TPTP Problem File: SLH0673^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FO_Theory_Rewriting/0068_Ground_MCtxt/prob_00942_038285__18528898_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1556 ( 636 unt; 294 typ; 0 def)
% Number of atoms : 3217 (1876 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11640 ( 409 ~; 35 |; 241 &;9612 @)
% ( 0 <=>;1343 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 41 ( 40 usr)
% Number of type conns : 1194 (1194 >; 0 *; 0 +; 0 <<)
% Number of symbols : 257 ( 254 usr; 25 con; 0-8 aty)
% Number of variables : 3624 ( 79 ^;3337 !; 208 ?;3624 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:01:37.156
%------------------------------------------------------------------------------
% Could-be-implicit typings (40)
thf(ty_n_t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
produc2459281416120113548term_a: $tType ).
thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
list_G5963651352136220186ctxt_a: $tType ).
thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
set_Gr4667965112854951924ctxt_a: $tType ).
thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_G6183871787581289948st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_Gr8007501428367021750st_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
list_l1578326660578982846ctxt_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
list_l4678906206682822203term_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
set_li3075106635084045668ctxt_a: $tType ).
thf(ty_n_t__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
ground723776419908861716ctxt_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Ground____MCtxt__Ogmctxt_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
ground2383972455873737935term_a: $tType ).
thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_It__String__Ochar_J_J,type,
list_G8740567006383293133t_char: $tType ).
thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_It__String__Ochar_J_J,type,
set_Gr8095363379391594151t_char: $tType ).
thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_It__Nat__Onat_J_J,type,
list_G1662964723573077452xt_nat: $tType ).
thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_It__Int__Oint_J_J,type,
list_G8396253615496126504xt_int: $tType ).
thf(ty_n_t__Ground____MCtxt__Ogmctxt_It__List__Olist_It__Nat__Onat_J_J,type,
ground2058972245365194582st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_It__Nat__Onat_J_J,type,
set_Gr4883853939966872486xt_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_It__Int__Oint_J_J,type,
set_Gr2393770795035145730xt_int: $tType ).
thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
list_Ground_gmctxt_a: $tType ).
thf(ty_n_t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
list_Ground_gterm_a: $tType ).
thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
set_Ground_gmctxt_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Ground____MCtxt__Ogmctxt_It__String__Ochar_J,type,
ground_gmctxt_char: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Ground____MCtxt__Ogmctxt_It__Nat__Onat_J,type,
ground_gmctxt_nat: $tType ).
thf(ty_n_t__Ground____MCtxt__Ogmctxt_It__Int__Oint_J,type,
ground_gmctxt_int: $tType ).
thf(ty_n_t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground_gmctxt_a: $tType ).
thf(ty_n_t__Ground____Terms__Ogterm_Itf__a_J,type,
ground_gterm_a: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__Ground____Ctxt__Ogctxt_Itf__a_J,type,
ground_gctxt_a: $tType ).
thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
set_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (254)
thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Nat__Onat,type,
basic_4874698711677410535ts_nat: nat > list_nat > list_list_nat ).
thf(sy_c_Basic__Utils_Olist__of__permutation__element__n_001t__Nat__Onat,type,
basic_7079635023375748421_n_nat: nat > nat > list_nat > list_list_nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
comp_G593817150734306193ctxt_a: ( ground723776419908861716ctxt_a > nat ) > ( ground723776419908861716ctxt_a > ground723776419908861716ctxt_a ) > ground723776419908861716ctxt_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
comp_G3324881245815093522term_a: ( ground723776419908861716ctxt_a > nat ) > ( ground2383972455873737935term_a > ground723776419908861716ctxt_a ) > ground2383972455873737935term_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_It__List__Olist_It__Nat__Onat_J_J,type,
comp_G3225351923774419985st_nat: ( ground2058972245365194582st_nat > nat ) > ( ground2058972245365194582st_nat > ground2058972245365194582st_nat ) > ground2058972245365194582st_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_It__Nat__Onat_J_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_It__Nat__Onat_J,type,
comp_G3928820918855900945xt_nat: ( ground_gmctxt_nat > nat ) > ( ground_gmctxt_nat > ground_gmctxt_nat ) > ground_gmctxt_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
comp_G2329228445783543871ctxt_a: ( ground_gmctxt_a > ground_gmctxt_a ) > ( ground_gmctxt_a > ground_gmctxt_a ) > ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____Terms__Ogterm_Itf__a_J,type,
comp_G3656465168958649678term_a: ( ground_gmctxt_a > ground_gmctxt_a ) > ( ground_gterm_a > ground_gmctxt_a ) > ground_gterm_a > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
comp_G3911404188448037873_a_nat: ( ground_gmctxt_a > ground_gmctxt_a ) > ( nat > ground_gmctxt_a ) > nat > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
comp_G4985101524424093169ctxt_a: ( ground_gmctxt_a > nat ) > ( ground_gmctxt_a > ground_gmctxt_a ) > ground_gmctxt_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat_001t__Ground____Terms__Ogterm_Itf__a_J,type,
comp_G8049203549856156444term_a: ( ground_gmctxt_a > nat ) > ( ground_gterm_a > ground_gmctxt_a ) > ground_gterm_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat_001t__Nat__Onat,type,
comp_G7635752065729029539at_nat: ( ground_gmctxt_a > nat ) > ( nat > ground_gmctxt_a ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
comp_G8862670755111606218ctxt_a: ( ground_gterm_a > ground_gmctxt_a ) > ( ground_gmctxt_a > ground_gterm_a ) > ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____Terms__Ogterm_Itf__a_J,type,
comp_G2980548778298907395term_a: ( ground_gterm_a > ground_gmctxt_a ) > ( ground_gterm_a > ground_gterm_a ) > ground_gterm_a > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
comp_G7903908942813652348_a_nat: ( ground_gterm_a > ground_gmctxt_a ) > ( nat > ground_gterm_a ) > nat > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
comp_G8977606278789707644ctxt_a: ( ground_gterm_a > nat ) > ( ground_gmctxt_a > ground_gterm_a ) > ground_gmctxt_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Nat__Onat_001t__Ground____Terms__Ogterm_Itf__a_J,type,
comp_G6116966634693717713term_a: ( ground_gterm_a > nat ) > ( ground_gterm_a > ground_gterm_a ) > ground_gterm_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Nat__Onat_001t__Nat__Onat,type,
comp_G5395315277947777326at_nat: ( ground_gterm_a > nat ) > ( nat > ground_gterm_a ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_l7223822213492037765st_nat: ( list_nat > list_nat ) > ( list_nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
comp_l7524593755204080629at_nat: ( list_nat > list_nat ) > ( nat > list_nat ) > nat > list_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
comp_l5285636708041795445st_nat: ( list_nat > nat ) > ( list_nat > list_nat ) > list_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
comp_l3539220536931113701at_nat: ( list_nat > nat ) > ( nat > list_nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
comp_n539216501031124977ctxt_a: ( nat > ground_gmctxt_a ) > ( ground_gmctxt_a > nat ) > ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____Terms__Ogterm_Itf__a_J,type,
comp_n7836545058038479132term_a: ( nat > ground_gmctxt_a ) > ( ground_gterm_a > nat ) > ground_gterm_a > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
comp_n5201041448192358307_a_nat: ( nat > ground_gmctxt_a ) > ( nat > nat ) > nat > ground_gmctxt_a ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_n1293334696235875061st_nat: ( nat > list_nat ) > ( list_nat > nat ) > list_nat > list_nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
comp_n6232535815235743333at_nat: ( nat > list_nat ) > ( nat > nat ) > nat > list_nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
comp_n6274738784168413603ctxt_a: ( nat > nat ) > ( ground_gmctxt_a > nat ) > ground_gmctxt_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Ground____Terms__Ogterm_Itf__a_J,type,
comp_n8259423687851516650term_a: ( nat > nat ) > ( ground_gterm_a > nat ) > ground_gterm_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
comp_n3993578768073458149st_nat: ( nat > nat ) > ( list_nat > nat ) > list_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001tf__a,type,
comp_a_nat_a: ( a > nat ) > ( a > a ) > a > nat ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Ground__Ctxt_Ogctxt_OGHole_001tf__a,type,
ground_GHole_a: ground_gctxt_a ).
thf(sy_c_Ground__Ctxt_Ogctxt_OGMore_001tf__a,type,
ground_GMore_a: a > list_Ground_gterm_a > ground_gctxt_a > list_Ground_gterm_a > ground_gctxt_a ).
thf(sy_c_Ground__Ctxt_Ogctxt__apply__term_001tf__a,type,
ground5122445521002937143term_a: ground_gctxt_a > ground_gterm_a > ground_gterm_a ).
thf(sy_c_Ground__MCtxt_Ocomp__gmctxtp_001tf__a,type,
ground4927988881900525590txtp_a: ground_gmctxt_a > ground_gmctxt_a > $o ).
thf(sy_c_Ground__MCtxt_Oeq__gfill_001tf__a,type,
ground_eq_gfill_a: ground_gterm_a > produc2459281416120113548term_a > $o ).
thf(sy_c_Ground__MCtxt_Ofill__gholes_001tf__a,type,
ground_fill_gholes_a: ground_gmctxt_a > list_Ground_gterm_a > ground_gterm_a ).
thf(sy_c_Ground__MCtxt_Ofill__gholes__gmctxt_001tf__a,type,
ground2530127064035342964ctxt_a: ground_gmctxt_a > list_Ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogctxt__of__gmctxt_001tf__a,type,
ground5333112502500237606ctxt_a: ground_gmctxt_a > ground_gctxt_a ).
thf(sy_c_Ground__MCtxt_Oghole__num__at__pos_001tf__a,type,
ground5706155212172713090_pos_a: list_nat > ground_gmctxt_a > nat ).
thf(sy_c_Ground__MCtxt_Oghole__num__bef__pos_001tf__a,type,
ground1934838843355771718_pos_a: list_nat > ground_gmctxt_a > nat ).
thf(sy_c_Ground__MCtxt_Oghole__poss_001tf__a,type,
ground_ghole_poss_a: ground_gmctxt_a > set_list_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground5462067128498283694ctxt_a: ground_gmctxt_a > list_G5963651352136220186ctxt_a > ground723776419908861716ctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001t__Int__Oint,type,
ground_GMFun_int: int > list_G8396253615496126504xt_int > ground_gmctxt_int ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001t__List__Olist_It__Nat__Onat_J,type,
ground940579722934967664st_nat: list_nat > list_G6183871787581289948st_nat > ground2058972245365194582st_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001t__Nat__Onat,type,
ground_GMFun_nat: nat > list_G1662964723573077452xt_nat > ground_gmctxt_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001t__String__Ochar,type,
ground_GMFun_char: char > list_G8740567006383293133t_char > ground_gmctxt_char ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001tf__a,type,
ground_GMFun_a: a > list_Ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMHole_001t__List__Olist_It__Nat__Onat_J,type,
ground8710035520252524849st_nat: ground2058972245365194582st_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMHole_001tf__a,type,
ground_GMHole_a: ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Ogfuns__mctxt_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground2642700214630189768ctxt_a: ground723776419908861716ctxt_a > set_Ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Ogfuns__mctxt_001t__Int__Oint,type,
ground8736743051440902870xt_int: ground_gmctxt_int > set_int ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Ogfuns__mctxt_001t__List__Olist_It__Nat__Onat_J,type,
ground1659619264735726218st_nat: ground2058972245365194582st_nat > set_list_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Ogfuns__mctxt_001t__Nat__Onat,type,
ground8739233521949953146xt_nat: ground_gmctxt_nat > set_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Ogfuns__mctxt_001t__String__Ochar,type,
ground3608886226622445179t_char: ground_gmctxt_char > set_char ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Ogfuns__mctxt_001tf__a,type,
ground_gfuns_mctxt_a: ground_gmctxt_a > set_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground5024537044114205507ctxt_a: ( ground_gmctxt_a > ground_gmctxt_a ) > ground723776419908861716ctxt_a > ground723776419908861716ctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
ground7700310005849181941_a_nat: ( ground_gmctxt_a > nat ) > ground723776419908861716ctxt_a > ground_gmctxt_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground8707887377537884296ctxt_a: ( ground_gterm_a > ground_gmctxt_a ) > ground2383972455873737935term_a > ground723776419908861716ctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Nat__Onat,type,
ground6644341496884830778_a_nat: ( ground_gterm_a > nat ) > ground2383972455873737935term_a > ground_gmctxt_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__Int__Oint_001t__Int__Oint,type,
ground4218667794956602719nt_int: ( int > int ) > ground_gmctxt_int > ground_gmctxt_int ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
ground8629520670591032391st_nat: ( list_nat > list_nat ) > ground2058972245365194582st_nat > ground2058972245365194582st_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
ground3379907019545862327at_nat: ( list_nat > nat ) > ground2058972245365194582st_nat > ground_gmctxt_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__Nat__Onat_001t__Nat__Onat,type,
ground3221804553564598439at_nat: ( nat > nat ) > ground_gmctxt_nat > ground_gmctxt_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001t__String__Ochar_001t__String__Ochar,type,
ground8079844605718550185r_char: ( char > char ) > ground_gmctxt_char > ground_gmctxt_char ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Omap__gmctxt_001tf__a_001tf__a,type,
ground2683026529610258053xt_a_a: ( a > a ) > ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Osize__gmctxt_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground2918769037360547453ctxt_a: ( ground_gmctxt_a > nat ) > ground723776419908861716ctxt_a > nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Osize__gmctxt_001t__Ground____Terms__Ogterm_Itf__a_J,type,
ground7767404987464290192term_a: ( ground_gterm_a > nat ) > ground2383972455873737935term_a > nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Osize__gmctxt_001t__List__Olist_It__Nat__Onat_J,type,
ground3062967014897493439st_nat: ( list_nat > nat ) > ground2058972245365194582st_nat > nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Osize__gmctxt_001t__Nat__Onat,type,
ground7473917417092711983xt_nat: ( nat > nat ) > ground_gmctxt_nat > nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_Osize__gmctxt_001tf__a,type,
ground_size_gmctxt_a: ( a > nat ) > ground_gmctxt_a > nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt__of__gctxt_001tf__a,type,
ground5442125528501538874ctxt_a: ground_gctxt_a > ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt__of__gterm_001tf__a,type,
ground582559344758063963term_a: ground_gterm_a > ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogterm__of__gmctxt_001tf__a,type,
ground2297065866753446215ctxt_a: ground_gmctxt_a > ground_gterm_a ).
thf(sy_c_Ground__MCtxt_Oless__eq__gmctxt_001tf__a,type,
ground6477899500129135038ctxt_a: ground_gmctxt_a > ground_gmctxt_a > $o ).
thf(sy_c_Ground__MCtxt_Onum__gholes_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground2597690549238950442ctxt_a: ground723776419908861716ctxt_a > nat ).
thf(sy_c_Ground__MCtxt_Onum__gholes_001tf__a,type,
ground_num_gholes_a: ground_gmctxt_a > nat ).
thf(sy_c_Ground__MCtxt_Onum__gholes__rel_001tf__a,type,
ground1147946743542812879_rel_a: ground_gmctxt_a > ground_gmctxt_a > $o ).
thf(sy_c_Ground__MCtxt_Osup__gmctxt__args_001tf__a,type,
ground7778985915038999110args_a: ground_gmctxt_a > ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_Ground__Terms_Ogterm_OGFun_001tf__a,type,
ground_GFun_a: a > list_Ground_gterm_a > ground_gterm_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
minus_4616606335744337325ctxt_a: set_Ground_gmctxt_a > set_Ground_gmctxt_a > set_Ground_gmctxt_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
uminus6214660439178182941ctxt_a: set_Ground_gmctxt_a > set_Ground_gmctxt_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Int__Oint,type,
groups4559388385066561235st_int: list_int > int ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__list_OF_001t__Int__Oint,type,
groups_monoid_F_int: ( int > int > int ) > int > list_int > int ).
thf(sy_c_Groups__List_Omonoid__list_OF_001t__Nat__Onat,type,
groups_monoid_F_nat: ( nat > nat > nat ) > nat > list_nat > nat ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_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_List_Oappend_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
append5238750768297430211ctxt_a: list_Ground_gmctxt_a > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Oconcat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
concat3993834769610107037ctxt_a: list_l1578326660578982846ctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Oconcat_001t__Ground____Terms__Ogterm_Itf__a_J,type,
concat1399541994378716528term_a: list_l4678906206682822203term_a > list_Ground_gterm_a ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Ocount__list_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
count_4307434365288015266ctxt_a: list_Ground_gmctxt_a > ground_gmctxt_a > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Odrop_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
drop_Ground_gmctxt_a: nat > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Odrop_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
drop_list_list_nat: nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Odrop_001t__List__Olist_It__Nat__Onat_J,type,
drop_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olist_OCons_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
cons_Ground_gmctxt_a: ground_gmctxt_a > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Olist_OCons_001t__Ground____Terms__Ogterm_Itf__a_J,type,
cons_Ground_gterm_a: ground_gterm_a > list_Ground_gterm_a > list_Ground_gterm_a ).
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__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
nil_Ground_gmctxt_a: list_Ground_gmctxt_a ).
thf(sy_c_List_Olist_ONil_001t__Ground____Terms__Ogterm_Itf__a_J,type,
nil_Ground_gterm_a: list_Ground_gterm_a ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
map_Gr980675717119874050ctxt_a: ( ground_gmctxt_a > ground_gmctxt_a ) > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____Terms__Ogterm_Itf__a_J,type,
map_Gr8458857686193410251term_a: ( ground_gmctxt_a > ground_gterm_a ) > list_Ground_gmctxt_a > list_Ground_gterm_a ).
thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Int__Oint,type,
map_Gr1677053652638677008_a_int: ( ground_gmctxt_a > int ) > list_Ground_gmctxt_a > list_int ).
thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
map_Gr1679544123147727284_a_nat: ( ground_gmctxt_a > nat ) > list_Ground_gmctxt_a > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Ground____Terms__Ogterm_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
map_Gr6591338906797447177ctxt_a: ( ground_gterm_a > ground_gmctxt_a ) > list_Ground_gterm_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
map_li3275209578935457798at_nat: ( list_list_nat > nat ) > list_list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_list_nat ).
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__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
map_nat_int: ( nat > int ) > list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__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__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
set_Gr2314466652747311913ctxt_a: list_G5963651352136220186ctxt_a > set_Gr4667965112854951924ctxt_a ).
thf(sy_c_List_Olist_Oset_001t__Ground____MCtxt__Ogmctxt_It__Int__Oint_J,type,
set_Gr5189245953941246007xt_int: list_G8396253615496126504xt_int > set_Gr2393770795035145730xt_int ).
thf(sy_c_List_Olist_Oset_001t__Ground____MCtxt__Ogmctxt_It__List__Olist_It__Nat__Onat_J_J,type,
set_Gr5001084564890852075st_nat: list_G6183871787581289948st_nat > set_Gr8007501428367021750st_nat ).
thf(sy_c_List_Olist_Oset_001t__Ground____MCtxt__Ogmctxt_It__Nat__Onat_J,type,
set_Gr143724936595666907xt_nat: list_G1662964723573077452xt_nat > set_Gr4883853939966872486xt_nat ).
thf(sy_c_List_Olist_Oset_001t__Ground____MCtxt__Ogmctxt_It__String__Ochar_J,type,
set_Gr5834957080589056860t_char: list_G8740567006383293133t_char > set_Gr8095363379391594151t_char ).
thf(sy_c_List_Olist_Oset_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
set_Ground_gmctxt_a2: list_Ground_gmctxt_a > set_Ground_gmctxt_a ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
set_li3057194059929822169ctxt_a: list_l1578326660578982846ctxt_a > set_li3075106635084045668ctxt_a ).
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__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__String__Ochar,type,
set_char2: list_char > set_char ).
thf(sy_c_List_Olist_Osize__list_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
size_l1580428505752620566ctxt_a: ( ground_gmctxt_a > nat ) > list_Ground_gmctxt_a > nat ).
thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__String__Ochar,type,
size_list_char: ( char > nat ) > list_char > 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_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omember_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
member3758482962427618243ctxt_a: list_Ground_gmctxt_a > ground_gmctxt_a > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
nth_Ground_gmctxt_a: list_Ground_gmctxt_a > nat > ground_gmctxt_a ).
thf(sy_c_List_Onth_001t__Ground____Terms__Ogterm_Itf__a_J,type,
nth_Ground_gterm_a: list_Ground_gterm_a > nat > ground_gterm_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
nth_li6678718213161987943ctxt_a: list_l1578326660578982846ctxt_a > nat > list_Ground_gmctxt_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
nth_li6172171122907299772term_a: list_l4678906206682822203term_a > nat > list_Ground_gterm_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nth_list_list_nat: list_list_list_nat > 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__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Oremove1_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
remove4701464168398194562ctxt_a: ground_gmctxt_a > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
replic9205591110366033502ctxt_a: nat > ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Otake_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
take_Ground_gmctxt_a: nat > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_List_Otake_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
take_list_list_nat: nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Otake_001t__List__Olist_It__Nat__Onat_J,type,
take_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Missing__List_Oadjust__idx,type,
missing_adjust_idx: nat > nat > nat ).
thf(sy_c_Missing__List_Oadjust__idx__rev,type,
missin3815256168798769645dx_rev: nat > nat > nat ).
thf(sy_c_Missing__List_Olist__diff_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
missin6797617701560608538ctxt_a: list_Ground_gmctxt_a > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).
thf(sy_c_Missing__List_Opermut_001t__List__Olist_It__Nat__Onat_J,type,
missin2960327169605006088st_nat: list_list_nat > ( nat > nat ) > list_list_nat ).
thf(sy_c_Missing__List_Opermut_001t__Nat__Onat,type,
missing_permut_nat: list_nat > ( nat > nat ) > list_nat ).
thf(sy_c_Missing__List_Opermut__aux_001t__List__Olist_It__Nat__Onat_J,type,
missin3204601078308456222st_nat: list_list_nat > ( nat > nat ) > list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Opermut__aux_001t__Nat__Onat,type,
missin1888654203714970382ux_nat: list_nat > ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_Missing__List_Oremove__nth_001t__List__Olist_It__Nat__Onat_J,type,
missin5133812807938250791st_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Oremove__nth_001t__Nat__Onat,type,
missin7175274867594579095th_nat: nat > list_nat > list_nat ).
thf(sy_c_Multihole__Context_Opartition__by_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
multih5172578161653122718ctxt_a: list_Ground_gmctxt_a > list_nat > list_l1578326660578982846ctxt_a ).
thf(sy_c_Multihole__Context_Opartition__by_001t__Ground____Terms__Ogterm_Itf__a_J,type,
multih1611307067487991599term_a: list_Ground_gterm_a > list_nat > list_l4678906206682822203term_a ).
thf(sy_c_Multihole__Context_Opartition__by_001t__List__Olist_It__Nat__Onat_J,type,
multih5630945896903731040st_nat: list_list_nat > list_nat > list_list_list_nat ).
thf(sy_c_Multihole__Context_Opartition__by_001t__Nat__Onat,type,
multih8544690682651757648by_nat: list_nat > list_nat > list_list_nat ).
thf(sy_c_Multihole__Context_Opartition__by__idx,type,
multih7279076602905066526by_idx: nat > list_nat > nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
size_s9020356940445107890ctxt_a: ground_gmctxt_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
size_s962548663205356226ctxt_a: list_Ground_gmctxt_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
size_s1893836384466666913term_a: list_Ground_gterm_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
size_s6994882312647770194ctxt_a: list_l1578326660578982846ctxt_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
size_s5321119603959729319term_a: list_l4678906206682822203term_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s6248950052170075156st_nat: list_list_list_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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Ground____MCtxt__Ogmctxt_Itf__a_J_M_Eo_J,type,
bot_bo868933149345519389xt_a_o: ground_gmctxt_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
bot_bo3757104637556896616ctxt_a: set_Ground_gmctxt_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__String__Ochar_J,type,
bot_bot_set_char: set_char ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
ord_le4651699007867869992ctxt_a: set_Ground_gmctxt_a > set_Ground_gmctxt_a > $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__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Product__Type_OPair_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
produc4720362732934906942term_a: ground_gmctxt_a > list_Ground_gterm_a > produc2459281416120113548term_a ).
thf(sy_c_Set_OCollect_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
collec8774204703398931443ctxt_a: ( ground_gmctxt_a > $o ) > set_Ground_gmctxt_a ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__String__Ochar,type,
collect_char: ( char > $o ) > set_char ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
insert2292343091172363342ctxt_a: ground_gmctxt_a > set_Ground_gmctxt_a > set_Ground_gmctxt_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Ois__singleton_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
is_sin4243146486607228458ctxt_a: set_Ground_gmctxt_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
is_sin2641923865335537900st_nat: set_list_nat > $o ).
thf(sy_c_Set_Othe__elem_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
the_el2851850405879629545ctxt_a: set_Ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Nat__Onat_J,type,
the_elem_list_nat: set_list_nat > list_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
accp_Ground_gmctxt_a: ( ground_gmctxt_a > ground_gmctxt_a > $o ) > ground_gmctxt_a > $o ).
thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
member7922894050923735741ctxt_a: ground723776419908861716ctxt_a > set_Gr4667965112854951924ctxt_a > $o ).
thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_It__Int__Oint_J,type,
member6278915634795464651xt_int: ground_gmctxt_int > set_Gr2393770795035145730xt_int > $o ).
thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_It__List__Olist_It__Nat__Onat_J_J,type,
member3016399515581905023st_nat: ground2058972245365194582st_nat > set_Gr8007501428367021750st_nat > $o ).
thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_It__Nat__Onat_J,type,
member1233394617449885551xt_nat: ground_gmctxt_nat > set_Gr4883853939966872486xt_nat > $o ).
thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_It__String__Ochar_J,type,
member539243600765869808t_char: ground_gmctxt_char > set_Gr8095363379391594151t_char > $o ).
thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
member3306386364193653941ctxt_a: ground_gmctxt_a > set_Ground_gmctxt_a > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
member2714729351964853061ctxt_a: list_Ground_gmctxt_a > set_li3075106635084045668ctxt_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__String__Ochar,type,
member_char: char > set_char > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_Ds____,type,
ds: list_Ground_gmctxt_a ).
thf(sy_v_f____,type,
f: a ).
% Relevant facts (1256)
thf(fact_0_GMFun_Ohyps,axiom,
! [X2a: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2a @ ( set_Ground_gmctxt_a2 @ ds ) )
=> ( ( ( ground_ghole_poss_a @ X2a )
= bot_bot_set_list_nat )
=> ( ( ground_num_gholes_a @ X2a )
= zero_zero_nat ) ) ) ).
% GMFun.hyps
thf(fact_1_gmctxt_Oinject,axiom,
! [X21: ground_gmctxt_a,X22: list_G5963651352136220186ctxt_a,Y21: ground_gmctxt_a,Y22: list_G5963651352136220186ctxt_a] :
( ( ( ground5462067128498283694ctxt_a @ X21 @ X22 )
= ( ground5462067128498283694ctxt_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% gmctxt.inject
thf(fact_2_gmctxt_Oinject,axiom,
! [X21: a,X22: list_Ground_gmctxt_a,Y21: a,Y22: list_Ground_gmctxt_a] :
( ( ( ground_GMFun_a @ X21 @ X22 )
= ( ground_GMFun_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% gmctxt.inject
thf(fact_3_GMFun_Oprems,axiom,
( ( ground_ghole_poss_a @ ( ground_GMFun_a @ f @ ds ) )
= bot_bot_set_list_nat ) ).
% GMFun.prems
thf(fact_4_num__gholes__zero__ghole__poss,axiom,
! [D: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ D )
= zero_zero_nat )
=> ( ( ground_ghole_poss_a @ D )
= bot_bot_set_list_nat ) ) ).
% num_gholes_zero_ghole_poss
thf(fact_5_num__holes__mctxt__of__term,axiom,
! [T: ground_gterm_a] :
( ( ground_num_gholes_a @ ( ground582559344758063963term_a @ T ) )
= zero_zero_nat ) ).
% num_holes_mctxt_of_term
thf(fact_6_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_7_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_8_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_9_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_10_no__gholes__gmctxt__of__gterm__gterm__of__gmctxt__id,axiom,
! [C2: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= zero_zero_nat )
=> ( ( ground582559344758063963term_a @ ( ground2297065866753446215ctxt_a @ C2 ) )
= C2 ) ) ).
% no_gholes_gmctxt_of_gterm_gterm_of_gmctxt_id
thf(fact_11_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_12_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_13_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_14_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_15_num__gholes__map__gmctxt,axiom,
! [F: ground_gmctxt_a > ground_gmctxt_a,C2: ground723776419908861716ctxt_a] :
( ( ground2597690549238950442ctxt_a @ ( ground5024537044114205507ctxt_a @ F @ C2 ) )
= ( ground2597690549238950442ctxt_a @ C2 ) ) ).
% num_gholes_map_gmctxt
thf(fact_16_num__gholes__map__gmctxt,axiom,
! [F: a > a,C2: ground_gmctxt_a] :
( ( ground_num_gholes_a @ ( ground2683026529610258053xt_a_a @ F @ C2 ) )
= ( ground_num_gholes_a @ C2 ) ) ).
% num_gholes_map_gmctxt
thf(fact_17_gterm__of__gmctxt__gmctxt__of__gterm__id,axiom,
! [T: ground_gterm_a] :
( ( ground2297065866753446215ctxt_a @ ( ground582559344758063963term_a @ T ) )
= T ) ).
% gterm_of_gmctxt_gmctxt_of_gterm_id
thf(fact_18_empty__iff,axiom,
! [C3: char] :
~ ( member_char @ C3 @ bot_bot_set_char ) ).
% empty_iff
thf(fact_19_empty__iff,axiom,
! [C3: int] :
~ ( member_int @ C3 @ bot_bot_set_int ) ).
% empty_iff
thf(fact_20_empty__iff,axiom,
! [C3: a] :
~ ( member_a @ C3 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_21_empty__iff,axiom,
! [C3: nat] :
~ ( member_nat2 @ C3 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_22_empty__iff,axiom,
! [C3: ground_gmctxt_a] :
~ ( member3306386364193653941ctxt_a @ C3 @ bot_bo3757104637556896616ctxt_a ) ).
% empty_iff
thf(fact_23_empty__iff,axiom,
! [C3: list_nat] :
~ ( member_list_nat @ C3 @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_24_all__not__in__conv,axiom,
! [A: set_char] :
( ( ! [X2: char] :
~ ( member_char @ X2 @ A ) )
= ( A = bot_bot_set_char ) ) ).
% all_not_in_conv
thf(fact_25_all__not__in__conv,axiom,
! [A: set_int] :
( ( ! [X2: int] :
~ ( member_int @ X2 @ A ) )
= ( A = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_26_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_27_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat2 @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_28_all__not__in__conv,axiom,
! [A: set_Ground_gmctxt_a] :
( ( ! [X2: ground_gmctxt_a] :
~ ( member3306386364193653941ctxt_a @ X2 @ A ) )
= ( A = bot_bo3757104637556896616ctxt_a ) ) ).
% all_not_in_conv
thf(fact_29_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X2: list_nat] :
~ ( member_list_nat @ X2 @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_30_Collect__empty__eq,axiom,
! [P: ground_gmctxt_a > $o] :
( ( ( collec8774204703398931443ctxt_a @ P )
= bot_bo3757104637556896616ctxt_a )
= ( ! [X2: ground_gmctxt_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_31_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_32_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_33_empty__Collect__eq,axiom,
! [P: ground_gmctxt_a > $o] :
( ( bot_bo3757104637556896616ctxt_a
= ( collec8774204703398931443ctxt_a @ P ) )
= ( ! [X2: ground_gmctxt_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_34_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_35_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_36_bot__apply,axiom,
( bot_bot_list_nat_o
= ( ^ [X2: list_nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_37_bot__apply,axiom,
( bot_bo868933149345519389xt_a_o
= ( ^ [X2: ground_gmctxt_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_38_num__gholes__o__gmctxt__of__gterm,axiom,
( ( comp_G8049203549856156444term_a @ ground_num_gholes_a @ ground582559344758063963term_a )
= ( ^ [X2: ground_gterm_a] : zero_zero_nat ) ) ).
% num_gholes_o_gmctxt_of_gterm
thf(fact_39_char_Osize__gen,axiom,
! [X1: $o,X23: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X23 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_40_gmctxt_Oset__cases,axiom,
! [E: char,A2: ground_gmctxt_char] :
( ( member_char @ E @ ( ground3608886226622445179t_char @ A2 ) )
=> ( ! [Z2: list_G8740567006383293133t_char] :
( A2
!= ( ground_GMFun_char @ E @ Z2 ) )
=> ~ ! [Z1: char,Z2: list_G8740567006383293133t_char] :
( ( A2
= ( ground_GMFun_char @ Z1 @ Z2 ) )
=> ! [X9: ground_gmctxt_char] :
( ( member539243600765869808t_char @ X9 @ ( set_Gr5834957080589056860t_char @ Z2 ) )
=> ~ ( member_char @ E @ ( ground3608886226622445179t_char @ X9 ) ) ) ) ) ) ).
% gmctxt.set_cases
thf(fact_41_gmctxt_Oset__cases,axiom,
! [E: nat,A2: ground_gmctxt_nat] :
( ( member_nat2 @ E @ ( ground8739233521949953146xt_nat @ A2 ) )
=> ( ! [Z2: list_G1662964723573077452xt_nat] :
( A2
!= ( ground_GMFun_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_G1662964723573077452xt_nat] :
( ( A2
= ( ground_GMFun_nat @ Z1 @ Z2 ) )
=> ! [X9: ground_gmctxt_nat] :
( ( member1233394617449885551xt_nat @ X9 @ ( set_Gr143724936595666907xt_nat @ Z2 ) )
=> ~ ( member_nat2 @ E @ ( ground8739233521949953146xt_nat @ X9 ) ) ) ) ) ) ).
% gmctxt.set_cases
thf(fact_42_gmctxt_Oset__cases,axiom,
! [E: int,A2: ground_gmctxt_int] :
( ( member_int @ E @ ( ground8736743051440902870xt_int @ A2 ) )
=> ( ! [Z2: list_G8396253615496126504xt_int] :
( A2
!= ( ground_GMFun_int @ E @ Z2 ) )
=> ~ ! [Z1: int,Z2: list_G8396253615496126504xt_int] :
( ( A2
= ( ground_GMFun_int @ Z1 @ Z2 ) )
=> ! [X9: ground_gmctxt_int] :
( ( member6278915634795464651xt_int @ X9 @ ( set_Gr5189245953941246007xt_int @ Z2 ) )
=> ~ ( member_int @ E @ ( ground8736743051440902870xt_int @ X9 ) ) ) ) ) ) ).
% gmctxt.set_cases
thf(fact_43_gmctxt_Oset__cases,axiom,
! [E: list_nat,A2: ground2058972245365194582st_nat] :
( ( member_list_nat @ E @ ( ground1659619264735726218st_nat @ A2 ) )
=> ( ! [Z2: list_G6183871787581289948st_nat] :
( A2
!= ( ground940579722934967664st_nat @ E @ Z2 ) )
=> ~ ! [Z1: list_nat,Z2: list_G6183871787581289948st_nat] :
( ( A2
= ( ground940579722934967664st_nat @ Z1 @ Z2 ) )
=> ! [X9: ground2058972245365194582st_nat] :
( ( member3016399515581905023st_nat @ X9 @ ( set_Gr5001084564890852075st_nat @ Z2 ) )
=> ~ ( member_list_nat @ E @ ( ground1659619264735726218st_nat @ X9 ) ) ) ) ) ) ).
% gmctxt.set_cases
thf(fact_44_gmctxt_Oset__cases,axiom,
! [E: ground_gmctxt_a,A2: ground723776419908861716ctxt_a] :
( ( member3306386364193653941ctxt_a @ E @ ( ground2642700214630189768ctxt_a @ A2 ) )
=> ( ! [Z2: list_G5963651352136220186ctxt_a] :
( A2
!= ( ground5462067128498283694ctxt_a @ E @ Z2 ) )
=> ~ ! [Z1: ground_gmctxt_a,Z2: list_G5963651352136220186ctxt_a] :
( ( A2
= ( ground5462067128498283694ctxt_a @ Z1 @ Z2 ) )
=> ! [X9: ground723776419908861716ctxt_a] :
( ( member7922894050923735741ctxt_a @ X9 @ ( set_Gr2314466652747311913ctxt_a @ Z2 ) )
=> ~ ( member3306386364193653941ctxt_a @ E @ ( ground2642700214630189768ctxt_a @ X9 ) ) ) ) ) ) ).
% gmctxt.set_cases
thf(fact_45_gmctxt_Oset__cases,axiom,
! [E: a,A2: ground_gmctxt_a] :
( ( member_a @ E @ ( ground_gfuns_mctxt_a @ A2 ) )
=> ( ! [Z2: list_Ground_gmctxt_a] :
( A2
!= ( ground_GMFun_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_Ground_gmctxt_a] :
( ( A2
= ( ground_GMFun_a @ Z1 @ Z2 ) )
=> ! [X9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ ( set_Ground_gmctxt_a2 @ Z2 ) )
=> ~ ( member_a @ E @ ( ground_gfuns_mctxt_a @ X9 ) ) ) ) ) ) ).
% gmctxt.set_cases
thf(fact_46_gmctxt_Oset__intros_I2_J,axiom,
! [Y: ground_gmctxt_char,X22: list_G8740567006383293133t_char,Ya: char,X21: char] :
( ( member539243600765869808t_char @ Y @ ( set_Gr5834957080589056860t_char @ X22 ) )
=> ( ( member_char @ Ya @ ( ground3608886226622445179t_char @ Y ) )
=> ( member_char @ Ya @ ( ground3608886226622445179t_char @ ( ground_GMFun_char @ X21 @ X22 ) ) ) ) ) ).
% gmctxt.set_intros(2)
thf(fact_47_gmctxt_Oset__intros_I2_J,axiom,
! [Y: ground_gmctxt_nat,X22: list_G1662964723573077452xt_nat,Ya: nat,X21: nat] :
( ( member1233394617449885551xt_nat @ Y @ ( set_Gr143724936595666907xt_nat @ X22 ) )
=> ( ( member_nat2 @ Ya @ ( ground8739233521949953146xt_nat @ Y ) )
=> ( member_nat2 @ Ya @ ( ground8739233521949953146xt_nat @ ( ground_GMFun_nat @ X21 @ X22 ) ) ) ) ) ).
% gmctxt.set_intros(2)
thf(fact_48_gmctxt_Oset__intros_I2_J,axiom,
! [Y: ground_gmctxt_int,X22: list_G8396253615496126504xt_int,Ya: int,X21: int] :
( ( member6278915634795464651xt_int @ Y @ ( set_Gr5189245953941246007xt_int @ X22 ) )
=> ( ( member_int @ Ya @ ( ground8736743051440902870xt_int @ Y ) )
=> ( member_int @ Ya @ ( ground8736743051440902870xt_int @ ( ground_GMFun_int @ X21 @ X22 ) ) ) ) ) ).
% gmctxt.set_intros(2)
thf(fact_49_gmctxt_Oset__intros_I2_J,axiom,
! [Y: ground2058972245365194582st_nat,X22: list_G6183871787581289948st_nat,Ya: list_nat,X21: list_nat] :
( ( member3016399515581905023st_nat @ Y @ ( set_Gr5001084564890852075st_nat @ X22 ) )
=> ( ( member_list_nat @ Ya @ ( ground1659619264735726218st_nat @ Y ) )
=> ( member_list_nat @ Ya @ ( ground1659619264735726218st_nat @ ( ground940579722934967664st_nat @ X21 @ X22 ) ) ) ) ) ).
% gmctxt.set_intros(2)
thf(fact_50_gmctxt_Oset__intros_I2_J,axiom,
! [Y: ground723776419908861716ctxt_a,X22: list_G5963651352136220186ctxt_a,Ya: ground_gmctxt_a,X21: ground_gmctxt_a] :
( ( member7922894050923735741ctxt_a @ Y @ ( set_Gr2314466652747311913ctxt_a @ X22 ) )
=> ( ( member3306386364193653941ctxt_a @ Ya @ ( ground2642700214630189768ctxt_a @ Y ) )
=> ( member3306386364193653941ctxt_a @ Ya @ ( ground2642700214630189768ctxt_a @ ( ground5462067128498283694ctxt_a @ X21 @ X22 ) ) ) ) ) ).
% gmctxt.set_intros(2)
thf(fact_51_gmctxt_Oset__intros_I2_J,axiom,
! [Y: ground_gmctxt_a,X22: list_Ground_gmctxt_a,Ya: a,X21: a] :
( ( member3306386364193653941ctxt_a @ Y @ ( set_Ground_gmctxt_a2 @ X22 ) )
=> ( ( member_a @ Ya @ ( ground_gfuns_mctxt_a @ Y ) )
=> ( member_a @ Ya @ ( ground_gfuns_mctxt_a @ ( ground_GMFun_a @ X21 @ X22 ) ) ) ) ) ).
% gmctxt.set_intros(2)
thf(fact_52_emptyE,axiom,
! [A2: char] :
~ ( member_char @ A2 @ bot_bot_set_char ) ).
% emptyE
thf(fact_53_emptyE,axiom,
! [A2: int] :
~ ( member_int @ A2 @ bot_bot_set_int ) ).
% emptyE
thf(fact_54_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_55_emptyE,axiom,
! [A2: nat] :
~ ( member_nat2 @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_56_emptyE,axiom,
! [A2: ground_gmctxt_a] :
~ ( member3306386364193653941ctxt_a @ A2 @ bot_bo3757104637556896616ctxt_a ) ).
% emptyE
thf(fact_57_emptyE,axiom,
! [A2: list_nat] :
~ ( member_list_nat @ A2 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_58_equals0D,axiom,
! [A: set_char,A2: char] :
( ( A = bot_bot_set_char )
=> ~ ( member_char @ A2 @ A ) ) ).
% equals0D
thf(fact_59_equals0D,axiom,
! [A: set_int,A2: int] :
( ( A = bot_bot_set_int )
=> ~ ( member_int @ A2 @ A ) ) ).
% equals0D
thf(fact_60_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_61_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_62_equals0D,axiom,
! [A: set_Ground_gmctxt_a,A2: ground_gmctxt_a] :
( ( A = bot_bo3757104637556896616ctxt_a )
=> ~ ( member3306386364193653941ctxt_a @ A2 @ A ) ) ).
% equals0D
thf(fact_63_equals0D,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_64_equals0I,axiom,
! [A: set_char] :
( ! [Y2: char] :
~ ( member_char @ Y2 @ A )
=> ( A = bot_bot_set_char ) ) ).
% equals0I
thf(fact_65_equals0I,axiom,
! [A: set_int] :
( ! [Y2: int] :
~ ( member_int @ Y2 @ A )
=> ( A = bot_bot_set_int ) ) ).
% equals0I
thf(fact_66_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_67_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_68_equals0I,axiom,
! [A: set_Ground_gmctxt_a] :
( ! [Y2: ground_gmctxt_a] :
~ ( member3306386364193653941ctxt_a @ Y2 @ A )
=> ( A = bot_bo3757104637556896616ctxt_a ) ) ).
% equals0I
thf(fact_69_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat @ Y2 @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_70_char_Oinject,axiom,
! [X1: $o,X23: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y23: $o,Y3: $o,Y4: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
( ( ( char2 @ X1 @ X23 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 )
= ( char2 @ Y1 @ Y23 @ Y3 @ Y4 @ Y5 @ Y6 @ Y7 @ Y8 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 )
& ( X3 = Y3 )
& ( X4 = Y4 )
& ( X5 = Y5 )
& ( X6 = Y6 )
& ( X7 = Y7 )
& ( X8 = Y8 ) ) ) ).
% char.inject
thf(fact_71_map__gmctxt__comp,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gterm_a > ground_gmctxt_a,C2: ground2383972455873737935term_a] :
( ( ground7700310005849181941_a_nat @ F @ ( ground8707887377537884296ctxt_a @ G @ C2 ) )
= ( ground6644341496884830778_a_nat @ ( comp_G8049203549856156444term_a @ F @ G ) @ C2 ) ) ).
% map_gmctxt_comp
thf(fact_72_map__gmctxt__comp,axiom,
! [F: list_nat > list_nat,G: list_nat > list_nat,C2: ground2058972245365194582st_nat] :
( ( ground8629520670591032391st_nat @ F @ ( ground8629520670591032391st_nat @ G @ C2 ) )
= ( ground8629520670591032391st_nat @ ( comp_l7223822213492037765st_nat @ F @ G ) @ C2 ) ) ).
% map_gmctxt_comp
thf(fact_73_map__gmctxt__comp,axiom,
! [F: nat > nat,G: nat > nat,C2: ground_gmctxt_nat] :
( ( ground3221804553564598439at_nat @ F @ ( ground3221804553564598439at_nat @ G @ C2 ) )
= ( ground3221804553564598439at_nat @ ( comp_nat_nat_nat @ F @ G ) @ C2 ) ) ).
% map_gmctxt_comp
thf(fact_74_map__gmctxt__comp,axiom,
! [F: list_nat > nat,G: list_nat > list_nat,C2: ground2058972245365194582st_nat] :
( ( ground3379907019545862327at_nat @ F @ ( ground8629520670591032391st_nat @ G @ C2 ) )
= ( ground3379907019545862327at_nat @ ( comp_l5285636708041795445st_nat @ F @ G ) @ C2 ) ) ).
% map_gmctxt_comp
thf(fact_75_map__gmctxt__comp,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gmctxt_a > ground_gmctxt_a,C2: ground723776419908861716ctxt_a] :
( ( ground7700310005849181941_a_nat @ F @ ( ground5024537044114205507ctxt_a @ G @ C2 ) )
= ( ground7700310005849181941_a_nat @ ( comp_G4985101524424093169ctxt_a @ F @ G ) @ C2 ) ) ).
% map_gmctxt_comp
thf(fact_76_map__gmctxt__comp,axiom,
! [F: a > a,G: a > a,C2: ground_gmctxt_a] :
( ( ground2683026529610258053xt_a_a @ F @ ( ground2683026529610258053xt_a_a @ G @ C2 ) )
= ( ground2683026529610258053xt_a_a @ ( comp_a_a_a @ F @ G ) @ C2 ) ) ).
% map_gmctxt_comp
thf(fact_77_map__gmctxt__comp,axiom,
! [F: ground_gmctxt_a > ground_gmctxt_a,G: ground_gmctxt_a > ground_gmctxt_a,C2: ground723776419908861716ctxt_a] :
( ( ground5024537044114205507ctxt_a @ F @ ( ground5024537044114205507ctxt_a @ G @ C2 ) )
= ( ground5024537044114205507ctxt_a @ ( comp_G2329228445783543871ctxt_a @ F @ G ) @ C2 ) ) ).
% map_gmctxt_comp
thf(fact_78_gmctxt_Omap__comp,axiom,
! [G: ground_gmctxt_a > nat,F: ground_gterm_a > ground_gmctxt_a,V: ground2383972455873737935term_a] :
( ( ground7700310005849181941_a_nat @ G @ ( ground8707887377537884296ctxt_a @ F @ V ) )
= ( ground6644341496884830778_a_nat @ ( comp_G8049203549856156444term_a @ G @ F ) @ V ) ) ).
% gmctxt.map_comp
thf(fact_79_gmctxt_Omap__comp,axiom,
! [G: list_nat > list_nat,F: list_nat > list_nat,V: ground2058972245365194582st_nat] :
( ( ground8629520670591032391st_nat @ G @ ( ground8629520670591032391st_nat @ F @ V ) )
= ( ground8629520670591032391st_nat @ ( comp_l7223822213492037765st_nat @ G @ F ) @ V ) ) ).
% gmctxt.map_comp
thf(fact_80_gmctxt_Omap__comp,axiom,
! [G: nat > nat,F: nat > nat,V: ground_gmctxt_nat] :
( ( ground3221804553564598439at_nat @ G @ ( ground3221804553564598439at_nat @ F @ V ) )
= ( ground3221804553564598439at_nat @ ( comp_nat_nat_nat @ G @ F ) @ V ) ) ).
% gmctxt.map_comp
thf(fact_81_gmctxt_Omap__comp,axiom,
! [G: list_nat > nat,F: list_nat > list_nat,V: ground2058972245365194582st_nat] :
( ( ground3379907019545862327at_nat @ G @ ( ground8629520670591032391st_nat @ F @ V ) )
= ( ground3379907019545862327at_nat @ ( comp_l5285636708041795445st_nat @ G @ F ) @ V ) ) ).
% gmctxt.map_comp
thf(fact_82_gmctxt_Omap__comp,axiom,
! [G: ground_gmctxt_a > nat,F: ground_gmctxt_a > ground_gmctxt_a,V: ground723776419908861716ctxt_a] :
( ( ground7700310005849181941_a_nat @ G @ ( ground5024537044114205507ctxt_a @ F @ V ) )
= ( ground7700310005849181941_a_nat @ ( comp_G4985101524424093169ctxt_a @ G @ F ) @ V ) ) ).
% gmctxt.map_comp
thf(fact_83_gmctxt_Omap__comp,axiom,
! [G: a > a,F: a > a,V: ground_gmctxt_a] :
( ( ground2683026529610258053xt_a_a @ G @ ( ground2683026529610258053xt_a_a @ F @ V ) )
= ( ground2683026529610258053xt_a_a @ ( comp_a_a_a @ G @ F ) @ V ) ) ).
% gmctxt.map_comp
thf(fact_84_gmctxt_Omap__comp,axiom,
! [G: ground_gmctxt_a > ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,V: ground723776419908861716ctxt_a] :
( ( ground5024537044114205507ctxt_a @ G @ ( ground5024537044114205507ctxt_a @ F @ V ) )
= ( ground5024537044114205507ctxt_a @ ( comp_G2329228445783543871ctxt_a @ G @ F ) @ V ) ) ).
% gmctxt.map_comp
thf(fact_85_char_Oexhaust,axiom,
! [Y: char] :
~ ! [X12: $o,X24: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( Y
!= ( char2 @ X12 @ X24 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) ) ).
% char.exhaust
thf(fact_86_bot__set__def,axiom,
( bot_bo3757104637556896616ctxt_a
= ( collec8774204703398931443ctxt_a @ bot_bo868933149345519389xt_a_o ) ) ).
% bot_set_def
thf(fact_87_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_88_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_89_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_90_gmctxt_Oset__intros_I1_J,axiom,
! [X21: char,X22: list_G8740567006383293133t_char] : ( member_char @ X21 @ ( ground3608886226622445179t_char @ ( ground_GMFun_char @ X21 @ X22 ) ) ) ).
% gmctxt.set_intros(1)
thf(fact_91_gmctxt_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_G1662964723573077452xt_nat] : ( member_nat2 @ X21 @ ( ground8739233521949953146xt_nat @ ( ground_GMFun_nat @ X21 @ X22 ) ) ) ).
% gmctxt.set_intros(1)
thf(fact_92_gmctxt_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_G8396253615496126504xt_int] : ( member_int @ X21 @ ( ground8736743051440902870xt_int @ ( ground_GMFun_int @ X21 @ X22 ) ) ) ).
% gmctxt.set_intros(1)
thf(fact_93_gmctxt_Oset__intros_I1_J,axiom,
! [X21: list_nat,X22: list_G6183871787581289948st_nat] : ( member_list_nat @ X21 @ ( ground1659619264735726218st_nat @ ( ground940579722934967664st_nat @ X21 @ X22 ) ) ) ).
% gmctxt.set_intros(1)
thf(fact_94_gmctxt_Oset__intros_I1_J,axiom,
! [X21: ground_gmctxt_a,X22: list_G5963651352136220186ctxt_a] : ( member3306386364193653941ctxt_a @ X21 @ ( ground2642700214630189768ctxt_a @ ( ground5462067128498283694ctxt_a @ X21 @ X22 ) ) ) ).
% gmctxt.set_intros(1)
thf(fact_95_gmctxt_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_Ground_gmctxt_a] : ( member_a @ X21 @ ( ground_gfuns_mctxt_a @ ( ground_GMFun_a @ X21 @ X22 ) ) ) ).
% gmctxt.set_intros(1)
thf(fact_96_gmctxt_Omap__ident__strong,axiom,
! [T: ground_gmctxt_char,F: char > char] :
( ! [Z: char] :
( ( member_char @ Z @ ( ground3608886226622445179t_char @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( ground8079844605718550185r_char @ F @ T )
= T ) ) ).
% gmctxt.map_ident_strong
thf(fact_97_gmctxt_Omap__ident__strong,axiom,
! [T: ground_gmctxt_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( ground8739233521949953146xt_nat @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( ground3221804553564598439at_nat @ F @ T )
= T ) ) ).
% gmctxt.map_ident_strong
thf(fact_98_gmctxt_Omap__ident__strong,axiom,
! [T: ground_gmctxt_int,F: int > int] :
( ! [Z: int] :
( ( member_int @ Z @ ( ground8736743051440902870xt_int @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( ground4218667794956602719nt_int @ F @ T )
= T ) ) ).
% gmctxt.map_ident_strong
thf(fact_99_gmctxt_Omap__ident__strong,axiom,
! [T: ground_gmctxt_a,F: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( ground_gfuns_mctxt_a @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( ground2683026529610258053xt_a_a @ F @ T )
= T ) ) ).
% gmctxt.map_ident_strong
thf(fact_100_gmctxt_Omap__ident__strong,axiom,
! [T: ground2058972245365194582st_nat,F: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat @ Z @ ( ground1659619264735726218st_nat @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( ground8629520670591032391st_nat @ F @ T )
= T ) ) ).
% gmctxt.map_ident_strong
thf(fact_101_gmctxt_Omap__ident__strong,axiom,
! [T: ground723776419908861716ctxt_a,F: ground_gmctxt_a > ground_gmctxt_a] :
( ! [Z: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Z @ ( ground2642700214630189768ctxt_a @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( ground5024537044114205507ctxt_a @ F @ T )
= T ) ) ).
% gmctxt.map_ident_strong
thf(fact_102_gmctxt_Oinj__map__strong,axiom,
! [X: ground_gmctxt_a,Xa: ground_gmctxt_a,F: a > a,Fa: a > a] :
( ! [Z: a,Za: a] :
( ( member_a @ Z @ ( ground_gfuns_mctxt_a @ X ) )
=> ( ( member_a @ Za @ ( ground_gfuns_mctxt_a @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( ground2683026529610258053xt_a_a @ F @ X )
= ( ground2683026529610258053xt_a_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% gmctxt.inj_map_strong
thf(fact_103_gmctxt_Oinj__map__strong,axiom,
! [X: ground723776419908861716ctxt_a,Xa: ground723776419908861716ctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,Fa: ground_gmctxt_a > ground_gmctxt_a] :
( ! [Z: ground_gmctxt_a,Za: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Z @ ( ground2642700214630189768ctxt_a @ X ) )
=> ( ( member3306386364193653941ctxt_a @ Za @ ( ground2642700214630189768ctxt_a @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( ground5024537044114205507ctxt_a @ F @ X )
= ( ground5024537044114205507ctxt_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% gmctxt.inj_map_strong
thf(fact_104_gmctxt_Omap__cong0,axiom,
! [X: ground_gmctxt_a,F: a > a,G: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( ground_gfuns_mctxt_a @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( ground2683026529610258053xt_a_a @ F @ X )
= ( ground2683026529610258053xt_a_a @ G @ X ) ) ) ).
% gmctxt.map_cong0
thf(fact_105_gmctxt_Omap__cong0,axiom,
! [X: ground723776419908861716ctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,G: ground_gmctxt_a > ground_gmctxt_a] :
( ! [Z: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Z @ ( ground2642700214630189768ctxt_a @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( ground5024537044114205507ctxt_a @ F @ X )
= ( ground5024537044114205507ctxt_a @ G @ X ) ) ) ).
% gmctxt.map_cong0
thf(fact_106_gmctxt_Omap__cong,axiom,
! [X: ground_gmctxt_a,Ya: ground_gmctxt_a,F: a > a,G: a > a] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( ground_gfuns_mctxt_a @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( ground2683026529610258053xt_a_a @ F @ X )
= ( ground2683026529610258053xt_a_a @ G @ Ya ) ) ) ) ).
% gmctxt.map_cong
thf(fact_107_gmctxt_Omap__cong,axiom,
! [X: ground723776419908861716ctxt_a,Ya: ground723776419908861716ctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,G: ground_gmctxt_a > ground_gmctxt_a] :
( ( X = Ya )
=> ( ! [Z: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Z @ ( ground2642700214630189768ctxt_a @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( ground5024537044114205507ctxt_a @ F @ X )
= ( ground5024537044114205507ctxt_a @ G @ Ya ) ) ) ) ).
% gmctxt.map_cong
thf(fact_108_bot__fun__def,axiom,
( bot_bot_list_nat_o
= ( ^ [X2: list_nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_109_bot__fun__def,axiom,
( bot_bo868933149345519389xt_a_o
= ( ^ [X2: ground_gmctxt_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_110_ex__in__conv,axiom,
! [A: set_char] :
( ( ? [X2: char] : ( member_char @ X2 @ A ) )
= ( A != bot_bot_set_char ) ) ).
% ex_in_conv
thf(fact_111_ex__in__conv,axiom,
! [A: set_int] :
( ( ? [X2: int] : ( member_int @ X2 @ A ) )
= ( A != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_112_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_113_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat2 @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_114_ex__in__conv,axiom,
! [A: set_Ground_gmctxt_a] :
( ( ? [X2: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X2 @ A ) )
= ( A != bot_bo3757104637556896616ctxt_a ) ) ).
% ex_in_conv
thf(fact_115_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_116_comp__apply,axiom,
( comp_G8049203549856156444term_a
= ( ^ [F2: ground_gmctxt_a > nat,G2: ground_gterm_a > ground_gmctxt_a,X2: ground_gterm_a] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_117_comp__apply,axiom,
( comp_G4985101524424093169ctxt_a
= ( ^ [F2: ground_gmctxt_a > nat,G2: ground_gmctxt_a > ground_gmctxt_a,X2: ground_gmctxt_a] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_118_comp__apply,axiom,
( comp_l7223822213492037765st_nat
= ( ^ [F2: list_nat > list_nat,G2: list_nat > list_nat,X2: list_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_119_comp__apply,axiom,
( comp_nat_nat_nat
= ( ^ [F2: nat > nat,G2: nat > nat,X2: nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_120_comp__apply,axiom,
( comp_l5285636708041795445st_nat
= ( ^ [F2: list_nat > nat,G2: list_nat > list_nat,X2: list_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_121_gmctxt_Osize__gen__o__map,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gterm_a > ground_gmctxt_a] :
( ( comp_G3324881245815093522term_a @ ( ground2918769037360547453ctxt_a @ F ) @ ( ground8707887377537884296ctxt_a @ G ) )
= ( ground7767404987464290192term_a @ ( comp_G8049203549856156444term_a @ F @ G ) ) ) ).
% gmctxt.size_gen_o_map
thf(fact_122_gmctxt_Osize__gen__o__map,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_G3928820918855900945xt_nat @ ( ground7473917417092711983xt_nat @ F ) @ ( ground3221804553564598439at_nat @ G ) )
= ( ground7473917417092711983xt_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% gmctxt.size_gen_o_map
thf(fact_123_gmctxt_Osize__gen__o__map,axiom,
! [F: list_nat > nat,G: list_nat > list_nat] :
( ( comp_G3225351923774419985st_nat @ ( ground3062967014897493439st_nat @ F ) @ ( ground8629520670591032391st_nat @ G ) )
= ( ground3062967014897493439st_nat @ ( comp_l5285636708041795445st_nat @ F @ G ) ) ) ).
% gmctxt.size_gen_o_map
thf(fact_124_gmctxt_Osize__gen__o__map,axiom,
! [F: a > nat,G: a > a] :
( ( comp_G4985101524424093169ctxt_a @ ( ground_size_gmctxt_a @ F ) @ ( ground2683026529610258053xt_a_a @ G ) )
= ( ground_size_gmctxt_a @ ( comp_a_nat_a @ F @ G ) ) ) ).
% gmctxt.size_gen_o_map
thf(fact_125_gmctxt_Osize__gen__o__map,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gmctxt_a > ground_gmctxt_a] :
( ( comp_G593817150734306193ctxt_a @ ( ground2918769037360547453ctxt_a @ F ) @ ( ground5024537044114205507ctxt_a @ G ) )
= ( ground2918769037360547453ctxt_a @ ( comp_G4985101524424093169ctxt_a @ F @ G ) ) ) ).
% gmctxt.size_gen_o_map
thf(fact_126_mem__Collect__eq,axiom,
! [A2: char,P: char > $o] :
( ( member_char @ A2 @ ( collect_char @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
! [A2: ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( member3306386364193653941ctxt_a @ A2 @ ( collec8774204703398931443ctxt_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A: set_char] :
( ( collect_char
@ ^ [X2: char] : ( member_char @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
! [A: set_Ground_gmctxt_a] :
( ( collec8774204703398931443ctxt_a
@ ^ [X2: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_138_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X9: list_nat] :
( ( P @ X9 )
= ( Q @ X9 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_139_Collect__cong,axiom,
! [P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o] :
( ! [X9: ground_gmctxt_a] :
( ( P @ X9 )
= ( Q @ X9 ) )
=> ( ( collec8774204703398931443ctxt_a @ P )
= ( collec8774204703398931443ctxt_a @ Q ) ) ) ).
% Collect_cong
thf(fact_140_char_Osize_I2_J,axiom,
! [X1: $o,X23: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X23 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_141_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: nat > nat,R1: nat > nat,R2: nat > nat,F: nat > nat,L: nat > nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= ( comp_nat_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_nat_nat_nat @ F @ R1 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_142_rewriteR__comp__comp2,axiom,
! [G: ground_gterm_a > ground_gmctxt_a,H: nat > ground_gterm_a,R1: nat > ground_gmctxt_a,R2: nat > nat,F: ground_gmctxt_a > nat,L: nat > nat] :
( ( ( comp_G7903908942813652348_a_nat @ G @ H )
= ( comp_n5201041448192358307_a_nat @ R1 @ R2 ) )
=> ( ( ( comp_G7635752065729029539at_nat @ F @ R1 )
= L )
=> ( ( comp_G5395315277947777326at_nat @ ( comp_G8049203549856156444term_a @ F @ G ) @ H )
= ( comp_nat_nat_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_143_rewriteR__comp__comp2,axiom,
! [G: ground_gmctxt_a > ground_gmctxt_a,H: nat > ground_gmctxt_a,R1: nat > ground_gmctxt_a,R2: nat > nat,F: ground_gmctxt_a > nat,L: nat > nat] :
( ( ( comp_G3911404188448037873_a_nat @ G @ H )
= ( comp_n5201041448192358307_a_nat @ R1 @ R2 ) )
=> ( ( ( comp_G7635752065729029539at_nat @ F @ R1 )
= L )
=> ( ( comp_G7635752065729029539at_nat @ ( comp_G4985101524424093169ctxt_a @ F @ G ) @ H )
= ( comp_nat_nat_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_144_rewriteR__comp__comp2,axiom,
! [G: list_nat > list_nat,H: nat > list_nat,R1: nat > list_nat,R2: nat > nat,F: list_nat > nat,L: nat > nat] :
( ( ( comp_l7524593755204080629at_nat @ G @ H )
= ( comp_n6232535815235743333at_nat @ R1 @ R2 ) )
=> ( ( ( comp_l3539220536931113701at_nat @ F @ R1 )
= L )
=> ( ( comp_l3539220536931113701at_nat @ ( comp_l5285636708041795445st_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_145_rewriteR__comp__comp2,axiom,
! [G: nat > ground_gmctxt_a,H: nat > nat,R1: ground_gterm_a > ground_gmctxt_a,R2: nat > ground_gterm_a,F: ground_gmctxt_a > nat,L: ground_gterm_a > nat] :
( ( ( comp_n5201041448192358307_a_nat @ G @ H )
= ( comp_G7903908942813652348_a_nat @ R1 @ R2 ) )
=> ( ( ( comp_G8049203549856156444term_a @ F @ R1 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_G7635752065729029539at_nat @ F @ G ) @ H )
= ( comp_G5395315277947777326at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_146_rewriteR__comp__comp2,axiom,
! [G: nat > ground_gmctxt_a,H: nat > nat,R1: ground_gmctxt_a > ground_gmctxt_a,R2: nat > ground_gmctxt_a,F: ground_gmctxt_a > nat,L: ground_gmctxt_a > nat] :
( ( ( comp_n5201041448192358307_a_nat @ G @ H )
= ( comp_G3911404188448037873_a_nat @ R1 @ R2 ) )
=> ( ( ( comp_G4985101524424093169ctxt_a @ F @ R1 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_G7635752065729029539at_nat @ F @ G ) @ H )
= ( comp_G7635752065729029539at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_147_rewriteR__comp__comp2,axiom,
! [G: nat > list_nat,H: nat > nat,R1: list_nat > list_nat,R2: nat > list_nat,F: list_nat > nat,L: list_nat > nat] :
( ( ( comp_n6232535815235743333at_nat @ G @ H )
= ( comp_l7524593755204080629at_nat @ R1 @ R2 ) )
=> ( ( ( comp_l5285636708041795445st_nat @ F @ R1 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_l3539220536931113701at_nat @ F @ G ) @ H )
= ( comp_l3539220536931113701at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_148_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: ground_gterm_a > nat,R1: ground_gmctxt_a > nat,R2: ground_gterm_a > ground_gmctxt_a,F: nat > nat,L: ground_gmctxt_a > nat] :
( ( ( comp_n8259423687851516650term_a @ G @ H )
= ( comp_G8049203549856156444term_a @ R1 @ R2 ) )
=> ( ( ( comp_n6274738784168413603ctxt_a @ F @ R1 )
= L )
=> ( ( comp_n8259423687851516650term_a @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_G8049203549856156444term_a @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_149_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: ground_gmctxt_a > nat,R1: ground_gmctxt_a > nat,R2: ground_gmctxt_a > ground_gmctxt_a,F: nat > nat,L: ground_gmctxt_a > nat] :
( ( ( comp_n6274738784168413603ctxt_a @ G @ H )
= ( comp_G4985101524424093169ctxt_a @ R1 @ R2 ) )
=> ( ( ( comp_n6274738784168413603ctxt_a @ F @ R1 )
= L )
=> ( ( comp_n6274738784168413603ctxt_a @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_G4985101524424093169ctxt_a @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_150_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: list_nat > nat,R1: list_nat > nat,R2: list_nat > list_nat,F: nat > nat,L: list_nat > nat] :
( ( ( comp_n3993578768073458149st_nat @ G @ H )
= ( comp_l5285636708041795445st_nat @ R1 @ R2 ) )
=> ( ( ( comp_n3993578768073458149st_nat @ F @ R1 )
= L )
=> ( ( comp_n3993578768073458149st_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_l5285636708041795445st_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_151_rewriteL__comp__comp2,axiom,
! [F: nat > nat,G: nat > nat,L1: nat > nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= ( comp_nat_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_152_rewriteL__comp__comp2,axiom,
! [F: nat > nat,G: ground_gterm_a > nat,L1: ground_gmctxt_a > nat,L2: ground_gterm_a > ground_gmctxt_a,H: nat > ground_gterm_a,R: nat > ground_gmctxt_a] :
( ( ( comp_n8259423687851516650term_a @ F @ G )
= ( comp_G8049203549856156444term_a @ L1 @ L2 ) )
=> ( ( ( comp_G7903908942813652348_a_nat @ L2 @ H )
= R )
=> ( ( comp_nat_nat_nat @ F @ ( comp_G5395315277947777326at_nat @ G @ H ) )
= ( comp_G7635752065729029539at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_153_rewriteL__comp__comp2,axiom,
! [F: nat > nat,G: ground_gmctxt_a > nat,L1: ground_gmctxt_a > nat,L2: ground_gmctxt_a > ground_gmctxt_a,H: nat > ground_gmctxt_a,R: nat > ground_gmctxt_a] :
( ( ( comp_n6274738784168413603ctxt_a @ F @ G )
= ( comp_G4985101524424093169ctxt_a @ L1 @ L2 ) )
=> ( ( ( comp_G3911404188448037873_a_nat @ L2 @ H )
= R )
=> ( ( comp_nat_nat_nat @ F @ ( comp_G7635752065729029539at_nat @ G @ H ) )
= ( comp_G7635752065729029539at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_154_rewriteL__comp__comp2,axiom,
! [F: ground_gmctxt_a > nat,G: nat > ground_gmctxt_a,L1: nat > nat,L2: nat > nat,H: ground_gterm_a > nat,R: ground_gterm_a > nat] :
( ( ( comp_G7635752065729029539at_nat @ F @ G )
= ( comp_nat_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_n8259423687851516650term_a @ L2 @ H )
= R )
=> ( ( comp_G8049203549856156444term_a @ F @ ( comp_n7836545058038479132term_a @ G @ H ) )
= ( comp_n8259423687851516650term_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_155_rewriteL__comp__comp2,axiom,
! [F: ground_gmctxt_a > nat,G: nat > ground_gmctxt_a,L1: nat > nat,L2: nat > nat,H: ground_gmctxt_a > nat,R: ground_gmctxt_a > nat] :
( ( ( comp_G7635752065729029539at_nat @ F @ G )
= ( comp_nat_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_n6274738784168413603ctxt_a @ L2 @ H )
= R )
=> ( ( comp_G4985101524424093169ctxt_a @ F @ ( comp_n539216501031124977ctxt_a @ G @ H ) )
= ( comp_n6274738784168413603ctxt_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_156_rewriteL__comp__comp2,axiom,
! [F: list_nat > nat,G: nat > list_nat,L1: nat > nat,L2: nat > nat,H: list_nat > nat,R: list_nat > nat] :
( ( ( comp_l3539220536931113701at_nat @ F @ G )
= ( comp_nat_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_n3993578768073458149st_nat @ L2 @ H )
= R )
=> ( ( comp_l5285636708041795445st_nat @ F @ ( comp_n1293334696235875061st_nat @ G @ H ) )
= ( comp_n3993578768073458149st_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_157_rewriteL__comp__comp2,axiom,
! [F: nat > nat,G: list_nat > nat,L1: list_nat > nat,L2: list_nat > list_nat,H: nat > list_nat,R: nat > list_nat] :
( ( ( comp_n3993578768073458149st_nat @ F @ G )
= ( comp_l5285636708041795445st_nat @ L1 @ L2 ) )
=> ( ( ( comp_l7524593755204080629at_nat @ L2 @ H )
= R )
=> ( ( comp_nat_nat_nat @ F @ ( comp_l3539220536931113701at_nat @ G @ H ) )
= ( comp_l3539220536931113701at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_158_rewriteL__comp__comp2,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gterm_a > ground_gmctxt_a,L1: nat > nat,L2: ground_gterm_a > nat,H: nat > ground_gterm_a,R: nat > nat] :
( ( ( comp_G8049203549856156444term_a @ F @ G )
= ( comp_n8259423687851516650term_a @ L1 @ L2 ) )
=> ( ( ( comp_G5395315277947777326at_nat @ L2 @ H )
= R )
=> ( ( comp_G7635752065729029539at_nat @ F @ ( comp_G7903908942813652348_a_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_159_rewriteL__comp__comp2,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gmctxt_a > ground_gmctxt_a,L1: nat > nat,L2: ground_gmctxt_a > nat,H: nat > ground_gmctxt_a,R: nat > nat] :
( ( ( comp_G4985101524424093169ctxt_a @ F @ G )
= ( comp_n6274738784168413603ctxt_a @ L1 @ L2 ) )
=> ( ( ( comp_G7635752065729029539at_nat @ L2 @ H )
= R )
=> ( ( comp_G7635752065729029539at_nat @ F @ ( comp_G3911404188448037873_a_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_160_rewriteL__comp__comp2,axiom,
! [F: nat > nat,G: nat > nat,L1: ground_gmctxt_a > nat,L2: nat > ground_gmctxt_a,H: ground_gterm_a > nat,R: ground_gterm_a > ground_gmctxt_a] :
( ( ( comp_nat_nat_nat @ F @ G )
= ( comp_G7635752065729029539at_nat @ L1 @ L2 ) )
=> ( ( ( comp_n7836545058038479132term_a @ L2 @ H )
= R )
=> ( ( comp_n8259423687851516650term_a @ F @ ( comp_n8259423687851516650term_a @ G @ H ) )
= ( comp_G8049203549856156444term_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_161_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: nat > nat,R: nat > nat,F: nat > nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= R )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_162_rewriteR__comp__comp,axiom,
! [G: ground_gmctxt_a > nat,H: ground_gterm_a > ground_gmctxt_a,R: ground_gterm_a > nat,F: nat > nat] :
( ( ( comp_G8049203549856156444term_a @ G @ H )
= R )
=> ( ( comp_G8049203549856156444term_a @ ( comp_n6274738784168413603ctxt_a @ F @ G ) @ H )
= ( comp_n8259423687851516650term_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_163_rewriteR__comp__comp,axiom,
! [G: ground_gmctxt_a > nat,H: ground_gmctxt_a > ground_gmctxt_a,R: ground_gmctxt_a > nat,F: nat > nat] :
( ( ( comp_G4985101524424093169ctxt_a @ G @ H )
= R )
=> ( ( comp_G4985101524424093169ctxt_a @ ( comp_n6274738784168413603ctxt_a @ F @ G ) @ H )
= ( comp_n6274738784168413603ctxt_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_164_rewriteR__comp__comp,axiom,
! [G: list_nat > nat,H: list_nat > list_nat,R: list_nat > nat,F: nat > nat] :
( ( ( comp_l5285636708041795445st_nat @ G @ H )
= R )
=> ( ( comp_l5285636708041795445st_nat @ ( comp_n3993578768073458149st_nat @ F @ G ) @ H )
= ( comp_n3993578768073458149st_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_165_rewriteR__comp__comp,axiom,
! [G: ground_gterm_a > ground_gmctxt_a,H: ground_gterm_a > ground_gterm_a,R: ground_gterm_a > ground_gmctxt_a,F: ground_gmctxt_a > nat] :
( ( ( comp_G2980548778298907395term_a @ G @ H )
= R )
=> ( ( comp_G6116966634693717713term_a @ ( comp_G8049203549856156444term_a @ F @ G ) @ H )
= ( comp_G8049203549856156444term_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_166_rewriteR__comp__comp,axiom,
! [G: ground_gterm_a > ground_gmctxt_a,H: ground_gmctxt_a > ground_gterm_a,R: ground_gmctxt_a > ground_gmctxt_a,F: ground_gmctxt_a > nat] :
( ( ( comp_G8862670755111606218ctxt_a @ G @ H )
= R )
=> ( ( comp_G8977606278789707644ctxt_a @ ( comp_G8049203549856156444term_a @ F @ G ) @ H )
= ( comp_G4985101524424093169ctxt_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_167_rewriteR__comp__comp,axiom,
! [G: ground_gmctxt_a > ground_gmctxt_a,H: ground_gterm_a > ground_gmctxt_a,R: ground_gterm_a > ground_gmctxt_a,F: ground_gmctxt_a > nat] :
( ( ( comp_G3656465168958649678term_a @ G @ H )
= R )
=> ( ( comp_G8049203549856156444term_a @ ( comp_G4985101524424093169ctxt_a @ F @ G ) @ H )
= ( comp_G8049203549856156444term_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_168_rewriteR__comp__comp,axiom,
! [G: ground_gmctxt_a > ground_gmctxt_a,H: ground_gmctxt_a > ground_gmctxt_a,R: ground_gmctxt_a > ground_gmctxt_a,F: ground_gmctxt_a > nat] :
( ( ( comp_G2329228445783543871ctxt_a @ G @ H )
= R )
=> ( ( comp_G4985101524424093169ctxt_a @ ( comp_G4985101524424093169ctxt_a @ F @ G ) @ H )
= ( comp_G4985101524424093169ctxt_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_169_rewriteR__comp__comp,axiom,
! [G: list_nat > list_nat,H: list_nat > list_nat,R: list_nat > list_nat,F: list_nat > nat] :
( ( ( comp_l7223822213492037765st_nat @ G @ H )
= R )
=> ( ( comp_l5285636708041795445st_nat @ ( comp_l5285636708041795445st_nat @ F @ G ) @ H )
= ( comp_l5285636708041795445st_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_170_rewriteR__comp__comp,axiom,
! [G: list_nat > nat,H: list_nat > list_nat,R: list_nat > nat,F: nat > list_nat] :
( ( ( comp_l5285636708041795445st_nat @ G @ H )
= R )
=> ( ( comp_l7223822213492037765st_nat @ ( comp_n1293334696235875061st_nat @ F @ G ) @ H )
= ( comp_n1293334696235875061st_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_171_rewriteL__comp__comp,axiom,
! [F: nat > nat,G: nat > nat,L: nat > nat,H: nat > nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= L )
=> ( ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_172_rewriteL__comp__comp,axiom,
! [F: nat > nat,G: ground_gmctxt_a > nat,L: ground_gmctxt_a > nat,H: ground_gterm_a > ground_gmctxt_a] :
( ( ( comp_n6274738784168413603ctxt_a @ F @ G )
= L )
=> ( ( comp_n8259423687851516650term_a @ F @ ( comp_G8049203549856156444term_a @ G @ H ) )
= ( comp_G8049203549856156444term_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_173_rewriteL__comp__comp,axiom,
! [F: nat > nat,G: ground_gmctxt_a > nat,L: ground_gmctxt_a > nat,H: ground_gmctxt_a > ground_gmctxt_a] :
( ( ( comp_n6274738784168413603ctxt_a @ F @ G )
= L )
=> ( ( comp_n6274738784168413603ctxt_a @ F @ ( comp_G4985101524424093169ctxt_a @ G @ H ) )
= ( comp_G4985101524424093169ctxt_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_174_rewriteL__comp__comp,axiom,
! [F: nat > nat,G: list_nat > nat,L: list_nat > nat,H: list_nat > list_nat] :
( ( ( comp_n3993578768073458149st_nat @ F @ G )
= L )
=> ( ( comp_n3993578768073458149st_nat @ F @ ( comp_l5285636708041795445st_nat @ G @ H ) )
= ( comp_l5285636708041795445st_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_175_rewriteL__comp__comp,axiom,
! [F: nat > list_nat,G: list_nat > nat,L: list_nat > list_nat,H: list_nat > list_nat] :
( ( ( comp_n1293334696235875061st_nat @ F @ G )
= L )
=> ( ( comp_n1293334696235875061st_nat @ F @ ( comp_l5285636708041795445st_nat @ G @ H ) )
= ( comp_l7223822213492037765st_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_176_rewriteL__comp__comp,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gterm_a > ground_gmctxt_a,L: ground_gterm_a > nat,H: ground_gterm_a > ground_gterm_a] :
( ( ( comp_G8049203549856156444term_a @ F @ G )
= L )
=> ( ( comp_G8049203549856156444term_a @ F @ ( comp_G2980548778298907395term_a @ G @ H ) )
= ( comp_G6116966634693717713term_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_177_rewriteL__comp__comp,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gterm_a > ground_gmctxt_a,L: ground_gterm_a > nat,H: ground_gmctxt_a > ground_gterm_a] :
( ( ( comp_G8049203549856156444term_a @ F @ G )
= L )
=> ( ( comp_G4985101524424093169ctxt_a @ F @ ( comp_G8862670755111606218ctxt_a @ G @ H ) )
= ( comp_G8977606278789707644ctxt_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_178_rewriteL__comp__comp,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gmctxt_a > ground_gmctxt_a,L: ground_gmctxt_a > nat,H: ground_gterm_a > ground_gmctxt_a] :
( ( ( comp_G4985101524424093169ctxt_a @ F @ G )
= L )
=> ( ( comp_G8049203549856156444term_a @ F @ ( comp_G3656465168958649678term_a @ G @ H ) )
= ( comp_G8049203549856156444term_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_179_rewriteL__comp__comp,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gmctxt_a > ground_gmctxt_a,L: ground_gmctxt_a > nat,H: ground_gmctxt_a > ground_gmctxt_a] :
( ( ( comp_G4985101524424093169ctxt_a @ F @ G )
= L )
=> ( ( comp_G4985101524424093169ctxt_a @ F @ ( comp_G2329228445783543871ctxt_a @ G @ H ) )
= ( comp_G4985101524424093169ctxt_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_180_rewriteL__comp__comp,axiom,
! [F: list_nat > nat,G: list_nat > list_nat,L: list_nat > nat,H: list_nat > list_nat] :
( ( ( comp_l5285636708041795445st_nat @ F @ G )
= L )
=> ( ( comp_l5285636708041795445st_nat @ F @ ( comp_l7223822213492037765st_nat @ G @ H ) )
= ( comp_l5285636708041795445st_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_181_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X: nat,N2: nat > nat,H: nat > nat,F: nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ M ) @ G @ X )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_182_type__copy__map__cong0,axiom,
! [M: ground_gterm_a > ground_gmctxt_a,G: nat > ground_gterm_a,X: nat,N2: nat > ground_gmctxt_a,H: nat > nat,F: ground_gmctxt_a > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_G5395315277947777326at_nat @ ( comp_G8049203549856156444term_a @ F @ M ) @ G @ X )
= ( comp_nat_nat_nat @ ( comp_G7635752065729029539at_nat @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_183_type__copy__map__cong0,axiom,
! [M: ground_gmctxt_a > ground_gmctxt_a,G: nat > ground_gmctxt_a,X: nat,N2: nat > ground_gmctxt_a,H: nat > nat,F: ground_gmctxt_a > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_G7635752065729029539at_nat @ ( comp_G4985101524424093169ctxt_a @ F @ M ) @ G @ X )
= ( comp_nat_nat_nat @ ( comp_G7635752065729029539at_nat @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_184_type__copy__map__cong0,axiom,
! [M: nat > nat,G: ground_gterm_a > nat,X: ground_gterm_a,N2: ground_gmctxt_a > nat,H: ground_gterm_a > ground_gmctxt_a,F: nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_n8259423687851516650term_a @ ( comp_nat_nat_nat @ F @ M ) @ G @ X )
= ( comp_G8049203549856156444term_a @ ( comp_n6274738784168413603ctxt_a @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_185_type__copy__map__cong0,axiom,
! [M: nat > nat,G: ground_gmctxt_a > nat,X: ground_gmctxt_a,N2: ground_gmctxt_a > nat,H: ground_gmctxt_a > ground_gmctxt_a,F: nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_n6274738784168413603ctxt_a @ ( comp_nat_nat_nat @ F @ M ) @ G @ X )
= ( comp_G4985101524424093169ctxt_a @ ( comp_n6274738784168413603ctxt_a @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_186_type__copy__map__cong0,axiom,
! [M: nat > nat,G: list_nat > nat,X: list_nat,N2: list_nat > nat,H: list_nat > list_nat,F: nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_n3993578768073458149st_nat @ ( comp_nat_nat_nat @ F @ M ) @ G @ X )
= ( comp_l5285636708041795445st_nat @ ( comp_n3993578768073458149st_nat @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_187_type__copy__map__cong0,axiom,
! [M: list_nat > list_nat,G: nat > list_nat,X: nat,N2: nat > list_nat,H: nat > nat,F: list_nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_l3539220536931113701at_nat @ ( comp_l5285636708041795445st_nat @ F @ M ) @ G @ X )
= ( comp_nat_nat_nat @ ( comp_l3539220536931113701at_nat @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_188_type__copy__map__cong0,axiom,
! [M: ground_gmctxt_a > nat,G: ground_gterm_a > ground_gmctxt_a,X: ground_gterm_a,N2: nat > nat,H: ground_gterm_a > nat,F: nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_G8049203549856156444term_a @ ( comp_n6274738784168413603ctxt_a @ F @ M ) @ G @ X )
= ( comp_n8259423687851516650term_a @ ( comp_nat_nat_nat @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_189_type__copy__map__cong0,axiom,
! [M: ground_gmctxt_a > nat,G: ground_gmctxt_a > ground_gmctxt_a,X: ground_gmctxt_a,N2: nat > nat,H: ground_gmctxt_a > nat,F: nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_G4985101524424093169ctxt_a @ ( comp_n6274738784168413603ctxt_a @ F @ M ) @ G @ X )
= ( comp_n6274738784168413603ctxt_a @ ( comp_nat_nat_nat @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_190_type__copy__map__cong0,axiom,
! [M: nat > ground_gmctxt_a,G: nat > nat,X: nat,N2: ground_gterm_a > ground_gmctxt_a,H: nat > ground_gterm_a,F: ground_gmctxt_a > nat] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp_nat_nat_nat @ ( comp_G7635752065729029539at_nat @ F @ M ) @ G @ X )
= ( comp_G5395315277947777326at_nat @ ( comp_G8049203549856156444term_a @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_191_comp__apply__eq,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gterm_a > ground_gmctxt_a,X: ground_gterm_a,H: ground_gmctxt_a > nat,K: ground_gterm_a > ground_gmctxt_a] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_G8049203549856156444term_a @ F @ G @ X )
= ( comp_G8049203549856156444term_a @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_192_comp__apply__eq,axiom,
! [F: ground_gmctxt_a > nat,G: ground_gmctxt_a > ground_gmctxt_a,X: ground_gmctxt_a,H: ground_gmctxt_a > nat,K: ground_gmctxt_a > ground_gmctxt_a] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_G4985101524424093169ctxt_a @ F @ G @ X )
= ( comp_G4985101524424093169ctxt_a @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_193_comp__apply__eq,axiom,
! [F: list_nat > list_nat,G: list_nat > list_nat,X: list_nat,H: list_nat > list_nat,K: list_nat > list_nat] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_l7223822213492037765st_nat @ F @ G @ X )
= ( comp_l7223822213492037765st_nat @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_194_comp__apply__eq,axiom,
! [F: nat > nat,G: nat > nat,X: nat,H: nat > nat,K: nat > nat] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_nat_nat_nat @ F @ G @ X )
= ( comp_nat_nat_nat @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_195_comp__apply__eq,axiom,
! [F: list_nat > nat,G: list_nat > list_nat,X: list_nat,H: list_nat > nat,K: list_nat > list_nat] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_l5285636708041795445st_nat @ F @ G @ X )
= ( comp_l5285636708041795445st_nat @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_196_comp__eq__dest__lhs,axiom,
! [A2: ground_gmctxt_a > nat,B: ground_gmctxt_a > ground_gmctxt_a,C3: ground_gmctxt_a > nat,V: ground_gmctxt_a] :
( ( ( comp_G4985101524424093169ctxt_a @ A2 @ B )
= C3 )
=> ( ( A2 @ ( B @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_197_comp__eq__dest__lhs,axiom,
! [A2: list_nat > list_nat,B: list_nat > list_nat,C3: list_nat > list_nat,V: list_nat] :
( ( ( comp_l7223822213492037765st_nat @ A2 @ B )
= C3 )
=> ( ( A2 @ ( B @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_198_comp__eq__dest__lhs,axiom,
! [A2: nat > nat,B: nat > nat,C3: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A2 @ B )
= C3 )
=> ( ( A2 @ ( B @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_199_comp__eq__dest__lhs,axiom,
! [A2: list_nat > nat,B: list_nat > list_nat,C3: list_nat > nat,V: list_nat] :
( ( ( comp_l5285636708041795445st_nat @ A2 @ B )
= C3 )
=> ( ( A2 @ ( B @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_200_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_201_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_202_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_203_bot__empty__eq,axiom,
( bot_bo868933149345519389xt_a_o
= ( ^ [X2: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X2 @ bot_bo3757104637556896616ctxt_a ) ) ) ).
% bot_empty_eq
thf(fact_204_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X2: list_nat] : ( member_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_205_Collect__empty__eq__bot,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( P = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_206_map__num__gholes__map__gmctxt,axiom,
! [F: a > a,Cs: list_Ground_gmctxt_a] :
( ( map_Gr1679544123147727284_a_nat @ ( comp_G4985101524424093169ctxt_a @ ground_num_gholes_a @ ( ground2683026529610258053xt_a_a @ F ) ) @ Cs )
= ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) ).
% map_num_gholes_map_gmctxt
thf(fact_207_is__singletonI_H,axiom,
! [A: set_Ground_gmctxt_a] :
( ( A != bot_bo3757104637556896616ctxt_a )
=> ( ! [X9: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ A )
=> ( ( member3306386364193653941ctxt_a @ Y2 @ A )
=> ( X9 = Y2 ) ) )
=> ( is_sin4243146486607228458ctxt_a @ A ) ) ) ).
% is_singletonI'
thf(fact_208_is__singletonI_H,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ! [X9: list_nat,Y2: list_nat] :
( ( member_list_nat @ X9 @ A )
=> ( ( member_list_nat @ Y2 @ A )
=> ( X9 = Y2 ) ) )
=> ( is_sin2641923865335537900st_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_209_gmctxt_Odistinct_I1_J,axiom,
! [X21: a,X22: list_Ground_gmctxt_a] :
( ground_GMHole_a
!= ( ground_GMFun_a @ X21 @ X22 ) ) ).
% gmctxt.distinct(1)
thf(fact_210_gmctxt_Oexhaust,axiom,
! [Y: ground_gmctxt_a] :
( ( Y != ground_GMHole_a )
=> ~ ! [X212: a,X222: list_Ground_gmctxt_a] :
( Y
!= ( ground_GMFun_a @ X212 @ X222 ) ) ) ).
% gmctxt.exhaust
thf(fact_211_funas__gmctxt_Ocases,axiom,
! [X: ground_gmctxt_a] :
( ! [F3: a,Cs2: list_Ground_gmctxt_a] :
( X
!= ( ground_GMFun_a @ F3 @ Cs2 ) )
=> ( X = ground_GMHole_a ) ) ).
% funas_gmctxt.cases
thf(fact_212_ghole__poss__list_Ocases,axiom,
! [X: ground_gmctxt_a] :
( ( X != ground_GMHole_a )
=> ~ ! [F3: a,Cs3: list_Ground_gmctxt_a] :
( X
!= ( ground_GMFun_a @ F3 @ Cs3 ) ) ) ).
% ghole_poss_list.cases
thf(fact_213_gmctxt_Osimps_I14_J,axiom,
( ( ground1659619264735726218st_nat @ ground8710035520252524849st_nat )
= bot_bot_set_list_nat ) ).
% gmctxt.simps(14)
thf(fact_214_map__comp__map,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_l7223822213492037765st_nat @ ( map_nat_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_215_map__map,axiom,
! [F: nat > nat,G: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( map_nat_nat @ G @ Xs ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_216_List_Omap_Ocompositionality,axiom,
! [F: nat > nat,G: nat > nat,List: list_nat] :
( ( map_nat_nat @ F @ ( map_nat_nat @ G @ List ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_217_list_Omap__comp,axiom,
! [G: nat > nat,F: nat > nat,V: list_nat] :
( ( map_nat_nat @ G @ ( map_nat_nat @ F @ V ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_218_List_Omap_Ocomp,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_l7223822213492037765st_nat @ ( map_nat_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_219_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_220_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_221_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ? [Y9: nat] :
( X2
= ( F @ Y9 ) ) ) ) ) ).
% ex_map_conv
thf(fact_222_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X9 )
= ( G @ X9 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_223_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X9 )
= X9 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_224_map__idI,axiom,
! [Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a] :
( ! [X9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( F @ X9 )
= X9 ) )
=> ( ( map_Gr980675717119874050ctxt_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_225_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_226_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_227_gmctxt_Osimps_I9_J,axiom,
! [F: a > a,X21: a,X22: list_Ground_gmctxt_a] :
( ( ground2683026529610258053xt_a_a @ F @ ( ground_GMFun_a @ X21 @ X22 ) )
= ( ground_GMFun_a @ ( F @ X21 ) @ ( map_Gr980675717119874050ctxt_a @ ( ground2683026529610258053xt_a_a @ F ) @ X22 ) ) ) ).
% gmctxt.simps(9)
thf(fact_228_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_229_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_230_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_231_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_232_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_233_list_Omap__ident__strong,axiom,
! [T: list_Ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a] :
( ! [Z: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Z @ ( set_Ground_gmctxt_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_Gr980675717119874050ctxt_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_234_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X9 )
= ( G @ X9 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_235_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_236_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_237_count__notin,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( count_4307434365288015266ctxt_a @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_238_size__list__map,axiom,
! [F: nat > nat,G: nat > nat,Xs: list_nat] :
( ( size_list_nat @ F @ ( map_nat_nat @ G @ Xs ) )
= ( size_list_nat @ ( comp_nat_nat_nat @ F @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_239_map__fun__upd,axiom,
! [Y: nat,Xs: list_nat,F: nat > nat,V: nat] :
( ~ ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ( map_nat_nat @ ( fun_upd_nat_nat @ F @ Y @ V ) @ Xs )
= ( map_nat_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_240_num__gholes_Osimps_I1_J,axiom,
( ( ground_num_gholes_a @ ground_GMHole_a )
= ( suc @ zero_zero_nat ) ) ).
% num_gholes.simps(1)
thf(fact_241_in__set__member,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
= ( member3758482962427618243ctxt_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_242_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_243_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_244_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_245_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_246_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_247_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_248_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_249_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_250_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_251_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_252_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X9: nat] : ( P @ X9 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X9: nat,Y2: nat] :
( ( P @ X9 @ Y2 )
=> ( P @ ( suc @ X9 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_253_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_254_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_255_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_256_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_257_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_258_list_Osize__gen__o__map,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_l5285636708041795445st_nat @ ( size_list_nat @ F ) @ ( map_nat_nat @ G ) )
= ( size_list_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_259_count__list__0__iff,axiom,
! [Xs: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
( ( ( count_4307434365288015266ctxt_a @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_260_gmctxt__of__gctxt__num__gholes,axiom,
! [C2: ground_gctxt_a] :
( ( ground_num_gholes_a @ ( ground5442125528501538874ctxt_a @ C2 ) )
= ( suc @ zero_zero_nat ) ) ).
% gmctxt_of_gctxt_num_gholes
thf(fact_261_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_262_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_263_num__gholes_Oelims,axiom,
! [X: ground_gmctxt_a,Y: nat] :
( ( ( ground_num_gholes_a @ X )
= Y )
=> ( ( ( X = ground_GMHole_a )
=> ( Y
!= ( suc @ zero_zero_nat ) ) )
=> ~ ! [Uu: a,Ctxts: list_Ground_gmctxt_a] :
( ( X
= ( ground_GMFun_a @ Uu @ Ctxts ) )
=> ( Y
!= ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Ctxts ) ) ) ) ) ) ).
% num_gholes.elims
thf(fact_264_gmctxt_Osize_I4_J,axiom,
! [X21: a,X22: list_Ground_gmctxt_a] :
( ( size_s9020356940445107890ctxt_a @ ( ground_GMFun_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_l1580428505752620566ctxt_a @ size_s9020356940445107890ctxt_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% gmctxt.size(4)
thf(fact_265_add__right__cancel,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C3 @ A2 ) )
= ( B = C3 ) ) ).
% add_right_cancel
thf(fact_266_add__left__cancel,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C3 ) )
= ( B = C3 ) ) ).
% add_left_cancel
thf(fact_267_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_268_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_269_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_270_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_271_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_272_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_273_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_274_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_275_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_276_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_277_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_278_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_279_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_280_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_281_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_282_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_283_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_284_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_285_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C3 @ A2 ) )
=> ( B = C3 ) ) ).
% add_right_imp_eq
thf(fact_286_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( B = C3 ) ) ).
% add_left_imp_eq
thf(fact_287_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C3 ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add.left_commute
thf(fact_288_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_289_add_Oassoc,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C3 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add.assoc
thf(fact_290_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_291_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_292_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_293_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C3 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_294_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_295_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_296_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_297_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_298_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_299_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_300_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_301_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_302_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_303_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_304_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_305_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_306_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N4: nat,Xs2: list_nat] : ( plus_plus_nat @ N4 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_307_num__gholes_Osimps_I2_J,axiom,
! [Uu2: a,Ctxts2: list_Ground_gmctxt_a] :
( ( ground_num_gholes_a @ ( ground_GMFun_a @ Uu2 @ Ctxts2 ) )
= ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Ctxts2 ) ) ) ).
% num_gholes.simps(2)
thf(fact_308_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ns ) )
=> ( X2 = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_309_size__list__conv__sum__list,axiom,
( size_list_nat
= ( ^ [F2: nat > nat,Xs2: list_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_310_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_311_Utility_Osum__list__0,axiom,
! [Xs: list_int] :
( ! [X9: int] :
( ( member_int @ X9 @ ( set_int2 @ Xs ) )
=> ( X9 = zero_zero_int ) )
=> ( ( groups4559388385066561235st_int @ Xs )
= zero_zero_int ) ) ).
% Utility.sum_list_0
thf(fact_312_Utility_Osum__list__0,axiom,
! [Xs: list_nat] :
( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( X9 = zero_zero_nat ) )
=> ( ( groups4561878855575611511st_nat @ Xs )
= zero_zero_nat ) ) ).
% Utility.sum_list_0
thf(fact_313_gmctxt__of__gctxt__gctxt__of__gmctxt,axiom,
! [C2: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= ( suc @ zero_zero_nat ) )
=> ( ( ground5442125528501538874ctxt_a @ ( ground5333112502500237606ctxt_a @ C2 ) )
= C2 ) ) ).
% gmctxt_of_gctxt_gctxt_of_gmctxt
thf(fact_314_num__gholes__greater0__fill__gholes__gmctxt,axiom,
! [C2: ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ? [X10: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X10 @ ( set_Ground_gmctxt_a2 @ Ds ) )
& ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_a @ X10 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Ds ) ) ) ) ) ).
% num_gholes_greater0_fill_gholes_gmctxt
thf(fact_315_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_316_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_317_add__0__iff,axiom,
! [B: int,A2: int] :
( ( B
= ( plus_plus_int @ B @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_318_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_319_add__less__cancel__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_320_add__less__cancel__left,axiom,
! [C3: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_321_add__less__cancel__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_322_add__less__cancel__right,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_323_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_324_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_325_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_326_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_327_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_328_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_329_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_330_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_331_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_332_add__less__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_333_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_334_add__less__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_335_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_336_less__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_337_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_338_less__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_339_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_340_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_341_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_342_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_343_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_344_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_345_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_346_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_347_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_348_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_349_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_350_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_351_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_352_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_353_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_354_gt__ex,axiom,
! [X: int] :
? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% gt_ex
thf(fact_355_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_356_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_357_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_358_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_359_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_360_ord__eq__less__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_361_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_362_ord__less__eq__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_363_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X9: nat] :
( ! [Y10: nat] :
( ( ord_less_nat @ Y10 @ X9 )
=> ( P @ Y10 ) )
=> ( P @ X9 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_364_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_365_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_366_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_367_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_368_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_369_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_370_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_371_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_372_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X11: nat] : ( P2 @ X11 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_373_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_374_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_375_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_376_order_Ostrict__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_377_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_378_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_379_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_380_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C3: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C3 @ B )
=> ( ord_less_int @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_381_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_382_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_383_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_384_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_385_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_386_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_387_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_388_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_389_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_390_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_391_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_392_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_393_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_394_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_395_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_396_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_397_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_398_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_399_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_400_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_401_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_402_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_403_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_404_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_405_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_406_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_407_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_408_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_409_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_410_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X9: nat,Y2: nat] :
( ( ord_less_nat @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_411_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_nat @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_412_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X9: int,Y2: int] :
( ( ord_less_int @ X9 @ Y2 )
=> ( ord_less_int @ ( F @ X9 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_413_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_414_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_415_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_416_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_417_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_418_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_419_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_420_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_421_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_422_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_423_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_424_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_425_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_426_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_427_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_428_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_429_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_430_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_431_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_432_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_433_bot_Onot__eq__extremum,axiom,
! [A2: set_list_nat] :
( ( A2 != bot_bot_set_list_nat )
= ( ord_le1190675801316882794st_nat @ bot_bot_set_list_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_434_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_435_bot_Oextremum__strict,axiom,
! [A2: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A2 @ bot_bot_set_list_nat ) ).
% bot.extremum_strict
thf(fact_436_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_437_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_438_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_439_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_440_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_441_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_442_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_443_add__strict__mono,axiom,
! [A2: nat,B: nat,C3: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C3 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_444_add__strict__mono,axiom,
! [A2: int,B: int,C3: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C3 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_445_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_446_add__strict__left__mono,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_447_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_448_add__strict__right__mono,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_449_add__less__imp__less__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_450_add__less__imp__less__left,axiom,
! [C3: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_451_add__less__imp__less__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_452_add__less__imp__less__right,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_453_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_454_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_455_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_456_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_457_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_458_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_459_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_460_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_461_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_462_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_463_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_464_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_465_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_466_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_467_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_468_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_469_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_470_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_471_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_472_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_473_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_474_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_475_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_476_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_477_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_478_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_479_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_480_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_481_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_482_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_483_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_484_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_485_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_486_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_487_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_488_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_489_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_490_add__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_491_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_492_add__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_493_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_494_pos__add__strict,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% pos_add_strict
thf(fact_495_pos__add__strict,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C3 ) ) ) ) ).
% pos_add_strict
thf(fact_496_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_497_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_498_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_499_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_500_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_501_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_502_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_503_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_504_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_505_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_506_size__list__estimation,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Y: nat,F: ground_gmctxt_a > nat] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_l1580428505752620566ctxt_a @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_507_length__pos__if__in__set,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_508_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_509_size__simp2,axiom,
! [T: ground_gmctxt_a,Ts: list_Ground_gmctxt_a,S: char] :
( ( member3306386364193653941ctxt_a @ T @ ( set_Ground_gmctxt_a2 @ Ts ) )
=> ( ord_less_nat @ ( size_s9020356940445107890ctxt_a @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_char @ S ) @ ( size_l1580428505752620566ctxt_a @ size_s9020356940445107890ctxt_a @ Ts ) ) ) ) ) ) ).
% size_simp2
thf(fact_510_size__simp2,axiom,
! [T: ground_gmctxt_a,Ts: list_Ground_gmctxt_a,S: list_nat] :
( ( member3306386364193653941ctxt_a @ T @ ( set_Ground_gmctxt_a2 @ Ts ) )
=> ( ord_less_nat @ ( size_s9020356940445107890ctxt_a @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_list_nat @ S ) @ ( size_l1580428505752620566ctxt_a @ size_s9020356940445107890ctxt_a @ Ts ) ) ) ) ) ) ).
% size_simp2
thf(fact_511_size__simp2,axiom,
! [T: char,Ts: list_char,S: char] :
( ( member_char @ T @ ( set_char2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_char @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_char @ S ) @ ( size_list_char @ size_size_char @ Ts ) ) ) ) ) ) ).
% size_simp2
thf(fact_512_size__simp2,axiom,
! [T: char,Ts: list_char,S: list_nat] :
( ( member_char @ T @ ( set_char2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_char @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_list_nat @ S ) @ ( size_list_char @ size_size_char @ Ts ) ) ) ) ) ) ).
% size_simp2
thf(fact_513_size__simp2,axiom,
! [T: list_nat,Ts: list_list_nat,S: char] :
( ( member_list_nat @ T @ ( set_list_nat2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_char @ S ) @ ( size_list_list_nat @ size_size_list_nat @ Ts ) ) ) ) ) ) ).
% size_simp2
thf(fact_514_size__simp2,axiom,
! [T: list_nat,Ts: list_list_nat,S: list_nat] :
( ( member_list_nat @ T @ ( set_list_nat2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_list_nat @ S ) @ ( size_list_list_nat @ size_size_list_nat @ Ts ) ) ) ) ) ) ).
% size_simp2
thf(fact_515_size__simp1,axiom,
! [T: ground_gmctxt_a,Ts: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ T @ ( set_Ground_gmctxt_a2 @ Ts ) )
=> ( ord_less_nat @ ( size_s9020356940445107890ctxt_a @ T ) @ ( suc @ ( size_l1580428505752620566ctxt_a @ size_s9020356940445107890ctxt_a @ Ts ) ) ) ) ).
% size_simp1
thf(fact_516_size__simp1,axiom,
! [T: char,Ts: list_char] :
( ( member_char @ T @ ( set_char2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_char @ T ) @ ( suc @ ( size_list_char @ size_size_char @ Ts ) ) ) ) ).
% size_simp1
thf(fact_517_size__simp1,axiom,
! [T: list_nat,Ts: list_list_nat] :
( ( member_list_nat @ T @ ( set_list_nat2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( size_list_list_nat @ size_size_list_nat @ Ts ) ) ) ) ).
% size_simp1
thf(fact_518_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_519_num__gholes__fill__gholes__gmctxt,axiom,
! [C2: ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ( ground_num_gholes_a @ ( ground2530127064035342964ctxt_a @ C2 @ Ds ) )
= ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Ds ) ) ) ) ).
% num_gholes_fill_gholes_gmctxt
thf(fact_520_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_521_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_522_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_523_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_524_num__gholes_Opelims,axiom,
! [X: ground_gmctxt_a,Y: nat] :
( ( ( ground_num_gholes_a @ X )
= Y )
=> ( ( accp_Ground_gmctxt_a @ ground1147946743542812879_rel_a @ X )
=> ( ( ( X = ground_GMHole_a )
=> ( ( Y
= ( suc @ zero_zero_nat ) )
=> ~ ( accp_Ground_gmctxt_a @ ground1147946743542812879_rel_a @ ground_GMHole_a ) ) )
=> ~ ! [Uu: a,Ctxts: list_Ground_gmctxt_a] :
( ( X
= ( ground_GMFun_a @ Uu @ Ctxts ) )
=> ( ( Y
= ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Ctxts ) ) )
=> ~ ( accp_Ground_gmctxt_a @ ground1147946743542812879_rel_a @ ( ground_GMFun_a @ Uu @ Ctxts ) ) ) ) ) ) ) ).
% num_gholes.pelims
thf(fact_525_gctxt__of__gmctxt__hole__dest,axiom,
! [C2: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= ( suc @ zero_zero_nat ) )
=> ( ( ( ground5333112502500237606ctxt_a @ C2 )
= ground_GHole_a )
=> ( C2 = ground_GMHole_a ) ) ) ).
% gctxt_of_gmctxt_hole_dest
thf(fact_526_not__psubset__empty,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A @ bot_bot_set_list_nat ) ).
% not_psubset_empty
thf(fact_527_fill__gholes__gmctxt__nHole,axiom,
! [C2: ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
( ( C2 != ground_GMHole_a )
=> ( ( ( ground_num_gholes_a @ C2 )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ( ground2530127064035342964ctxt_a @ C2 @ Ds )
!= ground_GMHole_a ) ) ) ).
% fill_gholes_gmctxt_nHole
thf(fact_528_fill__gholes__gmctxt__sup__mctxt__args,axiom,
! [C2: ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ( ground7778985915038999110args_a @ C2 @ ( ground2530127064035342964ctxt_a @ C2 @ Ds ) )
= Ds ) ) ).
% fill_gholes_gmctxt_sup_mctxt_args
thf(fact_529_fill__gholes__gmctxt__GMFun__replicate__length,axiom,
! [F: a,Cs: list_Ground_gmctxt_a] :
( ( ground2530127064035342964ctxt_a @ ( ground_GMFun_a @ F @ ( replic9205591110366033502ctxt_a @ ( size_s962548663205356226ctxt_a @ Cs ) @ ground_GMHole_a ) ) @ Cs )
= ( ground_GMFun_a @ F @ Cs ) ) ).
% fill_gholes_gmctxt_GMFun_replicate_length
thf(fact_530_sum__list__map__remove1,axiom,
! [X: nat,Xs: list_nat,F: nat > nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) )
= ( plus_plus_nat @ ( F @ X ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ ( remove1_nat @ X @ Xs ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_531_sum__list__map__remove1,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) )
= ( plus_plus_nat @ ( F @ X ) @ ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ F @ ( remove4701464168398194562ctxt_a @ X @ Xs ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_532_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_533_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_534_sum__list__def,axiom,
( groups4559388385066561235st_int
= ( groups_monoid_F_int @ plus_plus_int @ zero_zero_int ) ) ).
% sum_list_def
thf(fact_535_sum__list__def,axiom,
( groups4561878855575611511st_nat
= ( groups_monoid_F_nat @ plus_plus_nat @ zero_zero_nat ) ) ).
% sum_list_def
thf(fact_536_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_537_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_538_replicate__eq__replicate,axiom,
! [M2: nat,X: nat,N: nat,Y: nat] :
( ( ( replicate_nat @ M2 @ X )
= ( replicate_nat @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2 != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_539_map__replicate,axiom,
! [F: nat > nat,N: nat,X: nat] :
( ( map_nat_nat @ F @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ N @ ( F @ X ) ) ) ).
% map_replicate
thf(fact_540_in__set__remove1,axiom,
! [A2: ground_gmctxt_a,B: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( A2 != B )
=> ( ( member3306386364193653941ctxt_a @ A2 @ ( set_Ground_gmctxt_a2 @ ( remove4701464168398194562ctxt_a @ B @ Xs ) ) )
= ( member3306386364193653941ctxt_a @ A2 @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_541_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_542_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_543_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_544_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_545_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_546_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_547_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_548_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_549_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_550_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_551_in__set__replicate,axiom,
! [X: ground_gmctxt_a,N: nat,Y: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( replic9205591110366033502ctxt_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_552_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_553_Bex__set__replicate,axiom,
! [N: nat,A2: ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ? [X2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ ( replic9205591110366033502ctxt_a @ N @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_554_Bex__set__replicate,axiom,
! [N: nat,A2: nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_555_Ball__set__replicate,axiom,
! [N: nat,A2: ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ! [X2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ ( replic9205591110366033502ctxt_a @ N @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_556_Ball__set__replicate,axiom,
! [N: nat,A2: nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_557_fill__gholes__gmctxt__replicate__MHole,axiom,
! [C2: ground_gmctxt_a] :
( ( ground2530127064035342964ctxt_a @ C2 @ ( replic9205591110366033502ctxt_a @ ( ground_num_gholes_a @ C2 ) @ ground_GMHole_a ) )
= C2 ) ).
% fill_gholes_gmctxt_replicate_MHole
thf(fact_558_sup__gmctxt__args__MHole2,axiom,
! [C2: ground_gmctxt_a] :
( ( ground7778985915038999110args_a @ C2 @ ground_GMHole_a )
= ( replic9205591110366033502ctxt_a @ ( ground_num_gholes_a @ C2 ) @ ground_GMHole_a ) ) ).
% sup_gmctxt_args_MHole2
thf(fact_559_psubsetD,axiom,
! [A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a,C3: ground_gmctxt_a] :
( ( ord_le4651699007867869992ctxt_a @ A @ B3 )
=> ( ( member3306386364193653941ctxt_a @ C3 @ A )
=> ( member3306386364193653941ctxt_a @ C3 @ B3 ) ) ) ).
% psubsetD
thf(fact_560_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_561_notin__set__remove1,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Y: ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( remove4701464168398194562ctxt_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_562_remove1__idem,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( remove4701464168398194562ctxt_a @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_563_sup__gmctxt__args_Osimps_I2_J,axiom,
! [V: a,Va: list_Ground_gmctxt_a] :
( ( ground7778985915038999110args_a @ ( ground_GMFun_a @ V @ Va ) @ ground_GMHole_a )
= ( replic9205591110366033502ctxt_a @ ( ground_num_gholes_a @ ( ground_GMFun_a @ V @ Va ) ) @ ground_GMHole_a ) ) ).
% sup_gmctxt_args.simps(2)
thf(fact_564_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_565_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_566_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_567_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_568_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_569_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_570_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_571_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_572_replicate__eqI,axiom,
! [Xs: list_Ground_gmctxt_a,N: nat,X: ground_gmctxt_a] :
( ( ( size_s962548663205356226ctxt_a @ Xs )
= N )
=> ( ! [Y2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replic9205591110366033502ctxt_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_573_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_574_replicate__length__same,axiom,
! [Xs: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
( ! [X9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( X9 = X ) )
=> ( ( replic9205591110366033502ctxt_a @ ( size_s962548663205356226ctxt_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_575_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( X9 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_576_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_577_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_578_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% of_nat_mask_eq
thf(fact_579_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_nat_mask_eq
thf(fact_580_fill__gholes__gmctxt__GMFun__replicate__append,axiom,
! [Cs: list_Ground_gmctxt_a,N: nat,Ds: list_Ground_gmctxt_a,F: a] :
( ( ( size_s962548663205356226ctxt_a @ Cs )
= N )
=> ( ! [T2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ T2 @ ( set_Ground_gmctxt_a2 @ Ds ) )
=> ( ( ground_num_gholes_a @ T2 )
= zero_zero_nat ) )
=> ( ( ground2530127064035342964ctxt_a @ ( ground_GMFun_a @ F @ ( append5238750768297430211ctxt_a @ ( replic9205591110366033502ctxt_a @ N @ ground_GMHole_a ) @ Ds ) ) @ Cs )
= ( ground_GMFun_a @ F @ ( append5238750768297430211ctxt_a @ Cs @ Ds ) ) ) ) ) ).
% fill_gholes_gmctxt_GMFun_replicate_append
thf(fact_581_set__replicate,axiom,
! [N: nat,X: ground_gmctxt_a] :
( ( N != zero_zero_nat )
=> ( ( set_Ground_gmctxt_a2 @ ( replic9205591110366033502ctxt_a @ N @ X ) )
= ( insert2292343091172363342ctxt_a @ X @ bot_bo3757104637556896616ctxt_a ) ) ) ).
% set_replicate
thf(fact_582_set__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_583_set__replicate,axiom,
! [N: nat,X: list_nat] :
( ( N != zero_zero_nat )
=> ( ( set_list_nat2 @ ( replicate_list_nat @ N @ X ) )
= ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ).
% set_replicate
thf(fact_584_insert__iff,axiom,
! [A2: ground_gmctxt_a,B: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A2 @ ( insert2292343091172363342ctxt_a @ B @ A ) )
= ( ( A2 = B )
| ( member3306386364193653941ctxt_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_585_insertCI,axiom,
! [A2: ground_gmctxt_a,B3: set_Ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ~ ( member3306386364193653941ctxt_a @ A2 @ B3 )
=> ( A2 = B ) )
=> ( member3306386364193653941ctxt_a @ A2 @ ( insert2292343091172363342ctxt_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_586_singletonI,axiom,
! [A2: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ A2 @ ( insert2292343091172363342ctxt_a @ A2 @ bot_bo3757104637556896616ctxt_a ) ) ).
% singletonI
thf(fact_587_singletonI,axiom,
! [A2: list_nat] : ( member_list_nat @ A2 @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_588_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_589_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_590_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_591_is__singletonI,axiom,
! [X: list_nat] : ( is_sin2641923865335537900st_nat @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ).
% is_singletonI
thf(fact_592_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_593_mk__disjoint__insert,axiom,
! [A2: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A2 @ A )
=> ? [B5: set_Ground_gmctxt_a] :
( ( A
= ( insert2292343091172363342ctxt_a @ A2 @ B5 ) )
& ~ ( member3306386364193653941ctxt_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_594_insert__eq__iff,axiom,
! [A2: ground_gmctxt_a,A: set_Ground_gmctxt_a,B: ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ A2 @ A )
=> ( ~ ( member3306386364193653941ctxt_a @ B @ B3 )
=> ( ( ( insert2292343091172363342ctxt_a @ A2 @ A )
= ( insert2292343091172363342ctxt_a @ B @ B3 ) )
= ( ( ( A2 = B )
=> ( A = B3 ) )
& ( ( A2 != B )
=> ? [C5: set_Ground_gmctxt_a] :
( ( A
= ( insert2292343091172363342ctxt_a @ B @ C5 ) )
& ~ ( member3306386364193653941ctxt_a @ B @ C5 )
& ( B3
= ( insert2292343091172363342ctxt_a @ A2 @ C5 ) )
& ~ ( member3306386364193653941ctxt_a @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_595_insert__absorb,axiom,
! [A2: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A2 @ A )
=> ( ( insert2292343091172363342ctxt_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_596_insert__ident,axiom,
! [X: ground_gmctxt_a,A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ X @ A )
=> ( ~ ( member3306386364193653941ctxt_a @ X @ B3 )
=> ( ( ( insert2292343091172363342ctxt_a @ X @ A )
= ( insert2292343091172363342ctxt_a @ X @ B3 ) )
= ( A = B3 ) ) ) ) ).
% insert_ident
thf(fact_597_Set_Oset__insert,axiom,
! [X: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ A )
=> ~ ! [B5: set_Ground_gmctxt_a] :
( ( A
= ( insert2292343091172363342ctxt_a @ X @ B5 ) )
=> ( member3306386364193653941ctxt_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_598_insertI2,axiom,
! [A2: ground_gmctxt_a,B3: set_Ground_gmctxt_a,B: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A2 @ B3 )
=> ( member3306386364193653941ctxt_a @ A2 @ ( insert2292343091172363342ctxt_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_599_insertI1,axiom,
! [A2: ground_gmctxt_a,B3: set_Ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ A2 @ ( insert2292343091172363342ctxt_a @ A2 @ B3 ) ) ).
% insertI1
thf(fact_600_insertE,axiom,
! [A2: ground_gmctxt_a,B: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A2 @ ( insert2292343091172363342ctxt_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member3306386364193653941ctxt_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_601_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_602_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_603_singletonD,axiom,
! [B: ground_gmctxt_a,A2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ B @ ( insert2292343091172363342ctxt_a @ A2 @ bot_bo3757104637556896616ctxt_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_604_singletonD,axiom,
! [B: list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_605_singleton__iff,axiom,
! [B: ground_gmctxt_a,A2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ B @ ( insert2292343091172363342ctxt_a @ A2 @ bot_bo3757104637556896616ctxt_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_606_singleton__iff,axiom,
! [B: list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_607_doubleton__eq__iff,axiom,
! [A2: list_nat,B: list_nat,C3: list_nat,D2: list_nat] :
( ( ( insert_list_nat @ A2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
= ( insert_list_nat @ C3 @ ( insert_list_nat @ D2 @ bot_bot_set_list_nat ) ) )
= ( ( ( A2 = C3 )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_608_insert__not__empty,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat @ A2 @ A )
!= bot_bot_set_list_nat ) ).
% insert_not_empty
thf(fact_609_singleton__inject,axiom,
! [A2: list_nat,B: list_nat] :
( ( ( insert_list_nat @ A2 @ bot_bot_set_list_nat )
= ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_610_append__replicate__commute,axiom,
! [N: nat,X: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
= ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_611_replicate__add,axiom,
! [N: nat,M2: nat,X: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N @ M2 ) @ X )
= ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ M2 @ X ) ) ) ).
% replicate_add
thf(fact_612_remove1__append,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
( ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( remove4701464168398194562ctxt_a @ X @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
= ( append5238750768297430211ctxt_a @ ( remove4701464168398194562ctxt_a @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( remove4701464168398194562ctxt_a @ X @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
= ( append5238750768297430211ctxt_a @ Xs @ ( remove4701464168398194562ctxt_a @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_613_is__singleton__def,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A5: set_list_nat] :
? [X2: list_nat] :
( A5
= ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ).
% is_singleton_def
thf(fact_614_is__singletonE,axiom,
! [A: set_list_nat] :
( ( is_sin2641923865335537900st_nat @ A )
=> ~ ! [X9: list_nat] :
( A
!= ( insert_list_nat @ X9 @ bot_bot_set_list_nat ) ) ) ).
% is_singletonE
thf(fact_615_set__replicate__conv__if,axiom,
! [N: nat,X: ground_gmctxt_a] :
( ( ( N = zero_zero_nat )
=> ( ( set_Ground_gmctxt_a2 @ ( replic9205591110366033502ctxt_a @ N @ X ) )
= bot_bo3757104637556896616ctxt_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_Ground_gmctxt_a2 @ ( replic9205591110366033502ctxt_a @ N @ X ) )
= ( insert2292343091172363342ctxt_a @ X @ bot_bo3757104637556896616ctxt_a ) ) ) ) ).
% set_replicate_conv_if
thf(fact_616_set__replicate__conv__if,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_617_set__replicate__conv__if,axiom,
! [N: nat,X: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_list_nat2 @ ( replicate_list_nat @ N @ X ) )
= bot_bot_set_list_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_list_nat2 @ ( replicate_list_nat @ N @ X ) )
= ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_618_set__replicate__Suc,axiom,
! [N: nat,X: ground_gmctxt_a] :
( ( set_Ground_gmctxt_a2 @ ( replic9205591110366033502ctxt_a @ ( suc @ N ) @ X ) )
= ( insert2292343091172363342ctxt_a @ X @ bot_bo3757104637556896616ctxt_a ) ) ).
% set_replicate_Suc
thf(fact_619_set__replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_620_set__replicate__Suc,axiom,
! [N: nat,X: list_nat] :
( ( set_list_nat2 @ ( replicate_list_nat @ ( suc @ N ) @ X ) )
= ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ).
% set_replicate_Suc
thf(fact_621_sum__list__replicate__length,axiom,
! [Xs: list_nat] :
( ( groups4561878855575611511st_nat @ ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ zero_zero_nat ) ) )
= ( size_size_list_nat @ Xs ) ) ).
% sum_list_replicate_length
thf(fact_622_sum__list__replicate__length_H,axiom,
! [N: nat] :
( ( groups4561878855575611511st_nat @ ( replicate_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ).
% sum_list_replicate_length'
thf(fact_623_gmctxt__of__gctxt_Osimps_I2_J,axiom,
! [F: a,Ss: list_Ground_gterm_a,C2: ground_gctxt_a,Ts: list_Ground_gterm_a] :
( ( ground5442125528501538874ctxt_a @ ( ground_GMore_a @ F @ Ss @ C2 @ Ts ) )
= ( ground_GMFun_a @ F @ ( append5238750768297430211ctxt_a @ ( map_Gr6591338906797447177ctxt_a @ ground582559344758063963term_a @ Ss ) @ ( cons_Ground_gmctxt_a @ ( ground5442125528501538874ctxt_a @ C2 ) @ ( map_Gr6591338906797447177ctxt_a @ ground582559344758063963term_a @ Ts ) ) ) ) ) ).
% gmctxt_of_gctxt.simps(2)
thf(fact_624_is__singleton__the__elem,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A5: set_list_nat] :
( A5
= ( insert_list_nat @ ( the_elem_list_nat @ A5 ) @ bot_bot_set_list_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_625_the__elem__eq,axiom,
! [X: list_nat] :
( ( the_elem_list_nat @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
= X ) ).
% the_elem_eq
thf(fact_626_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_627_list_Osimps_I15_J,axiom,
! [X21: ground_gmctxt_a,X22: list_Ground_gmctxt_a] :
( ( set_Ground_gmctxt_a2 @ ( cons_Ground_gmctxt_a @ X21 @ X22 ) )
= ( insert2292343091172363342ctxt_a @ X21 @ ( set_Ground_gmctxt_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_628_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_629_set__ConsD,axiom,
! [Y: ground_gmctxt_a,X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y @ ( set_Ground_gmctxt_a2 @ ( cons_Ground_gmctxt_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member3306386364193653941ctxt_a @ Y @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_630_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_631_list_Oset__cases,axiom,
! [E: ground_gmctxt_a,A2: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ E @ ( set_Ground_gmctxt_a2 @ A2 ) )
=> ( ! [Z2: list_Ground_gmctxt_a] :
( A2
!= ( cons_Ground_gmctxt_a @ E @ Z2 ) )
=> ~ ! [Z1: ground_gmctxt_a,Z2: list_Ground_gmctxt_a] :
( ( A2
= ( cons_Ground_gmctxt_a @ Z1 @ Z2 ) )
=> ~ ( member3306386364193653941ctxt_a @ E @ ( set_Ground_gmctxt_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_632_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z2: list_nat] :
( A2
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_633_list_Oset__intros_I1_J,axiom,
! [X21: ground_gmctxt_a,X22: list_Ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X21 @ ( set_Ground_gmctxt_a2 @ ( cons_Ground_gmctxt_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_634_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_635_list_Oset__intros_I2_J,axiom,
! [Y: ground_gmctxt_a,X22: list_Ground_gmctxt_a,X21: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y @ ( set_Ground_gmctxt_a2 @ X22 ) )
=> ( member3306386364193653941ctxt_a @ Y @ ( set_Ground_gmctxt_a2 @ ( cons_Ground_gmctxt_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_636_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_637_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_638_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_639_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_640_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_641_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_642_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_643_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_644_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_645_remove1_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
= ( cons_nat @ Y @ ( remove1_nat @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_646_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_647_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y9: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y9 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_648_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y9: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y9 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_649_split__list__first__prop__iff,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ? [X2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
( ? [Zs2: list_Ground_gmctxt_a] :
( Xs
= ( append5238750768297430211ctxt_a @ Ys3 @ ( cons_Ground_gmctxt_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y9 @ ( set_Ground_gmctxt_a2 @ Ys3 ) )
=> ~ ( P @ Y9 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_650_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y9: nat] :
( ( member_nat2 @ Y9 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y9 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_651_split__list__last__prop__iff,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ? [X2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_Ground_gmctxt_a,X2: ground_gmctxt_a,Zs2: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ys3 @ ( cons_Ground_gmctxt_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y9 @ ( set_Ground_gmctxt_a2 @ Zs2 ) )
=> ~ ( P @ Y9 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_652_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y9: nat] :
( ( member_nat2 @ Y9 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Y9 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_653_in__set__conv__decomp__first,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
= ( ? [Ys3: list_Ground_gmctxt_a,Zs2: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ys3 @ ( cons_Ground_gmctxt_a @ X @ Zs2 ) ) )
& ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_654_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_655_in__set__conv__decomp__last,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
= ( ? [Ys3: list_Ground_gmctxt_a,Zs2: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ys3 @ ( cons_Ground_gmctxt_a @ X @ Zs2 ) ) )
& ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_656_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_657_split__list__first__propE,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ? [X10: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X10 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X10 ) )
=> ~ ! [Ys4: list_Ground_gmctxt_a,X9: ground_gmctxt_a] :
( ? [Zs3: list_Ground_gmctxt_a] :
( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X9 @ Zs3 ) ) )
=> ( ( P @ X9 )
=> ~ ! [Xa2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa2 @ ( set_Ground_gmctxt_a2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_658_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X10: nat] :
( ( member_nat2 @ X10 @ ( set_nat2 @ Xs ) )
& ( P @ X10 ) )
=> ~ ! [Ys4: list_nat,X9: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X9 @ Zs3 ) ) )
=> ( ( P @ X9 )
=> ~ ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_659_split__list__last__propE,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ? [X10: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X10 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X10 ) )
=> ~ ! [Ys4: list_Ground_gmctxt_a,X9: ground_gmctxt_a,Zs3: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X9 @ Zs3 ) ) )
=> ( ( P @ X9 )
=> ~ ! [Xa2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa2 @ ( set_Ground_gmctxt_a2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_660_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X10: nat] :
( ( member_nat2 @ X10 @ ( set_nat2 @ Xs ) )
& ( P @ X10 ) )
=> ~ ! [Ys4: list_nat,X9: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X9 @ Zs3 ) ) )
=> ( ( P @ X9 )
=> ~ ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_661_split__list__first__prop,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ? [X10: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X10 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X10 ) )
=> ? [Ys4: list_Ground_gmctxt_a,X9: ground_gmctxt_a] :
( ? [Zs3: list_Ground_gmctxt_a] :
( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X9 @ Zs3 ) ) )
& ( P @ X9 )
& ! [Xa2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa2 @ ( set_Ground_gmctxt_a2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_662_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X10: nat] :
( ( member_nat2 @ X10 @ ( set_nat2 @ Xs ) )
& ( P @ X10 ) )
=> ? [Ys4: list_nat,X9: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X9 @ Zs3 ) ) )
& ( P @ X9 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_663_split__list__last__prop,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ? [X10: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X10 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X10 ) )
=> ? [Ys4: list_Ground_gmctxt_a,X9: ground_gmctxt_a,Zs3: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X9 @ Zs3 ) ) )
& ( P @ X9 )
& ! [Xa2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa2 @ ( set_Ground_gmctxt_a2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_664_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X10: nat] :
( ( member_nat2 @ X10 @ ( set_nat2 @ Xs ) )
& ( P @ X10 ) )
=> ? [Ys4: list_nat,X9: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X9 @ Zs3 ) ) )
& ( P @ X9 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_665_in__set__conv__decomp,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
= ( ? [Ys3: list_Ground_gmctxt_a,Zs2: list_Ground_gmctxt_a] :
( Xs
= ( append5238750768297430211ctxt_a @ Ys3 @ ( cons_Ground_gmctxt_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_666_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_667_append__Cons__eq__iff,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,Xs4: list_Ground_gmctxt_a,Ys5: list_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Ys ) )
=> ( ( ( append5238750768297430211ctxt_a @ Xs @ ( cons_Ground_gmctxt_a @ X @ Ys ) )
= ( append5238750768297430211ctxt_a @ Xs4 @ ( cons_Ground_gmctxt_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_668_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_669_split__list__propE,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ? [X10: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X10 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X10 ) )
=> ~ ! [Ys4: list_Ground_gmctxt_a,X9: ground_gmctxt_a] :
( ? [Zs3: list_Ground_gmctxt_a] :
( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X9 @ Zs3 ) ) )
=> ~ ( P @ X9 ) ) ) ).
% split_list_propE
thf(fact_670_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X10: nat] :
( ( member_nat2 @ X10 @ ( set_nat2 @ Xs ) )
& ( P @ X10 ) )
=> ~ ! [Ys4: list_nat,X9: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X9 @ Zs3 ) ) )
=> ~ ( P @ X9 ) ) ) ).
% split_list_propE
thf(fact_671_split__list__first,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ? [Ys4: list_Ground_gmctxt_a,Zs3: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X @ Zs3 ) ) )
& ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_672_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_673_split__list__prop,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ? [X10: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X10 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X10 ) )
=> ? [Ys4: list_Ground_gmctxt_a,X9: ground_gmctxt_a] :
( ? [Zs3: list_Ground_gmctxt_a] :
( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X9 @ Zs3 ) ) )
& ( P @ X9 ) ) ) ).
% split_list_prop
thf(fact_674_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X10: nat] :
( ( member_nat2 @ X10 @ ( set_nat2 @ Xs ) )
& ( P @ X10 ) )
=> ? [Ys4: list_nat,X9: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X9 @ Zs3 ) ) )
& ( P @ X9 ) ) ) ).
% split_list_prop
thf(fact_675_split__list__last,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ? [Ys4: list_Ground_gmctxt_a,Zs3: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X @ Zs3 ) ) )
& ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_676_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_677_split__list,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ? [Ys4: list_Ground_gmctxt_a,Zs3: list_Ground_gmctxt_a] :
( Xs
= ( append5238750768297430211ctxt_a @ Ys4 @ ( cons_Ground_gmctxt_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_678_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_679_replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( replicate_nat @ ( suc @ N ) @ X )
= ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_680_replicate__app__Cons__same,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_681_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_682_remove1__split,axiom,
! [A2: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( ( remove4701464168398194562ctxt_a @ A2 @ Xs )
= Ys )
= ( ? [Ls: list_Ground_gmctxt_a,Rs: list_Ground_gmctxt_a] :
( ( Xs
= ( append5238750768297430211ctxt_a @ Ls @ ( cons_Ground_gmctxt_a @ A2 @ Rs ) ) )
& ~ ( member3306386364193653941ctxt_a @ A2 @ ( set_Ground_gmctxt_a2 @ Ls ) )
& ( Ys
= ( append5238750768297430211ctxt_a @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_683_remove1__split,axiom,
! [A2: nat,Xs: list_nat,Ys: list_nat] :
( ( member_nat2 @ A2 @ ( set_nat2 @ Xs ) )
=> ( ( ( remove1_nat @ A2 @ Xs )
= Ys )
= ( ? [Ls: list_nat,Rs: list_nat] :
( ( Xs
= ( append_nat @ Ls @ ( cons_nat @ A2 @ Rs ) ) )
& ~ ( member_nat2 @ A2 @ ( set_nat2 @ Ls ) )
& ( Ys
= ( append_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_684_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_685_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_686_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_687_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_688_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_689_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_690_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_691_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_692_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_693_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_694_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_695_add__minus__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_696_minus__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_697_minus__add__distrib,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_698_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_699_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_700_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_701_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_702_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_703_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_704_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_705_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_706_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_707_add_Oinverse__distrib__swap,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_708_group__cancel_Oneg1,axiom,
! [A: int,K: int,A2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_709_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_710_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_711_add__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_712_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_713_add_Oinverse__unique,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B ) ) ).
% add.inverse_unique
thf(fact_714_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_715_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( plus_plus_int @ A2 @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_716_gterm__of__gmctxt_Osimps,axiom,
! [F: a,Cs: list_Ground_gmctxt_a] :
( ( ground2297065866753446215ctxt_a @ ( ground_GMFun_a @ F @ Cs ) )
= ( ground_GFun_a @ F @ ( map_Gr8458857686193410251term_a @ ground2297065866753446215ctxt_a @ Cs ) ) ) ).
% gterm_of_gmctxt.simps
thf(fact_717_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_718_gmctxt__of__gterm_Osimps,axiom,
! [F: a,Ts: list_Ground_gterm_a] :
( ( ground582559344758063963term_a @ ( ground_GFun_a @ F @ Ts ) )
= ( ground_GMFun_a @ F @ ( map_Gr6591338906797447177ctxt_a @ ground582559344758063963term_a @ Ts ) ) ) ).
% gmctxt_of_gterm.simps
thf(fact_719_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_720_fill__gholes__replicate,axiom,
! [N: nat,Ss: list_Ground_gterm_a,F: a] :
( ( N
= ( size_s1893836384466666913term_a @ Ss ) )
=> ( ( ground_fill_gholes_a @ ( ground_GMFun_a @ F @ ( replic9205591110366033502ctxt_a @ N @ ground_GMHole_a ) ) @ Ss )
= ( ground_GFun_a @ F @ Ss ) ) ) ).
% fill_gholes_replicate
thf(fact_721_remove__nth__len,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ Xs )
= ( suc @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_722_adjust__idx__rev__length,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_723_adjust__idx__length,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_724_nth__Cons__0,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_725_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_726_nth__Cons__Suc,axiom,
! [X: list_nat,Xs: list_list_nat,N: nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_list_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_727_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_728_nth__replicate,axiom,
! [I: nat,N: nat,X: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_list_nat @ ( replicate_list_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_729_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_730_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_731_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_732_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_733_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_734_nth__append__length,axiom,
! [Xs: list_list_nat,X: list_nat,Ys: list_list_nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ Ys ) ) @ ( size_s3023201423986296836st_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_735_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_736_nth__append__length__plus,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,N: nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) )
= ( nth_list_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_737_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_738_nth__equalityE,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ~ ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I4 )
= ( nth_list_nat @ Ys @ I4 ) ) ) ) ) ).
% nth_equalityE
thf(fact_739_nth__equalityE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ~ ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Ys @ I4 ) ) ) ) ) ).
% nth_equalityE
thf(fact_740_Ex__list__of__length__P,axiom,
! [N: nat,P: list_nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X10: list_nat] : ( P @ X10 @ I3 ) )
=> ? [Xs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( nth_list_nat @ Xs3 @ I4 ) @ I4 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_741_Ex__list__of__length__P,axiom,
! [N: nat,P: nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X10: nat] : ( P @ X10 @ I3 ) )
=> ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( nth_nat @ Xs3 @ I4 ) @ I4 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_742_nth__equalityI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I3 )
= ( nth_list_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_743_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_744_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X13: list_nat] : ( P @ I2 @ X13 ) ) )
= ( ? [Xs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_745_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X13: nat] : ( P @ I2 @ X13 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_746_list__eq__iff__nth__eq,axiom,
( ( ^ [Y11: list_list_nat,Z5: list_list_nat] : ( Y11 = Z5 ) )
= ( ^ [Xs2: list_list_nat,Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ Xs2 @ I2 )
= ( nth_list_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_747_list__eq__iff__nth__eq,axiom,
( ( ^ [Y11: list_nat,Z5: list_nat] : ( Y11 = Z5 ) )
= ( ^ [Xs2: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_748_adjust__idx__nth,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ I @ Xs ) @ J )
= ( nth_list_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_749_adjust__idx__nth,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) @ J )
= ( nth_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_750_adjust__idx__rev__nth,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( J != I )
=> ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_list_nat @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_751_adjust__idx__rev__nth,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( J != I )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_nat @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_752_ex__set__conv__ex__nth,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( P @ ( nth_list_nat @ Xs @ I2 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_753_ex__set__conv__ex__nth,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ? [X2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
& ( P @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_754_ex__set__conv__ex__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_755_all__set__conv__all__nth,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_756_all__set__conv__all__nth,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ! [X2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( P @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_757_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_758_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X: list_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I3 ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_759_all__nth__imp__all__set,axiom,
! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o,X: ground_gmctxt_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( P @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) ) )
=> ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_760_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_761_in__set__conv__nth,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_762_in__set__conv__nth,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
& ( ( nth_Ground_gmctxt_a @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_763_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_764_list__ball__nth,axiom,
! [N: nat,Xs: list_list_nat,P: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [X9: list_nat] :
( ( member_list_nat @ X9 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X9 ) )
=> ( P @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_765_list__ball__nth,axiom,
! [N: nat,Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( ! [X9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( P @ X9 ) )
=> ( P @ ( nth_Ground_gmctxt_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_766_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( P @ X9 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_767_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_768_nth__mem,axiom,
! [N: nat,Xs: list_Ground_gmctxt_a] :
( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ N ) @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_769_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_770_in__set__idx,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_771_in__set__idx,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
& ( ( nth_Ground_gmctxt_a @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_772_in__set__idx,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_773_map__nth__eq__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Ys )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I2 ) )
= ( nth_list_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_nth_eq_conv
thf(fact_774_map__nth__eq__conv,axiom,
! [Xs: list_list_nat,Ys: list_nat,F: list_nat > nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_list_nat_nat @ F @ Xs )
= Ys )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I2 ) )
= ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_nth_eq_conv
thf(fact_775_map__nth__eq__conv,axiom,
! [Xs: list_nat,Ys: list_list_nat,F: nat > list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( map_nat_list_nat @ F @ Xs )
= Ys )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( nth_list_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_nth_eq_conv
thf(fact_776_map__nth__eq__conv,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_nat_nat @ F @ Xs )
= Ys )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_nth_eq_conv
thf(fact_777_map__nth__conv,axiom,
! [F: list_nat > nat,Ss: list_list_nat,G: nat > nat,Ts: list_nat] :
( ( ( map_list_nat_nat @ F @ Ss )
= ( map_nat_nat @ G @ Ts ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ss ) )
=> ( ( F @ ( nth_list_nat @ Ss @ I4 ) )
= ( G @ ( nth_nat @ Ts @ I4 ) ) ) ) ) ).
% map_nth_conv
thf(fact_778_map__nth__conv,axiom,
! [F: nat > nat,Ss: list_nat,G: list_nat > nat,Ts: list_list_nat] :
( ( ( map_nat_nat @ F @ Ss )
= ( map_list_nat_nat @ G @ Ts ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( ( F @ ( nth_nat @ Ss @ I4 ) )
= ( G @ ( nth_list_nat @ Ts @ I4 ) ) ) ) ) ).
% map_nth_conv
thf(fact_779_map__nth__conv,axiom,
! [F: nat > nat,Ss: list_nat,G: nat > nat,Ts: list_nat] :
( ( ( map_nat_nat @ F @ Ss )
= ( map_nat_nat @ G @ Ts ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( ( F @ ( nth_nat @ Ss @ I4 ) )
= ( G @ ( nth_nat @ Ts @ I4 ) ) ) ) ) ).
% map_nth_conv
thf(fact_780_map__eq__conv_H,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_eq_conv'
thf(fact_781_map__eq__conv_H,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_list_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_eq_conv'
thf(fact_782_map__eq__conv_H,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_eq_conv'
thf(fact_783_nth__map__conv,axiom,
! [Xs: list_list_nat,Ys: list_nat,F: list_nat > nat,G: nat > nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I3 ) )
= ( G @ ( nth_nat @ Ys @ I3 ) ) ) )
=> ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_784_nth__map__conv,axiom,
! [Xs: list_nat,Ys: list_list_nat,F: nat > nat,G: list_nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( F @ ( nth_nat @ Xs @ I3 ) )
= ( G @ ( nth_list_nat @ Ys @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_785_nth__map__conv,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( F @ ( nth_nat @ Xs @ I3 ) )
= ( G @ ( nth_nat @ Ys @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_786_map__eq__nth__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_eq_nth_conv
thf(fact_787_map__eq__nth__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_list_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_eq_nth_conv
thf(fact_788_map__eq__nth__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_eq_nth_conv
thf(fact_789_append__Cons__nth__middle,axiom,
! [I: nat,Xs: list_list_nat,Y: list_nat,Zs: list_list_nat] :
( ( I
= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ Y @ Zs ) ) @ I )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_790_append__Cons__nth__middle,axiom,
! [I: nat,Xs: list_nat,Y: nat,Zs: list_nat] :
( ( I
= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_791_append__Cons__nth__not__middle,axiom,
! [I: nat,Xs: list_list_nat,U: list_nat,Ys: list_list_nat,Z3: list_nat] :
( ( I
!= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ U @ Ys ) ) @ I )
= ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ Z3 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_792_append__Cons__nth__not__middle,axiom,
! [I: nat,Xs: list_nat,U: nat,Ys: list_nat,Z3: nat] :
( ( I
!= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
= ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z3 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_793_P__as__bs__extend,axiom,
! [As: list_list_nat,Bs: list_list_nat,Cs4: list_list_nat,Ds2: list_list_nat,P: list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ As )
= ( size_s3023201423986296836st_nat @ Bs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Cs4 )
= ( size_s3023201423986296836st_nat @ Ds2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Bs ) )
=> ( P @ ( nth_list_nat @ As @ I3 ) @ ( nth_list_nat @ Bs @ I3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ds2 ) )
=> ( P @ ( nth_list_nat @ Cs4 @ I3 ) @ ( nth_list_nat @ Ds2 @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( append_list_nat @ Bs @ Ds2 ) ) )
=> ( P @ ( nth_list_nat @ ( append_list_nat @ As @ Cs4 ) @ I4 ) @ ( nth_list_nat @ ( append_list_nat @ Bs @ Ds2 ) @ I4 ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_794_P__as__bs__extend,axiom,
! [As: list_list_nat,Bs: list_nat,Cs4: list_list_nat,Ds2: list_nat,P: list_nat > nat > $o] :
( ( ( size_s3023201423986296836st_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Cs4 )
= ( size_size_list_nat @ Ds2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
=> ( P @ ( nth_list_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ds2 ) )
=> ( P @ ( nth_list_nat @ Cs4 @ I3 ) @ ( nth_nat @ Ds2 @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds2 ) ) )
=> ( P @ ( nth_list_nat @ ( append_list_nat @ As @ Cs4 ) @ I4 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds2 ) @ I4 ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_795_P__as__bs__extend,axiom,
! [As: list_nat,Bs: list_list_nat,Cs4: list_nat,Ds2: list_list_nat,P: nat > list_nat > $o] :
( ( ( size_size_list_nat @ As )
= ( size_s3023201423986296836st_nat @ Bs ) )
=> ( ( ( size_size_list_nat @ Cs4 )
= ( size_s3023201423986296836st_nat @ Ds2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Bs ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_list_nat @ Bs @ I3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ds2 ) )
=> ( P @ ( nth_nat @ Cs4 @ I3 ) @ ( nth_list_nat @ Ds2 @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( append_list_nat @ Bs @ Ds2 ) ) )
=> ( P @ ( nth_nat @ ( append_nat @ As @ Cs4 ) @ I4 ) @ ( nth_list_nat @ ( append_list_nat @ Bs @ Ds2 ) @ I4 ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_796_P__as__bs__extend,axiom,
! [As: list_nat,Bs: list_nat,Cs4: list_nat,Ds2: list_nat,P: nat > nat > $o] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ( ( size_size_list_nat @ Cs4 )
= ( size_size_list_nat @ Ds2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ds2 ) )
=> ( P @ ( nth_nat @ Cs4 @ I3 ) @ ( nth_nat @ Ds2 @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds2 ) ) )
=> ( P @ ( nth_nat @ ( append_nat @ As @ Cs4 ) @ I4 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds2 ) @ I4 ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_797_nth__sum__listI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( ( groups4561878855575611511st_nat @ Xs )
= ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% nth_sum_listI
thf(fact_798_remove__nth__P__compat,axiom,
! [As: list_list_nat,Bs: list_list_nat,P: list_nat > list_nat > $o,P4: nat] :
( ( ( size_s3023201423986296836st_nat @ As )
= ( size_s3023201423986296836st_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ As ) )
=> ( P @ ( nth_list_nat @ As @ I3 ) @ ( nth_list_nat @ Bs @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ P4 @ As ) ) )
=> ( P @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P4 @ As ) @ I4 ) @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P4 @ Bs ) @ I4 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_799_remove__nth__P__compat,axiom,
! [As: list_list_nat,Bs: list_nat,P: list_nat > nat > $o,P4: nat] :
( ( ( size_s3023201423986296836st_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ As ) )
=> ( P @ ( nth_list_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ P4 @ As ) ) )
=> ( P @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P4 @ As ) @ I4 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P4 @ Bs ) @ I4 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_800_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_list_nat,P: nat > list_nat > $o,P4: nat] :
( ( ( size_size_list_nat @ As )
= ( size_s3023201423986296836st_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ As ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_list_nat @ Bs @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P4 @ As ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P4 @ As ) @ I4 ) @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P4 @ Bs ) @ I4 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_801_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_nat,P: nat > nat > $o,P4: nat] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ As ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P4 @ As ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P4 @ As ) @ I4 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P4 @ Bs ) @ I4 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_802_append__Cons__nth__left,axiom,
! [I: nat,Xs: list_list_nat,U: list_nat,Ys: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ U @ Ys ) ) @ I )
= ( nth_list_nat @ Xs @ I ) ) ) ).
% append_Cons_nth_left
thf(fact_803_append__Cons__nth__left,axiom,
! [I: nat,Xs: list_nat,U: nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% append_Cons_nth_left
thf(fact_804_append__Cons__nth__right,axiom,
! [Xs: list_list_nat,I: nat,U: list_nat,Ys: list_list_nat,Z3: list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ I )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ U @ Ys ) ) @ I )
= ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ Z3 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_right
thf(fact_805_append__Cons__nth__right,axiom,
! [Xs: list_nat,I: nat,U: nat,Ys: list_nat,Z3: nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
= ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z3 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_right
thf(fact_806_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ N @ Xs ) ) )
=> ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ N @ Xs ) @ I )
= ( nth_list_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_807_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_808_permut__sound,axiom,
! [I: nat,As: list_list_nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ As ) )
=> ( ( nth_list_nat @ ( missin2960327169605006088st_nat @ As @ F ) @ I )
= ( nth_list_nat @ As @ ( F @ I ) ) ) ) ).
% permut_sound
thf(fact_809_permut__sound,axiom,
! [I: nat,As: list_nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missing_permut_nat @ As @ F ) @ I )
= ( nth_nat @ As @ ( F @ I ) ) ) ) ).
% permut_sound
thf(fact_810_permut__aux__sound,axiom,
! [I: nat,As: list_list_nat,F: nat > nat,Bs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ As ) )
=> ( ( nth_list_nat @ ( missin3204601078308456222st_nat @ As @ F @ Bs ) @ I )
= ( nth_list_nat @ Bs @ ( F @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_811_permut__aux__sound,axiom,
! [I: nat,As: list_nat,F: nat > nat,Bs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As @ F @ Bs ) @ I )
= ( nth_nat @ Bs @ ( F @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_812_sum__list__1__E,axiom,
! [Xs: list_nat] :
( ( ( groups4561878855575611511st_nat @ Xs )
= ( suc @ zero_zero_nat ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I3 )
= ( suc @ zero_zero_nat ) )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_size_list_nat @ Xs ) )
=> ( ( J4 != I3 )
=> ( ( nth_nat @ Xs @ J4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_list_1_E
thf(fact_813_list__elem__to__partition__nth,axiom,
! [Cs: list_Ground_gmctxt_a,Xs: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
= ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Cs ) )
=> ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I3 ) ) ) ) ) ) ).
% list_elem_to_partition_nth
thf(fact_814_list__elem__to__partition__nth,axiom,
! [Cs: list_Ground_gmctxt_a,Xs: list_nat,X: nat] :
( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
= ( size_size_list_nat @ Xs ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Cs ) )
=> ~ ( member_nat2 @ X @ ( set_nat2 @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I3 ) ) ) ) ) ) ).
% list_elem_to_partition_nth
thf(fact_815_partition__gholes__subseteq,axiom,
! [Cs: list_Ground_gmctxt_a,Xs: list_Ground_gmctxt_a,I: nat,X: ground_gmctxt_a] :
( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
= ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
=> ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I ) ) )
=> ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ) ).
% partition_gholes_subseteq
thf(fact_816_partition__gholes__subseteq,axiom,
! [Cs: list_Ground_gmctxt_a,Xs: list_nat,I: nat,X: nat] :
( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
= ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ) ).
% partition_gholes_subseteq
thf(fact_817_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_818_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_819_list_Omap__disc__iff,axiom,
! [F: nat > nat,A2: list_nat] :
( ( ( map_nat_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_820_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_821_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_822_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_823_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_824_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_825_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_826_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_827_append_Oright__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ A2 @ nil_nat )
= A2 ) ).
% append.right_neutral
thf(fact_828_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_829_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_830_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_831_set__empty2,axiom,
! [Xs: list_Ground_gmctxt_a] :
( ( bot_bo3757104637556896616ctxt_a
= ( set_Ground_gmctxt_a2 @ Xs ) )
= ( Xs = nil_Ground_gmctxt_a ) ) ).
% set_empty2
thf(fact_832_set__empty2,axiom,
! [Xs: list_list_nat] :
( ( bot_bot_set_list_nat
= ( set_list_nat2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% set_empty2
thf(fact_833_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_834_set__empty,axiom,
! [Xs: list_Ground_gmctxt_a] :
( ( ( set_Ground_gmctxt_a2 @ Xs )
= bot_bo3757104637556896616ctxt_a )
= ( Xs = nil_Ground_gmctxt_a ) ) ).
% set_empty
thf(fact_835_set__empty,axiom,
! [Xs: list_list_nat] :
( ( ( set_list_nat2 @ Xs )
= bot_bot_set_list_nat )
= ( Xs = nil_list_nat ) ) ).
% set_empty
thf(fact_836_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_837_sum__list_ONil,axiom,
( ( groups4559388385066561235st_int @ nil_int )
= zero_zero_int ) ).
% sum_list.Nil
thf(fact_838_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_839_replicate__empty,axiom,
! [N: nat,X: nat] :
( ( ( replicate_nat @ N @ X )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_840_empty__replicate,axiom,
! [N: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_841_fill__gholes__gmctxt__of__gterm,axiom,
! [S: ground_gterm_a] :
( ( ground_fill_gholes_a @ ( ground582559344758063963term_a @ S ) @ nil_Ground_gterm_a )
= S ) ).
% fill_gholes_gmctxt_of_gterm
thf(fact_842_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_843_fill__gholes__no__holes,axiom,
! [C2: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= zero_zero_nat )
=> ( ( ground_fill_gholes_a @ C2 @ nil_Ground_gterm_a )
= ( ground2297065866753446215ctxt_a @ C2 ) ) ) ).
% fill_gholes_no_holes
thf(fact_844_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X9: nat] : ( P @ ( cons_nat @ X9 @ nil_nat ) )
=> ( ! [X9: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X9 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_845_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X9: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X9 @ Xs3 ) @ nil_nat )
=> ( ! [Y2: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys4 ) )
=> ( ! [X9: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X9 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_846_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y9: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y9 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_847_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X9: nat] :
( X
!= ( cons_nat @ X9 @ nil_nat ) )
=> ~ ! [X9: nat,Y2: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X9 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_848_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X9: nat,Xs3: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X9 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_849_List_Omin__list_Ocases,axiom,
! [X: list_nat] :
( ! [X9: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X9 @ Xs3 ) )
=> ( X = nil_nat ) ) ).
% List.min_list.cases
thf(fact_850_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_851_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_852_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_853_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_854_append_Oleft__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ nil_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_855_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_856_remove1_Osimps_I1_J,axiom,
! [X: nat] :
( ( remove1_nat @ X @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_857_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_858_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_859_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_860_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_861_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X9: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat,Z: 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 @ X9 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) @ ( cons_nat @ Z @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_862_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X9: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat,Z: 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 @ X9 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) @ ( cons_nat @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_863_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X9: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X9 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_864_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_865_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_866_empty__set,axiom,
( bot_bo3757104637556896616ctxt_a
= ( set_Ground_gmctxt_a2 @ nil_Ground_gmctxt_a ) ) ).
% empty_set
thf(fact_867_empty__set,axiom,
( bot_bot_set_list_nat
= ( set_list_nat2 @ nil_list_nat ) ) ).
% empty_set
thf(fact_868_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X9: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X9 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_869_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y2: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_870_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys6: list_nat] :
( ( ( cons_nat @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_871_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys6: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys6 ) )
& ( ( append_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_872_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X9: nat] : ( P @ ( cons_nat @ X9 @ nil_nat ) )
=> ( ! [X9: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X9 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_873_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_874_list_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_list_nat @ X @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_875_count__list_Osimps_I1_J,axiom,
! [Y: nat] :
( ( count_list_nat @ nil_nat @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_876_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X9: nat,Xs5: list_nat,Y2: nat,Ys7: list_nat] :
( ( X9 != Y2 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X9 @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_877_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_878_the__elem__set,axiom,
! [X: ground_gmctxt_a] :
( ( the_el2851850405879629545ctxt_a @ ( set_Ground_gmctxt_a2 @ ( cons_Ground_gmctxt_a @ X @ nil_Ground_gmctxt_a ) ) )
= X ) ).
% the_elem_set
thf(fact_879_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_880_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_881_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y9: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y9 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_882_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > int,G: nat > int] :
( ( Xs != nil_nat )
=> ( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( ord_less_int @ ( F @ X9 ) @ ( G @ X9 ) ) )
=> ( ord_less_int @ ( groups4559388385066561235st_int @ ( map_nat_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_nat_int @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_883_sum__list__strict__mono,axiom,
! [Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > int,G: ground_gmctxt_a > int] :
( ( Xs != nil_Ground_gmctxt_a )
=> ( ! [X9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ord_less_int @ ( F @ X9 ) @ ( G @ X9 ) ) )
=> ( ord_less_int @ ( groups4559388385066561235st_int @ ( map_Gr1677053652638677008_a_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_Gr1677053652638677008_a_int @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_884_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs != nil_nat )
=> ( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X9 ) @ ( G @ X9 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_885_sum__list__strict__mono,axiom,
! [Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,G: ground_gmctxt_a > nat] :
( ( Xs != nil_Ground_gmctxt_a )
=> ( ! [X9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X9 ) @ ( G @ X9 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_886_length__partition__gholes__nth,axiom,
! [Cs4: list_Ground_gmctxt_a,Ts: list_nat,I: nat] :
( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs4 ) )
= ( size_size_list_nat @ Ts ) )
=> ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs4 ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Ts @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs4 ) ) @ I ) )
= ( ground_num_gholes_a @ ( nth_Ground_gmctxt_a @ Cs4 @ I ) ) ) ) ) ).
% length_partition_gholes_nth
thf(fact_887_map__partition__by__nth,axiom,
! [I: nat,Ys: list_nat,F: nat > nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( map_nat_nat @ F @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) )
= ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( map_nat_nat @ F @ Xs ) @ Ys ) @ I ) ) ) ).
% map_partition_by_nth
thf(fact_888_partition__by__nth__nth__elem,axiom,
! [Ys: list_nat,Xs: list_list_nat,I: nat,J: nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
=> ( member_list_nat @ ( nth_list_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) @ J ) @ ( set_list_nat2 @ Xs ) ) ) ) ) ).
% partition_by_nth_nth_elem
thf(fact_889_partition__by__nth__nth__elem,axiom,
! [Ys: list_nat,Xs: list_Ground_gmctxt_a,I: nat,J: nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
=> ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) @ J ) @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ) ).
% partition_by_nth_nth_elem
thf(fact_890_partition__by__nth__nth__elem,axiom,
! [Ys: list_nat,Xs: list_nat,I: nat,J: nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
=> ( member_nat2 @ ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) @ J ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% partition_by_nth_nth_elem
thf(fact_891_partition__by__Nil,axiom,
! [Ys: list_nat] :
( ( multih8544690682651757648by_nat @ nil_nat @ Ys )
= ( replicate_list_nat @ ( size_size_list_nat @ Ys ) @ nil_nat ) ) ).
% partition_by_Nil
thf(fact_892_length__partition__by,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s3023201423986296836st_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
= ( size_size_list_nat @ Ys ) ) ).
% length_partition_by
thf(fact_893_map__length__partition__by,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( map_list_nat_nat @ size_size_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_length_partition_by
thf(fact_894_fill__gholes__gmctxt__no__holes,axiom,
! [C2: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C2 )
= zero_zero_nat )
=> ( ( ground2530127064035342964ctxt_a @ C2 @ nil_Ground_gmctxt_a )
= C2 ) ) ).
% fill_gholes_gmctxt_no_holes
thf(fact_895_ghole__poss_Osimps_I1_J,axiom,
( ( ground_ghole_poss_a @ ground_GMHole_a )
= ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).
% ghole_poss.simps(1)
thf(fact_896_map__map__partition__by,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
= ( multih8544690682651757648by_nat @ ( map_nat_nat @ F @ Xs ) @ Ys ) ) ).
% map_map_partition_by
thf(fact_897_Ball__set__partition__by,axiom,
! [Ys: list_nat,Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_s962548663205356226ctxt_a @ Xs ) )
=> ( ( ! [X2: list_Ground_gmctxt_a] :
( ( member2714729351964853061ctxt_a @ X2 @ ( set_li3057194059929822169ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) ) )
=> ! [Y9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y9 @ ( set_Ground_gmctxt_a2 @ X2 ) )
=> ( P @ Y9 ) ) ) )
= ( ! [X2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ) ).
% Ball_set_partition_by
thf(fact_898_Ball__set__partition__by,axiom,
! [Ys: list_nat,Xs: list_nat,P: nat > $o] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( multih8544690682651757648by_nat @ Xs @ Ys ) ) )
=> ! [Y9: nat] :
( ( member_nat2 @ Y9 @ ( set_nat2 @ X2 ) )
=> ( P @ Y9 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ) ).
% Ball_set_partition_by
thf(fact_899_length__partition__by__nth,axiom,
! [Ys: list_nat,Xs: list_nat,I: nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) )
= ( nth_nat @ Ys @ I ) ) ) ) ).
% length_partition_by_nth
thf(fact_900_partition__by__nth__nth_I1_J,axiom,
! [Ys: list_nat,Xs: list_list_nat,I: nat,J: nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
=> ( ( nth_list_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) @ J )
= ( nth_list_nat @ Xs @ ( multih7279076602905066526by_idx @ ( size_s3023201423986296836st_nat @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).
% partition_by_nth_nth(1)
thf(fact_901_partition__by__nth__nth_I1_J,axiom,
! [Ys: list_nat,Xs: list_nat,I: nat,J: nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
=> ( ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) @ J )
= ( nth_nat @ Xs @ ( multih7279076602905066526by_idx @ ( size_size_list_nat @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).
% partition_by_nth_nth(1)
thf(fact_902_partition__by__nth__nth_I2_J,axiom,
! [Ys: list_nat,Xs: list_nat,I: nat,J: nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
=> ( ord_less_nat @ ( multih7279076602905066526by_idx @ ( size_size_list_nat @ Xs ) @ Ys @ I @ J ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% partition_by_nth_nth(2)
thf(fact_903_fill__gholes__induct,axiom,
! [C2: ground_gmctxt_a,Xs: list_nat,P: ground_gmctxt_a > list_nat > $o] :
( ( ( ground_num_gholes_a @ C2 )
= ( size_size_list_nat @ Xs ) )
=> ( ! [X9: nat] : ( P @ ground_GMHole_a @ ( cons_nat @ X9 @ nil_nat ) )
=> ( ! [F3: a,Cs2: list_Ground_gmctxt_a,Xs3: list_nat] :
( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) )
= ( size_size_list_nat @ Xs3 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Cs2 ) )
=> ( P @ ( nth_Ground_gmctxt_a @ Cs2 @ I4 ) @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs3 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) @ I4 ) ) )
=> ( P @ ( ground_GMFun_a @ F3 @ Cs2 ) @ ( concat_nat @ ( multih8544690682651757648by_nat @ Xs3 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) ) ) ) )
=> ( P @ C2 @ Xs ) ) ) ) ).
% fill_gholes_induct
thf(fact_904_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_905_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_906_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_nat @ ( replicate_list_nat @ I @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_907_concat__partition__by,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( concat_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
= Xs ) ) ).
% concat_partition_by
thf(fact_908_partition__by__concat__id,axiom,
! [Xss2: list_list_nat,Ys: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xss2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xss2 @ I3 ) )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( ( multih8544690682651757648by_nat @ ( concat_nat @ Xss2 ) @ Ys )
= Xss2 ) ) ) ).
% partition_by_concat_id
thf(fact_909_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_910_sum__list__concat,axiom,
! [Ls2: list_list_nat] :
( ( groups4561878855575611511st_nat @ ( concat_nat @ Ls2 ) )
= ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ groups4561878855575611511st_nat @ Ls2 ) ) ) ).
% sum_list_concat
thf(fact_911_length__concat,axiom,
! [Xss2: list_list_nat] :
( ( size_size_list_nat @ ( concat_nat @ Xss2 ) )
= ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xss2 ) ) ) ).
% length_concat
thf(fact_912_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_913_eq__length__concat__nth,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ys @ I3 ) ) ) )
=> ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
= ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_914_concat__nth__nthI,axiom,
! [Ss: list_list_list_nat,Ts: list_list_list_nat,P: list_nat > list_nat > $o] :
( ( ( size_s6248950052170075156st_nat @ Ss )
= ( size_s6248950052170075156st_nat @ Ts ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Ts ) )
=> ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ss @ I3 ) )
= ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ts @ I3 ) ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Ts ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ts @ I3 ) ) )
=> ( P @ ( nth_list_nat @ ( nth_list_list_nat @ Ss @ I3 ) @ J2 ) @ ( nth_list_nat @ ( nth_list_list_nat @ Ts @ I3 ) @ J2 ) ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Ts ) ) )
=> ( P @ ( nth_list_nat @ ( concat_list_nat @ Ss ) @ I4 ) @ ( nth_list_nat @ ( concat_list_nat @ Ts ) @ I4 ) ) ) ) ) ) ).
% concat_nth_nthI
thf(fact_915_concat__nth__nthI,axiom,
! [Ss: list_list_list_nat,Ts: list_list_nat,P: list_nat > nat > $o] :
( ( ( size_s6248950052170075156st_nat @ Ss )
= ( size_s3023201423986296836st_nat @ Ts ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ts ) )
=> ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ss @ I3 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ts @ I3 ) ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ts ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I3 ) ) )
=> ( P @ ( nth_list_nat @ ( nth_list_list_nat @ Ss @ I3 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I3 ) @ J2 ) ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
=> ( P @ ( nth_list_nat @ ( concat_list_nat @ Ss ) @ I4 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I4 ) ) ) ) ) ) ).
% concat_nth_nthI
thf(fact_916_concat__nth__nthI,axiom,
! [Ss: list_list_nat,Ts: list_list_list_nat,P: nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Ss )
= ( size_s6248950052170075156st_nat @ Ts ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Ts ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I3 ) )
= ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ts @ I3 ) ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Ts ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ts @ I3 ) ) )
=> ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I3 ) @ J2 ) @ ( nth_list_nat @ ( nth_list_list_nat @ Ts @ I3 ) @ J2 ) ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Ts ) ) )
=> ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I4 ) @ ( nth_list_nat @ ( concat_list_nat @ Ts ) @ I4 ) ) ) ) ) ) ).
% concat_nth_nthI
thf(fact_917_concat__nth__nthI,axiom,
! [Ss: list_list_nat,Ts: list_list_nat,P: nat > nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Ss )
= ( size_s3023201423986296836st_nat @ Ts ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ts ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I3 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ts @ I3 ) ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ts ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I3 ) ) )
=> ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I3 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I3 ) @ J2 ) ) ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
=> ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I4 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I4 ) ) ) ) ) ) ).
% concat_nth_nthI
thf(fact_918_concat__all__nth,axiom,
! [Xs: list_list_list_nat,Ys: list_list_list_nat,P: list_nat > list_nat > $o] :
( ( ( size_s6248950052170075156st_nat @ Xs )
= ( size_s6248950052170075156st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I3 ) )
= ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ys @ I3 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I3 ) ) )
=> ( P @ ( nth_list_nat @ ( nth_list_list_nat @ Xs @ I3 ) @ J2 ) @ ( nth_list_nat @ ( nth_list_list_nat @ Ys @ I3 ) @ J2 ) ) ) )
=> ! [K4: nat] :
( ( ord_less_nat @ K4 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
=> ( P @ ( nth_list_nat @ ( concat_list_nat @ Xs ) @ K4 ) @ ( nth_list_nat @ ( concat_list_nat @ Ys ) @ K4 ) ) ) ) ) ) ).
% concat_all_nth
thf(fact_919_concat__all__nth,axiom,
! [Xs: list_list_list_nat,Ys: list_list_nat,P: list_nat > nat > $o] :
( ( ( size_s6248950052170075156st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I3 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ys @ I3 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I3 ) ) )
=> ( P @ ( nth_list_nat @ ( nth_list_list_nat @ Xs @ I3 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I3 ) @ J2 ) ) ) )
=> ! [K4: nat] :
( ( ord_less_nat @ K4 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
=> ( P @ ( nth_list_nat @ ( concat_list_nat @ Xs ) @ K4 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K4 ) ) ) ) ) ) ).
% concat_all_nth
thf(fact_920_concat__all__nth,axiom,
! [Xs: list_list_nat,Ys: list_list_list_nat,P: nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s6248950052170075156st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) )
= ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ys @ I3 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) ) )
=> ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I3 ) @ J2 ) @ ( nth_list_nat @ ( nth_list_list_nat @ Ys @ I3 ) @ J2 ) ) ) )
=> ! [K4: nat] :
( ( ord_less_nat @ K4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
=> ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K4 ) @ ( nth_list_nat @ ( concat_list_nat @ Ys ) @ K4 ) ) ) ) ) ) ).
% concat_all_nth
thf(fact_921_concat__all__nth,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,P: nat > nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Ys @ I3 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) ) )
=> ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I3 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I3 ) @ J2 ) ) ) )
=> ! [K4: nat] :
( ( ord_less_nat @ K4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
=> ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K4 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K4 ) ) ) ) ) ) ).
% concat_all_nth
thf(fact_922_nth__concat__split,axiom,
! [I: nat,Xs: list_list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s6248950052170075156st_nat @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J2 ) ) )
& ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
= ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J2 ) @ K2 ) ) ) ) ).
% nth_concat_split
thf(fact_923_nth__concat__split,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
& ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
= ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K2 ) ) ) ) ).
% nth_concat_split
thf(fact_924_nth__concat__two__lists,axiom,
! [I: nat,Xs: list_list_list_nat,Ys: list_list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
=> ( ( ( size_s6248950052170075156st_nat @ Ys )
= ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ys @ I3 ) )
= ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I3 ) ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s6248950052170075156st_nat @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J2 ) ) )
& ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
= ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J2 ) @ K2 ) )
& ( ( nth_list_nat @ ( concat_list_nat @ Ys ) @ I )
= ( nth_list_nat @ ( nth_list_list_nat @ Ys @ J2 ) @ K2 ) ) ) ) ) ) ).
% nth_concat_two_lists
thf(fact_925_nth__concat__two__lists,axiom,
! [I: nat,Xs: list_list_list_nat,Ys: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I3 ) )
= ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I3 ) ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s6248950052170075156st_nat @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J2 ) ) )
& ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
= ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J2 ) @ K2 ) )
& ( ( nth_nat @ ( concat_nat @ Ys ) @ I )
= ( nth_nat @ ( nth_list_nat @ Ys @ J2 ) @ K2 ) ) ) ) ) ) ).
% nth_concat_two_lists
thf(fact_926_nth__concat__two__lists,axiom,
! [I: nat,Xs: list_list_nat,Ys: list_list_list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
=> ( ( ( size_s6248950052170075156st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ys @ I3 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
& ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
= ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K2 ) )
& ( ( nth_list_nat @ ( concat_list_nat @ Ys ) @ I )
= ( nth_list_nat @ ( nth_list_list_nat @ Ys @ J2 ) @ K2 ) ) ) ) ) ) ).
% nth_concat_two_lists
thf(fact_927_nth__concat__two__lists,axiom,
! [I: nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I3 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
& ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
= ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K2 ) )
& ( ( nth_nat @ ( concat_nat @ Ys ) @ I )
= ( nth_nat @ ( nth_list_nat @ Ys @ J2 ) @ K2 ) ) ) ) ) ) ).
% nth_concat_two_lists
thf(fact_928_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs2: list_nat,Xs6: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs6 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_929_fill__gholes__gmctxt__MFun,axiom,
! [Cs: list_Ground_gmctxt_a,Ts: list_Ground_gmctxt_a,Ss: list_l1578326660578982846ctxt_a,F: a] :
( ( ( size_s962548663205356226ctxt_a @ Cs )
= ( size_s962548663205356226ctxt_a @ Ts ) )
=> ( ( ( size_s6994882312647770194ctxt_a @ Ss )
= ( size_s962548663205356226ctxt_a @ Ts ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ts ) )
=> ( ( ( ground_num_gholes_a @ ( nth_Ground_gmctxt_a @ Cs @ I3 ) )
= ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I3 ) ) )
& ( ( ground2530127064035342964ctxt_a @ ( nth_Ground_gmctxt_a @ Cs @ I3 ) @ ( nth_li6678718213161987943ctxt_a @ Ss @ I3 ) )
= ( nth_Ground_gmctxt_a @ Ts @ I3 ) ) ) )
=> ( ( ground2530127064035342964ctxt_a @ ( ground_GMFun_a @ F @ Cs ) @ ( concat3993834769610107037ctxt_a @ Ss ) )
= ( ground_GMFun_a @ F @ Ts ) ) ) ) ) ).
% fill_gholes_gmctxt_MFun
thf(fact_930_fill__gholes__induct2,axiom,
! [C2: ground_gmctxt_a,Xs: list_nat,Ys: list_nat,P: ground_gmctxt_a > list_nat > list_nat > $o] :
( ( ( ground_num_gholes_a @ C2 )
= ( size_size_list_nat @ Xs ) )
=> ( ( ( ground_num_gholes_a @ C2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [X9: nat,Y2: nat] : ( P @ ground_GMHole_a @ ( cons_nat @ X9 @ nil_nat ) @ ( cons_nat @ Y2 @ nil_nat ) )
=> ( ! [F3: a,Cs2: list_Ground_gmctxt_a,Xs3: list_nat,Ys4: list_nat] :
( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) )
= ( size_size_list_nat @ Xs3 ) )
=> ( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) )
= ( size_size_list_nat @ Ys4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Cs2 ) )
=> ( P @ ( nth_Ground_gmctxt_a @ Cs2 @ I4 ) @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs3 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) @ I4 ) @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Ys4 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) @ I4 ) ) )
=> ( P @ ( ground_GMFun_a @ F3 @ Cs2 ) @ ( concat_nat @ ( multih8544690682651757648by_nat @ Xs3 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) ) @ ( concat_nat @ ( multih8544690682651757648by_nat @ Ys4 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) ) ) ) ) )
=> ( P @ C2 @ Xs @ Ys ) ) ) ) ) ).
% fill_gholes_induct2
thf(fact_931_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_932_fill__gholes__apply__gctxt,axiom,
! [C2: ground_gmctxt_a,S: ground_gterm_a] :
( ( ( ground_num_gholes_a @ C2 )
= ( suc @ zero_zero_nat ) )
=> ( ( ground_fill_gholes_a @ C2 @ ( cons_Ground_gterm_a @ S @ nil_Ground_gterm_a ) )
= ( ground5122445521002937143term_a @ ( ground5333112502500237606ctxt_a @ C2 ) @ S ) ) ) ).
% fill_gholes_apply_gctxt
thf(fact_933_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_934_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_935_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_936_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_937_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list_nat,Q3: nat] :
( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
= ( upt @ M2 @ Q3 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_938_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_939_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_940_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_941_partition__by__idx__def,axiom,
( multih7279076602905066526by_idx
= ( ^ [L3: nat,Ys3: list_nat,I2: nat] : ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( upt @ zero_zero_nat @ L3 ) @ Ys3 ) @ I2 ) ) ) ) ).
% partition_by_idx_def
thf(fact_942_eq__gfill__induct,axiom,
! [T: ground_gterm_a,C2: ground_gmctxt_a,Ts: list_Ground_gterm_a,P: ground_gterm_a > ground_gmctxt_a > list_Ground_gterm_a > $o] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C2 @ Ts ) )
=> ( ! [T2: ground_gterm_a] : ( P @ T2 @ ground_GMHole_a @ ( cons_Ground_gterm_a @ T2 @ nil_Ground_gterm_a ) )
=> ( ! [F3: a,Ss2: list_Ground_gterm_a,Cs2: list_Ground_gmctxt_a,Ts2: list_Ground_gterm_a] :
( ( ( size_s962548663205356226ctxt_a @ Cs2 )
= ( size_s1893836384466666913term_a @ Ss2 ) )
=> ( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) )
= ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ss2 ) )
=> ( ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss2 @ I4 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Cs2 @ I4 ) @ ( nth_li6172171122907299772term_a @ ( multih1611307067487991599term_a @ Ts2 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) @ I4 ) ) )
& ( P @ ( nth_Ground_gterm_a @ Ss2 @ I4 ) @ ( nth_Ground_gmctxt_a @ Cs2 @ I4 ) @ ( nth_li6172171122907299772term_a @ ( multih1611307067487991599term_a @ Ts2 @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs2 ) ) @ I4 ) ) ) )
=> ( P @ ( ground_GFun_a @ F3 @ Ss2 ) @ ( ground_GMFun_a @ F3 @ Cs2 ) @ Ts2 ) ) ) )
=> ( P @ T @ C2 @ Ts ) ) ) ) ).
% eq_gfill_induct
thf(fact_943_partition__by__nth__nth__old,axiom,
! [I: nat,Xs: list_list_nat,Ys: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s6248950052170075156st_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) ) )
=> ( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) @ J )
= ( nth_list_nat @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li3275209578935457798at_nat @ size_s3023201423986296836st_nat @ ( take_list_list_nat @ I @ ( multih5630945896903731040st_nat @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).
% partition_by_nth_nth_old
thf(fact_944_partition__by__nth__nth__old,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) ) )
=> ( ( ( groups4561878855575611511st_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) @ J )
= ( nth_nat @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ I @ ( multih8544690682651757648by_nat @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).
% partition_by_nth_nth_old
thf(fact_945_nth__concat__split_H,axiom,
! [I: nat,Xs: list_list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ! [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J2 ) ) )
=> ( ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
= ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J2 ) @ K2 ) )
=> ( I
!= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li3275209578935457798at_nat @ size_s3023201423986296836st_nat @ ( take_list_list_nat @ J2 @ Xs ) ) ) @ K2 ) ) ) ) ) ) ).
% nth_concat_split'
thf(fact_946_nth__concat__split_H,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ! [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
=> ( ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
= ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K2 ) )
=> ( I
!= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ J2 @ Xs ) ) ) @ K2 ) ) ) ) ) ) ).
% nth_concat_split'
thf(fact_947_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_948_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_949_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_950_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs2: list_nat] : nil_nat ) ) ).
% take0
thf(fact_951_nth__take,axiom,
! [I: nat,N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_list_nat @ ( take_list_nat @ N @ Xs ) @ I )
= ( nth_list_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_952_nth__take,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_953_in__set__takeD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_954_in__set__takeD,axiom,
! [X: ground_gmctxt_a,N: nat,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( take_Ground_gmctxt_a @ N @ Xs ) ) )
=> ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_955_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_956_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_957_take__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ! [I3: nat] :
( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_958_eq__gfill_Ointros,axiom,
! [T: ground_gterm_a,D: ground_gmctxt_a,Ss: list_Ground_gterm_a] :
( ( T
= ( ground_fill_gholes_a @ D @ Ss ) )
=> ( ( ( ground_num_gholes_a @ D )
= ( size_s1893836384466666913term_a @ Ss ) )
=> ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss ) ) ) ) ).
% eq_gfill.intros
thf(fact_959_eq__gfill_Osimps,axiom,
( ground_eq_gfill_a
= ( ^ [A1: ground_gterm_a,A22: produc2459281416120113548term_a] :
? [T3: ground_gterm_a,D3: ground_gmctxt_a,Ss3: list_Ground_gterm_a] :
( ( A1 = T3 )
& ( A22
= ( produc4720362732934906942term_a @ D3 @ Ss3 ) )
& ( T3
= ( ground_fill_gholes_a @ D3 @ Ss3 ) )
& ( ( ground_num_gholes_a @ D3 )
= ( size_s1893836384466666913term_a @ Ss3 ) ) ) ) ) ).
% eq_gfill.simps
thf(fact_960_eq__gfill_Ocases,axiom,
! [A12: ground_gterm_a,A23: produc2459281416120113548term_a] :
( ( ground_eq_gfill_a @ A12 @ A23 )
=> ~ ! [D4: ground_gmctxt_a,Ss2: list_Ground_gterm_a] :
( ( A23
= ( produc4720362732934906942term_a @ D4 @ Ss2 ) )
=> ( ( A12
= ( ground_fill_gholes_a @ D4 @ Ss2 ) )
=> ( ( ground_num_gholes_a @ D4 )
!= ( size_s1893836384466666913term_a @ Ss2 ) ) ) ) ) ).
% eq_gfill.cases
thf(fact_961_eqgfE_I2_J,axiom,
! [T: ground_gterm_a,D: ground_gmctxt_a,Ss: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss ) )
=> ( ( ground_num_gholes_a @ D )
= ( size_s1893836384466666913term_a @ Ss ) ) ) ).
% eqgfE(2)
thf(fact_962_take__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_963_set__take__nth,axiom,
! [X: list_nat,I: nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ ( take_list_nat @ I @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ord_less_nat @ J2 @ I )
& ( ( nth_list_nat @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_964_set__take__nth,axiom,
! [X: ground_gmctxt_a,I: nat,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( take_Ground_gmctxt_a @ I @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
& ( ord_less_nat @ J2 @ I )
& ( ( nth_Ground_gmctxt_a @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_965_set__take__nth,axiom,
! [X: nat,I: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ I @ Xs ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
& ( ord_less_nat @ J2 @ I )
& ( ( nth_nat @ Xs @ J2 )
= X ) ) ) ).
% set_take_nth
thf(fact_966_fill__gholes_Ocases,axiom,
! [X: produc2459281416120113548term_a] :
( ! [T2: ground_gterm_a] :
( X
!= ( produc4720362732934906942term_a @ ground_GMHole_a @ ( cons_Ground_gterm_a @ T2 @ nil_Ground_gterm_a ) ) )
=> ( ! [F3: a,Cs3: list_Ground_gmctxt_a,Ts2: list_Ground_gterm_a] :
( X
!= ( produc4720362732934906942term_a @ ( ground_GMFun_a @ F3 @ Cs3 ) @ Ts2 ) )
=> ( ( X
!= ( produc4720362732934906942term_a @ ground_GMHole_a @ nil_Ground_gterm_a ) )
=> ~ ! [V2: ground_gterm_a,Vb: ground_gterm_a,Vc: list_Ground_gterm_a] :
( X
!= ( produc4720362732934906942term_a @ ground_GMHole_a @ ( cons_Ground_gterm_a @ V2 @ ( cons_Ground_gterm_a @ Vb @ Vc ) ) ) ) ) ) ) ).
% fill_gholes.cases
thf(fact_967_eqgf__GMFunI,axiom,
! [Cs: list_Ground_gmctxt_a,Ss: list_Ground_gterm_a,Ts: list_l4678906206682822203term_a,F: a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Cs ) )
=> ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss @ I3 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Cs @ I3 ) @ ( nth_li6172171122907299772term_a @ Ts @ I3 ) ) ) )
=> ( ( ( size_s962548663205356226ctxt_a @ Cs )
= ( size_s1893836384466666913term_a @ Ss ) )
=> ( ( ( size_s1893836384466666913term_a @ Ss )
= ( size_s5321119603959729319term_a @ Ts ) )
=> ( ground_eq_gfill_a @ ( ground_GFun_a @ F @ Ss ) @ ( produc4720362732934906942term_a @ ( ground_GMFun_a @ F @ Cs ) @ ( concat1399541994378716528term_a @ Ts ) ) ) ) ) ) ).
% eqgf_GMFunI
thf(fact_968_eqgf__GMFunE,axiom,
! [S: ground_gterm_a,F: a,Cs: list_Ground_gmctxt_a,Ss: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ ( ground_GMFun_a @ F @ Cs ) @ Ss ) )
=> ~ ! [Ts2: list_Ground_gterm_a] :
( ( S
= ( ground_GFun_a @ F @ Ts2 ) )
=> ( ( ( size_s1893836384466666913term_a @ Ts2 )
= ( size_s962548663205356226ctxt_a @ Cs ) )
=> ! [Sss: list_l4678906206682822203term_a] :
( ( ( size_s5321119603959729319term_a @ Sss )
= ( size_s962548663205356226ctxt_a @ Cs ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Cs ) )
=> ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts2 @ I4 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Cs @ I4 ) @ ( nth_li6172171122907299772term_a @ Sss @ I4 ) ) ) )
=> ( Ss
!= ( concat1399541994378716528term_a @ Sss ) ) ) ) ) ) ) ).
% eqgf_GMFunE
thf(fact_969_fill__gholes__gmctxt__sound,axiom,
! [Ds: list_Ground_gmctxt_a,C2: ground_gmctxt_a,Sss2: list_l4678906206682822203term_a,Ts: list_Ground_gterm_a] :
( ( ( size_s962548663205356226ctxt_a @ Ds )
= ( ground_num_gholes_a @ C2 ) )
=> ( ( ( size_s5321119603959729319term_a @ Sss2 )
= ( ground_num_gholes_a @ C2 ) )
=> ( ( ( size_s1893836384466666913term_a @ Ts )
= ( ground_num_gholes_a @ C2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts @ I3 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I3 ) @ ( nth_li6172171122907299772term_a @ Sss2 @ I3 ) ) ) )
=> ( ground_eq_gfill_a @ ( ground_fill_gholes_a @ C2 @ Ts ) @ ( produc4720362732934906942term_a @ ( ground2530127064035342964ctxt_a @ C2 @ Ds ) @ ( concat1399541994378716528term_a @ Sss2 ) ) ) ) ) ) ) ).
% fill_gholes_gmctxt_sound
thf(fact_970_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( take_list_nat @ ( suc @ I ) @ Xs )
= ( append_list_nat @ ( take_list_nat @ I @ Xs ) @ ( cons_list_nat @ ( nth_list_nat @ Xs @ I ) @ nil_list_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_971_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_972_sum__list__split,axiom,
! [I: nat,Xs: list_list_nat,J: nat,K: nat,L: nat] :
( ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ I @ Xs ) ) ) @ J )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ K @ Xs ) ) ) @ L ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ K @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) )
=> ( ( ord_less_nat @ L @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ K ) ) )
=> ( ( I = K )
& ( J = L ) ) ) ) ) ) ) ).
% sum_list_split
thf(fact_973_concat__nth__length,axiom,
! [I: nat,Uss: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Uss ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Uss @ I ) ) )
=> ( ord_less_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ I @ Uss ) ) ) @ J ) @ ( size_size_list_nat @ ( concat_nat @ Uss ) ) ) ) ) ).
% concat_nth_length
thf(fact_974_concat__nth,axiom,
! [M2: nat,Xs: list_list_list_nat,N: nat,I: nat] :
( ( ord_less_nat @ M2 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ M2 ) ) )
=> ( ( I
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li3275209578935457798at_nat @ size_s3023201423986296836st_nat @ ( take_list_list_nat @ M2 @ Xs ) ) ) @ N ) )
=> ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
= ( nth_list_nat @ ( nth_list_list_nat @ Xs @ M2 ) @ N ) ) ) ) ) ).
% concat_nth
thf(fact_975_concat__nth,axiom,
! [M2: nat,Xs: list_list_nat,N: nat,I: nat] :
( ( ord_less_nat @ M2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ M2 ) ) )
=> ( ( I
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ M2 @ Xs ) ) ) @ N ) )
=> ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
= ( nth_nat @ ( nth_list_nat @ Xs @ M2 ) @ N ) ) ) ) ) ).
% concat_nth
thf(fact_976_partition__by__nth__less,axiom,
! [K: nat,I: nat,Zs: list_nat,Xs: list_nat,J: nat,Y: nat,Ys: list_nat] :
( ( ord_less_nat @ K @ I )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Xs )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ I @ Zs ) ) @ J ) )
=> ( ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ Zs ) @ K )
= ( take_nat @ ( nth_nat @ Zs @ K ) @ ( drop_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ K @ Zs ) ) @ Xs ) ) ) ) ) ) ).
% partition_by_nth_less
thf(fact_977_comp__gmctxtp_Ocases,axiom,
! [A12: ground_gmctxt_a,A23: ground_gmctxt_a] :
( ( ground4927988881900525590txtp_a @ A12 @ A23 )
=> ( ( A12 != ground_GMHole_a )
=> ( ( A23 != ground_GMHole_a )
=> ~ ! [F3: a,G3: a,Cs2: list_Ground_gmctxt_a] :
( ( A12
= ( ground_GMFun_a @ F3 @ Cs2 ) )
=> ! [Ds3: list_Ground_gmctxt_a] :
( ( A23
= ( ground_GMFun_a @ G3 @ Ds3 ) )
=> ( ( F3 = G3 )
=> ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
= ( size_s962548663205356226ctxt_a @ Ds3 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ds3 ) )
=> ( ground4927988881900525590txtp_a @ ( nth_Ground_gmctxt_a @ Cs2 @ I4 ) @ ( nth_Ground_gmctxt_a @ Ds3 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% comp_gmctxtp.cases
thf(fact_978_comp__gmctxtp_Osimps,axiom,
( ground4927988881900525590txtp_a
= ( ^ [A1: ground_gmctxt_a,A22: ground_gmctxt_a] :
( ? [D3: ground_gmctxt_a] :
( ( A1 = ground_GMHole_a )
& ( A22 = D3 ) )
| ? [C5: ground_gmctxt_a] :
( ( A1 = C5 )
& ( A22 = ground_GMHole_a ) )
| ? [F2: a,G2: a,Cs5: list_Ground_gmctxt_a,Ds4: list_Ground_gmctxt_a] :
( ( A1
= ( ground_GMFun_a @ F2 @ Cs5 ) )
& ( A22
= ( ground_GMFun_a @ G2 @ Ds4 ) )
& ( F2 = G2 )
& ( ( size_s962548663205356226ctxt_a @ Cs5 )
= ( size_s962548663205356226ctxt_a @ Ds4 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ds4 ) )
=> ( ground4927988881900525590txtp_a @ ( nth_Ground_gmctxt_a @ Cs5 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ds4 @ I2 ) ) ) ) ) ) ) ).
% comp_gmctxtp.simps
thf(fact_979_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X2: list_nat] : X2 ) ) ).
% drop0
thf(fact_980_drop__upt,axiom,
! [M2: nat,I: nat,J: nat] :
( ( drop_nat @ M2 @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_981_drop__drop,axiom,
! [N: nat,M2: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M2 @ Xs ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ).
% drop_drop
thf(fact_982_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_983_append__take__drop__id,axiom,
! [N: nat,Xs: list_nat] :
( ( append_nat @ ( take_nat @ N @ Xs ) @ ( drop_nat @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_984_nth__via__drop,axiom,
! [N: nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( drop_list_nat @ N @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
=> ( ( nth_list_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_985_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_986_take__drop,axiom,
! [N: nat,M2: nat,Xs: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M2 @ Xs ) )
= ( drop_nat @ M2 @ ( take_nat @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ) ).
% take_drop
thf(fact_987_drop__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_map
thf(fact_988_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_989_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_990_in__set__dropD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_991_in__set__dropD,axiom,
! [X: ground_gmctxt_a,N: nat,Xs: list_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( drop_Ground_gmctxt_a @ N @ Xs ) ) )
=> ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_992_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_993_take__add,axiom,
! [I: nat,J: nat,Xs: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I @ J ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( take_nat @ J @ ( drop_nat @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_994_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( cons_list_nat @ ( nth_list_nat @ Xs @ I ) @ ( drop_list_nat @ ( suc @ I ) @ Xs ) )
= ( drop_list_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_995_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
= ( drop_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_996_take__nth__drop__concat,axiom,
! [I: nat,Xss2: list_list_list_nat,Ys: list_list_nat,J: nat,Z3: list_nat] :
( ( ord_less_nat @ I @ ( size_s6248950052170075156st_nat @ Xss2 ) )
=> ( ( ( nth_list_list_nat @ Xss2 @ I )
= Ys )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( nth_list_nat @ Ys @ J )
= Z3 )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xss2 ) ) )
& ( ( take_list_nat @ K2 @ ( concat_list_nat @ Xss2 ) )
= ( append_list_nat @ ( concat_list_nat @ ( take_list_list_nat @ I @ Xss2 ) ) @ ( take_list_nat @ J @ Ys ) ) )
& ( ( nth_list_nat @ ( concat_list_nat @ Xss2 ) @ K2 )
= ( nth_list_nat @ ( nth_list_list_nat @ Xss2 @ I ) @ J ) )
& ( ( drop_list_nat @ ( suc @ K2 ) @ ( concat_list_nat @ Xss2 ) )
= ( append_list_nat @ ( drop_list_nat @ ( suc @ J ) @ Ys ) @ ( concat_list_nat @ ( drop_list_list_nat @ ( suc @ I ) @ Xss2 ) ) ) ) ) ) ) ) ) ).
% take_nth_drop_concat
thf(fact_997_take__nth__drop__concat,axiom,
! [I: nat,Xss2: list_list_nat,Ys: list_nat,J: nat,Z3: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xss2 ) )
=> ( ( ( nth_list_nat @ Xss2 @ I )
= Ys )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ys ) )
=> ( ( ( nth_nat @ Ys @ J )
= Z3 )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xss2 ) ) )
& ( ( take_nat @ K2 @ ( concat_nat @ Xss2 ) )
= ( append_nat @ ( concat_nat @ ( take_list_nat @ I @ Xss2 ) ) @ ( take_nat @ J @ Ys ) ) )
& ( ( nth_nat @ ( concat_nat @ Xss2 ) @ K2 )
= ( nth_nat @ ( nth_list_nat @ Xss2 @ I ) @ J ) )
& ( ( drop_nat @ ( suc @ K2 ) @ ( concat_nat @ Xss2 ) )
= ( append_nat @ ( drop_nat @ ( suc @ J ) @ Ys ) @ ( concat_nat @ ( drop_list_nat @ ( suc @ I ) @ Xss2 ) ) ) ) ) ) ) ) ) ).
% take_nth_drop_concat
thf(fact_998_nth__drop__0,axiom,
! [Ss: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Ss ) )
=> ( ( cons_list_nat @ ( nth_list_nat @ Ss @ zero_zero_nat ) @ ( drop_list_nat @ ( suc @ zero_zero_nat ) @ Ss ) )
= Ss ) ) ).
% nth_drop_0
thf(fact_999_nth__drop__0,axiom,
! [Ss: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Ss ) )
=> ( ( cons_nat @ ( nth_nat @ Ss @ zero_zero_nat ) @ ( drop_nat @ ( suc @ zero_zero_nat ) @ Ss ) )
= Ss ) ) ).
% nth_drop_0
thf(fact_1000_id__take__nth__drop,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( Xs
= ( append_list_nat @ ( take_list_nat @ I @ Xs ) @ ( cons_list_nat @ ( nth_list_nat @ Xs @ I ) @ ( drop_list_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1001_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1002_partition__by__nth,axiom,
! [I: nat,Ys: list_nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I )
= ( take_nat @ ( nth_nat @ Ys @ I ) @ ( drop_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ I @ Ys ) ) @ Xs ) ) ) ) ).
% partition_by_nth
thf(fact_1003_comp__gmctxtp_OGMFun,axiom,
! [F: a,G: a,Cs: list_Ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
( ( F = G )
=> ( ( ( size_s962548663205356226ctxt_a @ Cs )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ground4927988881900525590txtp_a @ ( nth_Ground_gmctxt_a @ Cs @ I3 ) @ ( nth_Ground_gmctxt_a @ Ds @ I3 ) ) )
=> ( ground4927988881900525590txtp_a @ ( ground_GMFun_a @ F @ Cs ) @ ( ground_GMFun_a @ G @ Ds ) ) ) ) ) ).
% comp_gmctxtp.GMFun
thf(fact_1004_ghole__num__bef__pos_Osimps_I2_J,axiom,
! [I: nat,Q3: list_nat,F: a,Cs: list_Ground_gmctxt_a] :
( ( ground1934838843355771718_pos_a @ ( cons_nat @ I @ Q3 ) @ ( ground_GMFun_a @ F @ Cs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ ( take_Ground_gmctxt_a @ I @ Cs ) ) ) @ ( ground1934838843355771718_pos_a @ Q3 @ ( nth_Ground_gmctxt_a @ Cs @ I ) ) ) ) ).
% ghole_num_bef_pos.simps(2)
thf(fact_1005_less__eq__gmctxt_Oind,axiom,
! [Cs4: list_Ground_gmctxt_a,Ds2: list_Ground_gmctxt_a,F: a] :
( ( ( size_s962548663205356226ctxt_a @ Cs4 )
= ( size_s962548663205356226ctxt_a @ Ds2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Cs4 ) )
=> ( ground6477899500129135038ctxt_a @ ( nth_Ground_gmctxt_a @ Cs4 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ds2 @ I3 ) ) )
=> ( ground6477899500129135038ctxt_a @ ( ground_GMFun_a @ F @ Cs4 ) @ ( ground_GMFun_a @ F @ Ds2 ) ) ) ) ).
% less_eq_gmctxt.ind
thf(fact_1006_less__eq__gmctxt_Ocases,axiom,
! [A12: ground_gmctxt_a,A23: ground_gmctxt_a] :
( ( ground6477899500129135038ctxt_a @ A12 @ A23 )
=> ( ( A12 != ground_GMHole_a )
=> ~ ! [Cs3: list_Ground_gmctxt_a,Ds5: list_Ground_gmctxt_a,F3: a] :
( ( A12
= ( ground_GMFun_a @ F3 @ Cs3 ) )
=> ( ( A23
= ( ground_GMFun_a @ F3 @ Ds5 ) )
=> ( ( ( size_s962548663205356226ctxt_a @ Cs3 )
= ( size_s962548663205356226ctxt_a @ Ds5 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Cs3 ) )
=> ( ground6477899500129135038ctxt_a @ ( nth_Ground_gmctxt_a @ Cs3 @ I4 ) @ ( nth_Ground_gmctxt_a @ Ds5 @ I4 ) ) ) ) ) ) ) ) ).
% less_eq_gmctxt.cases
thf(fact_1007_less__eq__gmctxt_Osimps,axiom,
( ground6477899500129135038ctxt_a
= ( ^ [A1: ground_gmctxt_a,A22: ground_gmctxt_a] :
( ? [U2: ground_gmctxt_a] :
( ( A1 = ground_GMHole_a )
& ( A22 = U2 ) )
| ? [Cs6: list_Ground_gmctxt_a,Ds6: list_Ground_gmctxt_a,F2: a] :
( ( A1
= ( ground_GMFun_a @ F2 @ Cs6 ) )
& ( A22
= ( ground_GMFun_a @ F2 @ Ds6 ) )
& ( ( size_s962548663205356226ctxt_a @ Cs6 )
= ( size_s962548663205356226ctxt_a @ Ds6 ) )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s962548663205356226ctxt_a @ Cs6 ) )
=> ( ground6477899500129135038ctxt_a @ ( nth_Ground_gmctxt_a @ Cs6 @ X2 ) @ ( nth_Ground_gmctxt_a @ Ds6 @ X2 ) ) ) ) ) ) ) ).
% less_eq_gmctxt.simps
thf(fact_1008_ghole__num__bef__pos_Oelims,axiom,
! [X: list_nat,Xa: ground_gmctxt_a,Y: nat] :
( ( ( ground1934838843355771718_pos_a @ X @ Xa )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ( ! [I3: nat,Q2: list_nat] :
( ( X
= ( cons_nat @ I3 @ Q2 ) )
=> ! [F3: a,Cs2: list_Ground_gmctxt_a] :
( ( Xa
= ( ground_GMFun_a @ F3 @ Cs2 ) )
=> ( Y
!= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ ( take_Ground_gmctxt_a @ I3 @ Cs2 ) ) ) @ ( ground1934838843355771718_pos_a @ Q2 @ ( nth_Ground_gmctxt_a @ Cs2 @ I3 ) ) ) ) ) )
=> ~ ( ? [V2: nat,Va2: list_nat] :
( X
= ( cons_nat @ V2 @ Va2 ) )
=> ( ( Xa = ground_GMHole_a )
=> ( Y != undefined_nat ) ) ) ) ) ) ).
% ghole_num_bef_pos.elims
thf(fact_1009_sum__list__take__eq,axiom,
! [K: nat,I: nat,Xs: list_nat] :
( ( ord_less_nat @ K @ I )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( groups4561878855575611511st_nat @ ( take_nat @ I @ Xs ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ K @ Xs ) ) @ ( nth_nat @ Xs @ K ) ) @ ( groups4561878855575611511st_nat @ ( take_nat @ ( minus_minus_nat @ I @ ( suc @ K ) ) @ ( drop_nat @ ( suc @ K ) @ Xs ) ) ) ) ) ) ) ).
% sum_list_take_eq
thf(fact_1010_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1011_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1012_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_1013_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_1014_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1015_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_1016_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_1017_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_1018_add__diff__cancel__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_1019_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_1020_add__diff__cancel__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_1021_minus__diff__eq,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) )
= ( minus_minus_int @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_1022_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1023_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1024_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1025_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1026_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1027_diff__gt__0__iff__gt,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1028_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1029_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_1030_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1031_diff__minus__eq__add,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_1032_uminus__add__conv__diff,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( minus_minus_int @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1033_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1034_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_1035_drop__replicate,axiom,
! [I: nat,K: nat,X: nat] :
( ( drop_nat @ I @ ( replicate_nat @ K @ X ) )
= ( replicate_nat @ ( minus_minus_nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_1036_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1037_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_1038_take__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1039_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1040_drop__take,axiom,
! [N: nat,M2: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M2 @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ M2 @ N ) @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_take
thf(fact_1041_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1042_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1043_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1044_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1045_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1046_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1047_minus__diff__commute,axiom,
! [B: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_1048_eq__iff__diff__eq__0,axiom,
( ( ^ [Y11: int,Z5: int] : ( Y11 = Z5 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1049_diff__strict__right__mono,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C3 ) @ ( minus_minus_int @ B @ C3 ) ) ) ).
% diff_strict_right_mono
thf(fact_1050_diff__strict__left__mono,axiom,
! [B: int,A2: int,C3: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C3 @ A2 ) @ ( minus_minus_int @ C3 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1051_diff__eq__diff__less,axiom,
! [A2: int,B: int,C3: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C3 @ D2 ) )
=> ( ( ord_less_int @ A2 @ B )
= ( ord_less_int @ C3 @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_1052_diff__strict__mono,axiom,
! [A2: int,B: int,D2: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ D2 @ C3 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C3 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_1053_diff__diff__eq,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C3 )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% diff_diff_eq
thf(fact_1054_add__implies__diff,axiom,
! [C3: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C3 @ B )
= A2 )
=> ( C3
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_1055_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1056_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1057_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1058_diff__right__commute,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C3 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C3 ) ) ).
% diff_right_commute
thf(fact_1059_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1060_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1061_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1062_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1063_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1064_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1065_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1066_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1067_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1068_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A2: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A2 @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1069_diff__less__eq,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ C3 )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C3 @ B ) ) ) ).
% diff_less_eq
thf(fact_1070_less__diff__eq,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C3 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ C3 ) ) ).
% less_diff_eq
thf(fact_1071_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1072_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1073_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A2
= ( plus_plus_nat @ B @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1074_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A2
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1075_nth__append,axiom,
! [N: nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys ) @ N )
= ( nth_list_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys ) @ N )
= ( nth_list_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1076_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1077_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I2: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I2 ) @ J3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_1078_nth__map__upt,axiom,
! [I: nat,N: nat,M2: nat,F: nat > list_nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M2 ) )
=> ( ( nth_list_nat @ ( map_nat_list_nat @ F @ ( upt @ M2 @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_1079_nth__map__upt,axiom,
! [I: nat,N: nat,M2: nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M2 @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_1080_nth__append__Cons,axiom,
! [I: nat,Xs: list_list_nat,Y: list_nat,Zs: list_list_nat] :
( ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ Y @ Zs ) ) @ I )
= ( nth_list_nat @ Xs @ I ) ) )
& ( ~ ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( I
= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ Y @ Zs ) ) @ I )
= Y ) )
& ( ( I
!= ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ Y @ Zs ) ) @ I )
= ( nth_list_nat @ Zs @ ( minus_minus_nat @ I @ ( suc @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ) ) ) ).
% nth_append_Cons
thf(fact_1081_nth__append__Cons,axiom,
! [I: nat,Xs: list_nat,Y: nat,Zs: list_nat] :
( ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= ( nth_nat @ Xs @ I ) ) )
& ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I
= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= Y ) )
& ( ( I
!= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= ( nth_nat @ Zs @ ( minus_minus_nat @ I @ ( suc @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ) ) ) ).
% nth_append_Cons
thf(fact_1082_map__upt__eqI,axiom,
! [Xs: list_list_nat,N: nat,M2: nat,F: nat > list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( minus_minus_nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) )
=> ( ( map_nat_list_nat @ F @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_1083_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M2: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_1084_partition__by__nth__greater,axiom,
! [I: nat,K: nat,Zs: list_nat,J: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_nat @ I @ K )
=> ( ( ord_less_nat @ K @ ( size_size_list_nat @ Zs ) )
=> ( ( ord_less_nat @ J @ ( nth_nat @ Zs @ I ) )
=> ( ( ( size_size_list_nat @ Xs )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ I @ Zs ) ) @ J ) )
=> ( ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ Zs ) @ K )
= ( take_nat @ ( nth_nat @ Zs @ K ) @ ( drop_nat @ ( minus_minus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ K @ Zs ) ) @ one_one_nat ) @ ( append_nat @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% partition_by_nth_greater
thf(fact_1085_ghole__num__at__pos_Oelims,axiom,
! [X: list_nat,Xa: ground_gmctxt_a,Y: nat] :
( ( ( ground5706155212172713090_pos_a @ X @ Xa )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y
!= ( ground_num_gholes_a @ Xa ) ) )
=> ( ! [I3: nat,Q2: list_nat] :
( ( X
= ( cons_nat @ I3 @ Q2 ) )
=> ! [F3: a,Cs2: list_Ground_gmctxt_a] :
( ( Xa
= ( ground_GMFun_a @ F3 @ Cs2 ) )
=> ( Y
!= ( ground5706155212172713090_pos_a @ Q2 @ ( nth_Ground_gmctxt_a @ Cs2 @ I3 ) ) ) ) )
=> ~ ( ? [V2: nat,Va2: list_nat] :
( X
= ( cons_nat @ V2 @ Va2 ) )
=> ( ( Xa = ground_GMHole_a )
=> ( Y != undefined_nat ) ) ) ) ) ) ).
% ghole_num_at_pos.elims
thf(fact_1086_Diff__empty,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ bot_bot_set_list_nat )
= A ) ).
% Diff_empty
thf(fact_1087_empty__Diff,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ bot_bot_set_list_nat @ A )
= bot_bot_set_list_nat ) ).
% empty_Diff
thf(fact_1088_Diff__cancel,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ A )
= bot_bot_set_list_nat ) ).
% Diff_cancel
thf(fact_1089_Diff__insert0,axiom,
! [X: ground_gmctxt_a,A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ X @ A )
=> ( ( minus_4616606335744337325ctxt_a @ A @ ( insert2292343091172363342ctxt_a @ X @ B3 ) )
= ( minus_4616606335744337325ctxt_a @ A @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1090_insert__Diff1,axiom,
! [X: ground_gmctxt_a,B3: set_Ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ B3 )
=> ( ( minus_4616606335744337325ctxt_a @ ( insert2292343091172363342ctxt_a @ X @ A ) @ B3 )
= ( minus_4616606335744337325ctxt_a @ A @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1091_insert__Diff__single,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
= ( insert_list_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1092_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1093_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1094_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1095_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1096_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1097_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1098_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1099_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1100_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1101_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1102_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1103_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1104_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1105_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1106_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_1107_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_1108_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_1109_nth__Cons__pos,axiom,
! [N: nat,X: list_nat,Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= ( nth_list_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1110_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1111_Compl__insert,axiom,
! [X: list_nat,A: set_list_nat] :
( ( uminus3195874150345416415st_nat @ ( insert_list_nat @ X @ A ) )
= ( minus_7954133019191499631st_nat @ ( uminus3195874150345416415st_nat @ A ) @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ).
% Compl_insert
thf(fact_1112_Diff__insert,axiom,
! [A: set_list_nat,A2: list_nat,B3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ B3 ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ) ).
% Diff_insert
thf(fact_1113_insert__Diff,axiom,
! [A2: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A2 @ A )
=> ( ( insert2292343091172363342ctxt_a @ A2 @ ( minus_4616606335744337325ctxt_a @ A @ ( insert2292343091172363342ctxt_a @ A2 @ bot_bo3757104637556896616ctxt_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1114_insert__Diff,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ( ( insert_list_nat @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1115_Diff__insert2,axiom,
! [A: set_list_nat,A2: list_nat,B3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ B3 ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1116_Diff__insert__absorb,axiom,
! [X: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ X @ A )
=> ( ( minus_4616606335744337325ctxt_a @ ( insert2292343091172363342ctxt_a @ X @ A ) @ ( insert2292343091172363342ctxt_a @ X @ bot_bo3757104637556896616ctxt_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1117_Diff__insert__absorb,axiom,
! [X: list_nat,A: set_list_nat] :
( ~ ( member_list_nat @ X @ A )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A ) @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1118_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1119_psubset__imp__ex__mem,axiom,
! [A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ( ord_le4651699007867869992ctxt_a @ A @ B3 )
=> ? [B4: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ B4 @ ( minus_4616606335744337325ctxt_a @ B3 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1120_insert__Diff__if,axiom,
! [X: ground_gmctxt_a,B3: set_Ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( ( member3306386364193653941ctxt_a @ X @ B3 )
=> ( ( minus_4616606335744337325ctxt_a @ ( insert2292343091172363342ctxt_a @ X @ A ) @ B3 )
= ( minus_4616606335744337325ctxt_a @ A @ B3 ) ) )
& ( ~ ( member3306386364193653941ctxt_a @ X @ B3 )
=> ( ( minus_4616606335744337325ctxt_a @ ( insert2292343091172363342ctxt_a @ X @ A ) @ B3 )
= ( insert2292343091172363342ctxt_a @ X @ ( minus_4616606335744337325ctxt_a @ A @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1121_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1122_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1123_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1124_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1125_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1126_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1127_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1128_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1129_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1130_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1131_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1132_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1133_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1134_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1135_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1136_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1137_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1138_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_1139_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_1140_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_1141_count__list_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( count_list_nat @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_1142_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1143_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_1144_nth__Cons_H,axiom,
! [N: nat,X: list_nat,Xs: list_list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= ( nth_list_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1145_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1146_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1147_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1148_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1149_ghole__num__at__pos_Osimps_I1_J,axiom,
! [C2: ground_gmctxt_a] :
( ( ground5706155212172713090_pos_a @ nil_nat @ C2 )
= ( ground_num_gholes_a @ C2 ) ) ).
% ghole_num_at_pos.simps(1)
thf(fact_1150_length__remove1,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
( ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( size_s962548663205356226ctxt_a @ ( remove4701464168398194562ctxt_a @ X @ Xs ) )
= ( minus_minus_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( size_s962548663205356226ctxt_a @ ( remove4701464168398194562ctxt_a @ X @ Xs ) )
= ( size_s962548663205356226ctxt_a @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1151_length__remove1,axiom,
! [X: nat,Xs: list_nat] :
( ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1152_remove__nth__length,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_1153_nth__non__equal__first__eq,axiom,
! [X: list_nat,Y: list_nat,Xs: list_list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_list_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1154_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1155_take__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1156_Cons__replicate__eq,axiom,
! [X: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1157_comm__append__is__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [N3: nat,Zs3: list_nat] :
( ( ord_less_nat @ one_one_nat @ N3 )
& ( ( concat_nat @ ( replicate_list_nat @ N3 @ Zs3 ) )
= ( append_nat @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_1158_ghole__num__at__pos_Osimps_I2_J,axiom,
! [I: nat,Q3: list_nat,F: a,Cs: list_Ground_gmctxt_a] :
( ( ground5706155212172713090_pos_a @ ( cons_nat @ I @ Q3 ) @ ( ground_GMFun_a @ F @ Cs ) )
= ( ground5706155212172713090_pos_a @ Q3 @ ( nth_Ground_gmctxt_a @ Cs @ I ) ) ) ).
% ghole_num_at_pos.simps(2)
thf(fact_1159_list__of__permutation__element__n_Osimps_I1_J,axiom,
! [X: nat,L4: list_nat] :
( ( basic_7079635023375748421_n_nat @ X @ zero_zero_nat @ L4 )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% list_of_permutation_element_n.simps(1)
thf(fact_1160_DiffI,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ C3 @ A )
=> ( ~ ( member3306386364193653941ctxt_a @ C3 @ B3 )
=> ( member3306386364193653941ctxt_a @ C3 @ ( minus_4616606335744337325ctxt_a @ A @ B3 ) ) ) ) ).
% DiffI
thf(fact_1161_Diff__iff,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ C3 @ ( minus_4616606335744337325ctxt_a @ A @ B3 ) )
= ( ( member3306386364193653941ctxt_a @ C3 @ A )
& ~ ( member3306386364193653941ctxt_a @ C3 @ B3 ) ) ) ).
% Diff_iff
thf(fact_1162_ComplI,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ~ ( member3306386364193653941ctxt_a @ C3 @ A )
=> ( member3306386364193653941ctxt_a @ C3 @ ( uminus6214660439178182941ctxt_a @ A ) ) ) ).
% ComplI
thf(fact_1163_Compl__iff,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ C3 @ ( uminus6214660439178182941ctxt_a @ A ) )
= ( ~ ( member3306386364193653941ctxt_a @ C3 @ A ) ) ) ).
% Compl_iff
thf(fact_1164_DiffE,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ C3 @ ( minus_4616606335744337325ctxt_a @ A @ B3 ) )
=> ~ ( ( member3306386364193653941ctxt_a @ C3 @ A )
=> ( member3306386364193653941ctxt_a @ C3 @ B3 ) ) ) ).
% DiffE
thf(fact_1165_DiffD1,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ C3 @ ( minus_4616606335744337325ctxt_a @ A @ B3 ) )
=> ( member3306386364193653941ctxt_a @ C3 @ A ) ) ).
% DiffD1
thf(fact_1166_DiffD2,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ C3 @ ( minus_4616606335744337325ctxt_a @ A @ B3 ) )
=> ~ ( member3306386364193653941ctxt_a @ C3 @ B3 ) ) ).
% DiffD2
thf(fact_1167_ComplD,axiom,
! [C3: ground_gmctxt_a,A: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ C3 @ ( uminus6214660439178182941ctxt_a @ A ) )
=> ~ ( member3306386364193653941ctxt_a @ C3 @ A ) ) ).
% ComplD
thf(fact_1168_list__of__permutation__element__n_Oelims,axiom,
! [X: nat,Xa: nat,Xb: list_nat,Y: list_list_nat] :
( ( ( basic_7079635023375748421_n_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y
!= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
=> ~ ! [N3: nat] :
( ( Xa
= ( suc @ N3 ) )
=> ( Y
!= ( concat_list_nat @ ( map_li960784813134754710st_nat @ ( basic_4874698711677410535ts_nat @ X ) @ ( n_lists_nat @ N3 @ Xb ) ) ) ) ) ) ) ).
% list_of_permutation_element_n.elims
thf(fact_1169_set__list__diff,axiom,
! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
( ( set_Ground_gmctxt_a2 @ ( missin6797617701560608538ctxt_a @ Xs @ Ys ) )
= ( minus_4616606335744337325ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ ( set_Ground_gmctxt_a2 @ Ys ) ) ) ).
% set_list_diff
thf(fact_1170_length__add__elem__list__lists,axiom,
! [Ys: list_nat,X: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_1171_add__elem__list__lists_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Ys: list_nat] :
( ( basic_4874698711677410535ts_nat @ X @ ( cons_nat @ Y @ Ys ) )
= ( cons_list_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Ys ) ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ Y ) @ ( basic_4874698711677410535ts_nat @ X @ Ys ) ) ) ) ).
% add_elem_list_lists.simps(2)
thf(fact_1172_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1173_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I: nat,Xs: list_list_nat,Y: list_nat] :
( ( ord_less_nat @ J @ I )
=> ( ( ord_less_eq_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ ( take_list_nat @ J @ Xs ) @ ( cons_list_nat @ Y @ ( drop_list_nat @ ( suc @ J ) @ Xs ) ) ) @ I )
= ( nth_list_nat @ Xs @ I ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_1174_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs ) @ ( cons_nat @ Y @ ( drop_nat @ ( suc @ J ) @ Xs ) ) ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_1175_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1176_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1177_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1178_add__le__cancel__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_1179_add__le__cancel__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_1180_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_1181_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1182_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1183_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1184_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1185_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1186_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1187_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1188_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1189_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1190_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1191_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1192_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1193_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1194_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1195_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1196_diff__ge__0__iff__ge,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1197_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_1198_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_1199_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_1200_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_1201_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_1202_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_1203_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1204_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1205_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1206_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1207_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1208_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1209_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_1210_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1211_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1212_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1213_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1214_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1215_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1216_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1217_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_1218_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1219_take__upt,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
=> ( ( take_nat @ M2 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).
% take_upt
thf(fact_1220_nth__drop,axiom,
! [N: nat,Xs: list_list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( drop_list_nat @ N @ Xs ) @ I )
= ( nth_list_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_1221_nth__drop,axiom,
! [N: nat,Xs: list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_1222_count__list__map__ge,axiom,
! [Xs: list_nat,X: nat,F: nat > nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( count_list_nat @ ( map_nat_nat @ F @ Xs ) @ ( F @ X ) ) ) ).
% count_list_map_ge
thf(fact_1223_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1224_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1225_count__le__length,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_1226_remove__nth__id,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( missin7175274867594579095th_nat @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_1227_size__list__pointwise,axiom,
! [Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,G: ground_gmctxt_a > nat] :
( ! [X9: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X9 @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X9 ) @ ( G @ X9 ) ) )
=> ( ord_less_eq_nat @ ( size_l1580428505752620566ctxt_a @ F @ Xs ) @ ( size_l1580428505752620566ctxt_a @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_1228_size__list__estimation_H,axiom,
! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Y: nat,F: ground_gmctxt_a > nat] :
( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
=> ( ord_less_eq_nat @ Y @ ( size_l1580428505752620566ctxt_a @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_1229_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1230_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1231_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1232_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1233_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1234_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1235_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1236_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1237_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1238_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1239_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1240_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1241_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1242_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1243_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1244_add__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1245_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1246_add__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1247_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1248_add__increasing2,axiom,
! [C3: int,B: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C3 )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C3 ) ) ) ) ).
% add_increasing2
thf(fact_1249_add__increasing2,axiom,
! [C3: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C3 )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% add_increasing2
thf(fact_1250_add__decreasing2,axiom,
! [C3: int,A2: int,B: int] :
( ( ord_less_eq_int @ C3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C3 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1251_add__decreasing2,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C3 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1252_add__increasing,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C3 ) ) ) ) ).
% add_increasing
thf(fact_1253_add__increasing,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% add_increasing
thf(fact_1254_add__decreasing,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C3 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C3 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1255_add__decreasing,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C3 ) @ B ) ) ) ).
% add_decreasing
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
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,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ground_num_gholes_a @ ( ground_GMFun_a @ f @ ds ) )
= zero_zero_nat ) ).
%------------------------------------------------------------------------------