A: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B2 @ A )
= ( plus_p6334493942879108393et_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_432_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_433_add__0,axiom,
! [A: multis1201202736280713200et_nat] :
( ( plus_p8768199597779566713et_nat @ zero_z9085034013355480569et_nat @ A )
= A ) ).
% add_0
thf(fact_434_add__0,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A )
= A ) ).
% add_0
thf(fact_435_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_436_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_437_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_438_add__cancel__right__right,axiom,
! [A: multis1201202736280713200et_nat,B2: multis1201202736280713200et_nat] :
( ( A
= ( plus_p8768199597779566713et_nat @ A @ B2 ) )
= ( B2 = zero_z9085034013355480569et_nat ) ) ).
% add_cancel_right_right
thf(fact_439_add__cancel__right__right,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( A
= ( plus_p6334493942879108393et_nat @ A @ B2 ) )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_right
thf(fact_440_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_441_add__cancel__right__left,axiom,
! [A: multis1201202736280713200et_nat,B2: multis1201202736280713200et_nat] :
( ( A
= ( plus_p8768199597779566713et_nat @ B2 @ A ) )
= ( B2 = zero_z9085034013355480569et_nat ) ) ).
% add_cancel_right_left
thf(fact_442_add__cancel__right__left,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( A
= ( plus_p6334493942879108393et_nat @ B2 @ A ) )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_left
thf(fact_443_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_444_add__cancel__left__right,axiom,
! [A: multis1201202736280713200et_nat,B2: multis1201202736280713200et_nat] :
( ( ( plus_p8768199597779566713et_nat @ A @ B2 )
= A )
= ( B2 = zero_z9085034013355480569et_nat ) ) ).
% add_cancel_left_right
thf(fact_445_add__cancel__left__right,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ B2 )
= A )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_right
thf(fact_446_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_447_add__cancel__left__left,axiom,
! [B2: multis1201202736280713200et_nat,A: multis1201202736280713200et_nat] :
( ( ( plus_p8768199597779566713et_nat @ B2 @ A )
= A )
= ( B2 = zero_z9085034013355480569et_nat ) ) ).
% add_cancel_left_left
thf(fact_448_add__cancel__left__left,axiom,
! [B2: multiset_nat,A: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B2 @ A )
= A )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_left
thf(fact_449_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_450_add_Oright__neutral,axiom,
! [A: multis1201202736280713200et_nat] :
( ( plus_p8768199597779566713et_nat @ A @ zero_z9085034013355480569et_nat )
= A ) ).
% add.right_neutral
thf(fact_451_add_Oright__neutral,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% add.right_neutral
thf(fact_452_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_453_add__less__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_454_add__less__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_455_union__eq__empty,axiom,
! [M3: multis1201202736280713200et_nat,N4: multis1201202736280713200et_nat] :
( ( ( plus_p8768199597779566713et_nat @ M3 @ N4 )
= zero_z9085034013355480569et_nat )
= ( ( M3 = zero_z9085034013355480569et_nat )
& ( N4 = zero_z9085034013355480569et_nat ) ) ) ).
% union_eq_empty
thf(fact_456_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_457_empty__eq__union,axiom,
! [M3: multis1201202736280713200et_nat,N4: multis1201202736280713200et_nat] :
( ( zero_z9085034013355480569et_nat
= ( plus_p8768199597779566713et_nat @ M3 @ N4 ) )
= ( ( M3 = zero_z9085034013355480569et_nat )
& ( N4 = zero_z9085034013355480569et_nat ) ) ) ).
% empty_eq_union
thf(fact_458_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_459_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X2: multis1201202736280713200et_nat,Y3: multis1201202736280713200et_nat] :
( ( zero_z9085034013355480569et_nat
= ( plus_p8768199597779566713et_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_z9085034013355480569et_nat )
& ( Y3 = zero_z9085034013355480569et_nat ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_460_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_461_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X2: multis1201202736280713200et_nat,Y3: multis1201202736280713200et_nat] :
( ( ( plus_p8768199597779566713et_nat @ X2 @ Y3 )
= zero_z9085034013355480569et_nat )
= ( ( X2 = zero_z9085034013355480569et_nat )
& ( Y3 = zero_z9085034013355480569et_nat ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_462_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_463_add__less__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_464_add__less__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_465_less__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_466_less__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_467_mset__append,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( mset_multiset_nat @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( plus_p8768199597779566713et_nat @ ( mset_multiset_nat @ Xs ) @ ( mset_multiset_nat @ Ys2 ) ) ) ).
% mset_append
thf(fact_468_mset__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( mset_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ ( mset_nat @ Ys2 ) ) ) ).
% mset_append
thf(fact_469_size__mset,axiom,
! [Xs: list_multiset_nat] :
( ( size_s359445857611097220et_nat @ ( mset_multiset_nat @ Xs ) )
= ( size_s6386657463320973636et_nat @ Xs ) ) ).
% size_mset
thf(fact_470_size__mset,axiom,
! [Xs: list_list_nat] :
( ( size_s2759588557502552900st_nat @ ( mset_list_nat @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% size_mset
thf(fact_471_size__mset,axiom,
! [Xs: list_nat] :
( ( size_s5917832649809541300et_nat @ ( mset_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% size_mset
thf(fact_472_empty__neutral_I1_J,axiom,
! [X2: multis1201202736280713200et_nat] :
( ( plus_p8768199597779566713et_nat @ zero_z9085034013355480569et_nat @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_473_empty__neutral_I1_J,axiom,
! [X2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_474_empty__neutral_I2_J,axiom,
! [X2: multis1201202736280713200et_nat] :
( ( plus_p8768199597779566713et_nat @ X2 @ zero_z9085034013355480569et_nat )
= X2 ) ).
% empty_neutral(2)
thf(fact_475_empty__neutral_I2_J,axiom,
! [X2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ X2 @ zero_z7348594199698428585et_nat )
= X2 ) ).
% empty_neutral(2)
thf(fact_476_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) @ C )
= ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_477_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_478_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: multiset_nat,J2: multiset_nat,K: multiset_nat,L: multiset_nat] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_p6334493942879108393et_nat @ I2 @ K )
= ( plus_p6334493942879108393et_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_479_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_480_group__cancel_Oadd1,axiom,
! [A2: multiset_nat,K: multiset_nat,A: multiset_nat,B2: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ K @ A ) )
=> ( ( plus_p6334493942879108393et_nat @ A2 @ B2 )
= ( plus_p6334493942879108393et_nat @ K @ ( plus_p6334493942879108393et_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_481_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_482_group__cancel_Oadd2,axiom,
! [B: multiset_nat,K: multiset_nat,B2: multiset_nat,A: multiset_nat] :
( ( B
= ( plus_p6334493942879108393et_nat @ K @ B2 ) )
=> ( ( plus_p6334493942879108393et_nat @ A @ B )
= ( plus_p6334493942879108393et_nat @ K @ ( plus_p6334493942879108393et_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_483_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_484_add_Oassoc,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) @ C )
= ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_485_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_486_add_Ocommute,axiom,
( plus_p6334493942879108393et_nat
= ( ^ [A3: multiset_nat,B3: multiset_nat] : ( plus_p6334493942879108393et_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_487_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_488_add_Oleft__commute,axiom,
! [B2: multiset_nat,A: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A @ C ) )
= ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_489_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_490_add__left__imp__eq,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ B2 )
= ( plus_p6334493942879108393et_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_491_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_492_add__right__imp__eq,axiom,
! [B2: multiset_nat,A: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B2 @ A )
= ( plus_p6334493942879108393et_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_493_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_494_union__less__mono,axiom,
! [A2: multiset_nat,C2: multiset_nat,B: multiset_nat,D: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ A2 @ C2 )
=> ( ( ord_le5777773500796000884et_nat @ B @ D )
=> ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B ) @ ( plus_p6334493942879108393et_nat @ C2 @ D ) ) ) ) ).
% union_less_mono
thf(fact_495_union__le__mono2,axiom,
! [B: multiset_nat,D: multiset_nat,C2: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ B @ D )
=> ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ C2 @ B ) @ ( plus_p6334493942879108393et_nat @ C2 @ D ) ) ) ).
% union_le_mono2
thf(fact_496_union__le__mono1,axiom,
! [B: multiset_nat,D: multiset_nat,C2: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ B @ D )
=> ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ B @ C2 ) @ ( plus_p6334493942879108393et_nat @ D @ C2 ) ) ) ).
% union_le_mono1
thf(fact_497_psubset__trans,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_498_psubsetD,axiom,
! [A2: set_multiset_nat,B: set_multiset_nat,C: multiset_nat] :
( ( ord_le4494706898095093418et_nat @ A2 @ B )
=> ( ( member_multiset_nat @ C @ A2 )
=> ( member_multiset_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_499_psubsetD,axiom,
! [A2: set_list_nat,B: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B )
=> ( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_500_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_501_mset__eq__setD,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( ( mset_multiset_nat @ Xs )
= ( mset_multiset_nat @ Ys2 ) )
=> ( ( set_multiset_nat2 @ Xs )
= ( set_multiset_nat2 @ Ys2 ) ) ) ).
% mset_eq_setD
thf(fact_502_mset__eq__setD,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( mset_list_nat @ Xs )
= ( mset_list_nat @ Ys2 ) )
=> ( ( set_list_nat2 @ Xs )
= ( set_list_nat2 @ Ys2 ) ) ) ).
% mset_eq_setD
thf(fact_503_mset__eq__setD,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys2 ) )
=> ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys2 ) ) ) ).
% mset_eq_setD
thf(fact_504_mset__eq__length,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( ( mset_multiset_nat @ Xs )
= ( mset_multiset_nat @ Ys2 ) )
=> ( ( size_s6386657463320973636et_nat @ Xs )
= ( size_s6386657463320973636et_nat @ Ys2 ) ) ) ).
% mset_eq_length
thf(fact_505_mset__eq__length,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( mset_list_nat @ Xs )
= ( mset_list_nat @ Ys2 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% mset_eq_length
thf(fact_506_mset__eq__length,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% mset_eq_length
thf(fact_507_mset__eq__imp__distinct__iff,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( mset_list_nat @ Xs )
= ( mset_list_nat @ Ys2 ) )
=> ( ( distinct_list_nat @ Xs )
= ( distinct_list_nat @ Ys2 ) ) ) ).
% mset_eq_imp_distinct_iff
thf(fact_508_mset__eq__imp__distinct__iff,axiom,
! [Xs: list_set_nat,Ys2: list_set_nat] :
( ( ( mset_set_nat @ Xs )
= ( mset_set_nat @ Ys2 ) )
=> ( ( distinct_set_nat @ Xs )
= ( distinct_set_nat @ Ys2 ) ) ) ).
% mset_eq_imp_distinct_iff
thf(fact_509_mset__eq__imp__distinct__iff,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys2 ) )
=> ( ( distinct_nat @ Xs )
= ( distinct_nat @ Ys2 ) ) ) ).
% mset_eq_imp_distinct_iff
thf(fact_510_inj__on__add,axiom,
! [A: multiset_nat,A2: set_multiset_nat] : ( inj_on5670230764983331635et_nat @ ( plus_p6334493942879108393et_nat @ A ) @ A2 ) ).
% inj_on_add
thf(fact_511_inj__on__add,axiom,
! [A: nat,A2: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A ) @ A2 ) ).
% inj_on_add
thf(fact_512_add_Ocomm__neutral,axiom,
! [A: multis1201202736280713200et_nat] :
( ( plus_p8768199597779566713et_nat @ A @ zero_z9085034013355480569et_nat )
= A ) ).
% add.comm_neutral
thf(fact_513_add_Ocomm__neutral,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% add.comm_neutral
thf(fact_514_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_515_comm__monoid__add__class_Oadd__0,axiom,
! [A: multis1201202736280713200et_nat] :
( ( plus_p8768199597779566713et_nat @ zero_z9085034013355480569et_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_516_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_517_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_518_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_519_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_520_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_521_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_522_add__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_523_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_524_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_525_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_526_set__eq__iff__mset__eq__distinct,axiom,
! [X2: list_set_nat,Y3: list_set_nat] :
( ( distinct_set_nat @ X2 )
=> ( ( distinct_set_nat @ Y3 )
=> ( ( ( set_set_nat2 @ X2 )
= ( set_set_nat2 @ Y3 ) )
= ( ( mset_set_nat @ X2 )
= ( mset_set_nat @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_527_set__eq__iff__mset__eq__distinct,axiom,
! [X2: list_multiset_nat,Y3: list_multiset_nat] :
( ( distin6294748288989586407et_nat @ X2 )
=> ( ( distin6294748288989586407et_nat @ Y3 )
=> ( ( ( set_multiset_nat2 @ X2 )
= ( set_multiset_nat2 @ Y3 ) )
= ( ( mset_multiset_nat @ X2 )
= ( mset_multiset_nat @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_528_set__eq__iff__mset__eq__distinct,axiom,
! [X2: list_list_nat,Y3: list_list_nat] :
( ( distinct_list_nat @ X2 )
=> ( ( distinct_list_nat @ Y3 )
=> ( ( ( set_list_nat2 @ X2 )
= ( set_list_nat2 @ Y3 ) )
= ( ( mset_list_nat @ X2 )
= ( mset_list_nat @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_529_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_530_add__neg__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_531_add__pos__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_532_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ! [C4: nat] :
( ( B2
= ( plus_plus_nat @ A @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_533_pos__add__strict,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_534_mset__swap,axiom,
! [I2: nat,Ls: list_multiset_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6386657463320973636et_nat @ Ls ) )
=> ( ( ord_less_nat @ J2 @ ( size_s6386657463320973636et_nat @ Ls ) )
=> ( ( mset_multiset_nat @ ( list_u3438943574295160626et_nat @ ( list_u3438943574295160626et_nat @ Ls @ J2 @ ( nth_multiset_nat @ Ls @ I2 ) ) @ I2 @ ( nth_multiset_nat @ Ls @ J2 ) ) )
= ( mset_multiset_nat @ Ls ) ) ) ) ).
% mset_swap
thf(fact_535_mset__swap,axiom,
! [I2: nat,Ls: list_list_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ls ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Ls ) )
=> ( ( mset_list_nat @ ( list_update_list_nat @ ( list_update_list_nat @ Ls @ J2 @ ( nth_list_nat @ Ls @ I2 ) ) @ I2 @ ( nth_list_nat @ Ls @ J2 ) ) )
= ( mset_list_nat @ Ls ) ) ) ) ).
% mset_swap
thf(fact_536_mset__swap,axiom,
! [I2: nat,Ls: list_nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ls ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Ls ) )
=> ( ( mset_nat @ ( list_update_nat @ ( list_update_nat @ Ls @ J2 @ ( nth_nat @ Ls @ I2 ) ) @ I2 @ ( nth_nat @ Ls @ J2 ) ) )
= ( mset_nat @ Ls ) ) ) ) ).
% mset_swap
thf(fact_537_inj__on__inverseI,axiom,
! [A2: set_multiset_nat,G: multiset_nat > multiset_nat,F: multiset_nat > multiset_nat] :
( ! [X: multiset_nat] :
( ( member_multiset_nat @ X @ A2 )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on5670230764983331635et_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_538_inj__on__inverseI,axiom,
! [A2: set_list_nat,G: list_nat > list_nat,F: list_nat > list_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_539_inj__on__inverseI,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_540_inj__on__contraD,axiom,
! [F: multiset_nat > multiset_nat,A2: set_multiset_nat,X2: multiset_nat,Y3: multiset_nat] :
( ( inj_on5670230764983331635et_nat @ F @ A2 )
=> ( ( X2 != Y3 )
=> ( ( member_multiset_nat @ X2 @ A2 )
=> ( ( member_multiset_nat @ Y3 @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_541_inj__on__contraD,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,X2: list_nat,Y3: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( ( X2 != Y3 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_542_inj__on__contraD,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( X2 != Y3 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_543_inj__on__eq__iff,axiom,
! [F: multiset_nat > multiset_nat,A2: set_multiset_nat,X2: multiset_nat,Y3: multiset_nat] :
( ( inj_on5670230764983331635et_nat @ F @ A2 )
=> ( ( member_multiset_nat @ X2 @ A2 )
=> ( ( member_multiset_nat @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_544_inj__on__eq__iff,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,X2: list_nat,Y3: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_545_inj__on__eq__iff,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_546_inj__on__cong,axiom,
! [A2: set_multiset_nat,F: multiset_nat > multiset_nat,G: multiset_nat > multiset_nat] :
( ! [A5: multiset_nat] :
( ( member_multiset_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on5670230764983331635et_nat @ F @ A2 )
= ( inj_on5670230764983331635et_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_547_inj__on__cong,axiom,
! [A2: set_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on3049792774292151987st_nat @ F @ A2 )
= ( inj_on3049792774292151987st_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_548_inj__on__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_549_inj__on__def,axiom,
( inj_on5670230764983331635et_nat
= ( ^ [F2: multiset_nat > multiset_nat,A4: set_multiset_nat] :
! [X3: multiset_nat] :
( ( member_multiset_nat @ X3 @ A4 )
=> ! [Y4: multiset_nat] :
( ( member_multiset_nat @ Y4 @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_550_inj__on__def,axiom,
( inj_on3049792774292151987st_nat
= ( ^ [F2: list_nat > list_nat,A4: set_list_nat] :
! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_551_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_552_inj__onI,axiom,
! [A2: set_multiset_nat,F: multiset_nat > multiset_nat] :
( ! [X: multiset_nat,Y5: multiset_nat] :
( ( member_multiset_nat @ X @ A2 )
=> ( ( member_multiset_nat @ Y5 @ A2 )
=> ( ( ( F @ X )
= ( F @ Y5 ) )
=> ( X = Y5 ) ) ) )
=> ( inj_on5670230764983331635et_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_553_inj__onI,axiom,
! [A2: set_list_nat,F: list_nat > list_nat] :
( ! [X: list_nat,Y5: list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( member_list_nat @ Y5 @ A2 )
=> ( ( ( F @ X )
= ( F @ Y5 ) )
=> ( X = Y5 ) ) ) )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_554_inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X: nat,Y5: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y5 @ A2 )
=> ( ( ( F @ X )
= ( F @ Y5 ) )
=> ( X = Y5 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_555_inj__onD,axiom,
! [F: multiset_nat > multiset_nat,A2: set_multiset_nat,X2: multiset_nat,Y3: multiset_nat] :
( ( inj_on5670230764983331635et_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_multiset_nat @ X2 @ A2 )
=> ( ( member_multiset_nat @ Y3 @ A2 )
=> ( X2 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_556_inj__onD,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,X2: list_nat,Y3: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( X2 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_557_inj__onD,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( X2 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_558_nonempty__has__size,axiom,
! [S2: multiset_nat] :
( ( S2 != zero_z7348594199698428585et_nat )
= ( ord_less_nat @ zero_zero_nat @ ( size_s5917832649809541300et_nat @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_559_nonempty__has__size,axiom,
! [S2: multis1201202736280713200et_nat] :
( ( S2 != zero_z9085034013355480569et_nat )
= ( ord_less_nat @ zero_zero_nat @ ( size_s359445857611097220et_nat @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_560_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_561_mset__map__split,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat,Ys1: list_set_nat,Ys22: list_set_nat] :
( ( ( mset_set_nat @ ( map_list_nat_set_nat @ F @ Xs ) )
= ( plus_p8712254050562127327et_nat @ ( mset_set_nat @ Ys1 ) @ ( mset_set_nat @ Ys22 ) ) )
=> ? [Zs1: list_list_nat,Zs2: list_list_nat] :
( ( ( mset_list_nat @ Xs )
= ( plus_p6257111884068938041st_nat @ ( mset_list_nat @ Zs1 ) @ ( mset_list_nat @ Zs2 ) ) )
& ( Ys1
= ( map_list_nat_set_nat @ F @ Zs1 ) )
& ( Ys22
= ( map_list_nat_set_nat @ F @ Zs2 ) ) ) ) ).
% mset_map_split
thf(fact_562_mset__map__split,axiom,
! [F: nat > multiset_nat,Xs: list_nat,Ys1: list_multiset_nat,Ys22: list_multiset_nat] :
( ( ( mset_multiset_nat @ ( map_nat_multiset_nat @ F @ Xs ) )
= ( plus_p8768199597779566713et_nat @ ( mset_multiset_nat @ Ys1 ) @ ( mset_multiset_nat @ Ys22 ) ) )
=> ? [Zs1: list_nat,Zs2: list_nat] :
( ( ( mset_nat @ Xs )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
& ( Ys1
= ( map_nat_multiset_nat @ F @ Zs1 ) )
& ( Ys22
= ( map_nat_multiset_nat @ F @ Zs2 ) ) ) ) ).
% mset_map_split
thf(fact_563_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_564_Multiset_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: multiset_nat] : ( A4 = zero_z7348594199698428585et_nat ) ) ) ).
% Multiset.is_empty_def
thf(fact_565_Multiset_Ois__empty__def,axiom,
( is_emp1171713086576547631et_nat
= ( ^ [A4: multis1201202736280713200et_nat] : ( A4 = zero_z9085034013355480569et_nat ) ) ) ).
% Multiset.is_empty_def
thf(fact_566_add__0__iff,axiom,
! [B2: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_567_verit__sum__simplify,axiom,
! [A: multis1201202736280713200et_nat] :
( ( plus_p8768199597779566713et_nat @ A @ zero_z9085034013355480569et_nat )
= A ) ).
% verit_sum_simplify
thf(fact_568_verit__sum__simplify,axiom,
! [A: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
= A ) ).
% verit_sum_simplify
thf(fact_569_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_570_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_571_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_572_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_573_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_574_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_575_length__append,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( size_s6386657463320973636et_nat @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s6386657463320973636et_nat @ Xs ) @ ( size_s6386657463320973636et_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_576_length__append,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_577_length__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_578_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_579_nth__append__length__plus,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat,N: nat] :
( ( nth_multiset_nat @ ( append_multiset_nat @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_s6386657463320973636et_nat @ Xs ) @ N ) )
= ( nth_multiset_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_580_nth__append__length__plus,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,N: nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) )
= ( nth_list_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_581_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys2: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_582_nth__upt,axiom,
! [I2: nat,K: nat,J2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 )
=> ( ( nth_nat @ ( upt @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ K ) ) ) ).
% nth_upt
thf(fact_583_mset__le__not__refl,axiom,
! [M3: multiset_nat] :
~ ( ord_le5777773500796000884et_nat @ M3 @ M3 ) ).
% mset_le_not_refl
thf(fact_584_mset__le__not__sym,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ M3 @ N4 )
=> ~ ( ord_le5777773500796000884et_nat @ N4 @ M3 ) ) ).
% mset_le_not_sym
thf(fact_585_mset__le__irrefl,axiom,
! [M3: multiset_nat] :
~ ( ord_le5777773500796000884et_nat @ M3 @ M3 ) ).
% mset_le_irrefl
thf(fact_586_mset__le__trans,axiom,
! [K2: multiset_nat,M3: multiset_nat,N4: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ K2 @ M3 )
=> ( ( ord_le5777773500796000884et_nat @ M3 @ N4 )
=> ( ord_le5777773500796000884et_nat @ K2 @ N4 ) ) ) ).
% mset_le_trans
thf(fact_587_mset__le__asym,axiom,
! [M3: multiset_nat,N4: multiset_nat] :
( ( ord_le5777773500796000884et_nat @ M3 @ N4 )
=> ~ ( ord_le5777773500796000884et_nat @ N4 @ M3 ) ) ).
% mset_le_asym
thf(fact_588_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_589_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_590_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_591_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_592_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_593_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_594_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_595_trans__less__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_596_trans__less__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_597_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_598_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_599_verit__comp__simplify1_I1_J,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_600_verit__comp__simplify1_I1_J,axiom,
! [A: multiset_nat] :
~ ( ord_le5777773500796000884et_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_601_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_602_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_603_subset__mset_Osum__list__eq__0__iff,axiom,
! [Ns: list_m1471284410698777088et_nat] :
( ( ( groups8131247675274367973et_nat @ plus_p8768199597779566713et_nat @ zero_z9085034013355480569et_nat @ Ns )
= zero_z9085034013355480569et_nat )
= ( ! [X3: multis1201202736280713200et_nat] :
( ( member4301349206009146759et_nat @ X3 @ ( set_mu5931229344184163355et_nat @ Ns ) )
=> ( X3 = zero_z9085034013355480569et_nat ) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
thf(fact_604_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_605_length__code,axiom,
( size_s6386657463320973636et_nat
= ( gen_le4137597950846853325et_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_606_length__code,axiom,
( size_s3023201423986296836st_nat
= ( gen_length_list_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_607_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_608_PropVar_Osize__gen_I1_J,axiom,
! [X11: nat,X12: nat] :
( ( multis2955979900537361535ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(1)
thf(fact_609_mset__set__upto__eq__mset__upto,axiom,
! [N: nat] :
( ( mset_set_nat2 @ ( set_ord_lessThan_nat @ N ) )
= ( mset_nat @ ( upt @ zero_zero_nat @ N ) ) ) ).
% mset_set_upto_eq_mset_upto
thf(fact_610_map__eq__map__tailrec,axiom,
map_list_nat_set_nat = map_ta5245179185533627382et_nat ).
% map_eq_map_tailrec
thf(fact_611_map__eq__map__tailrec,axiom,
map_nat_multiset_nat = map_ta4907023928961249024et_nat ).
% map_eq_map_tailrec
thf(fact_612_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_613_remove__code_I1_J,axiom,
! [X2: multiset_nat,Xs: list_multiset_nat] :
( ( remove_multiset_nat @ X2 @ ( set_multiset_nat2 @ Xs ) )
= ( set_multiset_nat2 @ ( remove7538432002526776084et_nat @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_614_remove__code_I1_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( remove_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( set_list_nat2 @ ( removeAll_list_nat @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_615_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_616_member__remove,axiom,
! [X2: multiset_nat,Y3: multiset_nat,A2: set_multiset_nat] :
( ( member_multiset_nat @ X2 @ ( remove_multiset_nat @ Y3 @ A2 ) )
= ( ( member_multiset_nat @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_617_member__remove,axiom,
! [X2: list_nat,Y3: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X2 @ ( remove_list_nat @ Y3 @ A2 ) )
= ( ( member_list_nat @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_618_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_619_gen__length__def,axiom,
( gen_le4137597950846853325et_nat
= ( ^ [N2: nat,Xs2: list_multiset_nat] : ( plus_plus_nat @ N2 @ ( size_s6386657463320973636et_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_620_gen__length__def,axiom,
( gen_length_list_nat
= ( ^ [N2: nat,Xs2: list_list_nat] : ( plus_plus_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_621_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_622_mset__set__set,axiom,
! [Xs: list_set_nat] :
( ( distinct_set_nat @ Xs )
=> ( ( mset_set_set_nat @ ( set_set_nat2 @ Xs ) )
= ( mset_set_nat @ Xs ) ) ) ).
% mset_set_set
thf(fact_623_mset__set__set,axiom,
! [Xs: list_multiset_nat] :
( ( distin6294748288989586407et_nat @ Xs )
=> ( ( mset_s7006675203091291577et_nat @ ( set_multiset_nat2 @ Xs ) )
= ( mset_multiset_nat @ Xs ) ) ) ).
% mset_set_set
thf(fact_624_mset__set__set,axiom,
! [Xs: list_list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ( mset_set_list_nat @ ( set_list_nat2 @ Xs ) )
= ( mset_list_nat @ Xs ) ) ) ).
% mset_set_set
thf(fact_625_mset__set__set,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( mset_set_nat2 @ ( set_nat2 @ Xs ) )
= ( mset_nat @ Xs ) ) ) ).
% mset_set_set
thf(fact_626_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_627_mset__upt,axiom,
! [M: nat,N: nat] :
( ( mset_nat @ ( upt @ M @ N ) )
= ( mset_set_nat2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% mset_upt
thf(fact_628_map__upt__eqI,axiom,
! [Xs: list_multiset_nat,N: nat,M: nat,F: nat > multiset_nat] :
( ( ( size_s6386657463320973636et_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6386657463320973636et_nat @ Xs ) )
=> ( ( nth_multiset_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_nat_multiset_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_629_map__upt__eqI,axiom,
! [Xs: list_list_nat,N: nat,M: nat,F: nat > list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_nat_list_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_630_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_631_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_632_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_633_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_634_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_635_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_636_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_637_add__diff__cancel__right_H,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_638_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_639_add__diff__cancel__right,axiom,
! [A: multiset_nat,C: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ C ) )
= ( minus_8522176038001411705et_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_640_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_641_add__diff__cancel__left_H,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_642_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_643_add__diff__cancel__left,axiom,
! [C: multiset_nat,A: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ ( plus_p6334493942879108393et_nat @ C @ B2 ) )
= ( minus_8522176038001411705et_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_644_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_645_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_646_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_647_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_648_diff__add__zero,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A @ ( plus_p6334493942879108393et_nat @ A @ B2 ) )
= zero_z7348594199698428585et_nat ) ).
% diff_add_zero
thf(fact_649_diff__add__zero,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_650_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_651_subset__mset_Oadd__le__same__cancel1,axiom,
! [B2: multiset_nat,A: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ B2 @ A ) @ B2 )
= ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_652_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) @ B2 )
= ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_653_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ A @ B2 ) )
= ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B2 ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_654_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ B2 @ A ) )
= ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B2 ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_655_length__upt,axiom,
! [I2: nat,J2: nat] :
( ( size_size_list_nat @ ( upt @ I2 @ J2 ) )
= ( minus_minus_nat @ J2 @ I2 ) ) ).
% length_upt
thf(fact_656_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_657_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_658_atLeastLessThan__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_659_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_660_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_661_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_662_diff__diff__eq,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A @ B2 ) @ C )
= ( minus_8522176038001411705et_nat @ A @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_663_diff__diff__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_664_add__implies__diff,axiom,
! [C: multiset_nat,B2: multiset_nat,A: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ C @ B2 )
= A )
=> ( C
= ( minus_8522176038001411705et_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_665_add__implies__diff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A )
=> ( C
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_666_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_667_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_668_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_669_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_670_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_671_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_672_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_673_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_674_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B2: nat] :
( ~ ( ord_less_nat @ A @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_675_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_676_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_677_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_678_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_679_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_680_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_681_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_682_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_nat,C: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat )
=> ( ( subseteq_mset_nat @ C @ B2 )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ B2 ) ) ) ).
% subset_mset.add_decreasing
thf(fact_683_subset__mset_Oadd__increasing,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A )
=> ( ( subseteq_mset_nat @ B2 @ C )
=> ( subseteq_mset_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_684_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_nat,A: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ C @ zero_z7348594199698428585et_nat )
=> ( ( subseteq_mset_nat @ A @ B2 )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ B2 ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_685_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_nat,B2: multiset_nat,A: multiset_nat] :
( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ C )
=> ( ( subseteq_mset_nat @ B2 @ A )
=> ( subseteq_mset_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_686_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A )
=> ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B2 )
=> ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_687_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat )
=> ( ( subseteq_mset_nat @ B2 @ zero_z7348594199698428585et_nat )
=> ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) @ zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_688_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_689_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_690_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I: nat,J: nat] : ( set_nat2 @ ( upt @ I @ J ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_691_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_692_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ( ( ord_less_nat @ A @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_693_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_694_nth__map__upt,axiom,
! [I2: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I2 )
= ( F @ ( plus_plus_nat @ M @ I2 ) ) ) ) ).
% nth_map_upt
thf(fact_695_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat,X2: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X2 ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X2 )
= ( append_nat @ Xs @ ( list_update_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_696_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_697_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_698_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_699_bij__betw__nth,axiom,
! [Xs: list_nat,A2: set_nat,B: set_nat] :
( ( distinct_nat @ Xs )
=> ( ( A2
= ( set_ord_lessThan_nat @ ( size_size_list_nat @ Xs ) ) )
=> ( ( B
= ( set_nat2 @ Xs ) )
=> ( bij_betw_nat_nat @ ( nth_nat @ Xs ) @ A2 @ B ) ) ) ) ).
% bij_betw_nth
thf(fact_700_Un__Diff__cancel2,axiom,
! [B: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_701_Un__Diff__cancel,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_702_lessThan__minus__lessThan,axiom,
! [N: nat,M: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
= ( set_or4665077453230672383an_nat @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_703_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_704_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_705_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_706_bij__betw__imp__inj__on,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_707_Un__Diff,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_708_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_709_inj__on__diff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( inj_on_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% inj_on_diff
thf(fact_710_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_711_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_712_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_713_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_714_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_715_add__mono1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_716_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_717_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_718_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_719_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_720_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_721_last__upt,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( last_nat @ ( upt @ I2 @ J2 ) )
= ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).
% last_upt
thf(fact_722_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_723_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_724_Diff__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_725_DiffI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_726_list_Opred__inject_I2_J,axiom,
! [P: nat > $o,A: nat,Aa: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_nat @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_727_list__all__simps_I1_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_nat @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_728_list__ex__simps_I1_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_ex_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_nat @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_729_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_730_nth__append__length,axiom,
! [Xs: list_nat,X2: nat,Ys2: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_731_list__update__length,axiom,
! [Xs: list_nat,X2: nat,Ys2: list_nat,Y3: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) @ Y3 )
= ( append_nat @ Xs @ ( cons_nat @ Y3 @ Ys2 ) ) ) ).
% list_update_length
thf(fact_732_DiffD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_733_DiffD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_734_DiffE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_735_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_736_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_737_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_738_list_Oset__intros_I2_J,axiom,
! [Y3: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_739_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_740_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
=> ? [Z2: nat,Zs3: list_nat] :
( ( Ys2
= ( cons_nat @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_741_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys2 ) )
=> ? [Z2: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y3 )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_742_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
= ( ? [Z4: nat,Zs4: list_nat] :
( ( Ys2
= ( cons_nat @ Z4 @ Zs4 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs4 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_743_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys2 ) )
= ( ? [Z4: nat,Zs4: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs4 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_nat_nat @ F @ Zs4 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_744_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys2: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys2 )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_745_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys2: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_746_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_747_inj__on__Cons1,axiom,
! [X2: nat,A2: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X2 ) @ A2 ) ).
% inj_on_Cons1
thf(fact_748_distinct__length__2__or__more,axiom,
! [A: nat,B2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ A @ ( cons_nat @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_nat @ ( cons_nat @ A @ Xs ) )
& ( distinct_nat @ ( cons_nat @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_749_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_750_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_751_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_752_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_753_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_754_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_755_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys4: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_756_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_757_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys4: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_758_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys2: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys2 ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_759_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys: list_nat,Zs4: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs4 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_760_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys4: list_nat,X: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_761_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys4: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_762_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys4: list_nat,X: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_763_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys4: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_764_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys: list_nat,Zs4: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs4 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs4 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_765_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys: list_nat,Zs4: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs4 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_766_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys: list_nat,X3: nat,Zs4: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs4 ) ) )
& ( P @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Zs4 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_767_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys: list_nat,X3: nat] :
( ? [Zs4: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs4 ) ) )
& ( P @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_768_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_769_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_770_not__distinct__conv__prefix,axiom,
! [As: list_nat] :
( ( ~ ( distinct_nat @ As ) )
= ( ? [Xs2: list_nat,Y4: nat,Ys: list_nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 )
& ( As
= ( append_nat @ Xs2 @ ( cons_nat @ Y4 @ Ys ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_771_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_772_upt__eq__Cons__conv,axiom,
! [I2: nat,J2: nat,X2: nat,Xs: list_nat] :
( ( ( upt @ I2 @ J2 )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ord_less_nat @ I2 @ J2 )
& ( I2 = X2 )
& ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J2 )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_782_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_783_append__self__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_784_self__append__conv,axiom,
! [Y3: list_nat,Ys2: list_nat] :
( ( Y3
= ( append_nat @ Y3 @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_785_append__self__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_786_self__append__conv2,axiom,
! [Y3: list_nat,Xs: list_nat] :
( ( Y3
= ( append_nat @ Xs @ Y3 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_787_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_788_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_789_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_790_list__all__simps_I2_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list_all_simps(2)
thf(fact_791_in__set__remove1,axiom,
! [A: nat,B2: nat,Xs: list_nat] :
( ( A != B2 )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B2 @ Xs ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_792_list__ex__simps_I2_J,axiom,
! [P: nat > $o] :
~ ( list_ex_nat @ P @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_793_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_794_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_795_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys2: list_nat,Y3: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) )
= ( ( Xs = Ys2 )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_796_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_797_mset__zero__iff__right,axiom,
! [X2: list_nat] :
( ( zero_z7348594199698428585et_nat
= ( mset_nat @ X2 ) )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff_right
thf(fact_798_mset__zero__iff,axiom,
! [X2: list_nat] :
( ( ( mset_nat @ X2 )
= zero_z7348594199698428585et_nat )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff
thf(fact_799_set__mset__mset,axiom,
! [Xs: list_nat] :
( ( set_mset_nat @ ( mset_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_mset_mset
thf(fact_800_last__appendR,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_801_last__appendL,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_802_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_803_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_804_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_805_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_806_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_807_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_808_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_809_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_810_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_811_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y5 @ Ys4 ) )
=> ( ! [X: nat,Xs3: list_nat,Y5: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y5 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_812_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_813_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y5: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_814_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_815_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_816_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y3
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_817_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_818_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_819_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_820_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_821_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_822_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_823_remove1_Osimps_I1_J,axiom,
! [X2: nat] :
( ( remove1_nat @ X2 @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_824_multiset__nonemptyE,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ~ ! [X: nat] :
~ ( member_nat @ X @ ( set_mset_nat @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_825_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_826_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_827_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_828_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_829_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_830_remove1__idem,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_831_list_Opred__inject_I1_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list.pred_inject(1)
thf(fact_832_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_833_distinct__remove1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( remove1_nat @ X2 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_834_removeAll_Osimps_I1_J,axiom,
! [X2: nat] :
( ( removeAll_nat @ X2 @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_835_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_836_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_837_multi__member__last,axiom,
! [X2: nat] : ( member_nat @ X2 @ ( set_mset_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) ).
% multi_member_last
thf(fact_838_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_839_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y5: nat,Ys4: list_nat,Z2: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_840_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y5: nat,Ys4: list_nat,Z2: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y5 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_841_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y5: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_842_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_843_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y5: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y5 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_844_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys6: list_nat] :
( ( ( cons_nat @ X2 @ Ys6 )
= Ys2 )
& ( Xs
= ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_845_append__eq__Cons__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys2 = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys6: list_nat] :
( ( Ys2
= ( cons_nat @ X2 @ Ys6 ) )
& ( ( append_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_846_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_847_distinct__singleton,axiom,
! [X2: nat] : ( distinct_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_singleton
thf(fact_848_upt__0,axiom,
! [I2: nat] :
( ( upt @ I2 @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_849_mset_Osimps_I1_J,axiom,
( ( mset_nat @ nil_nat )
= zero_z7348594199698428585et_nat ) ).
% mset.simps(1)
thf(fact_850_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_851_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_852_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_853_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_854_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_855_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_856_last__append,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_857_longest__common__suffix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
? [Ss: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ( Xs
= ( append_nat @ Xs5 @ Ss ) )
& ( Ys2
= ( append_nat @ Ys7 @ Ss ) )
& ( ( Xs5 = nil_nat )
| ( Ys7 = nil_nat )
| ( ( last_nat @ Xs5 )
!= ( last_nat @ Ys7 ) ) ) ) ).
% longest_common_suffix
thf(fact_858_butlast__append,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_859_remove1__append,axiom,
! [X2: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( remove1_nat @ X2 @ Xs ) @ Ys2 ) ) )
& ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ Xs @ ( remove1_nat @ X2 @ Ys2 ) ) ) ) ) ).
% remove1_append
thf(fact_860_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_861_mset__subset__eq__single,axiom,
! [A: nat,B: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ B ) )
=> ( subseteq_mset_nat @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) @ B ) ) ).
% mset_subset_eq_single
thf(fact_862_multi__subset__induct,axiom,
! [F3: multiset_nat,A2: multiset_nat,P: multiset_nat > $o] :
( ( subseteq_mset_nat @ F3 @ A2 )
=> ( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [A5: nat,F4: multiset_nat] :
( ( member_nat @ A5 @ ( set_mset_nat @ A2 ) )
=> ( ( P @ F4 )
=> ( P @ ( add_mset_nat @ A5 @ F4 ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% multi_subset_induct
thf(fact_863_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_864_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_865_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_866_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_867_multi__drop__mem__not__eq,axiom,
! [C: nat,B: multiset_nat] :
( ( member_nat @ C @ ( set_mset_nat @ B ) )
=> ( ( minus_8522176038001411705et_nat @ B @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) )
!= B ) ) ).
% multi_drop_mem_not_eq
thf(fact_868_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_869_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_870_multiset__add__sub__el__shuffle,axiom,
! [C: nat,B: multiset_nat,B2: nat] :
( ( member_nat @ C @ ( set_mset_nat @ B ) )
=> ( ( B2 != C )
=> ( ( add_mset_nat @ B2 @ ( minus_8522176038001411705et_nat @ B @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) )
= ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B2 @ B ) @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_871_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_872_same__length__different,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ? [Pre: list_nat,X: nat,Xs5: list_nat,Y5: nat,Ys7: list_nat] :
( ( X != Y5 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X @ nil_nat ) @ Xs5 ) ) )
& ( Ys2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y5 @ nil_nat ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_873_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs3: list_nat,Ys4: list_nat,Zs3: list_nat,Y5: nat] :
( Ws
= ( append_nat @ Xs3 @ ( append_nat @ ( cons_nat @ Y5 @ nil_nat ) @ ( append_nat @ Ys4 @ ( append_nat @ ( cons_nat @ Y5 @ nil_nat ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_874_remove1__split,axiom,
! [A: nat,Xs: list_nat,Ys2: list_nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs ) )
=> ( ( ( remove1_nat @ A @ Xs )
= Ys2 )
= ( ? [Ls2: list_nat,Rs: list_nat] :
( ( Xs
= ( append_nat @ Ls2 @ ( cons_nat @ A @ Rs ) ) )
& ~ ( member_nat @ A @ ( set_nat2 @ Ls2 ) )
& ( Ys2
= ( append_nat @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_875_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_876_insert__subset__eq__iff,axiom,
! [A: nat,A2: multiset_nat,B: multiset_nat] :
( ( subseteq_mset_nat @ ( add_mset_nat @ A @ A2 ) @ B )
= ( ( member_nat @ A @ ( set_mset_nat @ B ) )
& ( subseteq_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ B @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_877_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_878_diff__union__single__conv,axiom,
! [A: nat,J4: multiset_nat,I5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ J4 ) )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ I5 @ J4 ) @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
= ( plus_p6334493942879108393et_nat @ I5 @ ( minus_8522176038001411705et_nat @ J4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_879_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X2: nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= Ys2 )
= ( ( Ys2 != nil_nat )
& ( ( butlast_nat @ Ys2 )
= Xs )
& ( ( last_nat @ Ys2 )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_888_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_889_sorted__list__of__multiset__empty,axiom,
( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
= nil_nat ) ).
% sorted_list_of_multiset_empty
thf(fact_890_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_891_length__n__lists__elem,axiom,
! [Ys2: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_901_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_902_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_903_sum__list__append,axiom,
! [Xs: list_multiset_nat,Ys2: list_multiset_nat] :
( ( groups8053510108761903431et_nat @ ( append_multiset_nat @ Xs @ Ys2 ) )
= ( plus_p6334493942879108393et_nat @ ( groups8053510108761903431et_nat @ Xs ) @ ( groups8053510108761903431et_nat @ Ys2 ) ) ) ).
% sum_list_append
thf(fact_904_sum__list__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( groups4561878855575611511st_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ).
% sum_list_append
thf(fact_905_sum__mset_Oempty,axiom,
( ( comm_m762188921832702859et_nat @ zero_z7348594199698428585et_nat )
= zero_zero_nat ) ).
% sum_mset.empty
thf(fact_906_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_907_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_908_distinct__set__subseqs,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_set_nat @ ( map_list_nat_set_nat @ set_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_909_Cons__in__subseqsD,axiom,
! [Y3: nat,Ys2: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y3 @ Ys2 ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_910_subseqs__distinctD,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( ( distinct_nat @ Xs )
=> ( distinct_nat @ Ys2 ) ) ) ).
% subseqs_distinctD
thf(fact_911_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_912_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_913_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_914_size__Suc__Diff1,axiom,
! [X2: nat,M3: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M3 ) )
=> ( ( suc @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) )
= ( size_s5917832649809541300et_nat @ M3 ) ) ) ).
% size_Suc_Diff1
thf(fact_915_subset__mset_Osum__mset__mono,axiom,
! [K2: multiset_nat,F: nat > multiset_nat,G: nat > multiset_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ ( set_mset_nat @ K2 ) )
=> ( subseteq_mset_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( subseteq_mset_nat @ ( comm_m5787568287065167983et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ ( image_1736967878604623021et_nat @ F @ K2 ) ) @ ( comm_m5787568287065167983et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ ( image_1736967878604623021et_nat @ G @ K2 ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_916_in__replicate__mset,axiom,
! [X2: nat,N: nat,Y3: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ ( replicate_mset_nat @ N @ Y3 ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( X2 = Y3 ) ) ) ).
% in_replicate_mset
thf(fact_917_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_918_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_919_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_920_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_921_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_922_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_923_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_924_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_925_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_926_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_927_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_928_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_929_mset__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( mset_nat @ ( map_nat_nat @ F @ Xs ) )
= ( image_mset_nat_nat @ F @ ( mset_nat @ Xs ) ) ) ).
% mset_map
thf(fact_930_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_931_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_932_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_933_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_934_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_935_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_936_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_937_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_938_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_939_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_940_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_941_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_942_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_943_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_944_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_945_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_946_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_947_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_948_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_949_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_950_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_951_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_952_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_953_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_954_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_955_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_956_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_957_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_958_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_959_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_960_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_961_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_962_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X: nat,Y5: nat] :
( ( P @ X @ Y5 )
=> ( P @ ( suc @ X ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_963_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_964_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_965_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_966_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_967_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_968_inj__Suc,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_969_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_970_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_971_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_972_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_973_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_974_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_975_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_976_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_977_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_978_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_979_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_980_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_981_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_982_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_983_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
? [K4: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_984_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_985_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_986_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_987_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_988_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_989_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_990_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_991_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_992_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_993_size__eq__Suc__imp__elem,axiom,
! [M3: multiset_nat,N: nat] :
( ( ( size_s5917832649809541300et_nat @ M3 )
= ( suc @ N ) )
=> ? [A5: nat] : ( member_nat @ A5 @ ( set_mset_nat @ M3 ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_994_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q3 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_995_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_996_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_997_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_998_upt__conv__Cons,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( upt @ I2 @ J2 )
= ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_999_image__mset__eq__plus__image__msetD,axiom,
! [F: nat > nat,A2: multiset_nat,B: multiset_nat,C2: multiset_nat] :
( ( ( image_mset_nat_nat @ F @ A2 )
= ( plus_p6334493942879108393et_nat @ B @ ( image_mset_nat_nat @ F @ C2 ) ) )
=> ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_mset_nat @ A2 ) @ ( set_mset_nat @ C2 ) ) )
=> ? [B6: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ B6 @ C2 ) )
& ( B
= ( image_mset_nat_nat @ F @ B6 ) ) ) ) ) ).
% image_mset_eq_plus_image_msetD
thf(fact_1000_image__mset__eq__image__mset__plusD,axiom,
! [F: nat > nat,A2: multiset_nat,B: multiset_nat,C2: multiset_nat] :
( ( ( image_mset_nat_nat @ F @ A2 )
= ( plus_p6334493942879108393et_nat @ ( image_mset_nat_nat @ F @ B ) @ C2 ) )
=> ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_mset_nat @ A2 ) @ ( set_mset_nat @ B ) ) )
=> ? [C5: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ B @ C5 ) )
& ( C2
= ( image_mset_nat_nat @ F @ C5 ) ) ) ) ) ).
% image_mset_eq_image_mset_plusD
thf(fact_1001_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1002_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1003_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1004_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1005_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1006_size__mset__SucE,axiom,
! [A2: multiset_nat,N: nat] :
( ( ( size_s5917832649809541300et_nat @ A2 )
= ( suc @ N ) )
=> ~ ! [A5: nat,B7: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ A5 @ zero_z7348594199698428585et_nat ) @ B7 ) )
=> ( ( size_s5917832649809541300et_nat @ B7 )
!= N ) ) ) ).
% size_mset_SucE
thf(fact_1007_upt__rec,axiom,
( upt
= ( ^ [I: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1008_length__append__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1009_length__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_1010_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_1011_sorted__list__of__set__range,axiom,
! [M: nat,N: nat] :
( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sorted_list_of_set_range
thf(fact_1012_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( distinct_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_1013_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_1014_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_1015_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_1016_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_1017_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J2: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ ( suc @ I2 ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_1018_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_1019_greaterThanLessThan__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1020_nth__Cons__numeral,axiom,
! [X2: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1021_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1022_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1023_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1024_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1025_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N2: nat,M4: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M4 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_1026_ivl__disj__un__one_I1_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_1027_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_1028_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_1029_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_1030_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J2: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ I2 ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_1031_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N2: nat,M4: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M4 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_1032_ivl__disj__un__two__touch_I1_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_1033_upt__Suc__append,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1034_upt__Suc,axiom,
! [I2: nat,J2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( suc @ J2 ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1035_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1036_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1037_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1038_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1039_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1040_le__sup__iff,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z3 )
= ( ( ord_less_eq_set_nat @ X2 @ Z3 )
& ( ord_less_eq_set_nat @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1041_le__sup__iff,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z3 )
= ( ( ord_less_eq_nat @ X2 @ Z3 )
& ( ord_less_eq_nat @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1042_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1043_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1044_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1045_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1046_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1047_lessThan__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_1048_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1049_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1050_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1051_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1052_atLeastLessThan__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1053_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1054_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1055_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1056_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1057_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1058_greaterThanAtMost__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1059_list__update__beyond,axiom,
! [Xs: list_nat,I2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I2 )
=> ( ( list_update_nat @ Xs @ I2 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_1060_upt__conv__Nil,axiom,
! [J2: nat,I2: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( upt @ I2 @ J2 )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1061_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1062_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1063_upt__eq__Nil__conv,axiom,
! [I2: nat,J2: nat] :
( ( ( upt @ I2 @ J2 )
= nil_nat )
= ( ( J2 = zero_zero_nat )
| ( ord_less_eq_nat @ J2 @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1064_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or6659071591806873216st_nat @ M @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_1065_length__removeAll__less__eq,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_1066_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1067_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1068_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1069_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1070_add__le__less__mono,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1071_add__less__le__mono,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1072_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K3 )
=> ~ ( P @ I6 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1073_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y5 @ A2 )
=> ( ( F @ X )
!= ( F @ Y5 ) ) ) ) )
=> ( ! [X: nat,Y5: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y5 @ A2 )
=> ( ( ord_less_eq_nat @ X @ Y5 )
| ( ord_less_eq_nat @ Y5 @ X ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_1074_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1075_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1076_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1077_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1078_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1079_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1080_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1081_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1082_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1083_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1084_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1085_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1086_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1087_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1088_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1089_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1090_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1091_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1092_verit__comp__simplify1_I3_J,axiom,
! [B8: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B8 @ A6 ) )
= ( ord_less_nat @ A6 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1093_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1094_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1095_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1096_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1097_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1098_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M6: nat] :
( M7
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_1099_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1100_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1101_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1102_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1103_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1104_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y5: nat,Z2: nat] :
( ( R @ X @ Y5 )
=> ( ( R @ Y5 @ Z2 )
=> ( R @ X @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1105_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1106_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1107_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: multiset_nat,J2: multiset_nat,K: multiset_nat,L: multiset_nat] :
( ( ( ord_le6602235886369790592et_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_le6602235886369790592et_nat @ ( plus_p6334493942879108393et_nat @ I2 @ K ) @ ( plus_p6334493942879108393et_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1108_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1109_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: multiset_nat,J2: multiset_nat,K: multiset_nat,L: multiset_nat] :
( ( ( I2 = J2 )
& ( ord_le6602235886369790592et_nat @ K @ L ) )
=> ( ord_le6602235886369790592et_nat @ ( plus_p6334493942879108393et_nat @ I2 @ K ) @ ( plus_p6334493942879108393et_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1110_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1111_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: multiset_nat,J2: multiset_nat,K: multiset_nat,L: multiset_nat] :
( ( ( ord_le6602235886369790592et_nat @ I2 @ J2 )
& ( ord_le6602235886369790592et_nat @ K @ L ) )
=> ( ord_le6602235886369790592et_nat @ ( plus_p6334493942879108393et_nat @ I2 @ K ) @ ( plus_p6334493942879108393et_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1112_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1113_add__mono,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat,D2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ A @ B2 )
=> ( ( ord_le6602235886369790592et_nat @ C @ D2 )
=> ( ord_le6602235886369790592et_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1114_add__mono,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1115_add__left__mono,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ A @ B2 )
=> ( ord_le6602235886369790592et_nat @ ( plus_p6334493942879108393et_nat @ C @ A ) @ ( plus_p6334493942879108393et_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1116_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1117_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_1118_add__right__mono,axiom,
! [A: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ A @ B2 )
=> ( ord_le6602235886369790592et_nat @ ( plus_p6334493942879108393et_nat @ A @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1119_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1120_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1121_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1122_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1123_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1124_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1125_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1126_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1127_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1128_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1129_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1130_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1131_sup_Ocobounded2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_1132_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_1133_sup_Ocobounded1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_1134_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_1135_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1136_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1137_sup_OboundedI,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1138_sup_OboundedI,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1139_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1140_sup_OboundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1141_sup__absorb2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_1142_sup__absorb2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_1143_sup__absorb1,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_1144_sup__absorb1,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_1145_sup_Oabsorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1146_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1147_sup_Oabsorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_1148_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_1149_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
( ! [X: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ X @ ( F @ X @ Y5 ) )
=> ( ! [X: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ ( F @ X @ Y5 ) )
=> ( ! [X: set_nat,Y5: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y5 @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( F @ Y5 @ Z2 ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_1150_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y5 ) )
=> ( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X @ Y5 ) )
=> ( ! [X: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y5 @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z2 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_1151_sup_OorderI,axiom,
! [A: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_1152_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_1153_sup_OorderE,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_1154_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_1155_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1156_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1157_sup__least,axiom,
! [Y3: set_nat,X2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_1158_sup__least,axiom,
! [Y3: nat,X2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_1159_sup__mono,axiom,
! [A: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1160_sup__mono,axiom,
! [A: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1161_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1162_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1163_le__supI2,axiom,
! [X2: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_1164_le__supI2,axiom,
! [X2: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_1165_le__supI1,axiom,
! [X2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_1166_le__supI1,axiom,
! [X2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_1167_sup__ge2,axiom,
! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_1168_sup__ge2,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_1169_sup__ge1,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_1170_sup__ge1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_1171_le__supI,axiom,
! [A: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_1172_le__supI,axiom,
! [A: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_1173_le__supE,axiom,
! [A: set_nat,B2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_1174_le__supE,axiom,
! [A: nat,B2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A @ X2 )
=> ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_1175_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_1176_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_1177_inf__sup__ord_I4_J,axiom,
! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_1178_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_1179_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1180_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1181_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1182_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1183_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1184_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1185_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1186_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1187_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1188_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1189_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1190_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1191_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1192_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1193_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1194_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1195_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1196_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1197_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1198_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1199_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1200_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1201_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1202_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1203_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1204_trans__le__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1205_trans__le__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1206_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M4 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1207_member__le__sum__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X2 @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% member_le_sum_list
thf(fact_1208_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1209_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1210_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1211_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1212_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1213_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_1214_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1215_impossible__Cons,axiom,
! [Xs: list_nat,Ys2: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1216_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A )
= C )
= ( B2
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1217_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1218_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1219_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1220_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1221_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1222_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1223_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1224_le__add__diff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1225_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1226_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1227_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1228_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1229_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1230_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1231_add__nonpos__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1232_add__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1233_add__increasing2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1234_add__decreasing2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1235_add__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1236_add__decreasing,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1237_add__strict__increasing2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1238_add__strict__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1239_add__pos__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1240_add__nonpos__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1241_add__nonneg__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1242_add__neg__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1243_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I6: nat] :
( ( ord_less_eq_nat @ I6 @ K3 )
=> ~ ( P @ I6 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1244_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1245_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1246_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_1247_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X ) )
=> ( ( ( groups4561878855575611511st_nat @ Xs )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = zero_zero_nat ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_1248_sum__list__nonpos,axiom,
! [Xs: list_nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ zero_zero_nat ) ) ).
% sum_list_nonpos
thf(fact_1249_multiset__induct__max,axiom,
! [P: multiset_nat > $o,M3: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X: nat,M8: multiset_nat] :
( ( P @ M8 )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_mset_nat @ M8 ) )
=> ( ord_less_eq_nat @ Xa2 @ X ) )
=> ( P @ ( add_mset_nat @ X @ M8 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_max
thf(fact_1250_multiset__induct__min,axiom,
! [P: multiset_nat > $o,M3: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X: nat,M8: multiset_nat] :
( ( P @ M8 )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_mset_nat @ M8 ) )
=> ( ord_less_eq_nat @ X @ Xa2 ) )
=> ( P @ ( add_mset_nat @ X @ M8 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_min
thf(fact_1251_ivl__disj__un__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_1252_ivl__disj__un__one_I3_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( set_ord_atMost_nat @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_1253_upt__add__eq__append,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( plus_plus_nat @ J2 @ K ) )
= ( append_nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1254_atLeastLessThan__add__Un,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) @ ( set_or4665077453230672383an_nat @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1255_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_1256_sum__list__mono2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_mono2
thf(fact_1257_ivl__disj__un__two_I1_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_1258_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or5834768355832116004an_nat @ M @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_1259_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1260_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ ( suc @ I2 ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_1261_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_1262_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1263_set__remove1__subset,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X2 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1264_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B ) ) ) ) ).
% subset_code(1)
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( k
= ( nth_nat @ ys1 @ ( pos_of @ ys1 @ k ) ) ) ).
%------------------------------------------------------------------------------