TPTP Problem File: SLH0578^1.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/0069_Context_Extensions/prob_00656_029531__18643902_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1571 ( 456 unt; 308 typ; 0 def)
% Number of atoms : 3676 (1059 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 13041 ( 362 ~; 55 |; 222 &;10591 @)
% ( 0 <=>;1811 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 7 avg)
% Number of types : 52 ( 51 usr)
% Number of type conns : 1103 (1103 >; 0 *; 0 +; 0 <<)
% Number of symbols : 260 ( 257 usr; 18 con; 0-4 aty)
% Number of variables : 4177 ( 399 ^;3640 !; 138 ?;4177 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:01:56.167
%------------------------------------------------------------------------------
% Could-be-implicit typings (51)
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
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thf(ty_n_t__Ground____Terms__Ogterm_Itf__a_J,type,
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thf(ty_n_t__Ground____Ctxt__Ogctxt_Itf__a_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (257)
thf(sy_c_Abstract__Rewriting_Orestrict_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
abstra1138918260165758998ctxt_a: set_Pr8705490787052945405ctxt_a > set_Ground_gmctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Abstract__Rewriting_Orestrict_001t__Ground____Terms__Ogterm_Itf__a_J,type,
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thf(sy_c_Abstract__Rewriting_Orestrict_001t__List__Olist_It__Nat__Onat_J,type,
abstra4205430435060868312st_nat: set_Pr3451248702717554689st_nat > set_list_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Abstract__Rewriting_Orestrict_001t__Nat__Onat,type,
abstra1219450721576609224ct_nat: set_Pr1261947904930325089at_nat > set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Abstract__Rewriting_Orestrict_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
abstra4135781082980823205ctxt_a: set_Pr490164551564945991ctxt_a > set_Pr8705490787052945405ctxt_a > set_Pr490164551564945991ctxt_a ).
thf(sy_c_Abstract__Rewriting_Orestrict_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
abstra6312477298767733231term_a: set_Pr5156918327056800967term_a > set_Pr8282532413760598727term_a > set_Pr5156918327056800967term_a ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
condit5323614196651773062ctxt_a: ( ground_gmctxt_a > ground_gmctxt_a > $o ) > set_Ground_gmctxt_a > $o ).
thf(sy_c_Context__Extensions_Ogmctxtex__onp_001tf__a,type,
contex3086340098647126761_onp_a: ( ground_gmctxt_a > $o ) > set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a ).
thf(sy_c_Euclidean__Division_Odivmod__nat,type,
euclidean_divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Ground__Closure_Otrancl__list_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
ground4295926921573292569ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8687807964536448157ctxt_a ).
thf(sy_c_Ground__Closure_Otrancl__list_001t__Ground____Terms__Ogterm_Itf__a_J,type,
ground4492235380174378228term_a: set_Pr8282532413760598727term_a > set_Pr2711430308268100935term_a ).
thf(sy_c_Ground__Closure_Otrancl__list_001t__Nat__Onat,type,
ground4984107042736062923st_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Ground__MCtxt_Ocomp__gmctxt_001tf__a,type,
ground_comp_gmctxt_a: set_Pr8705490787052945405ctxt_a ).
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_Oghole__poss__list_001tf__a,type,
ground1728035901660239936list_a: ground_gmctxt_a > list_list_nat ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMHole_001tf__a,type,
ground_GMHole_a: ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt__closing_001tf__a,type,
ground6954545657870034594sing_a: ground_gmctxt_a > ground_gmctxt_a > $o ).
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_Ogmctxt__subtgm__at__fill__args_001tf__a_001t__Ground____Terms__Ogterm_Itf__a_J,type,
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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_001tf__a,type,
ground_num_gholes_a: ground_gmctxt_a > nat ).
thf(sy_c_Ground__MCtxt_Oposs__gmctxt_001tf__a,type,
ground_poss_gmctxt_a: ground_gmctxt_a > set_list_nat ).
thf(sy_c_Ground__MCtxt_Osubgm__at_001tf__a,type,
ground_subgm_at_a: ground_gmctxt_a > list_nat > ground_gmctxt_a ).
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__MCtxt_Ounfill__gholes_001tf__a,type,
ground8214737007138946830oles_a: ground_gmctxt_a > ground_gterm_a > list_Ground_gterm_a ).
thf(sy_c_Ground__Terms_Ogsubt__at_001tf__a,type,
ground_gsubt_at_a: ground_gterm_a > list_nat > ground_gterm_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_M_Eo_J,type,
minus_8440876803814310393xt_a_o: ( produc8175205341597284509ctxt_a > $o ) > ( produc8175205341597284509ctxt_a > $o ) > produc8175205341597284509ctxt_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_M_Eo_J,type,
minus_7967641318176996911rm_a_o: ( produc1525979690750518759term_a > $o ) > ( produc1525979690750518759term_a > $o ) > produc1525979690750518759term_a > $o ).
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__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
minus_551215046067487790term_a: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
relcom896424912281157969at_nat: set_Pr6370437063884598352at_nat > set_Pr7682762132356531903at_nat > set_Pr6370437063884598352at_nat ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
relcom6405311860805859042at_nat: set_Pr6370437063884598352at_nat > set_Pr9093778441882193744at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
relcomp_nat_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Seq_Oinf__concat__simple,type,
inf_concat_simple: ( nat > nat ) > nat > product_prod_nat_nat ).
thf(sy_c_Seq_Oinf__concat__simple__rel,type,
inf_co2821209343098213861le_rel: produc8199716216217303280at_nat > produc8199716216217303280at_nat > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
collec4295779454384174240term_a: ( list_Ground_gterm_a > $o ) > set_li8588095516285831957term_a ).
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__List__Olist_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
collec7935380194126400472term_a: ( list_P2433084232546440685term_a > $o ) > set_li5439729168002949453term_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
collec5218855173738699400ctxt_a: ( produc8175205341597284509ctxt_a > $o ) > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
collec6713184107424615762term_a: ( produc1525979690750518759term_a > $o ) > set_Pr8282532413760598727term_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
collec1570431334306492044st_nat: ( produc1828647624359046049st_nat > $o ) > set_Pr3451248702717554689st_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
collec3796792608423614674ctxt_a: ( produc5976605159854024551ctxt_a > $o ) > set_Pr490164551564945991ctxt_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_Mt__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
collec7632642491840594258term_a: ( produc789695711980045543term_a > $o ) > set_Pr5156918327056800967term_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
transi3225167780237802813ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Ground____Terms__Ogterm_Itf__a_J,type,
transi8854406246868658384term_a: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
transi3146596849628481425at_nat: set_Pr7047708467701517959at_nat > set_Pr7047708467701517959at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
transi6952278968286904510ctxt_a: set_Pr490164551564945991ctxt_a > set_Pr490164551564945991ctxt_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
transi4994575220883992200term_a: set_Pr5156918327056800967term_a > set_Pr5156918327056800967term_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
transi243908449541399842at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Utils_Oconcat__index__split_001t__Ground____Terms__Ogterm_Itf__a_J,type,
concat4415380251885098797term_a: product_prod_nat_nat > list_l4678906206682822203term_a > product_prod_nat_nat ).
thf(sy_c_Utils_Oconcat__index__split_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
concat2033898366915437183at_nat: product_prod_nat_nat > list_l3264859301627795341at_nat > product_prod_nat_nat ).
thf(sy_c_Utils_Otrancl__list_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
trancl6028151697025156246ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8687807964536448157ctxt_a ).
thf(sy_c_Utils_Otrancl__list_001t__Ground____Terms__Ogterm_Itf__a_J,type,
trancl1902929335784340279term_a: set_Pr8282532413760598727term_a > set_Pr2711430308268100935term_a ).
thf(sy_c_Utils_Otrancl__list_001t__Nat__Onat,type,
trancl_list_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
accp_P6412015083367660281at_nat: ( produc8199716216217303280at_nat > produc8199716216217303280at_nat > $o ) > produc8199716216217303280at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $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__Ground____Terms__Ogterm_Itf__a_J,type,
member5261315023172567896term_a: ground_gterm_a > set_Ground_gterm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
member5011579089780872286term_a: list_Ground_gterm_a > set_li8588095516285831957term_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_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1096249278104964872at_nat: produc1932156733058919263at_nat > set_Pr7682762132356531903at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
member461722496487281478ctxt_a: produc8175205341597284509ctxt_a > set_Pr8705490787052945405ctxt_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
member4351891425000572176term_a: produc1525979690750518759term_a > set_Pr8282532413760598727term_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
member6597889099603620838ctxt_a: produc1017277696136915517ctxt_a > set_Pr8687807964536448157ctxt_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_Mt__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
member3518026002450860944term_a: produc6008132053207661927term_a > set_Pr2711430308268100935term_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8336108448496249625at_nat: produc85711943791777264at_nat > set_Pr6370437063884598352at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
member6622040750283622891term_a: produc7109479242990851284term_a > set_Pr1502840878978300170term_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member3795392313164512464at_nat: produc492020038885415591at_nat > set_Pr7047708467701517959at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
member4022424729989878928ctxt_a: produc5976605159854024551ctxt_a > set_Pr490164551564945991ctxt_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_Mt__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
member4810723946494196496term_a: produc789695711980045543term_a > set_Pr5156918327056800967term_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_v_C____,type,
c: ground_gmctxt_a ).
thf(sy_v_D____,type,
d: ground_gmctxt_a ).
thf(sy_v_P,type,
p: ground_gmctxt_a > $o ).
thf(sy_v_Q,type,
q: ground_gmctxt_a > $o ).
thf(sy_v__092_060R_062,type,
r: set_Pr8282532413760598727term_a ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_s____,type,
s: ground_gterm_a ).
thf(sy_v_t____,type,
t: ground_gterm_a ).
thf(sy_v_ta____,type,
ta: ground_gterm_a ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_u____,type,
u: ground_gterm_a ).
thf(sy_v_us____,type,
us: list_Ground_gterm_a ).
thf(sy_v_vs____,type,
vs: list_Ground_gterm_a ).
thf(sy_v_xs____,type,
xs: list_Ground_gterm_a ).
thf(sy_v_ys____,type,
ys: list_Ground_gterm_a ).
% Relevant facts (1256)
thf(fact_0__C_K_C_I1_J,axiom,
( s
= ( ground_fill_gholes_a @ c @ us ) ) ).
% "*"(1)
thf(fact_1__C_K_K_C_I2_J,axiom,
( u
= ( ground_fill_gholes_a @ d @ ys ) ) ).
% "**"(2)
thf(fact_2_inv_I1_J,axiom,
p @ c ).
% inv(1)
thf(fact_3_inv_H_I1_J,axiom,
p @ d ).
% inv'(1)
thf(fact_4_i,axiom,
ord_less_nat @ i @ ( size_s1893836384466666913term_a @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ u ) ) ).
% i
thf(fact_5_fill_I1_J,axiom,
( ( ground_fill_gholes_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ s ) )
= s ) ).
% fill(1)
thf(fact_6_fill_I2_J,axiom,
( ( ground_fill_gholes_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ u ) )
= u ) ).
% fill(2)
thf(fact_7_l_H_H_I2_J,axiom,
( ( size_s1893836384466666913term_a @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ s ) )
= ( size_s1893836384466666913term_a @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ u ) ) ) ).
% l''(2)
thf(fact_8__C_K_C_I2_J,axiom,
( ta
= ( ground_fill_gholes_a @ c @ vs ) ) ).
% "*"(2)
thf(fact_9__C_K_K_C_I1_J,axiom,
( ta
= ( ground_fill_gholes_a @ d @ xs ) ) ).
% "**"(1)
thf(fact_10_P,axiom,
p @ ( inf_in8301665843363081392ctxt_a @ c @ d ) ).
% P
thf(fact_11_l_H_H_I1_J,axiom,
( ( ground_num_gholes_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) )
= ( size_s1893836384466666913term_a @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ s ) ) ) ).
% l''(1)
thf(fact_12_inf__gmctxt__idem,axiom,
! [C: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ C @ C )
= C ) ).
% inf_gmctxt_idem
thf(fact_13_inf_Oidem,axiom,
! [A: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_14_inf__idem,axiom,
! [X: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_15_inf_Oleft__idem,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ A @ ( inf_in8301665843363081392ctxt_a @ A @ B ) )
= ( inf_in8301665843363081392ctxt_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_16_inf__left__idem,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ X @ ( inf_in8301665843363081392ctxt_a @ X @ Y ) )
= ( inf_in8301665843363081392ctxt_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_17_inf_Oright__idem,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ B )
= ( inf_in8301665843363081392ctxt_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_18_inf__right__idem,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ ( inf_in8301665843363081392ctxt_a @ X @ Y ) @ Y )
= ( inf_in8301665843363081392ctxt_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_19_l_H_I2_J,axiom,
( ( size_s1893836384466666913term_a @ xs )
= ( size_s1893836384466666913term_a @ ys ) ) ).
% l'(2)
thf(fact_20_l_I2_J,axiom,
( ( size_s1893836384466666913term_a @ us )
= ( size_s1893836384466666913term_a @ vs ) ) ).
% l(2)
thf(fact_21_l_H_I1_J,axiom,
( ( ground_num_gholes_a @ d )
= ( size_s1893836384466666913term_a @ xs ) ) ).
% l'(1)
thf(fact_22_l_I1_J,axiom,
( ( ground_num_gholes_a @ c )
= ( size_s1893836384466666913term_a @ us ) ) ).
% l(1)
thf(fact_23_inf_Ostrict__coboundedI2,axiom,
! [B: ground_gmctxt_a,C2: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ B @ C2 )
=> ( ord_le9048903087952742514ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_24_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_25_inf_Ostrict__coboundedI1,axiom,
! [A: ground_gmctxt_a,C2: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ C2 )
=> ( ord_le9048903087952742514ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_26_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_27_inf_Ostrict__order__iff,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( A2
= ( inf_in8301665843363081392ctxt_a @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_28_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_29_inf_Ostrict__boundedE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ ( inf_in8301665843363081392ctxt_a @ B @ C2 ) )
=> ~ ( ( ord_le9048903087952742514ctxt_a @ A @ B )
=> ~ ( ord_le9048903087952742514ctxt_a @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_30_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_31_inf_Oabsorb4,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ B @ A )
=> ( ( inf_in8301665843363081392ctxt_a @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_32_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_33_inf_Oabsorb3,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ B )
=> ( ( inf_in8301665843363081392ctxt_a @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_34_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_35_less__infI2,axiom,
! [B: ground_gmctxt_a,X: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ B @ X )
=> ( ord_le9048903087952742514ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_36_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_37_less__infI1,axiom,
! [A: ground_gmctxt_a,X: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ X )
=> ( ord_le9048903087952742514ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_38_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_39_inf__left__commute,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ X @ ( inf_in8301665843363081392ctxt_a @ Y @ Z ) )
= ( inf_in8301665843363081392ctxt_a @ Y @ ( inf_in8301665843363081392ctxt_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_40_inf_Oleft__commute,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ B @ ( inf_in8301665843363081392ctxt_a @ A @ C2 ) )
= ( inf_in8301665843363081392ctxt_a @ A @ ( inf_in8301665843363081392ctxt_a @ B @ C2 ) ) ) ).
% inf.left_commute
thf(fact_41_inf__commute,axiom,
( inf_in8301665843363081392ctxt_a
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( inf_in8301665843363081392ctxt_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_42_inf_Ocommute,axiom,
( inf_in8301665843363081392ctxt_a
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] : ( inf_in8301665843363081392ctxt_a @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_43_inf__assoc,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ ( inf_in8301665843363081392ctxt_a @ X @ Y ) @ Z )
= ( inf_in8301665843363081392ctxt_a @ X @ ( inf_in8301665843363081392ctxt_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_44_inf_Oassoc,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ C2 )
= ( inf_in8301665843363081392ctxt_a @ A @ ( inf_in8301665843363081392ctxt_a @ B @ C2 ) ) ) ).
% inf.assoc
thf(fact_45_inf__gmctxt__assoc,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a,E: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ ( inf_in8301665843363081392ctxt_a @ C @ D ) @ E )
= ( inf_in8301665843363081392ctxt_a @ C @ ( inf_in8301665843363081392ctxt_a @ D @ E ) ) ) ).
% inf_gmctxt_assoc
thf(fact_46_inf__gmctxt__comm,axiom,
( inf_in8301665843363081392ctxt_a
= ( ^ [C3: ground_gmctxt_a,D2: ground_gmctxt_a] : ( inf_in8301665843363081392ctxt_a @ D2 @ C3 ) ) ) ).
% inf_gmctxt_comm
thf(fact_47_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: produc1525979690750518759term_a,P: produc1525979690750518759term_a > $o] :
( ( member4351891425000572176term_a @ A @ ( collec6713184107424615762term_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: produc8175205341597284509ctxt_a,P: produc8175205341597284509ctxt_a > $o] :
( ( member461722496487281478ctxt_a @ A @ ( collec5218855173738699400ctxt_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A3: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A3: set_Pr8282532413760598727term_a] :
( ( collec6713184107424615762term_a
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A3: set_Pr8705490787052945405ctxt_a] :
( ( collec5218855173738699400ctxt_a
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_53_inv_H_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ ys ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ xs @ I ) @ ( nth_Ground_gterm_a @ ys @ I ) ) @ r ) ) ).
% inv'(2)
thf(fact_54_inv_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ vs ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ us @ I ) @ ( nth_Ground_gterm_a @ vs @ I ) ) @ r ) ) ).
% inv(2)
thf(fact_55_Ex__list__of__length__P,axiom,
! [N: nat,P: ground_gterm_a > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X3: ground_gterm_a] : ( P @ X3 @ I2 ) )
=> ? [Xs: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Xs )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( nth_Ground_gterm_a @ Xs @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_56_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_Ground_gterm_a,Z2: list_Ground_gterm_a] : ( Y3 = Z2 ) )
= ( ^ [Xs2: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Xs2 )
= ( size_s1893836384466666913term_a @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs2 ) )
=> ( ( nth_Ground_gterm_a @ Xs2 @ I3 )
= ( nth_Ground_gterm_a @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_57_Skolem__list__nth,axiom,
! [K: nat,P: nat > ground_gterm_a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: ground_gterm_a] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs2: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_Ground_gterm_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_58_nth__equalityE,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a] :
( ( Xs3 = Ys2 )
=> ~ ( ( ( size_s1893836384466666913term_a @ Xs3 )
= ( size_s1893836384466666913term_a @ Ys2 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( nth_Ground_gterm_a @ Xs3 @ I )
= ( nth_Ground_gterm_a @ Ys2 @ I ) ) ) ) ) ).
% nth_equalityE
thf(fact_59_nth__equalityI,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Xs3 )
= ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( nth_Ground_gterm_a @ Xs3 @ I2 )
= ( nth_Ground_gterm_a @ Ys2 @ I2 ) ) )
=> ( Xs3 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_60__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062D_Axs_Ays_O_A_092_060lbrakk_062t_A_061_Afill__gholes_AD_Axs_059_Au_A_061_Afill__gholes_AD_Ays_059_Anum__gholes_AD_A_061_Alength_Axs_059_Alength_Axs_A_061_Alength_Ays_059_AP_AD_059_A_092_060forall_062i_060length_Ays_O_A_Ixs_A_B_Ai_M_Ays_A_B_Ai_J_A_092_060in_062_A_092_060R_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D3: ground_gmctxt_a,Xs: list_Ground_gterm_a] :
( ( ta
= ( ground_fill_gholes_a @ D3 @ Xs ) )
=> ! [Ys3: list_Ground_gterm_a] :
( ( u
= ( ground_fill_gholes_a @ D3 @ Ys3 ) )
=> ( ( ( ground_num_gholes_a @ D3 )
= ( size_s1893836384466666913term_a @ Xs ) )
=> ( ( ( size_s1893836384466666913term_a @ Xs )
= ( size_s1893836384466666913term_a @ Ys3 ) )
=> ( ( p @ D3 )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys3 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys3 @ I ) ) @ r ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>D xs ys. \<lbrakk>t = fill_gholes D xs; u = fill_gholes D ys; num_gholes D = length xs; length xs = length ys; P D; \<forall>i<length ys. (xs ! i, ys ! i) \<in> \<R>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_61__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062C_Aus_Avs_O_A_092_060lbrakk_062s_A_061_Afill__gholes_AC_Aus_059_At_A_061_Afill__gholes_AC_Avs_059_Anum__gholes_AC_A_061_Alength_Aus_059_Alength_Aus_A_061_Alength_Avs_059_AP_AC_059_A_092_060forall_062i_060length_Avs_O_A_Ius_A_B_Ai_M_Avs_A_B_Ai_J_A_092_060in_062_A_092_060R_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [C4: ground_gmctxt_a,Us: list_Ground_gterm_a] :
( ( s
= ( ground_fill_gholes_a @ C4 @ Us ) )
=> ! [Vs: list_Ground_gterm_a] :
( ( ta
= ( ground_fill_gholes_a @ C4 @ Vs ) )
=> ( ( ( ground_num_gholes_a @ C4 )
= ( size_s1893836384466666913term_a @ Us ) )
=> ( ( ( size_s1893836384466666913term_a @ Us )
= ( size_s1893836384466666913term_a @ Vs ) )
=> ( ( p @ C4 )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Vs ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Us @ I ) @ ( nth_Ground_gterm_a @ Vs @ I ) ) @ r ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>C us vs. \<lbrakk>s = fill_gholes C us; t = fill_gholes C vs; num_gholes C = length us; length us = length vs; P C; \<forall>i<length vs. (us ! i, vs ! i) \<in> \<R>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_62_less_I2_J,axiom,
ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ ( ground582559344758063963term_a @ u ) ).
% less(2)
thf(fact_63_less_I1_J,axiom,
ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ ( ground582559344758063963term_a @ s ) ).
% less(1)
thf(fact_64_le__inf__iff,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ ( inf_in8301665843363081392ctxt_a @ Y @ Z ) )
= ( ( ord_le1940079914304802430ctxt_a @ X @ Y )
& ( ord_le1940079914304802430ctxt_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_65_le__inf__iff,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a,Z: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ ( inf_in4315656883606118581term_a @ Y @ Z ) )
= ( ( ord_le7556488698670074471term_a @ X @ Y )
& ( ord_le7556488698670074471term_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_66_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_67_inf_Obounded__iff,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ ( inf_in8301665843363081392ctxt_a @ B @ C2 ) )
= ( ( ord_le1940079914304802430ctxt_a @ A @ B )
& ( ord_le1940079914304802430ctxt_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_68_inf_Obounded__iff,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ ( inf_in4315656883606118581term_a @ B @ C2 ) )
= ( ( ord_le7556488698670074471term_a @ A @ B )
& ( ord_le7556488698670074471term_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_69_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_70_length__unfill__gholes,axiom,
! [C: ground_gmctxt_a,T: ground_gterm_a] :
( ( ord_le1940079914304802430ctxt_a @ C @ ( ground582559344758063963term_a @ T ) )
=> ( ( size_s1893836384466666913term_a @ ( ground8214737007138946830oles_a @ C @ T ) )
= ( ground_num_gholes_a @ C ) ) ) ).
% length_unfill_gholes
thf(fact_71_step_Ohyps_I2_J,axiom,
member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ta @ u ) @ ( contex3086340098647126761_onp_a @ p @ r ) ).
% step.hyps(2)
thf(fact_72_less__gmctxt__def,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [C3: ground_gmctxt_a,D2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C3 @ D2 )
& ~ ( ord_le1940079914304802430ctxt_a @ D2 @ C3 ) ) ) ) ).
% less_gmctxt_def
thf(fact_73_inf__sup__ord_I2_J,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_74_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_75_inf__sup__ord_I1_J,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_76_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_77_inf__le1,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_78_inf__le1,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_79_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_80_inf__le2,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_81_inf__le2,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_82_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_83_le__infE,axiom,
! [X: ground_gmctxt_a,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ ( inf_in8301665843363081392ctxt_a @ A @ B ) )
=> ~ ( ( ord_le1940079914304802430ctxt_a @ X @ A )
=> ~ ( ord_le1940079914304802430ctxt_a @ X @ B ) ) ) ).
% le_infE
thf(fact_84_le__infE,axiom,
! [X: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ ( inf_in4315656883606118581term_a @ A @ B ) )
=> ~ ( ( ord_le7556488698670074471term_a @ X @ A )
=> ~ ( ord_le7556488698670074471term_a @ X @ B ) ) ) ).
% le_infE
thf(fact_85_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_86_le__infI,axiom,
! [X: ground_gmctxt_a,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ A )
=> ( ( ord_le1940079914304802430ctxt_a @ X @ B )
=> ( ord_le1940079914304802430ctxt_a @ X @ ( inf_in8301665843363081392ctxt_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_87_le__infI,axiom,
! [X: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ A )
=> ( ( ord_le7556488698670074471term_a @ X @ B )
=> ( ord_le7556488698670074471term_a @ X @ ( inf_in4315656883606118581term_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_88_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_89_inf__mono,axiom,
! [A: ground_gmctxt_a,C2: ground_gmctxt_a,B: ground_gmctxt_a,D4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ C2 )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ D4 )
=> ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ ( inf_in8301665843363081392ctxt_a @ C2 @ D4 ) ) ) ) ).
% inf_mono
thf(fact_90_inf__mono,axiom,
! [A: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,D4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ C2 )
=> ( ( ord_le7556488698670074471term_a @ B @ D4 )
=> ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A @ B ) @ ( inf_in4315656883606118581term_a @ C2 @ D4 ) ) ) ) ).
% inf_mono
thf(fact_91_inf__mono,axiom,
! [A: nat,C2: nat,B: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ D4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C2 @ D4 ) ) ) ) ).
% inf_mono
thf(fact_92_le__infI1,axiom,
! [A: ground_gmctxt_a,X: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ X )
=> ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_93_le__infI1,axiom,
! [A: set_Pr8282532413760598727term_a,X: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ X )
=> ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_94_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_95_le__infI2,axiom,
! [B: ground_gmctxt_a,X: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B @ X )
=> ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_96_le__infI2,axiom,
! [B: set_Pr8282532413760598727term_a,X: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B @ X )
=> ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_97_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_98_inf_OorderE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( A
= ( inf_in8301665843363081392ctxt_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_99_inf_OorderE,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( A
= ( inf_in4315656883606118581term_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_100_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_101_inf_OorderI,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( A
= ( inf_in8301665843363081392ctxt_a @ A @ B ) )
=> ( ord_le1940079914304802430ctxt_a @ A @ B ) ) ).
% inf.orderI
thf(fact_102_inf_OorderI,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( A
= ( inf_in4315656883606118581term_a @ A @ B ) )
=> ( ord_le7556488698670074471term_a @ A @ B ) ) ).
% inf.orderI
thf(fact_103_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_104_inf__unique,axiom,
! [F: ground_gmctxt_a > ground_gmctxt_a > ground_gmctxt_a,X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ( F @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ( F @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ( ord_le1940079914304802430ctxt_a @ X5 @ Z3 )
=> ( ord_le1940079914304802430ctxt_a @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in8301665843363081392ctxt_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_105_inf__unique,axiom,
! [F: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a,X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( F @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( F @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a,Z3: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ( ord_le7556488698670074471term_a @ X5 @ Z3 )
=> ( ord_le7556488698670074471term_a @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in4315656883606118581term_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_106_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_107_le__iff__inf,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_108_le__iff__inf,axiom,
( ord_le7556488698670074471term_a
= ( ^ [X2: set_Pr8282532413760598727term_a,Y2: set_Pr8282532413760598727term_a] :
( ( inf_in4315656883606118581term_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_109_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_110_inf_Oabsorb1,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( inf_in8301665843363081392ctxt_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_111_inf_Oabsorb1,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( inf_in4315656883606118581term_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_112_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_113_inf_Oabsorb2,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B @ A )
=> ( ( inf_in8301665843363081392ctxt_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_114_inf_Oabsorb2,axiom,
! [B: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B @ A )
=> ( ( inf_in4315656883606118581term_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_115_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_116_inf__absorb1,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ Y )
=> ( ( inf_in8301665843363081392ctxt_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_117_inf__absorb1,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ Y )
=> ( ( inf_in4315656883606118581term_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_118_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_119_inf__absorb2,axiom,
! [Y: ground_gmctxt_a,X: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ Y @ X )
=> ( ( inf_in8301665843363081392ctxt_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_120_inf__absorb2,axiom,
! [Y: set_Pr8282532413760598727term_a,X: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ Y @ X )
=> ( ( inf_in4315656883606118581term_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_121_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_122_inf_OboundedE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ ( inf_in8301665843363081392ctxt_a @ B @ C2 ) )
=> ~ ( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ~ ( ord_le1940079914304802430ctxt_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_123_inf_OboundedE,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ ( inf_in4315656883606118581term_a @ B @ C2 ) )
=> ~ ( ( ord_le7556488698670074471term_a @ A @ B )
=> ~ ( ord_le7556488698670074471term_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_124_inf_OboundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_125_inf_OboundedI,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ A @ C2 )
=> ( ord_le1940079914304802430ctxt_a @ A @ ( inf_in8301665843363081392ctxt_a @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_126_inf_OboundedI,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ A @ C2 )
=> ( ord_le7556488698670074471term_a @ A @ ( inf_in4315656883606118581term_a @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_127_inf_OboundedI,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_128_inf__greatest,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ Y )
=> ( ( ord_le1940079914304802430ctxt_a @ X @ Z )
=> ( ord_le1940079914304802430ctxt_a @ X @ ( inf_in8301665843363081392ctxt_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_129_inf__greatest,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a,Z: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ Y )
=> ( ( ord_le7556488698670074471term_a @ X @ Z )
=> ( ord_le7556488698670074471term_a @ X @ ( inf_in4315656883606118581term_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_130_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_131_inf_Oorder__iff,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( A2
= ( inf_in8301665843363081392ctxt_a @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_132_inf_Oorder__iff,axiom,
( ord_le7556488698670074471term_a
= ( ^ [A2: set_Pr8282532413760598727term_a,B2: set_Pr8282532413760598727term_a] :
( A2
= ( inf_in4315656883606118581term_a @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_133_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_134_inf_Ocobounded1,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_135_inf_Ocobounded1,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_136_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_137_inf_Ocobounded2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_138_inf_Ocobounded2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_139_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_140_inf_Oabsorb__iff1,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_141_inf_Oabsorb__iff1,axiom,
( ord_le7556488698670074471term_a
= ( ^ [A2: set_Pr8282532413760598727term_a,B2: set_Pr8282532413760598727term_a] :
( ( inf_in4315656883606118581term_a @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_142_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_143_inf_Oabsorb__iff2,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [B2: ground_gmctxt_a,A2: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_144_inf_Oabsorb__iff2,axiom,
( ord_le7556488698670074471term_a
= ( ^ [B2: set_Pr8282532413760598727term_a,A2: set_Pr8282532413760598727term_a] :
( ( inf_in4315656883606118581term_a @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_145_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_146_inf_OcoboundedI1,axiom,
! [A: ground_gmctxt_a,C2: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ C2 )
=> ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_147_inf_OcoboundedI1,axiom,
! [A: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ C2 )
=> ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_148_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_149_inf_OcoboundedI2,axiom,
! [B: ground_gmctxt_a,C2: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ord_le1940079914304802430ctxt_a @ ( inf_in8301665843363081392ctxt_a @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_150_inf_OcoboundedI2,axiom,
! [B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_151_inf_OcoboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_152_less__eq__gmctxt__def,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [C3: ground_gmctxt_a,D2: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ C3 @ D2 )
= C3 ) ) ) ).
% less_eq_gmctxt_def
thf(fact_153_fill__unfill__gholes,axiom,
! [C: ground_gmctxt_a,T: ground_gterm_a] :
( ( ord_le1940079914304802430ctxt_a @ C @ ( ground582559344758063963term_a @ T ) )
=> ( ( ground_fill_gholes_a @ C @ ( ground8214737007138946830oles_a @ C @ T ) )
= T ) ) ).
% fill_unfill_gholes
thf(fact_154_neq__if__length__neq,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Xs3 )
!= ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( Xs3 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_155_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_Ground_gterm_a] :
( ( size_s1893836384466666913term_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_156_length__induct,axiom,
! [P: list_Ground_gterm_a > $o,Xs3: list_Ground_gterm_a] :
( ! [Xs: list_Ground_gterm_a] :
( ! [Ys4: list_Ground_gterm_a] :
( ( ord_less_nat @ ( size_s1893836384466666913term_a @ Ys4 ) @ ( size_s1893836384466666913term_a @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs3 ) ) ).
% length_induct
thf(fact_157_step_Ohyps_I1_J,axiom,
member4351891425000572176term_a @ ( produc7419039783125785943term_a @ s @ ta ) @ ( transi8854406246868658384term_a @ ( contex3086340098647126761_onp_a @ p @ r ) ) ).
% step.hyps(1)
thf(fact_158_permut__sound,axiom,
! [I4: nat,As: list_Ground_gterm_a,F: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ As ) )
=> ( ( nth_Ground_gterm_a @ ( missin5607496018787941767term_a @ As @ F ) @ I4 )
= ( nth_Ground_gterm_a @ As @ ( F @ I4 ) ) ) ) ).
% permut_sound
thf(fact_159_gmctxtex__onpI,axiom,
! [P: ground_gmctxt_a > $o,C: ground_gmctxt_a,Us2: list_Ground_gterm_a,Vs2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
( ( P @ C )
=> ( ( ( ground_num_gholes_a @ C )
= ( size_s1893836384466666913term_a @ Us2 ) )
=> ( ( ( size_s1893836384466666913term_a @ Us2 )
= ( size_s1893836384466666913term_a @ Vs2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Vs2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Us2 @ I2 ) @ ( nth_Ground_gterm_a @ Vs2 @ I2 ) ) @ R ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_fill_gholes_a @ C @ Us2 ) @ ( ground_fill_gholes_a @ C @ Vs2 ) ) @ ( contex3086340098647126761_onp_a @ P @ R ) ) ) ) ) ) ).
% gmctxtex_onpI
thf(fact_160_old_Oprod_Oinject,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,A4: ground_gterm_a,B3: ground_gterm_a] :
( ( ( produc7419039783125785943term_a @ A @ B )
= ( produc7419039783125785943term_a @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_161_old_Oprod_Oinject,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,A4: ground_gmctxt_a,B3: ground_gmctxt_a] :
( ( ( produc6347872285926497493ctxt_a @ A @ B )
= ( produc6347872285926497493ctxt_a @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_162_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A4: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_163_old_Oprod_Oinject,axiom,
! [A: nat > nat,B: nat,A4: nat > nat,B3: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_164_prod_Oinject,axiom,
! [X1: ground_gterm_a,X22: ground_gterm_a,Y1: ground_gterm_a,Y22: ground_gterm_a] :
( ( ( produc7419039783125785943term_a @ X1 @ X22 )
= ( produc7419039783125785943term_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_165_prod_Oinject,axiom,
! [X1: ground_gmctxt_a,X22: ground_gmctxt_a,Y1: ground_gmctxt_a,Y22: ground_gmctxt_a] :
( ( ( produc6347872285926497493ctxt_a @ X1 @ X22 )
= ( produc6347872285926497493ctxt_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_166_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_167_prod_Oinject,axiom,
! [X1: nat > nat,X22: nat,Y1: nat > nat,Y22: nat] :
( ( ( produc72220940542539688at_nat @ X1 @ X22 )
= ( produc72220940542539688at_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_168_dual__order_Orefl,axiom,
! [A: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ A @ A ) ).
% dual_order.refl
thf(fact_169_dual__order_Orefl,axiom,
! [A: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ A @ A ) ).
% dual_order.refl
thf(fact_170_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_171_order__refl,axiom,
! [X: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ X @ X ) ).
% order_refl
thf(fact_172_order__refl,axiom,
! [X: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ X @ X ) ).
% order_refl
thf(fact_173_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_174_gmctxt__order__bot_Obdd__below__bot,axiom,
! [A3: set_Ground_gmctxt_a] : ( condit5323614196651773062ctxt_a @ ord_le1940079914304802430ctxt_a @ A3 ) ).
% gmctxt_order_bot.bdd_below_bot
thf(fact_175_gmctxtex__onpE,axiom,
! [S: ground_gterm_a,T: ground_gterm_a,P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ P @ R ) )
=> ~ ! [C4: ground_gmctxt_a,Us: list_Ground_gterm_a] :
( ( S
= ( ground_fill_gholes_a @ C4 @ Us ) )
=> ! [Vs: list_Ground_gterm_a] :
( ( T
= ( ground_fill_gholes_a @ C4 @ Vs ) )
=> ( ( ( ground_num_gholes_a @ C4 )
= ( size_s1893836384466666913term_a @ Us ) )
=> ( ( ( size_s1893836384466666913term_a @ Us )
= ( size_s1893836384466666913term_a @ Vs ) )
=> ( ( P @ C4 )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Vs ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Us @ I ) @ ( nth_Ground_gterm_a @ Vs @ I ) ) @ R ) ) ) ) ) ) ) ) ).
% gmctxtex_onpE
thf(fact_176__092_060open_062_Is_M_At_____J_A_092_060in_062_A_Igmctxtex__onp_AP_A_092_060R_062_J_092_060_094sup_062_L_092_060close_062,axiom,
member4351891425000572176term_a @ ( produc7419039783125785943term_a @ s @ t ) @ ( transi8854406246868658384term_a @ ( contex3086340098647126761_onp_a @ p @ r ) ) ).
% \<open>(s, t__) \<in> (gmctxtex_onp P \<R>)\<^sup>+\<close>
thf(fact_177_order__le__imp__less__or__eq,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ Y )
=> ( ( ord_le9048903087952742514ctxt_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_178_order__le__imp__less__or__eq,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ Y )
=> ( ( ord_le2815447592711331187term_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_179_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_180_gmctxtex__onp__mono,axiom,
! [P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ( ord_le8768141855491817553xt_a_o @ P @ Q )
=> ( ord_le7556488698670074471term_a @ ( contex3086340098647126761_onp_a @ P @ R ) @ ( contex3086340098647126761_onp_a @ Q @ R ) ) ) ).
% gmctxtex_onp_mono
thf(fact_181_gmctxtex__eqI,axiom,
! [P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a,L: set_Pr8282532413760598727term_a] :
( ( P = Q )
=> ( ( R = L )
=> ( ( contex3086340098647126761_onp_a @ P @ R )
= ( contex3086340098647126761_onp_a @ Q @ L ) ) ) ) ).
% gmctxtex_eqI
thf(fact_182_gmctxtex__onp__rel__mono,axiom,
! [L: set_Pr8282532413760598727term_a,R: set_Pr8282532413760598727term_a,P: ground_gmctxt_a > $o] :
( ( ord_le7556488698670074471term_a @ L @ R )
=> ( ord_le7556488698670074471term_a @ ( contex3086340098647126761_onp_a @ P @ L ) @ ( contex3086340098647126761_onp_a @ P @ R ) ) ) ).
% gmctxtex_onp_rel_mono
thf(fact_183_gmctxtex__onp__mem,axiom,
! [P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o,S: ground_gterm_a,T: ground_gterm_a,R: set_Pr8282532413760598727term_a] :
( ( ord_le8768141855491817553xt_a_o @ P @ Q )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ P @ R ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ Q @ R ) ) ) ) ).
% gmctxtex_onp_mem
thf(fact_184_prefix__and__fewer__gholes__implies__equal__gmctxt,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C @ D )
=> ( ( ord_le6045566169113846134st_nat @ ( ground_ghole_poss_a @ C ) @ ( ground_ghole_poss_a @ D ) )
=> ( C = D ) ) ) ).
% prefix_and_fewer_gholes_implies_equal_gmctxt
thf(fact_185_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_186_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_187_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: ground_gmctxt_a,Z2: ground_gmctxt_a] : ( Y3 = Z2 ) )
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X2 @ Y2 )
& ( ord_le1940079914304802430ctxt_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_188_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr8282532413760598727term_a,Z2: set_Pr8282532413760598727term_a] : ( Y3 = Z2 ) )
= ( ^ [X2: set_Pr8282532413760598727term_a,Y2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X2 @ Y2 )
& ( ord_le7556488698670074471term_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_189_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_190_ord__eq__le__trans,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( A = B )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ord_le1940079914304802430ctxt_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_191_ord__eq__le__trans,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( A = B )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ord_le7556488698670074471term_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_192_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_193_ord__le__eq__trans,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le1940079914304802430ctxt_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_194_ord__le__eq__trans,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le7556488698670074471term_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_195_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_196_order__antisym,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ Y )
=> ( ( ord_le1940079914304802430ctxt_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_197_order__antisym,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ Y )
=> ( ( ord_le7556488698670074471term_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_198_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_199_order_Otrans,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ord_le1940079914304802430ctxt_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_200_order_Otrans,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ord_le7556488698670074471term_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_201_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_202_order__trans,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ Y )
=> ( ( ord_le1940079914304802430ctxt_a @ Y @ Z )
=> ( ord_le1940079914304802430ctxt_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_203_order__trans,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a,Z: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ Y )
=> ( ( ord_le7556488698670074471term_a @ Y @ Z )
=> ( ord_le7556488698670074471term_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_204_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_205_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_206_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: ground_gmctxt_a,Z2: ground_gmctxt_a] : ( Y3 = Z2 ) )
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B2 @ A2 )
& ( ord_le1940079914304802430ctxt_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_207_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Pr8282532413760598727term_a,Z2: set_Pr8282532413760598727term_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Pr8282532413760598727term_a,B2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B2 @ A2 )
& ( ord_le7556488698670074471term_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_208_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_209_dual__order_Oantisym,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B @ A )
=> ( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_210_dual__order_Oantisym,axiom,
! [B: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B @ A )
=> ( ( ord_le7556488698670074471term_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_211_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_212_dual__order_Otrans,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B @ A )
=> ( ( ord_le1940079914304802430ctxt_a @ C2 @ B )
=> ( ord_le1940079914304802430ctxt_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_213_dual__order_Otrans,axiom,
! [B: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B @ A )
=> ( ( ord_le7556488698670074471term_a @ C2 @ B )
=> ( ord_le7556488698670074471term_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_214_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_215_antisym,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_216_antisym,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_217_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_218_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: ground_gmctxt_a,Z2: ground_gmctxt_a] : ( Y3 = Z2 ) )
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A2 @ B2 )
& ( ord_le1940079914304802430ctxt_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_219_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr8282532413760598727term_a,Z2: set_Pr8282532413760598727term_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Pr8282532413760598727term_a,B2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A2 @ B2 )
& ( ord_le7556488698670074471term_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_220_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_221_order__subst1,axiom,
! [A: ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_222_order__subst1,axiom,
! [A: ground_gmctxt_a,F: set_Pr8282532413760598727term_a > ground_gmctxt_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_223_order__subst1,axiom,
! [A: ground_gmctxt_a,F: nat > ground_gmctxt_a,B: nat,C2: nat] :
( ( ord_le1940079914304802430ctxt_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_224_order__subst1,axiom,
! [A: set_Pr8282532413760598727term_a,F: ground_gmctxt_a > set_Pr8282532413760598727term_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le7556488698670074471term_a @ A @ ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_225_order__subst1,axiom,
! [A: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_226_order__subst1,axiom,
! [A: set_Pr8282532413760598727term_a,F: nat > set_Pr8282532413760598727term_a,B: nat,C2: nat] :
( ( ord_le7556488698670074471term_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_227_order__subst1,axiom,
! [A: nat,F: ground_gmctxt_a > nat,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_228_order__subst1,axiom,
! [A: nat,F: set_Pr8282532413760598727term_a > nat,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_229_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_230_order__subst2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ ( F @ B ) @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_231_order__subst2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ ( F @ B ) @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_232_order__subst2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > nat,C2: nat] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_233_order__subst2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ ( F @ B ) @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_234_order__subst2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ ( F @ B ) @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_235_order__subst2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > nat,C2: nat] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_236_order__subst2,axiom,
! [A: nat,B: nat,F: nat > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_237_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_238_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_239_order__eq__refl,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( X = Y )
=> ( ord_le1940079914304802430ctxt_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_240_order__eq__refl,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ( X = Y )
=> ( ord_le7556488698670074471term_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_241_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_242_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_243_ord__eq__le__subst,axiom,
! [A: ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_244_ord__eq__le__subst,axiom,
! [A: set_Pr8282532413760598727term_a,F: ground_gmctxt_a > set_Pr8282532413760598727term_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_245_ord__eq__le__subst,axiom,
! [A: nat,F: ground_gmctxt_a > nat,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_246_ord__eq__le__subst,axiom,
! [A: ground_gmctxt_a,F: set_Pr8282532413760598727term_a > ground_gmctxt_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_247_ord__eq__le__subst,axiom,
! [A: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_248_ord__eq__le__subst,axiom,
! [A: nat,F: set_Pr8282532413760598727term_a > nat,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_249_ord__eq__le__subst,axiom,
! [A: ground_gmctxt_a,F: nat > ground_gmctxt_a,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_250_ord__eq__le__subst,axiom,
! [A: set_Pr8282532413760598727term_a,F: nat > set_Pr8282532413760598727term_a,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_251_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_252_ord__le__eq__subst,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_253_ord__le__eq__subst,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_254_ord__le__eq__subst,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > nat,C2: nat] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_255_ord__le__eq__subst,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_256_ord__le__eq__subst,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_257_ord__le__eq__subst,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > nat,C2: nat] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_258_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_259_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le7556488698670074471term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_260_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_261_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_262_order__antisym__conv,axiom,
! [Y: ground_gmctxt_a,X: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ Y @ X )
=> ( ( ord_le1940079914304802430ctxt_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_263_order__antisym__conv,axiom,
! [Y: set_Pr8282532413760598727term_a,X: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ Y @ X )
=> ( ( ord_le7556488698670074471term_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_264_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_265_old_Oprod_Oexhaust,axiom,
! [Y: produc1525979690750518759term_a] :
~ ! [A5: ground_gterm_a,B4: ground_gterm_a] :
( Y
!= ( produc7419039783125785943term_a @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_266_old_Oprod_Oexhaust,axiom,
! [Y: produc8175205341597284509ctxt_a] :
~ ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a] :
( Y
!= ( produc6347872285926497493ctxt_a @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_267_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A5: nat,B4: nat] :
( Y
!= ( product_Pair_nat_nat @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_268_old_Oprod_Oexhaust,axiom,
! [Y: produc8199716216217303280at_nat] :
~ ! [A5: nat > nat,B4: nat] :
( Y
!= ( produc72220940542539688at_nat @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_269_surj__pair,axiom,
! [P2: produc1525979690750518759term_a] :
? [X5: ground_gterm_a,Y4: ground_gterm_a] :
( P2
= ( produc7419039783125785943term_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_270_surj__pair,axiom,
! [P2: produc8175205341597284509ctxt_a] :
? [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( P2
= ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_271_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X5: nat,Y4: nat] :
( P2
= ( product_Pair_nat_nat @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_272_surj__pair,axiom,
! [P2: produc8199716216217303280at_nat] :
? [X5: nat > nat,Y4: nat] :
( P2
= ( produc72220940542539688at_nat @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_273_prod__cases,axiom,
! [P: produc1525979690750518759term_a > $o,P2: produc1525979690750518759term_a] :
( ! [A5: ground_gterm_a,B4: ground_gterm_a] : ( P @ ( produc7419039783125785943term_a @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_274_prod__cases,axiom,
! [P: produc8175205341597284509ctxt_a > $o,P2: produc8175205341597284509ctxt_a] :
( ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a] : ( P @ ( produc6347872285926497493ctxt_a @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_275_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A5: nat,B4: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_276_prod__cases,axiom,
! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
( ! [A5: nat > nat,B4: nat] : ( P @ ( produc72220940542539688at_nat @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_277_Pair__inject,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,A4: ground_gterm_a,B3: ground_gterm_a] :
( ( ( produc7419039783125785943term_a @ A @ B )
= ( produc7419039783125785943term_a @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_278_Pair__inject,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,A4: ground_gmctxt_a,B3: ground_gmctxt_a] :
( ( ( produc6347872285926497493ctxt_a @ A @ B )
= ( produc6347872285926497493ctxt_a @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_279_Pair__inject,axiom,
! [A: nat,B: nat,A4: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_280_Pair__inject,axiom,
! [A: nat > nat,B: nat,A4: nat > nat,B3: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_281_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_282_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_283_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_284_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_285_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_286_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X5: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_287_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_288_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_289_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_290_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_291_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_292_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_293_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_294_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_295_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_296_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_297_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_298_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_299_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_300_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_301_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_302_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_303_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_304_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_305_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_306_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_307_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_308_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_309_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_310_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_311_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_312_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_313_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_314_gmctxtex__onp__closure__predI,axiom,
! [P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ! [C4: ground_gmctxt_a,Ss: list_Ground_gterm_a,Ts: list_Ground_gterm_a] :
( ( P @ C4 )
=> ( ( ( ground_num_gholes_a @ C4 )
= ( size_s1893836384466666913term_a @ Ss ) )
=> ( ( ( size_s1893836384466666913term_a @ Ss )
= ( size_s1893836384466666913term_a @ Ts ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ts ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( nth_Ground_gterm_a @ Ts @ I ) ) @ R ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_fill_gholes_a @ C4 @ Ss ) @ ( ground_fill_gholes_a @ C4 @ Ts ) ) @ R ) ) ) ) )
=> ( ord_le7556488698670074471term_a @ ( contex3086340098647126761_onp_a @ P @ R ) @ R ) ) ).
% gmctxtex_onp_closure_predI
thf(fact_315_leD,axiom,
! [Y: ground_gmctxt_a,X: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ Y @ X )
=> ~ ( ord_le9048903087952742514ctxt_a @ X @ Y ) ) ).
% leD
thf(fact_316_leD,axiom,
! [Y: set_Pr8282532413760598727term_a,X: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ Y @ X )
=> ~ ( ord_le2815447592711331187term_a @ X @ Y ) ) ).
% leD
thf(fact_317_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_318_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_319_nless__le,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ~ ( ord_le9048903087952742514ctxt_a @ A @ B ) )
= ( ~ ( ord_le1940079914304802430ctxt_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_320_nless__le,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ~ ( ord_le2815447592711331187term_a @ A @ B ) )
= ( ~ ( ord_le7556488698670074471term_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_321_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_322_antisym__conv1,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ~ ( ord_le9048903087952742514ctxt_a @ X @ Y )
=> ( ( ord_le1940079914304802430ctxt_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_323_antisym__conv1,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ~ ( ord_le2815447592711331187term_a @ X @ Y )
=> ( ( ord_le7556488698670074471term_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_324_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_325_antisym__conv2,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ Y )
=> ( ( ~ ( ord_le9048903087952742514ctxt_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_326_antisym__conv2,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ Y )
=> ( ( ~ ( ord_le2815447592711331187term_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_327_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_328_less__le__not__le,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X2 @ Y2 )
& ~ ( ord_le1940079914304802430ctxt_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_329_less__le__not__le,axiom,
( ord_le2815447592711331187term_a
= ( ^ [X2: set_Pr8282532413760598727term_a,Y2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X2 @ Y2 )
& ~ ( ord_le7556488698670074471term_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_330_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_331_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_332_order_Oorder__iff__strict,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_333_order_Oorder__iff__strict,axiom,
( ord_le7556488698670074471term_a
= ( ^ [A2: set_Pr8282532413760598727term_a,B2: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_334_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_335_order_Ostrict__iff__order,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_336_order_Ostrict__iff__order,axiom,
( ord_le2815447592711331187term_a
= ( ^ [A2: set_Pr8282532413760598727term_a,B2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_337_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_338_order_Ostrict__trans1,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le9048903087952742514ctxt_a @ B @ C2 )
=> ( ord_le9048903087952742514ctxt_a @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_339_order_Ostrict__trans1,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le2815447592711331187term_a @ B @ C2 )
=> ( ord_le2815447592711331187term_a @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_340_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_341_order_Ostrict__trans2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ord_le9048903087952742514ctxt_a @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_342_order_Ostrict__trans2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ord_le2815447592711331187term_a @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_343_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_344_order_Ostrict__iff__not,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A2 @ B2 )
& ~ ( ord_le1940079914304802430ctxt_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_345_order_Ostrict__iff__not,axiom,
( ord_le2815447592711331187term_a
= ( ^ [A2: set_Pr8282532413760598727term_a,B2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A2 @ B2 )
& ~ ( ord_le7556488698670074471term_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_346_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_347_dual__order_Oorder__iff__strict,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [B2: ground_gmctxt_a,A2: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_348_dual__order_Oorder__iff__strict,axiom,
( ord_le7556488698670074471term_a
= ( ^ [B2: set_Pr8282532413760598727term_a,A2: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_349_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_350_dual__order_Ostrict__iff__order,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [B2: ground_gmctxt_a,A2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_351_dual__order_Ostrict__iff__order,axiom,
( ord_le2815447592711331187term_a
= ( ^ [B2: set_Pr8282532413760598727term_a,A2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_352_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_353_dual__order_Ostrict__trans1,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B @ A )
=> ( ( ord_le9048903087952742514ctxt_a @ C2 @ B )
=> ( ord_le9048903087952742514ctxt_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_354_dual__order_Ostrict__trans1,axiom,
! [B: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B @ A )
=> ( ( ord_le2815447592711331187term_a @ C2 @ B )
=> ( ord_le2815447592711331187term_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_355_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_356_dual__order_Ostrict__trans2,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ B @ A )
=> ( ( ord_le1940079914304802430ctxt_a @ C2 @ B )
=> ( ord_le9048903087952742514ctxt_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_357_dual__order_Ostrict__trans2,axiom,
! [B: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ B @ A )
=> ( ( ord_le7556488698670074471term_a @ C2 @ B )
=> ( ord_le2815447592711331187term_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_358_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_359_dual__order_Ostrict__iff__not,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [B2: ground_gmctxt_a,A2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ B2 @ A2 )
& ~ ( ord_le1940079914304802430ctxt_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_360_dual__order_Ostrict__iff__not,axiom,
( ord_le2815447592711331187term_a
= ( ^ [B2: set_Pr8282532413760598727term_a,A2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B2 @ A2 )
& ~ ( ord_le7556488698670074471term_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_361_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_362_order_Ostrict__implies__order,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ B )
=> ( ord_le1940079914304802430ctxt_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_363_order_Ostrict__implies__order,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A @ B )
=> ( ord_le7556488698670074471term_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_364_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_365_dual__order_Ostrict__implies__order,axiom,
! [B: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ B @ A )
=> ( ord_le1940079914304802430ctxt_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_366_dual__order_Ostrict__implies__order,axiom,
! [B: set_Pr8282532413760598727term_a,A: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ B @ A )
=> ( ord_le7556488698670074471term_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_367_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_368_order__le__less,axiom,
( ord_le1940079914304802430ctxt_a
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_369_order__le__less,axiom,
( ord_le7556488698670074471term_a
= ( ^ [X2: set_Pr8282532413760598727term_a,Y2: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_370_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_371_order__less__le,axiom,
( ord_le9048903087952742514ctxt_a
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_372_order__less__le,axiom,
( ord_le2815447592711331187term_a
= ( ^ [X2: set_Pr8282532413760598727term_a,Y2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_373_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_374_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_375_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_376_order__less__imp__le,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ X @ Y )
=> ( ord_le1940079914304802430ctxt_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_377_order__less__imp__le,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ X @ Y )
=> ( ord_le7556488698670074471term_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_378_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_379_order__le__neq__trans,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( A != B )
=> ( ord_le9048903087952742514ctxt_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_380_order__le__neq__trans,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( A != B )
=> ( ord_le2815447592711331187term_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_381_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_382_order__neq__le__trans,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( A != B )
=> ( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ord_le9048903087952742514ctxt_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_383_order__neq__le__trans,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a] :
( ( A != B )
=> ( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ord_le2815447592711331187term_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_384_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_385_order__le__less__trans,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X @ Y )
=> ( ( ord_le9048903087952742514ctxt_a @ Y @ Z )
=> ( ord_le9048903087952742514ctxt_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_386_order__le__less__trans,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a,Z: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X @ Y )
=> ( ( ord_le2815447592711331187term_a @ Y @ Z )
=> ( ord_le2815447592711331187term_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_387_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_388_order__less__le__trans,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ X @ Y )
=> ( ( ord_le1940079914304802430ctxt_a @ Y @ Z )
=> ( ord_le9048903087952742514ctxt_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_389_order__less__le__trans,axiom,
! [X: set_Pr8282532413760598727term_a,Y: set_Pr8282532413760598727term_a,Z: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ X @ Y )
=> ( ( ord_le7556488698670074471term_a @ Y @ Z )
=> ( ord_le2815447592711331187term_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_390_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_391_order__le__less__subst1,axiom,
! [A: ground_gmctxt_a,F: nat > ground_gmctxt_a,B: nat,C2: nat] :
( ( ord_le1940079914304802430ctxt_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_le9048903087952742514ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_392_order__le__less__subst1,axiom,
! [A: set_Pr8282532413760598727term_a,F: nat > set_Pr8282532413760598727term_a,B: nat,C2: nat] :
( ( ord_le7556488698670074471term_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_le2815447592711331187term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_393_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_394_order__le__less__subst2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le9048903087952742514ctxt_a @ ( F @ B ) @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_395_order__le__less__subst2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_le2815447592711331187term_a @ ( F @ B ) @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_396_order__le__less__subst2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,F: ground_gmctxt_a > nat,C2: nat] :
( ( ord_le1940079914304802430ctxt_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_397_order__le__less__subst2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le9048903087952742514ctxt_a @ ( F @ B ) @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_398_order__le__less__subst2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_le2815447592711331187term_a @ ( F @ B ) @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_399_order__le__less__subst2,axiom,
! [A: set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > nat,C2: nat] :
( ( ord_le7556488698670074471term_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_400_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9048903087952742514ctxt_a @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_401_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2815447592711331187term_a @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_402_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_403_order__less__le__subst1,axiom,
! [A: ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_404_order__less__le__subst1,axiom,
! [A: set_Pr8282532413760598727term_a,F: ground_gmctxt_a > set_Pr8282532413760598727term_a,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_le2815447592711331187term_a @ A @ ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_405_order__less__le__subst1,axiom,
! [A: nat,F: ground_gmctxt_a > nat,B: ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le1940079914304802430ctxt_a @ B @ C2 )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_406_order__less__le__subst1,axiom,
! [A: ground_gmctxt_a,F: set_Pr8282532413760598727term_a > ground_gmctxt_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le9048903087952742514ctxt_a @ A @ ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_407_order__less__le__subst1,axiom,
! [A: set_Pr8282532413760598727term_a,F: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A @ ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_408_order__less__le__subst1,axiom,
! [A: nat,F: set_Pr8282532413760598727term_a > nat,B: set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le7556488698670074471term_a @ B @ C2 )
=> ( ! [X5: set_Pr8282532413760598727term_a,Y4: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_409_order__less__le__subst1,axiom,
! [A: ground_gmctxt_a,F: nat > ground_gmctxt_a,B: nat,C2: nat] :
( ( ord_le9048903087952742514ctxt_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_410_order__less__le__subst1,axiom,
! [A: set_Pr8282532413760598727term_a,F: nat > set_Pr8282532413760598727term_a,B: nat,C2: nat] :
( ( ord_le2815447592711331187term_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_le7556488698670074471term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_411_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_412_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > ground_gmctxt_a,C2: ground_gmctxt_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le1940079914304802430ctxt_a @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_le9048903087952742514ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le9048903087952742514ctxt_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_413_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr8282532413760598727term_a,C2: set_Pr8282532413760598727term_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le7556488698670074471term_a @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_le2815447592711331187term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_le2815447592711331187term_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_414_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_415_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_416_comp,axiom,
member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ c @ d ) @ ground_comp_gmctxt_a ).
% comp
thf(fact_417_assms_I5_J,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( p @ C )
=> ( ( p @ D )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ D ) @ ground_comp_gmctxt_a )
=> ( p @ ( inf_in8301665843363081392ctxt_a @ C @ D ) ) ) ) ) ).
% assms(5)
thf(fact_418_trancl__list__all__step__induct,axiom,
! [Ss2: list_Ground_gmctxt_a,Ts2: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o] :
( ( ( size_s962548663205356226ctxt_a @ Ss2 )
= ( size_s962548663205356226ctxt_a @ Ts2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ts2 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ss2 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ts2 @ I2 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
=> ( ! [Xs: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
( ( ( size_s962548663205356226ctxt_a @ Xs )
= ( size_s962548663205356226ctxt_a @ Ys3 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I ) ) @ R ) )
=> ( P @ Xs @ Ys3 ) ) )
=> ( ! [Xs: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a,Zs: list_Ground_gmctxt_a] :
( ( ( size_s962548663205356226ctxt_a @ Xs )
= ( size_s962548663205356226ctxt_a @ Ys3 ) )
=> ( ( ( size_s962548663205356226ctxt_a @ Ys3 )
= ( size_s962548663205356226ctxt_a @ Zs ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Zs ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Zs ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys3 @ I ) @ ( nth_Ground_gmctxt_a @ Zs @ I ) ) @ R )
| ( ( nth_Ground_gmctxt_a @ Ys3 @ I )
= ( nth_Ground_gmctxt_a @ Zs @ I ) ) ) )
=> ( ( P @ Xs @ Ys3 )
=> ( P @ Xs @ Zs ) ) ) ) ) )
=> ( P @ Ss2 @ Ts2 ) ) ) ) ) ).
% trancl_list_all_step_induct
thf(fact_419_trancl__list__all__step__induct,axiom,
! [Ss2: list_nat,Ts2: list_nat,R: set_Pr1261947904930325089at_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Ss2 )
= ( size_size_list_nat @ Ts2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ts2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ss2 @ I2 ) @ ( nth_nat @ Ts2 @ I2 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
=> ( ! [Xs: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys3 @ I ) ) @ R ) )
=> ( P @ Xs @ Ys3 ) ) )
=> ( ! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys3 @ I ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys3 @ I ) @ ( nth_nat @ Zs @ I ) ) @ R )
| ( ( nth_nat @ Ys3 @ I )
= ( nth_nat @ Zs @ I ) ) ) )
=> ( ( P @ Xs @ Ys3 )
=> ( P @ Xs @ Zs ) ) ) ) ) )
=> ( P @ Ss2 @ Ts2 ) ) ) ) ) ).
% trancl_list_all_step_induct
thf(fact_420_trancl__list__all__step__induct,axiom,
! [Ss2: list_Ground_gterm_a,Ts2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a,P: list_Ground_gterm_a > list_Ground_gterm_a > $o] :
( ( ( size_s1893836384466666913term_a @ Ss2 )
= ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss2 @ I2 ) @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) ) @ ( transi8854406246868658384term_a @ R ) ) )
=> ( ! [Xs: list_Ground_gterm_a,Ys3: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Xs )
= ( size_s1893836384466666913term_a @ Ys3 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys3 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys3 @ I ) ) @ R ) )
=> ( P @ Xs @ Ys3 ) ) )
=> ( ! [Xs: list_Ground_gterm_a,Ys3: list_Ground_gterm_a,Zs: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Xs )
= ( size_s1893836384466666913term_a @ Ys3 ) )
=> ( ( ( size_s1893836384466666913term_a @ Ys3 )
= ( size_s1893836384466666913term_a @ Zs ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Zs ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys3 @ I ) ) @ ( transi8854406246868658384term_a @ R ) ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Zs ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys3 @ I ) @ ( nth_Ground_gterm_a @ Zs @ I ) ) @ R )
| ( ( nth_Ground_gterm_a @ Ys3 @ I )
= ( nth_Ground_gterm_a @ Zs @ I ) ) ) )
=> ( ( P @ Xs @ Ys3 )
=> ( P @ Xs @ Zs ) ) ) ) ) )
=> ( P @ Ss2 @ Ts2 ) ) ) ) ) ).
% trancl_list_all_step_induct
thf(fact_421_step_OIH,axiom,
member4351891425000572176term_a @ ( produc7419039783125785943term_a @ s @ ta ) @ ( contex3086340098647126761_onp_a @ p @ r ) ).
% step.IH
thf(fact_422_assms_I3_J,axiom,
ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ r @ ( contex3086340098647126761_onp_a @ q @ r ) ) @ r ).
% assms(3)
thf(fact_423_assms_I2_J,axiom,
ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ ( contex3086340098647126761_onp_a @ q @ r ) @ r ) @ r ).
% assms(2)
thf(fact_424_gmctxtex__onpI2,axiom,
! [P: ground_gmctxt_a > $o,C: ground_gmctxt_a,S: ground_gterm_a,Ss2: list_Ground_gterm_a,T: ground_gterm_a,Ts2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
( ( P @ C )
=> ( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ C @ Ss2 ) )
=> ( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C @ Ts2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss2 @ I2 ) @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) ) @ R ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ P @ R ) ) ) ) ) ) ).
% gmctxtex_onpI2
thf(fact_425_Utils_Otrancl__list__conv,axiom,
! [Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
= ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
= ( size_s962548663205356226ctxt_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys2 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys2 @ I3 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ) ).
% Utils.trancl_list_conv
thf(fact_426_Utils_Otrancl__list__conv,axiom,
! [Xs3: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( trancl_list_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).
% Utils.trancl_list_conv
thf(fact_427_Utils_Otrancl__list__conv,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( trancl1902929335784340279term_a @ R ) )
= ( ( ( size_s1893836384466666913term_a @ Xs3 )
= ( size_s1893836384466666913term_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ I3 ) @ ( nth_Ground_gterm_a @ Ys2 @ I3 ) ) @ ( transi8854406246868658384term_a @ R ) ) ) ) ) ).
% Utils.trancl_list_conv
thf(fact_428_comp__gmctxt__sym,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ D ) @ ground_comp_gmctxt_a )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ D @ C ) @ ground_comp_gmctxt_a ) ) ).
% comp_gmctxt_sym
thf(fact_429_comp__gmctxt__refl,axiom,
! [C: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ C ) @ ground_comp_gmctxt_a ) ).
% comp_gmctxt_refl
thf(fact_430_eqgf__comp__gmctxt,axiom,
! [S: ground_gterm_a,C: ground_gmctxt_a,Ss2: list_Ground_gterm_a,D: ground_gmctxt_a,Ts2: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ C @ Ss2 ) )
=> ( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ D @ Ts2 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ D ) @ ground_comp_gmctxt_a ) ) ) ).
% eqgf_comp_gmctxt
thf(fact_431_eqgfE_I1_J,axiom,
! [T: ground_gterm_a,D: ground_gmctxt_a,Ss2: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss2 ) )
=> ( T
= ( ground_fill_gholes_a @ D @ Ss2 ) ) ) ).
% eqgfE(1)
thf(fact_432_less__eq__comp__gmctxt,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C @ D )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ D ) @ ground_comp_gmctxt_a ) ) ).
% less_eq_comp_gmctxt
thf(fact_433_eqgfE_I2_J,axiom,
! [T: ground_gterm_a,D: ground_gmctxt_a,Ss2: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss2 ) )
=> ( ( ground_num_gholes_a @ D )
= ( size_s1893836384466666913term_a @ Ss2 ) ) ) ).
% eqgfE(2)
thf(fact_434_eqgf__less__eq,axiom,
! [S: ground_gterm_a,C: ground_gmctxt_a,Ss2: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ C @ Ss2 ) )
=> ( ord_le1940079914304802430ctxt_a @ C @ ( ground582559344758063963term_a @ S ) ) ) ).
% eqgf_less_eq
thf(fact_435_eq__gfill_Ointros,axiom,
! [T: ground_gterm_a,D: ground_gmctxt_a,Ss2: list_Ground_gterm_a] :
( ( T
= ( ground_fill_gholes_a @ D @ Ss2 ) )
=> ( ( ( ground_num_gholes_a @ D )
= ( size_s1893836384466666913term_a @ Ss2 ) )
=> ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss2 ) ) ) ) ).
% eq_gfill.intros
thf(fact_436_eq__gfill_Osimps,axiom,
( ground_eq_gfill_a
= ( ^ [A1: ground_gterm_a,A22: produc2459281416120113548term_a] :
? [T2: ground_gterm_a,D2: ground_gmctxt_a,Ss3: list_Ground_gterm_a] :
( ( A1 = T2 )
& ( A22
= ( produc4720362732934906942term_a @ D2 @ Ss3 ) )
& ( T2
= ( ground_fill_gholes_a @ D2 @ Ss3 ) )
& ( ( ground_num_gholes_a @ D2 )
= ( size_s1893836384466666913term_a @ Ss3 ) ) ) ) ) ).
% eq_gfill.simps
thf(fact_437_eq__gfill_Ocases,axiom,
! [A12: ground_gterm_a,A23: produc2459281416120113548term_a] :
( ( ground_eq_gfill_a @ A12 @ A23 )
=> ~ ! [D3: ground_gmctxt_a,Ss: list_Ground_gterm_a] :
( ( A23
= ( produc4720362732934906942term_a @ D3 @ Ss ) )
=> ( ( A12
= ( ground_fill_gholes_a @ D3 @ Ss ) )
=> ( ( ground_num_gholes_a @ D3 )
!= ( size_s1893836384466666913term_a @ Ss ) ) ) ) ) ).
% eq_gfill.cases
thf(fact_438_Utils_Otrancl__list_Obase,axiom,
! [Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( ( size_s962548663205356226ctxt_a @ Xs3 )
= ( size_s962548663205356226ctxt_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys2 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys2 @ I2 ) ) @ R ) )
=> ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( trancl6028151697025156246ctxt_a @ R ) ) ) ) ).
% Utils.trancl_list.base
thf(fact_439_Utils_Otrancl__list_Obase,axiom,
! [Xs3: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( trancl_list_nat @ R ) ) ) ) ).
% Utils.trancl_list.base
thf(fact_440_Utils_Otrancl__list_Obase,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
( ( ( size_s1893836384466666913term_a @ Xs3 )
= ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ I2 ) @ ( nth_Ground_gterm_a @ Ys2 @ I2 ) ) @ R ) )
=> ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( trancl1902929335784340279term_a @ R ) ) ) ) ).
% Utils.trancl_list.base
thf(fact_441_gmctxtex__onpE2,axiom,
! [S: ground_gterm_a,T: ground_gterm_a,P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ P @ R ) )
=> ~ ! [C4: ground_gmctxt_a,Us: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ C4 @ Us ) )
=> ! [Vs: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C4 @ Vs ) )
=> ( ( P @ C4 )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Vs ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Us @ I ) @ ( nth_Ground_gterm_a @ Vs @ I ) ) @ R ) ) ) ) ) ) ).
% gmctxtex_onpE2
thf(fact_442_ghole__poss__nth__subt__at,axiom,
! [T: ground_gterm_a,C: ground_gmctxt_a,Ts2: list_Ground_gterm_a,P2: list_nat] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C @ Ts2 ) )
=> ( ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ C ) )
=> ( ( ord_less_nat @ ( ground1934838843355771718_pos_a @ P2 @ C ) @ ( size_s1893836384466666913term_a @ Ts2 ) )
& ( ( ground_gsubt_at_a @ T @ P2 )
= ( nth_Ground_gterm_a @ Ts2 @ ( ground1934838843355771718_pos_a @ P2 @ C ) ) ) ) ) ) ).
% ghole_poss_nth_subt_at
thf(fact_443_fill__gholes__ghole__poss,axiom,
! [T: ground_gterm_a,C: ground_gmctxt_a,Ts2: list_Ground_gterm_a,I4: nat] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C @ Ts2 ) )
=> ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( ( ground_gsubt_at_a @ T @ ( nth_list_nat @ ( ground1728035901660239936list_a @ C ) @ I4 ) )
= ( nth_Ground_gterm_a @ Ts2 @ I4 ) ) ) ) ).
% fill_gholes_ghole_poss
thf(fact_444_comp__trancl,axiom,
! [R2: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ R2 @ S2 ) @ S2 )
=> ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ R2 @ ( transi8854406246868658384term_a @ S2 ) ) @ ( transi8854406246868658384term_a @ S2 ) ) ) ).
% comp_trancl
thf(fact_445_kleene__trancl__induct,axiom,
! [A3: set_Pr8282532413760598727term_a,X7: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,C: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ X7 )
=> ( ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ B5 @ X7 ) @ X7 )
=> ( ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ X7 @ C ) @ X7 )
=> ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ ( transi8854406246868658384term_a @ B5 ) @ ( relcom4905991001586507054term_a @ A3 @ ( transi8854406246868658384term_a @ C ) ) ) @ X7 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_446_trancl__Int__subset,axiom,
! [R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ R3 @ S )
=> ( ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ ( inf_in4315656883606118581term_a @ ( transi8854406246868658384term_a @ R3 ) @ S ) @ R3 ) @ S )
=> ( ord_le7556488698670074471term_a @ ( transi8854406246868658384term_a @ R3 ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_447_Ground__Closure_Otrancl__list__conv,axiom,
! [Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( ground4295926921573292569ctxt_a @ R ) )
= ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
= ( size_s962548663205356226ctxt_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys2 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys2 @ I3 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ) ).
% Ground_Closure.trancl_list_conv
thf(fact_448_Ground__Closure_Otrancl__list__conv,axiom,
! [Xs3: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( ground4984107042736062923st_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).
% Ground_Closure.trancl_list_conv
thf(fact_449_Ground__Closure_Otrancl__list__conv,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( ground4492235380174378228term_a @ R ) )
= ( ( ( size_s1893836384466666913term_a @ Xs3 )
= ( size_s1893836384466666913term_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ I3 ) @ ( nth_Ground_gterm_a @ Ys2 @ I3 ) ) @ ( transi8854406246868658384term_a @ R ) ) ) ) ) ).
% Ground_Closure.trancl_list_conv
thf(fact_450_gmctxtex__onp__doubleE,axiom,
! [S: ground_gterm_a,T: ground_gterm_a,P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ P @ ( contex3086340098647126761_onp_a @ Q @ R ) ) )
=> ~ ! [C4: ground_gmctxt_a,Ds: list_Ground_gmctxt_a,Ss: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ C4 @ Ss ) )
=> ! [Ts: list_Ground_gterm_a] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C4 @ Ts ) )
=> ( ( P @ C4 )
=> ( ! [X3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Ds ) )
=> ( Q @ X3 ) )
=> ( ( ( ground_num_gholes_a @ C4 )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ( ( size_s962548663205356226ctxt_a @ Ds )
= ( size_s1893836384466666913term_a @ Ss ) )
=> ( ( ( size_s1893836384466666913term_a @ Ss )
= ( size_s1893836384466666913term_a @ Ts ) )
=> ! [Us: list_l4678906206682822203term_a] :
( ( ( size_s1893836384466666913term_a @ Ts )
= ( size_s5321119603959729319term_a @ Us ) )
=> ! [Vs: list_l4678906206682822203term_a] :
( ( ( size_s5321119603959729319term_a @ Us )
= ( size_s5321119603959729319term_a @ Vs ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I ) @ ( nth_li6172171122907299772term_a @ Us @ I ) ) )
& ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I ) @ ( nth_li6172171122907299772term_a @ Vs @ I ) ) ) ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Vs @ I ) ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Us @ I ) @ J ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Vs @ I ) @ J ) ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gmctxtex_onp_doubleE
thf(fact_451_subset__code_I1_J,axiom,
! [Xs3: list_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ B5 )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs3 ) )
=> ( member_list_nat @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_452_subset__code_I1_J,axiom,
! [Xs3: list_P2170180374705713187ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) @ B5 )
= ( ! [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) )
=> ( member461722496487281478ctxt_a @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_453_subset__code_I1_J,axiom,
! [Xs3: list_P2433084232546440685term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Xs3 ) @ B5 )
= ( ! [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ ( set_Pr5405221470436175740term_a @ Xs3 ) )
=> ( member4351891425000572176term_a @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_454_in__set__idx,axiom,
! [X: list_nat,Xs3: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
& ( ( nth_list_nat @ Xs3 @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_455_in__set__idx,axiom,
! [X: produc1525979690750518759term_a,Xs3: list_P2433084232546440685term_a] :
( ( member4351891425000572176term_a @ X @ ( set_Pr5405221470436175740term_a @ Xs3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ Xs3 ) )
& ( ( nth_Pr5943965302187034734term_a @ Xs3 @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_456_in__set__idx,axiom,
! [X: produc8175205341597284509ctxt_a,Xs3: list_P2170180374705713187ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s274307637295186063ctxt_a @ Xs3 ) )
& ( ( nth_Pr2917159649707247524ctxt_a @ Xs3 @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_457_in__set__idx,axiom,
! [X: ground_gterm_a,Xs3: list_Ground_gterm_a] :
( ( member5261315023172567896term_a @ X @ ( set_Ground_gterm_a2 @ Xs3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Xs3 ) )
& ( ( nth_Ground_gterm_a @ Xs3 @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_458_length__ghole__poss__list__num__gholes,axiom,
( ground_num_gholes_a
= ( ^ [C3: ground_gmctxt_a] : ( size_s3023201423986296836st_nat @ ( ground1728035901660239936list_a @ C3 ) ) ) ) ).
% length_ghole_poss_list_num_gholes
thf(fact_459_set__list__subset__eq__nth__conv,axiom,
! [Xs3: list_list_nat,A3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ A3 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs3 @ I3 ) @ A3 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_460_set__list__subset__eq__nth__conv,axiom,
! [Xs3: list_P2170180374705713187ctxt_a,A3: set_Pr8705490787052945405ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) @ A3 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s274307637295186063ctxt_a @ Xs3 ) )
=> ( member461722496487281478ctxt_a @ ( nth_Pr2917159649707247524ctxt_a @ Xs3 @ I3 ) @ A3 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_461_set__list__subset__eq__nth__conv,axiom,
! [Xs3: list_Ground_gterm_a,A3: set_Ground_gterm_a] :
( ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ Xs3 ) @ A3 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Xs3 @ I3 ) @ A3 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_462_set__list__subset__eq__nth__conv,axiom,
! [Xs3: list_P2433084232546440685term_a,A3: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Xs3 ) @ A3 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Xs3 ) )
=> ( member4351891425000572176term_a @ ( nth_Pr5943965302187034734term_a @ Xs3 @ I3 ) @ A3 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_463_nth__mem,axiom,
! [N: nat,Xs3: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs3 @ N ) @ ( set_list_nat2 @ Xs3 ) ) ) ).
% nth_mem
thf(fact_464_nth__mem,axiom,
! [N: nat,Xs3: list_P2433084232546440685term_a] :
( ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs3 ) )
=> ( member4351891425000572176term_a @ ( nth_Pr5943965302187034734term_a @ Xs3 @ N ) @ ( set_Pr5405221470436175740term_a @ Xs3 ) ) ) ).
% nth_mem
thf(fact_465_nth__mem,axiom,
! [N: nat,Xs3: list_P2170180374705713187ctxt_a] :
( ( ord_less_nat @ N @ ( size_s274307637295186063ctxt_a @ Xs3 ) )
=> ( member461722496487281478ctxt_a @ ( nth_Pr2917159649707247524ctxt_a @ Xs3 @ N ) @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) ) ) ).
% nth_mem
thf(fact_466_nth__mem,axiom,
! [N: nat,Xs3: list_Ground_gterm_a] :
( ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Xs3 @ N ) @ ( set_Ground_gterm_a2 @ Xs3 ) ) ) ).
% nth_mem
thf(fact_467_list__ball__nth,axiom,
! [N: nat,Xs3: list_Ground_gterm_a,P: ground_gterm_a > $o] :
( ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ! [X5: ground_gterm_a] :
( ( member5261315023172567896term_a @ X5 @ ( set_Ground_gterm_a2 @ Xs3 ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_Ground_gterm_a @ Xs3 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_468_in__set__conv__nth,axiom,
! [X: list_nat,Xs3: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
& ( ( nth_list_nat @ Xs3 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_469_in__set__conv__nth,axiom,
! [X: produc1525979690750518759term_a,Xs3: list_P2433084232546440685term_a] :
( ( member4351891425000572176term_a @ X @ ( set_Pr5405221470436175740term_a @ Xs3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Xs3 ) )
& ( ( nth_Pr5943965302187034734term_a @ Xs3 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_470_in__set__conv__nth,axiom,
! [X: produc8175205341597284509ctxt_a,Xs3: list_P2170180374705713187ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s274307637295186063ctxt_a @ Xs3 ) )
& ( ( nth_Pr2917159649707247524ctxt_a @ Xs3 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_471_in__set__conv__nth,axiom,
! [X: ground_gterm_a,Xs3: list_Ground_gterm_a] :
( ( member5261315023172567896term_a @ X @ ( set_Ground_gterm_a2 @ Xs3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs3 ) )
& ( ( nth_Ground_gterm_a @ Xs3 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_472_all__nth__imp__all__set,axiom,
! [Xs3: list_list_nat,P: list_nat > $o,X: list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( P @ ( nth_list_nat @ Xs3 @ I2 ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs3 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_473_all__nth__imp__all__set,axiom,
! [Xs3: list_P2433084232546440685term_a,P: produc1525979690750518759term_a > $o,X: produc1525979690750518759term_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ Xs3 ) )
=> ( P @ ( nth_Pr5943965302187034734term_a @ Xs3 @ I2 ) ) )
=> ( ( member4351891425000572176term_a @ X @ ( set_Pr5405221470436175740term_a @ Xs3 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_474_all__nth__imp__all__set,axiom,
! [Xs3: list_P2170180374705713187ctxt_a,P: produc8175205341597284509ctxt_a > $o,X: produc8175205341597284509ctxt_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s274307637295186063ctxt_a @ Xs3 ) )
=> ( P @ ( nth_Pr2917159649707247524ctxt_a @ Xs3 @ I2 ) ) )
=> ( ( member461722496487281478ctxt_a @ X @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_475_all__nth__imp__all__set,axiom,
! [Xs3: list_Ground_gterm_a,P: ground_gterm_a > $o,X: ground_gterm_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( P @ ( nth_Ground_gterm_a @ Xs3 @ I2 ) ) )
=> ( ( member5261315023172567896term_a @ X @ ( set_Ground_gterm_a2 @ Xs3 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_476_all__set__conv__all__nth,axiom,
! [Xs3: list_Ground_gterm_a,P: ground_gterm_a > $o] :
( ( ! [X2: ground_gterm_a] :
( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs3 ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( P @ ( nth_Ground_gterm_a @ Xs3 @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_477_ex__set__conv__ex__nth,axiom,
! [Xs3: list_Ground_gterm_a,P: ground_gterm_a > $o] :
( ( ? [X2: ground_gterm_a] :
( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs3 ) )
& ( P @ X2 ) ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs3 ) )
& ( P @ ( nth_Ground_gterm_a @ Xs3 @ I3 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_478_trancl__induct2,axiom,
! [Ax: ground_gterm_a,Ay: ground_gterm_a,Bx: ground_gterm_a,By: ground_gterm_a,R3: set_Pr5156918327056800967term_a,P: ground_gterm_a > ground_gterm_a > $o] :
( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ Ax @ Ay ) @ ( produc7419039783125785943term_a @ Bx @ By ) ) @ ( transi4994575220883992200term_a @ R3 ) )
=> ( ! [A5: ground_gterm_a,B4: ground_gterm_a] :
( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ Ax @ Ay ) @ ( produc7419039783125785943term_a @ A5 @ B4 ) ) @ R3 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: ground_gterm_a,B4: ground_gterm_a,Aa: ground_gterm_a,Ba: ground_gterm_a] :
( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ Ax @ Ay ) @ ( produc7419039783125785943term_a @ A5 @ B4 ) ) @ ( transi4994575220883992200term_a @ R3 ) )
=> ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ A5 @ B4 ) @ ( produc7419039783125785943term_a @ Aa @ Ba ) ) @ R3 )
=> ( ( P @ A5 @ B4 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_479_trancl__induct2,axiom,
! [Ax: ground_gmctxt_a,Ay: ground_gmctxt_a,Bx: ground_gmctxt_a,By: ground_gmctxt_a,R3: set_Pr490164551564945991ctxt_a,P: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ Bx @ By ) ) @ ( transi6952278968286904510ctxt_a @ R3 ) )
=> ( ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ A5 @ B4 ) ) @ R3 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a,Aa: ground_gmctxt_a,Ba: ground_gmctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ A5 @ B4 ) ) @ ( transi6952278968286904510ctxt_a @ R3 ) )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ A5 @ B4 ) @ ( produc6347872285926497493ctxt_a @ Aa @ Ba ) ) @ R3 )
=> ( ( P @ A5 @ B4 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_480_trancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R3: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R3 ) )
=> ( ! [A5: nat,B4: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A5 @ B4 ) ) @ R3 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A5 @ B4 ) ) @ ( transi243908449541399842at_nat @ R3 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R3 )
=> ( ( P @ A5 @ B4 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_481_trancl__induct2,axiom,
! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R3: set_Pr7047708467701517959at_nat,P: ( nat > nat ) > nat > $o] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3146596849628481425at_nat @ R3 ) )
=> ( ! [A5: nat > nat,B4: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A5 @ B4 ) ) @ R3 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat > nat,B4: nat,Aa: nat > nat,Ba: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A5 @ B4 ) ) @ ( transi3146596849628481425at_nat @ R3 ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A5 @ B4 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R3 )
=> ( ( P @ A5 @ B4 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_482_r__into__trancl_H,axiom,
! [P2: produc8175205341597284509ctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ P2 @ R3 )
=> ( member461722496487281478ctxt_a @ P2 @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ).
% r_into_trancl'
thf(fact_483_r__into__trancl_H,axiom,
! [P2: produc1525979690750518759term_a,R3: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ P2 @ R3 )
=> ( member4351891425000572176term_a @ P2 @ ( transi8854406246868658384term_a @ R3 ) ) ) ).
% r_into_trancl'
thf(fact_484_subseteq__set__conv__nth,axiom,
! [Ss2: list_list_nat,T3: set_list_nat] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ss2 ) )
=> ( member_list_nat @ ( nth_list_nat @ Ss2 @ I3 ) @ T3 ) ) )
= ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ss2 ) @ T3 ) ) ).
% subseteq_set_conv_nth
thf(fact_485_subseteq__set__conv__nth,axiom,
! [Ss2: list_P2170180374705713187ctxt_a,T3: set_Pr8705490787052945405ctxt_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s274307637295186063ctxt_a @ Ss2 ) )
=> ( member461722496487281478ctxt_a @ ( nth_Pr2917159649707247524ctxt_a @ Ss2 @ I3 ) @ T3 ) ) )
= ( ord_le2690539916662146973ctxt_a @ ( set_Pr5641390345007089586ctxt_a @ Ss2 ) @ T3 ) ) ).
% subseteq_set_conv_nth
thf(fact_486_subseteq__set__conv__nth,axiom,
! [Ss2: list_Ground_gterm_a,T3: set_Ground_gterm_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ss2 ) )
=> ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Ss2 @ I3 ) @ T3 ) ) )
= ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ Ss2 ) @ T3 ) ) ).
% subseteq_set_conv_nth
thf(fact_487_subseteq__set__conv__nth,axiom,
! [Ss2: list_P2433084232546440685term_a,T3: set_Pr8282532413760598727term_a] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ss2 ) )
=> ( member4351891425000572176term_a @ ( nth_Pr5943965302187034734term_a @ Ss2 @ I3 ) @ T3 ) ) )
= ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Ss2 ) @ T3 ) ) ).
% subseteq_set_conv_nth
thf(fact_488_relcomp3__I,axiom,
! [T: nat,U: nat > nat,A3: set_Pr6370437063884598352at_nat,S: nat > nat,B5: set_Pr9093778441882193744at_nat,V: nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ T @ U ) @ A3 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T ) @ B5 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U @ V ) @ B5 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ V ) @ ( relcom1859588284133128290at_nat @ B5 @ ( relcom6405311860805859042at_nat @ A3 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_489_relcomp3__I,axiom,
! [T: ground_gterm_a,U: ground_gterm_a,A3: set_Pr8282532413760598727term_a,S: ground_gterm_a,B5: set_Pr8282532413760598727term_a,V: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ T @ U ) @ A3 )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ B5 )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ U @ V ) @ B5 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ V ) @ ( relcom4905991001586507054term_a @ B5 @ ( relcom4905991001586507054term_a @ A3 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_490_relcomp3__I,axiom,
! [T: ground_gmctxt_a,U: ground_gmctxt_a,A3: set_Pr8705490787052945405ctxt_a,S: ground_gmctxt_a,B5: set_Pr8705490787052945405ctxt_a,V: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ T @ U ) @ A3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ T ) @ B5 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ U @ V ) @ B5 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ V ) @ ( relcom6868745818274832221ctxt_a @ B5 @ ( relcom6868745818274832221ctxt_a @ A3 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_491_relcomp3__I,axiom,
! [T: nat,U: nat,A3: set_Pr1261947904930325089at_nat,S: nat,B5: set_Pr1261947904930325089at_nat,V: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ A3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ B5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ B5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ V ) @ ( relcomp_nat_nat_nat @ B5 @ ( relcomp_nat_nat_nat @ A3 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_492_relcomp3__I,axiom,
! [T: nat > nat,U: nat,A3: set_Pr9093778441882193744at_nat,S: nat,B5: set_Pr6370437063884598352at_nat,V: nat > nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ T @ U ) @ A3 )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ T ) @ B5 )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ U @ V ) @ B5 )
=> ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ V ) @ ( relcom896424912281157969at_nat @ B5 @ ( relcom7598182128343433937at_nat @ A3 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_493_trancl_Ocases,axiom,
! [A12: ground_gmctxt_a,A23: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A12 @ A23 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A12 @ A23 ) @ R3 )
=> ~ ! [B4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A12 @ B4 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B4 @ A23 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_494_trancl_Ocases,axiom,
! [A12: nat,A23: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ R3 )
=> ~ ! [B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ B4 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A23 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_495_trancl_Ocases,axiom,
! [A12: ground_gterm_a,A23: ground_gterm_a,R3: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ R3 )
=> ~ ! [B4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ B4 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B4 @ A23 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_496_trancl_Osimps,axiom,
! [A12: ground_gmctxt_a,A23: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A12 @ A23 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
= ( ? [A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( A12 = A2 )
& ( A23 = B2 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A2 @ B2 ) @ R3 ) )
| ? [A2: ground_gmctxt_a,B2: ground_gmctxt_a,C5: ground_gmctxt_a] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A2 @ B2 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B2 @ C5 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_497_trancl_Osimps,axiom,
! [A12: nat,A23: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
= ( ? [A2: nat,B2: nat] :
( ( A12 = A2 )
& ( A23 = B2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R3 ) )
| ? [A2: nat,B2: nat,C5: nat] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C5 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_498_trancl_Osimps,axiom,
! [A12: ground_gterm_a,A23: ground_gterm_a,R3: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( transi8854406246868658384term_a @ R3 ) )
= ( ? [A2: ground_gterm_a,B2: ground_gterm_a] :
( ( A12 = A2 )
& ( A23 = B2 )
& ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A2 @ B2 ) @ R3 ) )
| ? [A2: ground_gterm_a,B2: ground_gterm_a,C5: ground_gterm_a] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A2 @ B2 ) @ ( transi8854406246868658384term_a @ R3 ) )
& ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B2 @ C5 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_499_trancl_Or__into__trancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_500_trancl_Or__into__trancl,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_501_trancl_Or__into__trancl,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R3: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R3 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_502_trancl__map,axiom,
! [R3: set_Pr8705490787052945405ctxt_a,F: ground_gmctxt_a > ground_gmctxt_a,S: set_Pr8705490787052945405ctxt_a,X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ R3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ X ) @ ( F @ Y ) ) @ ( transi3225167780237802813ctxt_a @ S ) ) ) ) ).
% trancl_map
thf(fact_503_trancl__map,axiom,
! [R3: set_Pr8705490787052945405ctxt_a,F: ground_gmctxt_a > nat,S: set_Pr1261947904930325089at_nat,X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_504_trancl__map,axiom,
! [R3: set_Pr1261947904930325089at_nat,F: nat > ground_gmctxt_a,S: set_Pr8705490787052945405ctxt_a,X: nat,Y: nat] :
( ! [X5: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ X ) @ ( F @ Y ) ) @ ( transi3225167780237802813ctxt_a @ S ) ) ) ) ).
% trancl_map
thf(fact_505_trancl__map,axiom,
! [R3: set_Pr1261947904930325089at_nat,F: nat > nat,S: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ! [X5: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_506_trancl__map,axiom,
! [R3: set_Pr8705490787052945405ctxt_a,F: ground_gmctxt_a > ground_gterm_a,S: set_Pr8282532413760598727term_a,X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ R3 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X ) @ ( F @ Y ) ) @ ( transi8854406246868658384term_a @ S ) ) ) ) ).
% trancl_map
thf(fact_507_trancl__map,axiom,
! [R3: set_Pr1261947904930325089at_nat,F: nat > ground_gterm_a,S: set_Pr8282532413760598727term_a,X: nat,Y: nat] :
( ! [X5: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R3 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X ) @ ( F @ Y ) ) @ ( transi8854406246868658384term_a @ S ) ) ) ) ).
% trancl_map
thf(fact_508_trancl__map,axiom,
! [R3: set_Pr8282532413760598727term_a,F: ground_gterm_a > ground_gmctxt_a,S: set_Pr8705490787052945405ctxt_a,X: ground_gterm_a,Y: ground_gterm_a] :
( ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ R3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ X ) @ ( F @ Y ) ) @ ( transi3225167780237802813ctxt_a @ S ) ) ) ) ).
% trancl_map
thf(fact_509_trancl__map,axiom,
! [R3: set_Pr8282532413760598727term_a,F: ground_gterm_a > nat,S: set_Pr1261947904930325089at_nat,X: ground_gterm_a,Y: ground_gterm_a] :
( ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_510_trancl__map,axiom,
! [R3: set_Pr8282532413760598727term_a,F: ground_gterm_a > ground_gterm_a,S: set_Pr8282532413760598727term_a,X: ground_gterm_a,Y: ground_gterm_a] :
( ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ R3 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X5 ) @ ( F @ Y4 ) ) @ S ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X ) @ ( F @ Y ) ) @ ( transi8854406246868658384term_a @ S ) ) ) ) ).
% trancl_map
thf(fact_511_tranclE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R3 )
=> ~ ! [C6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C6 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C6 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_512_tranclE,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ~ ! [C6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C6 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C6 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_513_tranclE,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R3: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R3 )
=> ~ ! [C6: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C6 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ C6 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_514_trancl__trans,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,Z: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ Z ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Z ) @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_515_trancl__trans,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_516_trancl__trans,axiom,
! [X: ground_gterm_a,Y: ground_gterm_a,R3: set_Pr8282532413760598727term_a,Z: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y @ Z ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Z ) @ ( transi8854406246868658384term_a @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_517_trancl__induct,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,P: ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ! [Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ Y4 ) @ R3 )
=> ( P @ Y4 ) )
=> ( ! [Y4: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ Y4 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ Z3 ) @ R3 )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_518_trancl__induct,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ! [Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y4 ) @ R3 )
=> ( P @ Y4 ) )
=> ( ! [Y4: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y4 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ R3 )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_519_trancl__induct,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R3: set_Pr8282532413760598727term_a,P: ground_gterm_a > $o] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ! [Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ Y4 ) @ R3 )
=> ( P @ Y4 ) )
=> ( ! [Y4: ground_gterm_a,Z3: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ Y4 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z3 ) @ R3 )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_520_r__r__into__trancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R2 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ R2 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R2 ) ) ) ) ).
% r_r_into_trancl
thf(fact_521_r__r__into__trancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% r_r_into_trancl
thf(fact_522_r__r__into__trancl,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R2: set_Pr8282532413760598727term_a,C2: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R2 )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B @ C2 ) @ R2 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C2 ) @ ( transi8854406246868658384term_a @ R2 ) ) ) ) ).
% r_r_into_trancl
thf(fact_523_converse__tranclE,axiom,
! [X: ground_gmctxt_a,Z: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Z ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Z ) @ R3 )
=> ~ ! [Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y4 ) @ R3 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ Z ) @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_524_converse__tranclE,axiom,
! [X: nat,Z: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R3 )
=> ~ ! [Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_525_converse__tranclE,axiom,
! [X: ground_gterm_a,Z: ground_gterm_a,R3: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Z ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Z ) @ R3 )
=> ~ ! [Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y4 ) @ R3 )
=> ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z ) @ ( transi8854406246868658384term_a @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_526_irrefl__trancl__rD,axiom,
! [R3: set_Pr8705490787052945405ctxt_a,X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ! [X5: ground_gmctxt_a] :
~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ X5 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_527_irrefl__trancl__rD,axiom,
! [R3: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ! [X5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_528_irrefl__trancl__rD,axiom,
! [R3: set_Pr8282532413760598727term_a,X: ground_gterm_a,Y: ground_gterm_a] :
( ! [X5: ground_gterm_a] :
~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ X5 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_529_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ R3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_530_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_531_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R3: set_Pr8282532413760598727term_a,C2: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B @ C2 ) @ R3 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C2 ) @ ( transi8854406246868658384term_a @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_532_trancl__into__trancl2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_533_trancl__into__trancl2,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_534_trancl__into__trancl2,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R3: set_Pr8282532413760598727term_a,C2: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R3 )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B @ C2 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C2 ) @ ( transi8854406246868658384term_a @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_535_trancl__trans__induct,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,P: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ R3 )
=> ( P @ X5 @ Y4 ) )
=> ( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( P @ X5 @ Y4 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ Z3 ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( P @ Y4 @ Z3 )
=> ( P @ X5 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_536_trancl__trans__induct,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ! [X5: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R3 )
=> ( P @ X5 @ Y4 ) )
=> ( ! [X5: nat,Y4: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ( P @ X5 @ Y4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ( P @ Y4 @ Z3 )
=> ( P @ X5 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_537_trancl__trans__induct,axiom,
! [X: ground_gterm_a,Y: ground_gterm_a,R3: set_Pr8282532413760598727term_a,P: ground_gterm_a > ground_gterm_a > $o] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ R3 )
=> ( P @ X5 @ Y4 ) )
=> ( ! [X5: ground_gterm_a,Y4: ground_gterm_a,Z3: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( P @ X5 @ Y4 )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z3 ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( P @ Y4 @ Z3 )
=> ( P @ X5 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_538_converse__trancl__induct,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,P: ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ! [Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ B ) @ R3 )
=> ( P @ Y4 ) )
=> ( ! [Y4: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ Z3 ) @ R3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Z3 @ B ) @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_539_converse__trancl__induct,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ! [Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ B ) @ R3 )
=> ( P @ Y4 ) )
=> ( ! [Y4: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_540_converse__trancl__induct,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R3: set_Pr8282532413760598727term_a,P: ground_gterm_a > $o] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ! [Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ B ) @ R3 )
=> ( P @ Y4 ) )
=> ( ! [Y4: ground_gterm_a,Z3: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z3 ) @ R3 )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Z3 @ B ) @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_541_O__mono1,axiom,
! [R2: set_Pr8282532413760598727term_a,R4: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ R2 @ R4 )
=> ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ S2 @ R2 ) @ ( relcom4905991001586507054term_a @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_542_O__mono2,axiom,
! [R2: set_Pr8282532413760598727term_a,R4: set_Pr8282532413760598727term_a,T3: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ R2 @ R4 )
=> ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ R2 @ T3 ) @ ( relcom4905991001586507054term_a @ R4 @ T3 ) ) ) ).
% O_mono2
thf(fact_543_trancl__mono__set,axiom,
! [R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ R3 @ S )
=> ( ord_le7556488698670074471term_a @ ( transi8854406246868658384term_a @ R3 ) @ ( transi8854406246868658384term_a @ S ) ) ) ).
% trancl_mono_set
thf(fact_544_trancl__mono,axiom,
! [P2: produc8175205341597284509ctxt_a,R3: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ P2 @ ( transi3225167780237802813ctxt_a @ R3 ) )
=> ( ( ord_le2690539916662146973ctxt_a @ R3 @ S )
=> ( member461722496487281478ctxt_a @ P2 @ ( transi3225167780237802813ctxt_a @ S ) ) ) ) ).
% trancl_mono
thf(fact_545_trancl__mono,axiom,
! [P2: produc1525979690750518759term_a,R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ P2 @ ( transi8854406246868658384term_a @ R3 ) )
=> ( ( ord_le7556488698670074471term_a @ R3 @ S )
=> ( member4351891425000572176term_a @ P2 @ ( transi8854406246868658384term_a @ S ) ) ) ) ).
% trancl_mono
thf(fact_546_Ground__Closure_Otrancl__list_Obase,axiom,
! [Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( ( size_s962548663205356226ctxt_a @ Xs3 )
= ( size_s962548663205356226ctxt_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys2 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys2 @ I2 ) ) @ R ) )
=> ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( ground4295926921573292569ctxt_a @ R ) ) ) ) ).
% Ground_Closure.trancl_list.base
thf(fact_547_Ground__Closure_Otrancl__list_Obase,axiom,
! [Xs3: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( ground4984107042736062923st_nat @ R ) ) ) ) ).
% Ground_Closure.trancl_list.base
thf(fact_548_Ground__Closure_Otrancl__list_Obase,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
( ( ( size_s1893836384466666913term_a @ Xs3 )
= ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ I2 ) @ ( nth_Ground_gterm_a @ Ys2 @ I2 ) ) @ R ) )
=> ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( ground4492235380174378228term_a @ R ) ) ) ) ).
% Ground_Closure.trancl_list.base
thf(fact_549_set__generate__lists,axiom,
! [N: nat,Xs3: list_Ground_gterm_a] :
( ( set_li7546925467519454922term_a @ ( missin8875442748449356934term_a @ N @ Xs3 ) )
= ( collec4295779454384174240term_a
@ ^ [As2: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ As2 )
= N )
& ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ As2 ) @ ( set_Ground_gterm_a2 @ Xs3 ) ) ) ) ) ).
% set_generate_lists
thf(fact_550_set__generate__lists,axiom,
! [N: nat,Xs3: list_P2433084232546440685term_a] :
( ( set_li5508857501565128962term_a @ ( missin2084454622452852542term_a @ N @ Xs3 ) )
= ( collec7935380194126400472term_a
@ ^ [As2: list_P2433084232546440685term_a] :
( ( ( size_s281034491502894169term_a @ As2 )
= N )
& ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ As2 ) @ ( set_Pr5405221470436175740term_a @ Xs3 ) ) ) ) ) ).
% set_generate_lists
thf(fact_551_Int__subset__iff,axiom,
! [C: set_Pr8282532413760598727term_a,A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ C @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) )
= ( ( ord_le7556488698670074471term_a @ C @ A3 )
& ( ord_le7556488698670074471term_a @ C @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_552_gmctxtex__onp__listE,axiom,
! [Ts2: list_Ground_gterm_a,Ss2: list_Ground_gterm_a,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss2 @ I2 ) @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) ) @ ( contex3086340098647126761_onp_a @ Q @ R ) ) )
=> ( ( ( size_s1893836384466666913term_a @ Ss2 )
= ( size_s1893836384466666913term_a @ Ts2 ) )
=> ~ ! [Ds: list_Ground_gmctxt_a] :
( ( ( size_s1893836384466666913term_a @ Ts2 )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ! [Sss: list_l4678906206682822203term_a] :
( ( ( size_s962548663205356226ctxt_a @ Ds )
= ( size_s5321119603959729319term_a @ Sss ) )
=> ! [Tss: list_l4678906206682822203term_a] :
( ( ( size_s5321119603959729319term_a @ Sss )
= ( size_s5321119603959729319term_a @ Tss ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Tss ) )
=> ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Sss @ I ) )
= ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Tss @ I ) ) ) )
=> ( ! [X3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Ds ) )
=> ( Q @ X3 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Tss ) )
=> ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss2 @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I ) @ ( nth_li6172171122907299772term_a @ Sss @ I ) ) ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Tss ) )
=> ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts2 @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I ) @ ( nth_li6172171122907299772term_a @ Tss @ I ) ) ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Tss ) ) )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Sss ) @ I ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Tss ) @ I ) ) @ R ) ) ) ) ) ) ) ) ) ) ) ).
% gmctxtex_onp_listE
thf(fact_553_Int__iff,axiom,
! [C2: list_nat,A3: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A3 @ B5 ) )
= ( ( member_list_nat @ C2 @ A3 )
& ( member_list_nat @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_554_Int__iff,axiom,
! [C2: produc1525979690750518759term_a,A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ C2 @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) )
= ( ( member4351891425000572176term_a @ C2 @ A3 )
& ( member4351891425000572176term_a @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_555_Int__iff,axiom,
! [C2: produc8175205341597284509ctxt_a,A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( inf_in3429776173938890475ctxt_a @ A3 @ B5 ) )
= ( ( member461722496487281478ctxt_a @ C2 @ A3 )
& ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_556_IntI,axiom,
! [C2: list_nat,A3: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ A3 )
=> ( ( member_list_nat @ C2 @ B5 )
=> ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A3 @ B5 ) ) ) ) ).
% IntI
thf(fact_557_IntI,axiom,
! [C2: produc1525979690750518759term_a,A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ C2 @ A3 )
=> ( ( member4351891425000572176term_a @ C2 @ B5 )
=> ( member4351891425000572176term_a @ C2 @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) ) ) ) ).
% IntI
thf(fact_558_IntI,axiom,
! [C2: produc8175205341597284509ctxt_a,A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ A3 )
=> ( ( member461722496487281478ctxt_a @ C2 @ B5 )
=> ( member461722496487281478ctxt_a @ C2 @ ( inf_in3429776173938890475ctxt_a @ A3 @ B5 ) ) ) ) ).
% IntI
thf(fact_559_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A6 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_560_less__eq__set__def,axiom,
( ord_le2690539916662146973ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
( ord_le4188747423230057856xt_a_o
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ A6 )
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_561_less__eq__set__def,axiom,
( ord_le7556488698670074471term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( ord_le2091684138676743478rm_a_o
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ A6 )
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_562_pred__subset__eq,axiom,
! [R2: set_list_nat,S2: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ R2 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ S2 ) )
= ( ord_le6045566169113846134st_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_563_pred__subset__eq,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,S2: set_Pr8705490787052945405ctxt_a] :
( ( ord_le4188747423230057856xt_a_o
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ R2 )
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ S2 ) )
= ( ord_le2690539916662146973ctxt_a @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_564_pred__subset__eq,axiom,
! [R2: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a] :
( ( ord_le2091684138676743478rm_a_o
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ R2 )
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ S2 ) )
= ( ord_le7556488698670074471term_a @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_565_psubsetI,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_le2815447592711331187term_a @ A3 @ B5 ) ) ) ).
% psubsetI
thf(fact_566_subset__antisym,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( ord_le7556488698670074471term_a @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_567_subsetI,axiom,
! [A3: set_list_nat,B5: set_list_nat] :
( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ A3 )
=> ( member_list_nat @ X5 @ B5 ) )
=> ( ord_le6045566169113846134st_nat @ A3 @ B5 ) ) ).
% subsetI
thf(fact_568_subsetI,axiom,
! [A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ! [X5: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X5 @ A3 )
=> ( member461722496487281478ctxt_a @ X5 @ B5 ) )
=> ( ord_le2690539916662146973ctxt_a @ A3 @ B5 ) ) ).
% subsetI
thf(fact_569_subsetI,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ! [X5: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X5 @ A3 )
=> ( member4351891425000572176term_a @ X5 @ B5 ) )
=> ( ord_le7556488698670074471term_a @ A3 @ B5 ) ) ).
% subsetI
thf(fact_570_ghole__poss__ghole__poss__list__conv,axiom,
( ground_ghole_poss_a
= ( ^ [C3: ground_gmctxt_a] : ( set_list_nat2 @ ( ground1728035901660239936list_a @ C3 ) ) ) ) ).
% ghole_poss_ghole_poss_list_conv
thf(fact_571_psubsetE,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A3 @ B5 )
=> ~ ( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ord_le7556488698670074471term_a @ B5 @ A3 ) ) ) ).
% psubsetE
thf(fact_572_psubset__eq,axiom,
( ord_le2815447592711331187term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_573_psubset__imp__subset,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A3 @ B5 )
=> ( ord_le7556488698670074471term_a @ A3 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_574_psubset__subset__trans,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,C: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A3 @ B5 )
=> ( ( ord_le7556488698670074471term_a @ B5 @ C )
=> ( ord_le2815447592711331187term_a @ A3 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_575_subset__not__subset__eq,axiom,
( ord_le2815447592711331187term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A6 @ B6 )
& ~ ( ord_le7556488698670074471term_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_576_subset__psubset__trans,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,C: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( ord_le2815447592711331187term_a @ B5 @ C )
=> ( ord_le2815447592711331187term_a @ A3 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_577_subset__iff__psubset__eq,axiom,
( ord_le7556488698670074471term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_578_Collect__mono__iff,axiom,
! [P: produc1525979690750518759term_a > $o,Q: produc1525979690750518759term_a > $o] :
( ( ord_le7556488698670074471term_a @ ( collec6713184107424615762term_a @ P ) @ ( collec6713184107424615762term_a @ Q ) )
= ( ! [X2: produc1525979690750518759term_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_579_set__eq__subset,axiom,
( ( ^ [Y3: set_Pr8282532413760598727term_a,Z2: set_Pr8282532413760598727term_a] : ( Y3 = Z2 ) )
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A6 @ B6 )
& ( ord_le7556488698670074471term_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_580_subset__trans,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,C: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( ord_le7556488698670074471term_a @ B5 @ C )
=> ( ord_le7556488698670074471term_a @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_581_Collect__mono,axiom,
! [P: produc1525979690750518759term_a > $o,Q: produc1525979690750518759term_a > $o] :
( ! [X5: produc1525979690750518759term_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le7556488698670074471term_a @ ( collec6713184107424615762term_a @ P ) @ ( collec6713184107424615762term_a @ Q ) ) ) ).
% Collect_mono
thf(fact_582_subset__refl,axiom,
! [A3: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_583_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A6 )
=> ( member_list_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_584_subset__iff,axiom,
( ord_le2690539916662146973ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
! [T2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ T2 @ A6 )
=> ( member461722496487281478ctxt_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_585_subset__iff,axiom,
( ord_le7556488698670074471term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
! [T2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ T2 @ A6 )
=> ( member4351891425000572176term_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_586_equalityD2,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( A3 = B5 )
=> ( ord_le7556488698670074471term_a @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_587_equalityD1,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( A3 = B5 )
=> ( ord_le7556488698670074471term_a @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_588_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
! [X2: list_nat] :
( ( member_list_nat @ X2 @ A6 )
=> ( member_list_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_589_subset__eq,axiom,
( ord_le2690539916662146973ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
! [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ A6 )
=> ( member461722496487281478ctxt_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_590_subset__eq,axiom,
( ord_le7556488698670074471term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
! [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ A6 )
=> ( member4351891425000572176term_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_591_equalityE,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( A3 = B5 )
=> ~ ( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ~ ( ord_le7556488698670074471term_a @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_592_subsetD,axiom,
! [A3: set_list_nat,B5: set_list_nat,C2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B5 )
=> ( ( member_list_nat @ C2 @ A3 )
=> ( member_list_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_593_subsetD,axiom,
! [A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,C2: produc8175205341597284509ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ A3 @ B5 )
=> ( ( member461722496487281478ctxt_a @ C2 @ A3 )
=> ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_594_subsetD,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,C2: produc1525979690750518759term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( member4351891425000572176term_a @ C2 @ A3 )
=> ( member4351891425000572176term_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_595_in__mono,axiom,
! [A3: set_list_nat,B5: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B5 )
=> ( ( member_list_nat @ X @ A3 )
=> ( member_list_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_596_in__mono,axiom,
! [A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,X: produc8175205341597284509ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ A3 @ B5 )
=> ( ( member461722496487281478ctxt_a @ X @ A3 )
=> ( member461722496487281478ctxt_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_597_in__mono,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,X: produc1525979690750518759term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( member4351891425000572176term_a @ X @ A3 )
=> ( member4351891425000572176term_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_598_IntE,axiom,
! [C2: list_nat,A3: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A3 @ B5 ) )
=> ~ ( ( member_list_nat @ C2 @ A3 )
=> ~ ( member_list_nat @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_599_IntE,axiom,
! [C2: produc1525979690750518759term_a,A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ C2 @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) )
=> ~ ( ( member4351891425000572176term_a @ C2 @ A3 )
=> ~ ( member4351891425000572176term_a @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_600_IntE,axiom,
! [C2: produc8175205341597284509ctxt_a,A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( inf_in3429776173938890475ctxt_a @ A3 @ B5 ) )
=> ~ ( ( member461722496487281478ctxt_a @ C2 @ A3 )
=> ~ ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_601_IntD1,axiom,
! [C2: list_nat,A3: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A3 @ B5 ) )
=> ( member_list_nat @ C2 @ A3 ) ) ).
% IntD1
thf(fact_602_IntD1,axiom,
! [C2: produc1525979690750518759term_a,A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ C2 @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) )
=> ( member4351891425000572176term_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_603_IntD1,axiom,
! [C2: produc8175205341597284509ctxt_a,A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( inf_in3429776173938890475ctxt_a @ A3 @ B5 ) )
=> ( member461722496487281478ctxt_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_604_IntD2,axiom,
! [C2: list_nat,A3: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A3 @ B5 ) )
=> ( member_list_nat @ C2 @ B5 ) ) ).
% IntD2
thf(fact_605_IntD2,axiom,
! [C2: produc1525979690750518759term_a,A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ C2 @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) )
=> ( member4351891425000572176term_a @ C2 @ B5 ) ) ).
% IntD2
thf(fact_606_IntD2,axiom,
! [C2: produc8175205341597284509ctxt_a,A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( inf_in3429776173938890475ctxt_a @ A3 @ B5 ) )
=> ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ).
% IntD2
thf(fact_607_O__assoc,axiom,
! [R2: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a,T3: set_Pr8282532413760598727term_a] :
( ( relcom4905991001586507054term_a @ ( relcom4905991001586507054term_a @ R2 @ S2 ) @ T3 )
= ( relcom4905991001586507054term_a @ R2 @ ( relcom4905991001586507054term_a @ S2 @ T3 ) ) ) ).
% O_assoc
thf(fact_608_eq__length__concat__nth,axiom,
! [Xs3: list_l4678906206682822203term_a,Ys2: list_l4678906206682822203term_a] :
( ( ( size_s5321119603959729319term_a @ Xs3 )
= ( size_s5321119603959729319term_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Xs3 ) )
=> ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) )
= ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys2 @ I2 ) ) ) )
=> ( ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) )
= ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Ys2 ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_609_pred__equals__eq2,axiom,
! [R2: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a] :
( ( ( ^ [X2: ground_gterm_a,Y2: ground_gterm_a] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ R2 ) )
= ( ^ [X2: ground_gterm_a,Y2: ground_gterm_a] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_610_pred__equals__eq2,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,S2: set_Pr8705490787052945405ctxt_a] :
( ( ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ R2 ) )
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_611_pred__equals__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R2 ) )
= ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_612_pred__equals__eq2,axiom,
! [R2: set_Pr9093778441882193744at_nat,S2: set_Pr9093778441882193744at_nat] :
( ( ( ^ [X2: nat > nat,Y2: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ R2 ) )
= ( ^ [X2: nat > nat,Y2: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_613_Collect__subset,axiom,
! [A3: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_614_Collect__subset,axiom,
! [A3: set_Pr8705490787052945405ctxt_a,P: produc8175205341597284509ctxt_a > $o] :
( ord_le2690539916662146973ctxt_a
@ ( collec5218855173738699400ctxt_a
@ ^ [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_615_Collect__subset,axiom,
! [A3: set_Pr8282532413760598727term_a,P: produc1525979690750518759term_a > $o] :
( ord_le7556488698670074471term_a
@ ( collec6713184107424615762term_a
@ ^ [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_616_inf__Int__eq,axiom,
! [R2: set_list_nat,S2: set_list_nat] :
( ( inf_inf_list_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ R2 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ S2 ) )
= ( ^ [X2: list_nat] : ( member_list_nat @ X2 @ ( inf_inf_set_list_nat @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_617_inf__Int__eq,axiom,
! [R2: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a] :
( ( inf_in1656062065083243624rm_a_o
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ R2 )
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ S2 ) )
= ( ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ ( inf_in4315656883606118581term_a @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_618_inf__Int__eq,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,S2: set_Pr8705490787052945405ctxt_a] :
( ( inf_in204731089199125426xt_a_o
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ R2 )
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ S2 ) )
= ( ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ ( inf_in3429776173938890475ctxt_a @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_619_inf__set__def,axiom,
( inf_inf_set_list_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( collect_list_nat
@ ( inf_inf_list_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A6 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_620_inf__set__def,axiom,
( inf_in4315656883606118581term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( collec6713184107424615762term_a
@ ( inf_in1656062065083243624rm_a_o
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ A6 )
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_621_inf__set__def,axiom,
( inf_in3429776173938890475ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
( collec5218855173738699400ctxt_a
@ ( inf_in204731089199125426xt_a_o
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ A6 )
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_622_Int__Collect,axiom,
! [X: list_nat,A3: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) )
= ( ( member_list_nat @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_623_Int__Collect,axiom,
! [X: produc1525979690750518759term_a,A3: set_Pr8282532413760598727term_a,P: produc1525979690750518759term_a > $o] :
( ( member4351891425000572176term_a @ X @ ( inf_in4315656883606118581term_a @ A3 @ ( collec6713184107424615762term_a @ P ) ) )
= ( ( member4351891425000572176term_a @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_624_Int__Collect,axiom,
! [X: produc8175205341597284509ctxt_a,A3: set_Pr8705490787052945405ctxt_a,P: produc8175205341597284509ctxt_a > $o] :
( ( member461722496487281478ctxt_a @ X @ ( inf_in3429776173938890475ctxt_a @ A3 @ ( collec5218855173738699400ctxt_a @ P ) ) )
= ( ( member461722496487281478ctxt_a @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_625_Int__def,axiom,
( inf_inf_set_list_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A6 )
& ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_626_Int__def,axiom,
( inf_in4315656883606118581term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( collec6713184107424615762term_a
@ ^ [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ A6 )
& ( member4351891425000572176term_a @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_627_Int__def,axiom,
( inf_in3429776173938890475ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
( collec5218855173738699400ctxt_a
@ ^ [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ A6 )
& ( member461722496487281478ctxt_a @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_628_nth__concat__two__lists,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,Ys2: list_l4678906206682822203term_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( ( size_s5321119603959729319term_a @ Ys2 )
= ( size_s5321119603959729319term_a @ Xs3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Xs3 ) )
=> ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys2 @ I2 ) )
= ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s5321119603959729319term_a @ Xs3 ) )
& ( ord_less_nat @ K2 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J2 ) ) )
& ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ I4 )
= ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J2 ) @ K2 ) )
& ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ys2 ) @ I4 )
= ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ys2 @ J2 ) @ K2 ) ) ) ) ) ) ).
% nth_concat_two_lists
thf(fact_629_nth__concat__split,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s5321119603959729319term_a @ Xs3 ) )
& ( ord_less_nat @ K2 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J2 ) ) )
& ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ I4 )
= ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J2 ) @ K2 ) ) ) ) ).
% nth_concat_split
thf(fact_630_concat__all__nth,axiom,
! [Xs3: list_l4678906206682822203term_a,Ys2: list_l4678906206682822203term_a,P: ground_gterm_a > ground_gterm_a > $o] :
( ( ( size_s5321119603959729319term_a @ Xs3 )
= ( size_s5321119603959729319term_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Xs3 ) )
=> ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) )
= ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys2 @ I2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Xs3 ) )
=> ( ( ord_less_nat @ J2 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) ) )
=> ( P @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) @ J2 ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ys2 @ I2 ) @ J2 ) ) ) )
=> ! [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( P @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ K3 ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ys2 ) @ K3 ) ) ) ) ) ) ).
% concat_all_nth
thf(fact_631_concat__nth__nthI,axiom,
! [Ss2: list_l4678906206682822203term_a,Ts2: list_l4678906206682822203term_a,P: ground_gterm_a > ground_gterm_a > $o] :
( ( ( size_s5321119603959729319term_a @ Ss2 )
= ( size_s5321119603959729319term_a @ Ts2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Ts2 ) )
=> ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ss2 @ I2 ) )
= ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ts2 @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Ts2 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ts2 @ I2 ) ) )
=> ( P @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ss2 @ I2 ) @ J2 ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ts2 @ I2 ) @ J2 ) ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Ts2 ) ) )
=> ( P @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ss2 ) @ I ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ts2 ) @ I ) ) ) ) ) ) ).
% concat_nth_nthI
thf(fact_632_nth__concat__diff,axiom,
! [I1: nat,Xs3: list_l4678906206682822203term_a,I22: nat] :
( ( ord_less_nat @ I1 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( ord_less_nat @ I22 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( I1 != I22 )
=> ? [J1: nat,K1: nat,J22: nat,K22: nat] :
( ( ( product_Pair_nat_nat @ J1 @ K1 )
!= ( product_Pair_nat_nat @ J22 @ K22 ) )
& ( ord_less_nat @ J1 @ ( size_s5321119603959729319term_a @ Xs3 ) )
& ( ord_less_nat @ J22 @ ( size_s5321119603959729319term_a @ Xs3 ) )
& ( ord_less_nat @ K1 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J1 ) ) )
& ( ord_less_nat @ K22 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J22 ) ) )
& ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ I1 )
= ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J1 ) @ K1 ) )
& ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ I22 )
= ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ J22 ) @ K22 ) ) ) ) ) ) ).
% nth_concat_diff
thf(fact_633_subrelI,axiom,
! [R3: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ R3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ S ) )
=> ( ord_le2690539916662146973ctxt_a @ R3 @ S ) ) ).
% subrelI
thf(fact_634_subrelI,axiom,
! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X5: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R3 @ S ) ) ).
% subrelI
thf(fact_635_subrelI,axiom,
! [R3: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ! [X5: nat > nat,Y4: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y4 ) @ R3 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y4 ) @ S ) )
=> ( ord_le3678578370064672496at_nat @ R3 @ S ) ) ).
% subrelI
thf(fact_636_subrelI,axiom,
! [R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ R3 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ S ) )
=> ( ord_le7556488698670074471term_a @ R3 @ S ) ) ).
% subrelI
thf(fact_637_Int__mono,axiom,
! [A3: set_Pr8282532413760598727term_a,C: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,D: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ C )
=> ( ( ord_le7556488698670074471term_a @ B5 @ D )
=> ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) @ ( inf_in4315656883606118581term_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_638_Int__lower1,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) @ A3 ) ).
% Int_lower1
thf(fact_639_Int__lower2,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_640_Int__absorb1,axiom,
! [B5: set_Pr8282532413760598727term_a,A3: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ B5 @ A3 )
=> ( ( inf_in4315656883606118581term_a @ A3 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_641_Int__absorb2,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( inf_in4315656883606118581term_a @ A3 @ B5 )
= A3 ) ) ).
% Int_absorb2
thf(fact_642_Int__greatest,axiom,
! [C: set_Pr8282532413760598727term_a,A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ C @ A3 )
=> ( ( ord_le7556488698670074471term_a @ C @ B5 )
=> ( ord_le7556488698670074471term_a @ C @ ( inf_in4315656883606118581term_a @ A3 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_643_Int__Collect__mono,axiom,
! [A3: set_list_nat,B5: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A3 @ B5 )
=> ( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B5 @ ( collect_list_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_644_Int__Collect__mono,axiom,
! [A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,P: produc8175205341597284509ctxt_a > $o,Q: produc8175205341597284509ctxt_a > $o] :
( ( ord_le2690539916662146973ctxt_a @ A3 @ B5 )
=> ( ! [X5: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le2690539916662146973ctxt_a @ ( inf_in3429776173938890475ctxt_a @ A3 @ ( collec5218855173738699400ctxt_a @ P ) ) @ ( inf_in3429776173938890475ctxt_a @ B5 @ ( collec5218855173738699400ctxt_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_645_Int__Collect__mono,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,P: produc1525979690750518759term_a > $o,Q: produc1525979690750518759term_a > $o] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ! [X5: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le7556488698670074471term_a @ ( inf_in4315656883606118581term_a @ A3 @ ( collec6713184107424615762term_a @ P ) ) @ ( inf_in4315656883606118581term_a @ B5 @ ( collec6713184107424615762term_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_646_relcomp_Ocases,axiom,
! [A12: ground_gterm_a,A23: ground_gterm_a,R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( relcom4905991001586507054term_a @ R3 @ S ) )
=> ~ ! [B4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ B4 ) @ R3 )
=> ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_647_relcomp_Ocases,axiom,
! [A12: ground_gmctxt_a,A23: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A12 @ A23 ) @ ( relcom6868745818274832221ctxt_a @ R3 @ S ) )
=> ~ ! [B4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A12 @ B4 ) @ R3 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_648_relcomp_Ocases,axiom,
! [A12: nat,A23: nat,R3: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( relcom6405311860805859042at_nat @ R3 @ S ) )
=> ~ ! [B4: nat > nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A12 @ B4 ) @ R3 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_649_relcomp_Ocases,axiom,
! [A12: nat,A23: nat,R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( relcomp_nat_nat_nat @ R3 @ S ) )
=> ~ ! [B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ B4 ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_650_relcomp_Ocases,axiom,
! [A12: nat > nat,A23: nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A12 @ A23 ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
=> ~ ! [B4: nat > nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A12 @ B4 ) @ R3 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_651_relcomp_Ocases,axiom,
! [A12: nat > nat,A23: nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A12 @ A23 ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
=> ~ ! [B4: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A12 @ B4 ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_652_relcomp_Osimps,axiom,
! [A12: ground_gterm_a,A23: ground_gterm_a,R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( relcom4905991001586507054term_a @ R3 @ S ) )
= ( ? [A2: ground_gterm_a,B2: ground_gterm_a,C5: ground_gterm_a] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A2 @ B2 ) @ R3 )
& ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B2 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_653_relcomp_Osimps,axiom,
! [A12: ground_gmctxt_a,A23: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A12 @ A23 ) @ ( relcom6868745818274832221ctxt_a @ R3 @ S ) )
= ( ? [A2: ground_gmctxt_a,B2: ground_gmctxt_a,C5: ground_gmctxt_a] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A2 @ B2 ) @ R3 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B2 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_654_relcomp_Osimps,axiom,
! [A12: nat,A23: nat,R3: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( relcom6405311860805859042at_nat @ R3 @ S ) )
= ( ? [A2: nat,B2: nat > nat,C5: nat] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A2 @ B2 ) @ R3 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_655_relcomp_Osimps,axiom,
! [A12: nat,A23: nat,R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( relcomp_nat_nat_nat @ R3 @ S ) )
= ( ? [A2: nat,B2: nat,C5: nat] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_656_relcomp_Osimps,axiom,
! [A12: nat > nat,A23: nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A12 @ A23 ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
= ( ? [A2: nat > nat,B2: nat > nat,C5: nat] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A2 @ B2 ) @ R3 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_657_relcomp_Osimps,axiom,
! [A12: nat > nat,A23: nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A12 @ A23 ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
= ( ? [A2: nat > nat,B2: nat,C5: nat] :
( ( A12 = A2 )
& ( A23 = C5 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ R3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_658_relcomp_OrelcompI,axiom,
! [A: nat,B: nat > nat,R3: set_Pr6370437063884598352at_nat,C2: nat,S: set_Pr9093778441882193744at_nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A @ B ) @ R3 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B @ C2 ) @ S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( relcom6405311860805859042at_nat @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_659_relcomp_OrelcompI,axiom,
! [A: nat > nat,B: nat > nat,R3: set_Pr7682762132356531903at_nat,C2: nat,S: set_Pr9093778441882193744at_nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B ) @ R3 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B @ C2 ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C2 ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_660_relcomp_OrelcompI,axiom,
! [A: ground_gterm_a,B: ground_gterm_a,R3: set_Pr8282532413760598727term_a,C2: ground_gterm_a,S: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R3 )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B @ C2 ) @ S )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C2 ) @ ( relcom4905991001586507054term_a @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_661_relcomp_OrelcompI,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ S )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( relcom6868745818274832221ctxt_a @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_662_relcomp_OrelcompI,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C2: nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( relcomp_nat_nat_nat @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_663_relcomp_OrelcompI,axiom,
! [A: nat > nat,B: nat,R3: set_Pr9093778441882193744at_nat,C2: nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C2 ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_664_relcompE,axiom,
! [Xz: produc1525979690750518759term_a,R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ Xz @ ( relcom4905991001586507054term_a @ R3 @ S ) )
=> ~ ! [X5: ground_gterm_a,Y4: ground_gterm_a,Z3: ground_gterm_a] :
( ( Xz
= ( produc7419039783125785943term_a @ X5 @ Z3 ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X5 @ Y4 ) @ R3 )
=> ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_665_relcompE,axiom,
! [Xz: produc8175205341597284509ctxt_a,R3: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ Xz @ ( relcom6868745818274832221ctxt_a @ R3 @ S ) )
=> ~ ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( Xz
= ( produc6347872285926497493ctxt_a @ X5 @ Z3 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) @ R3 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_666_relcompE,axiom,
! [Xz: product_prod_nat_nat,R3: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ Xz @ ( relcom6405311860805859042at_nat @ R3 @ S ) )
=> ~ ! [X5: nat,Y4: nat > nat,Z3: nat] :
( ( Xz
= ( product_Pair_nat_nat @ X5 @ Z3 ) )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ X5 @ Y4 ) @ R3 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_667_relcompE,axiom,
! [Xz: product_prod_nat_nat,R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R3 @ S ) )
=> ~ ! [X5: nat,Y4: nat,Z3: nat] :
( ( Xz
= ( product_Pair_nat_nat @ X5 @ Z3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_668_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
=> ~ ! [X5: nat > nat,Y4: nat > nat,Z3: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X5 @ Z3 ) )
=> ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X5 @ Y4 ) @ R3 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_669_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
=> ~ ! [X5: nat > nat,Y4: nat,Z3: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X5 @ Z3 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y4 ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_670_relcompEpair,axiom,
! [A: ground_gterm_a,C2: ground_gterm_a,R3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C2 ) @ ( relcom4905991001586507054term_a @ R3 @ S ) )
=> ~ ! [B4: ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B4 ) @ R3 )
=> ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B4 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_671_relcompEpair,axiom,
! [A: ground_gmctxt_a,C2: ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( relcom6868745818274832221ctxt_a @ R3 @ S ) )
=> ~ ! [B4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B4 ) @ R3 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B4 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_672_relcompEpair,axiom,
! [A: nat,C2: nat,R3: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( relcom6405311860805859042at_nat @ R3 @ S ) )
=> ~ ! [B4: nat > nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A @ B4 ) @ R3 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B4 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_673_relcompEpair,axiom,
! [A: nat,C2: nat,R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( relcomp_nat_nat_nat @ R3 @ S ) )
=> ~ ! [B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B4 ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_674_relcompEpair,axiom,
! [A: nat > nat,C2: nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C2 ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
=> ~ ! [B4: nat > nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B4 ) @ R3 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B4 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_675_relcompEpair,axiom,
! [A: nat > nat,C2: nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C2 ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
=> ~ ! [B4: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B4 ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_676_relcomp__mono,axiom,
! [R5: set_Pr8282532413760598727term_a,R3: set_Pr8282532413760598727term_a,S3: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ R5 @ R3 )
=> ( ( ord_le7556488698670074471term_a @ S3 @ S )
=> ( ord_le7556488698670074471term_a @ ( relcom4905991001586507054term_a @ R5 @ S3 ) @ ( relcom4905991001586507054term_a @ R3 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_677_pred__subset__eq2,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,S2: set_Pr8705490787052945405ctxt_a] :
( ( ord_le5512775298915626986xt_a_o
@ ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ R2 )
@ ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ S2 ) )
= ( ord_le2690539916662146973ctxt_a @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_678_pred__subset__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R2 )
@ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S2 ) )
= ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_679_pred__subset__eq2,axiom,
! [R2: set_Pr9093778441882193744at_nat,S2: set_Pr9093778441882193744at_nat] :
( ( ord_le5974136086019949911_nat_o
@ ^ [X2: nat > nat,Y2: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ R2 )
@ ^ [X2: nat > nat,Y2: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ S2 ) )
= ( ord_le3678578370064672496at_nat @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_680_pred__subset__eq2,axiom,
! [R2: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a] :
( ( ord_le1594183596158294862rm_a_o
@ ^ [X2: ground_gterm_a,Y2: ground_gterm_a] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ R2 )
@ ^ [X2: ground_gterm_a,Y2: ground_gterm_a] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ S2 ) )
= ( ord_le7556488698670074471term_a @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_681_inf__Int__eq2,axiom,
! [R2: set_Pr8282532413760598727term_a,S2: set_Pr8282532413760598727term_a] :
( ( inf_in8046810946836202908rm_a_o
@ ^ [X2: ground_gterm_a,Y2: ground_gterm_a] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ R2 )
@ ^ [X2: ground_gterm_a,Y2: ground_gterm_a] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ S2 ) )
= ( ^ [X2: ground_gterm_a,Y2: ground_gterm_a] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ ( inf_in4315656883606118581term_a @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_682_inf__Int__eq2,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,S2: set_Pr8705490787052945405ctxt_a] :
( ( inf_in8884196872908837688xt_a_o
@ ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ R2 )
@ ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ S2 ) )
= ( ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ ( inf_in3429776173938890475ctxt_a @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_683_inf__Int__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( inf_inf_nat_nat_o
@ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R2 )
@ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S2 ) )
= ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( inf_in2572325071724192079at_nat @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_684_inf__Int__eq2,axiom,
! [R2: set_Pr9093778441882193744at_nat,S2: set_Pr9093778441882193744at_nat] :
( ( inf_in2844074045483549605_nat_o
@ ^ [X2: nat > nat,Y2: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ R2 )
@ ^ [X2: nat > nat,Y2: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ S2 ) )
= ( ^ [X2: nat > nat,Y2: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ ( inf_in907833683887804734at_nat @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_685_fill__gholes__gmctxt__sound,axiom,
! [Ds2: list_Ground_gmctxt_a,C: ground_gmctxt_a,Sss2: list_l4678906206682822203term_a,Ts2: list_Ground_gterm_a] :
( ( ( size_s962548663205356226ctxt_a @ Ds2 )
= ( ground_num_gholes_a @ C ) )
=> ( ( ( size_s5321119603959729319term_a @ Sss2 )
= ( ground_num_gholes_a @ C ) )
=> ( ( ( size_s1893836384466666913term_a @ Ts2 )
= ( ground_num_gholes_a @ C ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ds2 ) )
=> ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds2 @ I2 ) @ ( nth_li6172171122907299772term_a @ Sss2 @ I2 ) ) ) )
=> ( ground_eq_gfill_a @ ( ground_fill_gholes_a @ C @ Ts2 ) @ ( produc4720362732934906942term_a @ ( ground2530127064035342964ctxt_a @ C @ Ds2 ) @ ( concat1399541994378716528term_a @ Sss2 ) ) ) ) ) ) ) ).
% fill_gholes_gmctxt_sound
thf(fact_686_set__n__lists,axiom,
! [N: nat,Xs3: list_Ground_gterm_a] :
( ( set_li7546925467519454922term_a @ ( n_list1804630026390934205term_a @ N @ Xs3 ) )
= ( collec4295779454384174240term_a
@ ^ [Ys: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Ys )
= N )
& ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ Ys ) @ ( set_Ground_gterm_a2 @ Xs3 ) ) ) ) ) ).
% set_n_lists
thf(fact_687_set__n__lists,axiom,
! [N: nat,Xs3: list_P2433084232546440685term_a] :
( ( set_li5508857501565128962term_a @ ( n_list7796480980296868341term_a @ N @ Xs3 ) )
= ( collec7935380194126400472term_a
@ ^ [Ys: list_P2433084232546440685term_a] :
( ( ( size_s281034491502894169term_a @ Ys )
= N )
& ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Ys ) @ ( set_Pr5405221470436175740term_a @ Xs3 ) ) ) ) ) ).
% set_n_lists
thf(fact_688_comp__gmctxt__inf__ghole__poss__cases,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a,P2: list_nat] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ D ) @ ground_comp_gmctxt_a )
=> ( ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ ( inf_in8301665843363081392ctxt_a @ C @ D ) ) )
=> ( ( ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ C ) )
& ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ D ) ) )
| ( ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ C ) )
& ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ D ) ) )
| ( ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ D ) )
& ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ C ) ) ) ) ) ) ).
% comp_gmctxt_inf_ghole_poss_cases
thf(fact_689_concat__index__split__unique,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,Ys2: list_l4678906206682822203term_a,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( ( size_s5321119603959729319term_a @ Xs3 )
= ( size_s5321119603959729319term_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Xs3 ) )
=> ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) )
= ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys2 @ I2 ) ) ) )
=> ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs3 )
= ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys2 ) ) ) ) ) ).
% concat_index_split_unique
thf(fact_690_assms_I4_J,axiom,
! [C: ground_gmctxt_a,P2: list_nat] :
( ( p @ C )
=> ( ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ C ) )
=> ( q @ ( ground_subgm_at_a @ C @ P2 ) ) ) ) ).
% assms(4)
thf(fact_691_gmctxtex__pred__cmp__subseteq,axiom,
! [P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ! [C4: ground_gmctxt_a,D3: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C4 @ D3 )
=> ( ( P @ C4 )
=> ( ! [X3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ ( ground7778985915038999110args_a @ C4 @ D3 ) ) )
=> ( Q @ X3 ) )
=> ( Q @ D3 ) ) ) )
=> ( ord_le7556488698670074471term_a @ ( contex3086340098647126761_onp_a @ P @ ( contex3086340098647126761_onp_a @ Q @ R ) ) @ ( contex3086340098647126761_onp_a @ Q @ R ) ) ) ).
% gmctxtex_pred_cmp_subseteq
thf(fact_692_fill__gholes__gmctxt__sup__mctxt__args,axiom,
! [C: ground_gmctxt_a,Ds2: list_Ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C )
= ( size_s962548663205356226ctxt_a @ Ds2 ) )
=> ( ( ground7778985915038999110args_a @ C @ ( ground2530127064035342964ctxt_a @ C @ Ds2 ) )
= Ds2 ) ) ).
% fill_gholes_gmctxt_sup_mctxt_args
thf(fact_693_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A6 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_694_less__set__def,axiom,
( ord_le2815447592711331187term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( ord_le8880671632563163434rm_a_o
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ A6 )
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_695_less__set__def,axiom,
( ord_le254475249434039465ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
( ord_le6000898540501895028xt_a_o
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ A6 )
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_696_psubsetD,axiom,
! [A3: set_list_nat,B5: set_list_nat,C2: list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B5 )
=> ( ( member_list_nat @ C2 @ A3 )
=> ( member_list_nat @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_697_psubsetD,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,C2: produc1525979690750518759term_a] :
( ( ord_le2815447592711331187term_a @ A3 @ B5 )
=> ( ( member4351891425000572176term_a @ C2 @ A3 )
=> ( member4351891425000572176term_a @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_698_psubsetD,axiom,
! [A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,C2: produc8175205341597284509ctxt_a] :
( ( ord_le254475249434039465ctxt_a @ A3 @ B5 )
=> ( ( member461722496487281478ctxt_a @ C2 @ A3 )
=> ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_699_ghole__poss__not__in__poss__gmctxt,axiom,
! [P2: list_nat,C: ground_gmctxt_a] :
( ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ C ) )
=> ~ ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ C ) ) ) ).
% ghole_poss_not_in_poss_gmctxt
thf(fact_700_length__n__lists__elem,axiom,
! [Ys2: list_nat,N: nat,Xs3: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs3 ) ) )
=> ( ( size_size_list_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_701_length__n__lists__elem,axiom,
! [Ys2: list_Ground_gterm_a,N: nat,Xs3: list_Ground_gterm_a] :
( ( member5011579089780872286term_a @ Ys2 @ ( set_li7546925467519454922term_a @ ( n_list1804630026390934205term_a @ N @ Xs3 ) ) )
=> ( ( size_s1893836384466666913term_a @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_702_fill__gholes__gmctxt__less__eq,axiom,
! [C: ground_gmctxt_a,Ds2: list_Ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C )
= ( size_s962548663205356226ctxt_a @ Ds2 ) )
=> ( ord_le1940079914304802430ctxt_a @ C @ ( ground2530127064035342964ctxt_a @ C @ Ds2 ) ) ) ).
% fill_gholes_gmctxt_less_eq
thf(fact_703_less__eq__to__sup__mctxt__args,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C @ D )
=> ~ ! [Ds: list_Ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C )
= ( size_s962548663205356226ctxt_a @ Ds ) )
=> ( D
!= ( ground2530127064035342964ctxt_a @ C @ Ds ) ) ) ) ).
% less_eq_to_sup_mctxt_args
thf(fact_704_num__gholes__sup__gmctxt__args,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ D ) @ ground_comp_gmctxt_a )
=> ( ( ground_num_gholes_a @ C )
= ( size_s962548663205356226ctxt_a @ ( ground7778985915038999110args_a @ C @ D ) ) ) ) ).
% num_gholes_sup_gmctxt_args
thf(fact_705_gmctxtex__pred__cmp__subseteq2,axiom,
! [P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ! [C4: ground_gmctxt_a,D3: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C4 @ D3 )
=> ( ( P @ C4 )
=> ( ! [X3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ ( ground7778985915038999110args_a @ C4 @ D3 ) ) )
=> ( Q @ X3 ) )
=> ( P @ D3 ) ) ) )
=> ( ord_le7556488698670074471term_a @ ( contex3086340098647126761_onp_a @ P @ ( contex3086340098647126761_onp_a @ Q @ R ) ) @ ( contex3086340098647126761_onp_a @ P @ R ) ) ) ).
% gmctxtex_pred_cmp_subseteq2
thf(fact_706_poss__gmctxt__fill__gholes__split,axiom,
! [T: ground_gterm_a,C: ground_gmctxt_a,Ts2: list_Ground_gterm_a,P2: list_nat] :
( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C @ Ts2 ) )
=> ( ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ C ) )
=> ( ground_eq_gfill_a @ ( ground_gsubt_at_a @ T @ P2 ) @ ( produc4720362732934906942term_a @ ( ground_subgm_at_a @ C @ P2 ) @ ( ground8448531477516352152term_a @ P2 @ C @ Ts2 ) ) ) ) ) ).
% poss_gmctxt_fill_gholes_split
thf(fact_707_concat__index__split__less__length__concat_I4_J,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,M2: nat,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 )
= ( product_Pair_nat_nat @ M2 @ N ) )
=> ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ I4 )
= ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ M2 ) @ N ) ) ) ) ).
% concat_index_split_less_length_concat(4)
thf(fact_708_concat__index__split__less__length__concat_I2_J,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,M2: nat,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 )
= ( product_Pair_nat_nat @ M2 @ N ) )
=> ( ord_less_nat @ M2 @ ( size_s5321119603959729319term_a @ Xs3 ) ) ) ) ).
% concat_index_split_less_length_concat(2)
thf(fact_709_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_710_boolean__algebra__cancel_Oinf1,axiom,
! [A3: ground_gmctxt_a,K: ground_gmctxt_a,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( A3
= ( inf_in8301665843363081392ctxt_a @ K @ A ) )
=> ( ( inf_in8301665843363081392ctxt_a @ A3 @ B )
= ( inf_in8301665843363081392ctxt_a @ K @ ( inf_in8301665843363081392ctxt_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_711_boolean__algebra__cancel_Oinf2,axiom,
! [B5: ground_gmctxt_a,K: ground_gmctxt_a,B: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( B5
= ( inf_in8301665843363081392ctxt_a @ K @ B ) )
=> ( ( inf_in8301665843363081392ctxt_a @ A @ B5 )
= ( inf_in8301665843363081392ctxt_a @ K @ ( inf_in8301665843363081392ctxt_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_712_length__pos__if__in__set,axiom,
! [X: list_nat,Xs3: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs3 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ).
% length_pos_if_in_set
thf(fact_713_length__pos__if__in__set,axiom,
! [X: produc1525979690750518759term_a,Xs3: list_P2433084232546440685term_a] :
( ( member4351891425000572176term_a @ X @ ( set_Pr5405221470436175740term_a @ Xs3 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s281034491502894169term_a @ Xs3 ) ) ) ).
% length_pos_if_in_set
thf(fact_714_length__pos__if__in__set,axiom,
! [X: produc8175205341597284509ctxt_a,Xs3: list_P2170180374705713187ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s274307637295186063ctxt_a @ Xs3 ) ) ) ).
% length_pos_if_in_set
thf(fact_715_length__pos__if__in__set,axiom,
! [X: ground_gterm_a,Xs3: list_Ground_gterm_a] :
( ( member5261315023172567896term_a @ X @ ( set_Ground_gterm_a2 @ Xs3 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1893836384466666913term_a @ Xs3 ) ) ) ).
% length_pos_if_in_set
thf(fact_716_num__gholes__at__least1,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_a @ C ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_a @ ( inf_in8301665843363081392ctxt_a @ C @ D ) ) ) ) ).
% num_gholes_at_least1
thf(fact_717_concat__index__split__less__length__concat_I3_J,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,M2: nat,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 )
= ( product_Pair_nat_nat @ M2 @ N ) )
=> ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ M2 ) ) ) ) ) ).
% concat_index_split_less_length_concat(3)
thf(fact_718_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_719_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_720_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_721_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_722_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_723_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_724_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_725_size__neq__size__imp__neq,axiom,
! [X: list_Ground_gterm_a,Y: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ X )
!= ( size_s1893836384466666913term_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_726_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_727_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_728_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_729_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_730_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_731_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_732_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_733_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_734_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_735_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_736_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J3: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_737_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_738_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_739_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_740_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_741_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_742_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_743_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_744_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_745_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_746_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_747_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_748_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_749_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_750_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_751_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_752_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_753_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_754_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_755_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_756_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_757_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_758_fill__gholes__gmctxt__ghole__poss__subseteq,axiom,
! [C: ground_gmctxt_a,Ds2: list_Ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C )
= ( size_s962548663205356226ctxt_a @ Ds2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ds2 ) )
=> ( ( ( nth_Ground_gmctxt_a @ Ds2 @ I2 )
= ground_GMHole_a )
| ( ( ground_num_gholes_a @ ( nth_Ground_gmctxt_a @ Ds2 @ I2 ) )
= zero_zero_nat ) ) )
=> ( ord_le6045566169113846134st_nat @ ( ground_ghole_poss_a @ ( ground2530127064035342964ctxt_a @ C @ Ds2 ) ) @ ( ground_ghole_poss_a @ C ) ) ) ) ).
% fill_gholes_gmctxt_ghole_poss_subseteq
thf(fact_759_ghole__num__at__pos__fill__args__length,axiom,
! [P2: list_nat,C: ground_gmctxt_a,Ts2: list_Ground_gterm_a] :
( ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ C ) )
=> ( ( ( ground_num_gholes_a @ C )
= ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( ( ground5706155212172713090_pos_a @ P2 @ C )
= ( size_s1893836384466666913term_a @ ( ground8448531477516352152term_a @ P2 @ C @ Ts2 ) ) ) ) ) ).
% ghole_num_at_pos_fill_args_length
thf(fact_760_concat__index__split__sound,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ I4 )
= ( produc5250376639289121761term_a
@ ^ [K4: nat] : ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ K4 ) )
@ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 ) ) ) ) ).
% concat_index_split_sound
thf(fact_761_concat__index__split__sound,axiom,
! [I4: nat,Xs3: list_l3264859301627795341at_nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ ( concat7691415812945658306at_nat @ Xs3 ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( concat7691415812945658306at_nat @ Xs3 ) @ I4 )
= ( produc2626176000494625587at_nat
@ ^ [K4: nat] : ( nth_Pr7617993195940197384at_nat @ ( nth_li852852631024258574at_nat @ Xs3 @ K4 ) )
@ ( concat2033898366915437183at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 ) ) ) ) ).
% concat_index_split_sound
thf(fact_762_inv__imples__ghole__poss__subseteq,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C @ D )
=> ( ! [X5: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X5 @ ( set_Ground_gmctxt_a2 @ ( ground7778985915038999110args_a @ C @ D ) ) )
=> ( ( X5 = ground_GMHole_a )
| ( ( ground_num_gholes_a @ X5 )
= zero_zero_nat ) ) )
=> ( ord_le6045566169113846134st_nat @ ( ground_ghole_poss_a @ D ) @ ( ground_ghole_poss_a @ C ) ) ) ) ).
% inv_imples_ghole_poss_subseteq
thf(fact_763_no__gholes__gmctxt__of__gterm__gterm__of__gmctxt__id,axiom,
! [C: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C )
= zero_zero_nat )
=> ( ( ground582559344758063963term_a @ ( ground2297065866753446215ctxt_a @ C ) )
= C ) ) ).
% no_gholes_gmctxt_of_gterm_gterm_of_gmctxt_id
thf(fact_764_inf__gmctxt__GMHole2,axiom,
! [C: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ C @ ground_GMHole_a )
= ground_GMHole_a ) ).
% inf_gmctxt_GMHole2
thf(fact_765_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_766_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_767_gmctxtex__onp__arg__monoI,axiom,
! [P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ( P @ ground_GMHole_a )
=> ( ord_le7556488698670074471term_a @ R @ ( contex3086340098647126761_onp_a @ P @ R ) ) ) ).
% gmctxtex_onp_arg_monoI
thf(fact_768_subgm__at__hole__poss,axiom,
! [P2: list_nat,C: ground_gmctxt_a] :
( ( member_list_nat @ P2 @ ( ground_ghole_poss_a @ C ) )
=> ( ( ground_subgm_at_a @ C @ P2 )
= ground_GMHole_a ) ) ).
% subgm_at_hole_poss
thf(fact_769_gmctxtex__onp__idem,axiom,
! [P: ground_gmctxt_a > $o,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
( ( P @ ground_GMHole_a )
=> ( ! [C4: ground_gmctxt_a,D3: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ C4 @ D3 )
=> ( ( P @ C4 )
=> ( ! [X3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ ( ground7778985915038999110args_a @ C4 @ D3 ) ) )
=> ( Q @ X3 ) )
=> ( Q @ D3 ) ) ) )
=> ( ( contex3086340098647126761_onp_a @ P @ ( contex3086340098647126761_onp_a @ Q @ R ) )
= ( contex3086340098647126761_onp_a @ Q @ R ) ) ) ) ).
% gmctxtex_onp_idem
thf(fact_770_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_771_le__trans,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_772_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_773_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_774_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_775_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_776_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat,X1: nat,X22: nat] :
( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_777_prod_Ocase__distrib,axiom,
! [H: product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( H @ ( produc2626176000494625587at_nat @ F @ Prod ) )
= ( produc2626176000494625587at_nat
@ ^ [X12: nat,X23: nat] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_778_less__eq__gmctxt__intros_I1_J,axiom,
! [U: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ground_GMHole_a @ U ) ).
% less_eq_gmctxt_intros(1)
thf(fact_779_gmctxt__order__bot_Obot_Oextremum,axiom,
! [A: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ ground_GMHole_a @ A ) ).
% gmctxt_order_bot.bot.extremum
thf(fact_780_gmctxt__order__bot_Obot_Oextremum__unique,axiom,
! [A: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ ground_GMHole_a )
= ( A = ground_GMHole_a ) ) ).
% gmctxt_order_bot.bot.extremum_unique
thf(fact_781_gmctxt__order__bot_Obot_Oextremum__uniqueI,axiom,
! [A: ground_gmctxt_a] :
( ( ord_le1940079914304802430ctxt_a @ A @ ground_GMHole_a )
=> ( A = ground_GMHole_a ) ) ).
% gmctxt_order_bot.bot.extremum_uniqueI
thf(fact_782_inf__gmctxt_Osimps_I1_J,axiom,
! [D: ground_gmctxt_a] :
( ( inf_in8301665843363081392ctxt_a @ ground_GMHole_a @ D )
= ground_GMHole_a ) ).
% inf_gmctxt.simps(1)
thf(fact_783_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X5: nat,Y4: nat] :
( ( F @ X5 @ Y4 )
= ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
=> ( ( produc2626176000494625587at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_784_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat] :
( ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_785_case__prodE2,axiom,
! [Q: product_prod_nat_nat > $o,P: nat > nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( Q @ ( produc2626176000494625587at_nat @ P @ Z ) )
=> ~ ! [X5: nat,Y4: nat] :
( ( Z
= ( product_Pair_nat_nat @ X5 @ Y4 ) )
=> ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_786_comp__gmctxt_OGMHole1,axiom,
! [D: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ground_GMHole_a @ D ) @ ground_comp_gmctxt_a ) ).
% comp_gmctxt.GMHole1
thf(fact_787_comp__gmctxt_OGMHole2,axiom,
! [C: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ ground_GMHole_a ) @ ground_comp_gmctxt_a ) ).
% comp_gmctxt.GMHole2
thf(fact_788_fill__gholes__gmctxt__nHole,axiom,
! [C: ground_gmctxt_a,Ds2: list_Ground_gmctxt_a] :
( ( C != ground_GMHole_a )
=> ( ( ( ground_num_gholes_a @ C )
= ( size_s962548663205356226ctxt_a @ Ds2 ) )
=> ( ( ground2530127064035342964ctxt_a @ C @ Ds2 )
!= ground_GMHole_a ) ) ) ).
% fill_gholes_gmctxt_nHole
thf(fact_789_num__gholes__subgm__at,axiom,
! [P2: list_nat,C: ground_gmctxt_a] :
( ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ C ) )
=> ( ( ground_num_gholes_a @ ( ground_subgm_at_a @ C @ P2 ) )
= ( ground5706155212172713090_pos_a @ P2 @ C ) ) ) ).
% num_gholes_subgm_at
thf(fact_790_gmctxt__closing__sup__args__ghole__or__gterm,axiom,
! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
( ( ground6954545657870034594sing_a @ C @ D )
=> ! [X3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ ( ground7778985915038999110args_a @ C @ D ) ) )
=> ( ( X3 = ground_GMHole_a )
| ( ( ground_num_gholes_a @ X3 )
= zero_zero_nat ) ) ) ) ).
% gmctxt_closing_sup_args_ghole_or_gterm
thf(fact_791_concat__index__split__split__sound__aux,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs3 ) @ I4 )
= ( produc5250376639289121761term_a
@ ^ [K4: nat] : ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs3 @ ( minus_minus_nat @ K4 @ N ) ) )
@ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs3 ) ) ) ) ).
% concat_index_split_split_sound_aux
thf(fact_792_concat__index__split__split__sound__aux,axiom,
! [I4: nat,Xs3: list_l3264859301627795341at_nat,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ ( concat7691415812945658306at_nat @ Xs3 ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( concat7691415812945658306at_nat @ Xs3 ) @ I4 )
= ( produc2626176000494625587at_nat
@ ^ [K4: nat] : ( nth_Pr7617993195940197384at_nat @ ( nth_li852852631024258574at_nat @ Xs3 @ ( minus_minus_nat @ K4 @ N ) ) )
@ ( concat2033898366915437183at_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs3 ) ) ) ) ).
% concat_index_split_split_sound_aux
thf(fact_793_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C5: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_794_gmctxt__order__bot_Omin__bot2,axiom,
! [X: ground_gmctxt_a] :
( ( min_Ground_gmctxt_a @ ord_le1940079914304802430ctxt_a @ X @ ground_GMHole_a )
= ground_GMHole_a ) ).
% gmctxt_order_bot.min_bot2
thf(fact_795_gmctxt__order__bot_Omax__bot2,axiom,
! [X: ground_gmctxt_a] :
( ( max_Ground_gmctxt_a @ ord_le1940079914304802430ctxt_a @ X @ ground_GMHole_a )
= X ) ).
% gmctxt_order_bot.max_bot2
thf(fact_796_case__prodI,axiom,
! [F: ground_gterm_a > ground_gterm_a > $o,A: ground_gterm_a,B: ground_gterm_a] :
( ( F @ A @ B )
=> ( produc7608081248467820332rm_a_o @ F @ ( produc7419039783125785943term_a @ A @ B ) ) ) ).
% case_prodI
thf(fact_797_case__prodI,axiom,
! [F: ground_gmctxt_a > ground_gmctxt_a > $o,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( F @ A @ B )
=> ( produc4116768209347204654xt_a_o @ F @ ( produc6347872285926497493ctxt_a @ A @ B ) ) ) ).
% case_prodI
thf(fact_798_case__prodI,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( F @ A @ B )
=> ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_799_case__prodI,axiom,
! [F: ( nat > nat ) > nat > $o,A: nat > nat,B: nat] :
( ( F @ A @ B )
=> ( produc1191635805960096411_nat_o @ F @ ( produc72220940542539688at_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_800_case__prodI2,axiom,
! [P2: produc1525979690750518759term_a,C2: ground_gterm_a > ground_gterm_a > $o] :
( ! [A5: ground_gterm_a,B4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ A5 @ B4 ) )
=> ( C2 @ A5 @ B4 ) )
=> ( produc7608081248467820332rm_a_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_801_case__prodI2,axiom,
! [P2: produc8175205341597284509ctxt_a,C2: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ A5 @ B4 ) )
=> ( C2 @ A5 @ B4 ) )
=> ( produc4116768209347204654xt_a_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_802_case__prodI2,axiom,
! [P2: product_prod_nat_nat,C2: nat > nat > $o] :
( ! [A5: nat,B4: nat] :
( ( P2
= ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ( C2 @ A5 @ B4 ) )
=> ( produc6081775807080527818_nat_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_803_case__prodI2,axiom,
! [P2: produc8199716216217303280at_nat,C2: ( nat > nat ) > nat > $o] :
( ! [A5: nat > nat,B4: nat] :
( ( P2
= ( produc72220940542539688at_nat @ A5 @ B4 ) )
=> ( C2 @ A5 @ B4 ) )
=> ( produc1191635805960096411_nat_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_804_mem__case__prodI,axiom,
! [Z: list_nat,C2: nat > nat > set_list_nat,A: nat,B: nat] :
( ( member_list_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_list_nat @ Z @ ( produc5054468372610444900st_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_805_mem__case__prodI,axiom,
! [Z: list_nat,C2: ground_gterm_a > ground_gterm_a > set_list_nat,A: ground_gterm_a,B: ground_gterm_a] :
( ( member_list_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_list_nat @ Z @ ( produc4288096804835968130st_nat @ C2 @ ( produc7419039783125785943term_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_806_mem__case__prodI,axiom,
! [Z: list_nat,C2: ground_gmctxt_a > ground_gmctxt_a > set_list_nat,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( member_list_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_list_nat @ Z @ ( produc7782890255398601472st_nat @ C2 @ ( produc6347872285926497493ctxt_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_807_mem__case__prodI,axiom,
! [Z: list_nat,C2: ( nat > nat ) > nat > set_list_nat,A: nat > nat,B: nat] :
( ( member_list_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_list_nat @ Z @ ( produc6565588688694178387st_nat @ C2 @ ( produc72220940542539688at_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_808_mem__case__prodI,axiom,
! [Z: produc1525979690750518759term_a,C2: nat > nat > set_Pr8282532413760598727term_a,A: nat,B: nat] :
( ( member4351891425000572176term_a @ Z @ ( C2 @ A @ B ) )
=> ( member4351891425000572176term_a @ Z @ ( produc2502414538954563065term_a @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_809_mem__case__prodI,axiom,
! [Z: produc8175205341597284509ctxt_a,C2: nat > nat > set_Pr8705490787052945405ctxt_a,A: nat,B: nat] :
( ( member461722496487281478ctxt_a @ Z @ ( C2 @ A @ B ) )
=> ( member461722496487281478ctxt_a @ Z @ ( produc962762980789875247ctxt_a @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_810_mem__case__prodI,axiom,
! [Z: produc1525979690750518759term_a,C2: ground_gterm_a > ground_gterm_a > set_Pr8282532413760598727term_a,A: ground_gterm_a,B: ground_gterm_a] :
( ( member4351891425000572176term_a @ Z @ ( C2 @ A @ B ) )
=> ( member4351891425000572176term_a @ Z @ ( produc281830614751001691term_a @ C2 @ ( produc7419039783125785943term_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_811_mem__case__prodI,axiom,
! [Z: produc8175205341597284509ctxt_a,C2: ground_gterm_a > ground_gterm_a > set_Pr8705490787052945405ctxt_a,A: ground_gterm_a,B: ground_gterm_a] :
( ( member461722496487281478ctxt_a @ Z @ ( C2 @ A @ B ) )
=> ( member461722496487281478ctxt_a @ Z @ ( produc8348828404203306897ctxt_a @ C2 @ ( produc7419039783125785943term_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_812_mem__case__prodI,axiom,
! [Z: produc1525979690750518759term_a,C2: ground_gmctxt_a > ground_gmctxt_a > set_Pr8282532413760598727term_a,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( member4351891425000572176term_a @ Z @ ( C2 @ A @ B ) )
=> ( member4351891425000572176term_a @ Z @ ( produc4728407230358510173term_a @ C2 @ ( produc6347872285926497493ctxt_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_813_mem__case__prodI,axiom,
! [Z: produc8175205341597284509ctxt_a,C2: ground_gmctxt_a > ground_gmctxt_a > set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ Z @ ( C2 @ A @ B ) )
=> ( member461722496487281478ctxt_a @ Z @ ( produc4706414670047607955ctxt_a @ C2 @ ( produc6347872285926497493ctxt_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_814_mem__case__prodI2,axiom,
! [P2: product_prod_nat_nat,Z: list_nat,C2: nat > nat > set_list_nat] :
( ! [A5: nat,B4: nat] :
( ( P2
= ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ( member_list_nat @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member_list_nat @ Z @ ( produc5054468372610444900st_nat @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_815_mem__case__prodI2,axiom,
! [P2: produc1525979690750518759term_a,Z: list_nat,C2: ground_gterm_a > ground_gterm_a > set_list_nat] :
( ! [A5: ground_gterm_a,B4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ A5 @ B4 ) )
=> ( member_list_nat @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member_list_nat @ Z @ ( produc4288096804835968130st_nat @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_816_mem__case__prodI2,axiom,
! [P2: produc8175205341597284509ctxt_a,Z: list_nat,C2: ground_gmctxt_a > ground_gmctxt_a > set_list_nat] :
( ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ A5 @ B4 ) )
=> ( member_list_nat @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member_list_nat @ Z @ ( produc7782890255398601472st_nat @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_817_mem__case__prodI2,axiom,
! [P2: produc8199716216217303280at_nat,Z: list_nat,C2: ( nat > nat ) > nat > set_list_nat] :
( ! [A5: nat > nat,B4: nat] :
( ( P2
= ( produc72220940542539688at_nat @ A5 @ B4 ) )
=> ( member_list_nat @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member_list_nat @ Z @ ( produc6565588688694178387st_nat @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_818_mem__case__prodI2,axiom,
! [P2: product_prod_nat_nat,Z: produc1525979690750518759term_a,C2: nat > nat > set_Pr8282532413760598727term_a] :
( ! [A5: nat,B4: nat] :
( ( P2
= ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ( member4351891425000572176term_a @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member4351891425000572176term_a @ Z @ ( produc2502414538954563065term_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_819_mem__case__prodI2,axiom,
! [P2: product_prod_nat_nat,Z: produc8175205341597284509ctxt_a,C2: nat > nat > set_Pr8705490787052945405ctxt_a] :
( ! [A5: nat,B4: nat] :
( ( P2
= ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ( member461722496487281478ctxt_a @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member461722496487281478ctxt_a @ Z @ ( produc962762980789875247ctxt_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_820_mem__case__prodI2,axiom,
! [P2: produc1525979690750518759term_a,Z: produc1525979690750518759term_a,C2: ground_gterm_a > ground_gterm_a > set_Pr8282532413760598727term_a] :
( ! [A5: ground_gterm_a,B4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ A5 @ B4 ) )
=> ( member4351891425000572176term_a @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member4351891425000572176term_a @ Z @ ( produc281830614751001691term_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_821_mem__case__prodI2,axiom,
! [P2: produc1525979690750518759term_a,Z: produc8175205341597284509ctxt_a,C2: ground_gterm_a > ground_gterm_a > set_Pr8705490787052945405ctxt_a] :
( ! [A5: ground_gterm_a,B4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ A5 @ B4 ) )
=> ( member461722496487281478ctxt_a @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member461722496487281478ctxt_a @ Z @ ( produc8348828404203306897ctxt_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_822_mem__case__prodI2,axiom,
! [P2: produc8175205341597284509ctxt_a,Z: produc1525979690750518759term_a,C2: ground_gmctxt_a > ground_gmctxt_a > set_Pr8282532413760598727term_a] :
( ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ A5 @ B4 ) )
=> ( member4351891425000572176term_a @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member4351891425000572176term_a @ Z @ ( produc4728407230358510173term_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_823_mem__case__prodI2,axiom,
! [P2: produc8175205341597284509ctxt_a,Z: produc8175205341597284509ctxt_a,C2: ground_gmctxt_a > ground_gmctxt_a > set_Pr8705490787052945405ctxt_a] :
( ! [A5: ground_gmctxt_a,B4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ A5 @ B4 ) )
=> ( member461722496487281478ctxt_a @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member461722496487281478ctxt_a @ Z @ ( produc4706414670047607955ctxt_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_824_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_825_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_826_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_827_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_828_gmctxt__order__bot_Omax__bot,axiom,
! [X: ground_gmctxt_a] :
( ( max_Ground_gmctxt_a @ ord_le1940079914304802430ctxt_a @ ground_GMHole_a @ X )
= X ) ).
% gmctxt_order_bot.max_bot
thf(fact_829_gmctxt__order__bot_Omin__bot,axiom,
! [X: ground_gmctxt_a] :
( ( min_Ground_gmctxt_a @ ord_le1940079914304802430ctxt_a @ ground_GMHole_a @ X )
= ground_GMHole_a ) ).
% gmctxt_order_bot.min_bot
thf(fact_830_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_831_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_832_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_833_diff__le__mono,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_834_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_835_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_836_diff__le__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_837_mem__case__prodE,axiom,
! [Z: list_nat,C2: nat > nat > set_list_nat,P2: product_prod_nat_nat] :
( ( member_list_nat @ Z @ ( produc5054468372610444900st_nat @ C2 @ P2 ) )
=> ~ ! [X5: nat,Y4: nat] :
( ( P2
= ( product_Pair_nat_nat @ X5 @ Y4 ) )
=> ~ ( member_list_nat @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_838_mem__case__prodE,axiom,
! [Z: list_nat,C2: ground_gterm_a > ground_gterm_a > set_list_nat,P2: produc1525979690750518759term_a] :
( ( member_list_nat @ Z @ ( produc4288096804835968130st_nat @ C2 @ P2 ) )
=> ~ ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ X5 @ Y4 ) )
=> ~ ( member_list_nat @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_839_mem__case__prodE,axiom,
! [Z: list_nat,C2: ground_gmctxt_a > ground_gmctxt_a > set_list_nat,P2: produc8175205341597284509ctxt_a] :
( ( member_list_nat @ Z @ ( produc7782890255398601472st_nat @ C2 @ P2 ) )
=> ~ ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) )
=> ~ ( member_list_nat @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_840_mem__case__prodE,axiom,
! [Z: list_nat,C2: ( nat > nat ) > nat > set_list_nat,P2: produc8199716216217303280at_nat] :
( ( member_list_nat @ Z @ ( produc6565588688694178387st_nat @ C2 @ P2 ) )
=> ~ ! [X5: nat > nat,Y4: nat] :
( ( P2
= ( produc72220940542539688at_nat @ X5 @ Y4 ) )
=> ~ ( member_list_nat @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_841_mem__case__prodE,axiom,
! [Z: produc1525979690750518759term_a,C2: nat > nat > set_Pr8282532413760598727term_a,P2: product_prod_nat_nat] :
( ( member4351891425000572176term_a @ Z @ ( produc2502414538954563065term_a @ C2 @ P2 ) )
=> ~ ! [X5: nat,Y4: nat] :
( ( P2
= ( product_Pair_nat_nat @ X5 @ Y4 ) )
=> ~ ( member4351891425000572176term_a @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_842_mem__case__prodE,axiom,
! [Z: produc8175205341597284509ctxt_a,C2: nat > nat > set_Pr8705490787052945405ctxt_a,P2: product_prod_nat_nat] :
( ( member461722496487281478ctxt_a @ Z @ ( produc962762980789875247ctxt_a @ C2 @ P2 ) )
=> ~ ! [X5: nat,Y4: nat] :
( ( P2
= ( product_Pair_nat_nat @ X5 @ Y4 ) )
=> ~ ( member461722496487281478ctxt_a @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_843_mem__case__prodE,axiom,
! [Z: produc1525979690750518759term_a,C2: ground_gterm_a > ground_gterm_a > set_Pr8282532413760598727term_a,P2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ Z @ ( produc281830614751001691term_a @ C2 @ P2 ) )
=> ~ ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ X5 @ Y4 ) )
=> ~ ( member4351891425000572176term_a @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_844_mem__case__prodE,axiom,
! [Z: produc8175205341597284509ctxt_a,C2: ground_gterm_a > ground_gterm_a > set_Pr8705490787052945405ctxt_a,P2: produc1525979690750518759term_a] :
( ( member461722496487281478ctxt_a @ Z @ ( produc8348828404203306897ctxt_a @ C2 @ P2 ) )
=> ~ ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ X5 @ Y4 ) )
=> ~ ( member461722496487281478ctxt_a @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_845_mem__case__prodE,axiom,
! [Z: produc1525979690750518759term_a,C2: ground_gmctxt_a > ground_gmctxt_a > set_Pr8282532413760598727term_a,P2: produc8175205341597284509ctxt_a] :
( ( member4351891425000572176term_a @ Z @ ( produc4728407230358510173term_a @ C2 @ P2 ) )
=> ~ ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) )
=> ~ ( member4351891425000572176term_a @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_846_mem__case__prodE,axiom,
! [Z: produc8175205341597284509ctxt_a,C2: ground_gmctxt_a > ground_gmctxt_a > set_Pr8705490787052945405ctxt_a,P2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Z @ ( produc4706414670047607955ctxt_a @ C2 @ P2 ) )
=> ~ ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) )
=> ~ ( member461722496487281478ctxt_a @ Z @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_847_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N: nat] :
( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_848_diff__less__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_849_case__prodD,axiom,
! [F: ground_gterm_a > ground_gterm_a > $o,A: ground_gterm_a,B: ground_gterm_a] :
( ( produc7608081248467820332rm_a_o @ F @ ( produc7419039783125785943term_a @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_850_case__prodD,axiom,
! [F: ground_gmctxt_a > ground_gmctxt_a > $o,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( produc4116768209347204654xt_a_o @ F @ ( produc6347872285926497493ctxt_a @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_851_case__prodD,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_852_case__prodD,axiom,
! [F: ( nat > nat ) > nat > $o,A: nat > nat,B: nat] :
( ( produc1191635805960096411_nat_o @ F @ ( produc72220940542539688at_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_853_case__prodE,axiom,
! [C2: ground_gterm_a > ground_gterm_a > $o,P2: produc1525979690750518759term_a] :
( ( produc7608081248467820332rm_a_o @ C2 @ P2 )
=> ~ ! [X5: ground_gterm_a,Y4: ground_gterm_a] :
( ( P2
= ( produc7419039783125785943term_a @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_854_case__prodE,axiom,
! [C2: ground_gmctxt_a > ground_gmctxt_a > $o,P2: produc8175205341597284509ctxt_a] :
( ( produc4116768209347204654xt_a_o @ C2 @ P2 )
=> ~ ! [X5: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( P2
= ( produc6347872285926497493ctxt_a @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_855_case__prodE,axiom,
! [C2: nat > nat > $o,P2: product_prod_nat_nat] :
( ( produc6081775807080527818_nat_o @ C2 @ P2 )
=> ~ ! [X5: nat,Y4: nat] :
( ( P2
= ( product_Pair_nat_nat @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_856_case__prodE,axiom,
! [C2: ( nat > nat ) > nat > $o,P2: produc8199716216217303280at_nat] :
( ( produc1191635805960096411_nat_o @ C2 @ P2 )
=> ~ ! [X5: nat > nat,Y4: nat] :
( ( P2
= ( produc72220940542539688at_nat @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_857_Collect__case__prod__mono,axiom,
! [A3: ground_gterm_a > ground_gterm_a > $o,B5: ground_gterm_a > ground_gterm_a > $o] :
( ( ord_le1594183596158294862rm_a_o @ A3 @ B5 )
=> ( ord_le7556488698670074471term_a @ ( collec6713184107424615762term_a @ ( produc7608081248467820332rm_a_o @ A3 ) ) @ ( collec6713184107424615762term_a @ ( produc7608081248467820332rm_a_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_858_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_859_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_860_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_861_gmctxt__closing__def,axiom,
( ground6954545657870034594sing_a
= ( ^ [C3: ground_gmctxt_a,D2: ground_gmctxt_a] :
( ( ground6477899500129135038ctxt_a @ C3 @ D2 )
& ( ord_le6045566169113846134st_nat @ ( ground_ghole_poss_a @ D2 ) @ ( ground_ghole_poss_a @ C3 ) ) ) ) ) ).
% gmctxt_closing_def
thf(fact_862_concat__index__split__sound__fst__arg,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ord_less_nat @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 ) ) @ ( size_s5321119603959729319term_a @ Xs3 ) ) ) ).
% concat_index_split_sound_fst_arg
thf(fact_863_gmctxt__order__bot_Obot_Oordering__top__axioms,axiom,
( orderi7909243889287335981ctxt_a
@ ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ Y2 @ X2 )
@ ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] : ( ord_le9048903087952742514ctxt_a @ Y2 @ X2 )
@ ground_GMHole_a ) ).
% gmctxt_order_bot.bot.ordering_top_axioms
thf(fact_864_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P @ I ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_865_Diff__mono,axiom,
! [A3: set_Pr8282532413760598727term_a,C: set_Pr8282532413760598727term_a,D: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ C )
=> ( ( ord_le7556488698670074471term_a @ D @ B5 )
=> ( ord_le7556488698670074471term_a @ ( minus_551215046067487790term_a @ A3 @ B5 ) @ ( minus_551215046067487790term_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_866_Diff__subset,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ ( minus_551215046067487790term_a @ A3 @ B5 ) @ A3 ) ).
% Diff_subset
thf(fact_867_double__diff,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a,C: set_Pr8282532413760598727term_a] :
( ( ord_le7556488698670074471term_a @ A3 @ B5 )
=> ( ( ord_le7556488698670074471term_a @ B5 @ C )
=> ( ( minus_551215046067487790term_a @ B5 @ ( minus_551215046067487790term_a @ C @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_868_psubset__imp__ex__mem,axiom,
! [A3: set_list_nat,B5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B5 )
=> ? [B4: list_nat] : ( member_list_nat @ B4 @ ( minus_7954133019191499631st_nat @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_869_psubset__imp__ex__mem,axiom,
! [A3: set_Pr8282532413760598727term_a,B5: set_Pr8282532413760598727term_a] :
( ( ord_le2815447592711331187term_a @ A3 @ B5 )
=> ? [B4: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ B4 @ ( minus_551215046067487790term_a @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_870_psubset__imp__ex__mem,axiom,
! [A3: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ord_le254475249434039465ctxt_a @ A3 @ B5 )
=> ? [B4: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ B4 @ ( minus_3912115830687649252ctxt_a @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_871_fst__eqD,axiom,
! [X: ground_gterm_a,Y: ground_gterm_a,A: ground_gterm_a] :
( ( ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_872_fst__eqD,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_873_fst__eqD,axiom,
! [X: nat > nat,Y: nat,A: nat > nat] :
( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_874_fst__eqD,axiom,
! [X: nat,Y: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_875_fst__conv,axiom,
! [X1: ground_gterm_a,X22: ground_gterm_a] :
( ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_876_fst__conv,axiom,
! [X1: ground_gmctxt_a,X22: ground_gmctxt_a] :
( ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_877_fst__conv,axiom,
! [X1: nat > nat,X22: nat] :
( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_878_fst__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_879_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_880_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_881_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_882_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_883_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_884_fst__def,axiom,
( product_fst_nat_nat
= ( produc6842872674320459806at_nat
@ ^ [X12: nat,X23: nat] : X12 ) ) ).
% fst_def
thf(fact_885_less__eq__gmctxt__prime,axiom,
ord_le1940079914304802430ctxt_a = ground6477899500129135038ctxt_a ).
% less_eq_gmctxt_prime
thf(fact_886_concat__index__split__mono__first__arg,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,O_idx: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs3 ) ) ) ) ).
% concat_index_split_mono_first_arg
thf(fact_887_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat,P2: product_prod_nat_nat] :
( ! [X5: nat,Y4: nat] :
( ( ( product_Pair_nat_nat @ X5 @ Y4 )
= Q2 )
=> ( ( F @ X5 @ Y4 )
= ( G @ X5 @ Y4 ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc2626176000494625587at_nat @ F @ P2 )
= ( produc2626176000494625587at_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_888_concat__index__split__sound__snd__arg__aux,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ord_less_nat @ ( product_snd_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs3 ) ) @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs3 ) ) @ N ) ) ) ) ) ).
% concat_index_split_sound_snd_arg_aux
thf(fact_889_concat__index__split__sound__snd__arg,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ord_less_nat @ ( product_snd_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 ) ) @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs3 ) ) ) ) ) ) ).
% concat_index_split_sound_snd_arg
thf(fact_890_concat__index__split__sound__fst__arg__aux,axiom,
! [I4: nat,Xs3: list_l4678906206682822203term_a,O_idx: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs3 ) ) )
=> ( ord_less_nat @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs3 ) ) @ ( plus_plus_nat @ ( size_s5321119603959729319term_a @ Xs3 ) @ O_idx ) ) ) ).
% concat_index_split_sound_fst_arg_aux
thf(fact_891_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_892_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_893_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_894_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_895_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_896_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_897_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_898_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_899_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_900_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_901_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_902_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_903_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_904_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_905_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_906_prod_Ocollapse,axiom,
! [Prod: produc1525979690750518759term_a] :
( ( produc7419039783125785943term_a @ ( produc5097907526672912195term_a @ Prod ) @ ( produc5323275678763266181term_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_907_prod_Ocollapse,axiom,
! [Prod: produc8175205341597284509ctxt_a] :
( ( produc6347872285926497493ctxt_a @ ( produc844137399493384385ctxt_a @ Prod ) @ ( produc3308216634374912259ctxt_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_908_prod_Ocollapse,axiom,
! [Prod: produc8199716216217303280at_nat] :
( ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_909_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_910_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_911_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_912_Nat_Odiff__diff__right,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_913_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A6 )
& ~ ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_914_set__diff__eq,axiom,
( minus_551215046067487790term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( collec6713184107424615762term_a
@ ^ [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ A6 )
& ~ ( member4351891425000572176term_a @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_915_set__diff__eq,axiom,
( minus_3912115830687649252ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
( collec5218855173738699400ctxt_a
@ ^ [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ A6 )
& ~ ( member461722496487281478ctxt_a @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_916_minus__set__def,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( collect_list_nat
@ ( minus_1139252259498527702_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A6 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_917_minus__set__def,axiom,
( minus_551215046067487790term_a
= ( ^ [A6: set_Pr8282532413760598727term_a,B6: set_Pr8282532413760598727term_a] :
( collec6713184107424615762term_a
@ ( minus_7967641318176996911rm_a_o
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ A6 )
@ ^ [X2: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_918_minus__set__def,axiom,
( minus_3912115830687649252ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
( collec5218855173738699400ctxt_a
@ ( minus_8440876803814310393xt_a_o
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ A6 )
@ ^ [X2: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_919_prod_Oexpand,axiom,
! [Prod: product_prod_nat_nat,Prod2: product_prod_nat_nat] :
( ( ( ( product_fst_nat_nat @ Prod )
= ( product_fst_nat_nat @ Prod2 ) )
& ( ( product_snd_nat_nat @ Prod )
= ( product_snd_nat_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_920_prod__eqI,axiom,
! [P2: product_prod_nat_nat,Q2: product_prod_nat_nat] :
( ( ( product_fst_nat_nat @ P2 )
= ( product_fst_nat_nat @ Q2 ) )
=> ( ( ( product_snd_nat_nat @ P2 )
= ( product_snd_nat_nat @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_921_prod__eq__iff,axiom,
( ( ^ [Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [S4: product_prod_nat_nat,T2: product_prod_nat_nat] :
( ( ( product_fst_nat_nat @ S4 )
= ( product_fst_nat_nat @ T2 ) )
& ( ( product_snd_nat_nat @ S4 )
= ( product_snd_nat_nat @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_922_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_923_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_924_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C5: nat] :
( B2
= ( plus_plus_nat @ A2 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_925_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_926_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C6: nat] :
( B
!= ( plus_plus_nat @ A @ C6 ) ) ) ).
% less_eqE
thf(fact_927_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_928_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_mono
thf(fact_929_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I4 @ J3 )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_930_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( I4 = J3 )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_931_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I4 @ J3 )
& ( K = L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_932_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_933_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_934_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_935_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_936_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_937_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N3: nat] :
( L2
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_938_add__le__mono,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_939_add__le__mono1,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_le_mono1
thf(fact_940_trans__le__add1,axiom,
! [I4: nat,J3: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J3 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_941_trans__le__add2,axiom,
! [I4: nat,J3: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M2 @ J3 ) ) ) ).
% trans_le_add2
thf(fact_942_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_943_less__add__eq__less,axiom,
! [K: nat,L2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M2 @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_944_trans__less__add2,axiom,
! [I4: nat,J3: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M2 @ J3 ) ) ) ).
% trans_less_add2
thf(fact_945_trans__less__add1,axiom,
! [I4: nat,J3: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J3 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_946_add__less__mono1,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_less_mono1
thf(fact_947_not__add__less2,axiom,
! [J3: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_948_not__add__less1,axiom,
! [I4: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ I4 ) ).
% not_add_less1
thf(fact_949_add__less__mono,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_950_add__lessD1,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K )
=> ( ord_less_nat @ I4 @ K ) ) ).
% add_lessD1
thf(fact_951_snd__conv,axiom,
! [X1: ground_gterm_a,X22: ground_gterm_a] :
( ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_952_snd__conv,axiom,
! [X1: ground_gmctxt_a,X22: ground_gmctxt_a] :
( ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_953_snd__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_954_snd__conv,axiom,
! [X1: nat > nat,X22: nat] :
( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_955_snd__eqD,axiom,
! [X: ground_gterm_a,Y: ground_gterm_a,A: ground_gterm_a] :
( ( ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_956_snd__eqD,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_957_snd__eqD,axiom,
! [X: nat,Y: nat,A: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_958_snd__eqD,axiom,
! [X: nat > nat,Y: nat,A: nat] :
( ( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_959_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_960_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_961_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_962_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_963_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_strict_mono
thf(fact_964_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I4 @ J3 )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_965_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( I4 = J3 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_966_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I4 @ J3 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_967_prod_Oexhaust__sel,axiom,
! [Prod: produc1525979690750518759term_a] :
( Prod
= ( produc7419039783125785943term_a @ ( produc5097907526672912195term_a @ Prod ) @ ( produc5323275678763266181term_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_968_prod_Oexhaust__sel,axiom,
! [Prod: produc8175205341597284509ctxt_a] :
( Prod
= ( produc6347872285926497493ctxt_a @ ( produc844137399493384385ctxt_a @ Prod ) @ ( produc3308216634374912259ctxt_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_969_prod_Oexhaust__sel,axiom,
! [Prod: produc8199716216217303280at_nat] :
( Prod
= ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_970_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_971_surjective__pairing,axiom,
! [T: produc1525979690750518759term_a] :
( T
= ( produc7419039783125785943term_a @ ( produc5097907526672912195term_a @ T ) @ ( produc5323275678763266181term_a @ T ) ) ) ).
% surjective_pairing
thf(fact_972_surjective__pairing,axiom,
! [T: produc8175205341597284509ctxt_a] :
( T
= ( produc6347872285926497493ctxt_a @ ( produc844137399493384385ctxt_a @ T ) @ ( produc3308216634374912259ctxt_a @ T ) ) ) ).
% surjective_pairing
thf(fact_973_surjective__pairing,axiom,
! [T: produc8199716216217303280at_nat] :
( T
= ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ T ) @ ( produc1852801350702243542at_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_974_surjective__pairing,axiom,
! [T: product_prod_nat_nat] :
( T
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_975_case__prod__beta,axiom,
( produc2626176000494625587at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat,P5: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P5 ) @ ( product_snd_nat_nat @ P5 ) ) ) ) ).
% case_prod_beta
thf(fact_976_split__beta,axiom,
( produc2626176000494625587at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat,Prod3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_977_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_978_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_979_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_980_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_981_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_982_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_983_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_984_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_985_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_less_le_mono
thf(fact_986_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_le_less_mono
thf(fact_987_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I4 @ J3 )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_988_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: nat,J3: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I4 @ J3 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_989_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_990_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C6: nat] :
( ( B
= ( plus_plus_nat @ A @ C6 ) )
=> ( C6 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_991_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_992_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_993_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_994_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_995_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_996_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_997_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_998_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_999_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1000_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1001_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1002_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1003_Product__Type_OCollect__case__prodD,axiom,
! [X: produc1525979690750518759term_a,A3: ground_gterm_a > ground_gterm_a > $o] :
( ( member4351891425000572176term_a @ X @ ( collec6713184107424615762term_a @ ( produc7608081248467820332rm_a_o @ A3 ) ) )
=> ( A3 @ ( produc5097907526672912195term_a @ X ) @ ( produc5323275678763266181term_a @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_1004_Product__Type_OCollect__case__prodD,axiom,
! [X: produc8175205341597284509ctxt_a,A3: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ X @ ( collec5218855173738699400ctxt_a @ ( produc4116768209347204654xt_a_o @ A3 ) ) )
=> ( A3 @ ( produc844137399493384385ctxt_a @ X ) @ ( produc3308216634374912259ctxt_a @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_1005_Product__Type_OCollect__case__prodD,axiom,
! [X: product_prod_nat_nat,A3: nat > nat > $o] :
( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A3 ) ) )
=> ( A3 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_1006_less__imp__add__positive,axiom,
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I4 @ K2 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_1007_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1008_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_1009_less__diff__conv,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ).
% less_diff_conv
thf(fact_1010_le__diff__conv,axiom,
! [J3: nat,K: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I4 @ K ) ) ) ).
% le_diff_conv
thf(fact_1011_Nat_Ole__diff__conv2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1012_Nat_Odiff__add__assoc,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1013_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1014_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I4 )
= K )
= ( J3
= ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1015_case__prod__beta_H,axiom,
( produc2626176000494625587at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat,X2: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X2 ) @ ( product_snd_nat_nat @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_1016_case__prod__unfold,axiom,
( produc2626176000494625587at_nat
= ( ^ [C5: nat > nat > product_prod_nat_nat,P5: product_prod_nat_nat] : ( C5 @ ( product_fst_nat_nat @ P5 ) @ ( product_snd_nat_nat @ P5 ) ) ) ) ).
% case_prod_unfold
thf(fact_1017_prod_Osplit__sel,axiom,
! [P: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( P @ ( produc2626176000494625587at_nat @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
=> ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_1018_prod_Osplit__sel__asm,axiom,
! [P: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( P @ ( produc2626176000494625587at_nat @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
& ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_1019_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1020_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1021_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1022_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1023_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1024_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1025_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1026_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1027_less__diff__conv2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1028_case__prod__Pair__iden,axiom,
! [P2: produc1525979690750518759term_a] :
( ( produc3782323831342481275term_a @ produc7419039783125785943term_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_1029_case__prod__Pair__iden,axiom,
! [P2: produc8175205341597284509ctxt_a] :
( ( produc8334113781445266739ctxt_a @ produc6347872285926497493ctxt_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_1030_case__prod__Pair__iden,axiom,
! [P2: produc8199716216217303280at_nat] :
( ( produc3780431343476367475at_nat @ produc72220940542539688at_nat @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_1031_case__prod__Pair__iden,axiom,
! [P2: product_prod_nat_nat] :
( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_1032_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
@ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1033_ghole__num__bef__at__pos__num__gholes__less__eq,axiom,
! [P2: list_nat,C: ground_gmctxt_a] :
( ( member_list_nat @ P2 @ ( ground_poss_gmctxt_a @ C ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( ground1934838843355771718_pos_a @ P2 @ C ) @ ( ground5706155212172713090_pos_a @ P2 @ C ) ) @ ( ground_num_gholes_a @ C ) ) ) ).
% ghole_num_bef_at_pos_num_gholes_less_eq
thf(fact_1034_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1035_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1036_in__set__enumerate__eq,axiom,
! [P2: produc7109479242990851284term_a,N: nat,Xs3: list_Ground_gterm_a] :
( ( member6622040750283622891term_a @ P2 @ ( set_Pr3203312087732499071term_a @ ( enumer8595146184022354964term_a @ N @ Xs3 ) ) )
= ( ( ord_less_eq_nat @ N @ ( produc6672059348947727898term_a @ P2 ) )
& ( ord_less_nat @ ( produc6672059348947727898term_a @ P2 ) @ ( plus_plus_nat @ ( size_s1893836384466666913term_a @ Xs3 ) @ N ) )
& ( ( nth_Ground_gterm_a @ Xs3 @ ( minus_minus_nat @ ( produc6672059348947727898term_a @ P2 ) @ N ) )
= ( produc4207926146178914648term_a @ P2 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_1037_in__set__enumerate__eq,axiom,
! [P2: product_prod_nat_nat,N: nat,Xs3: list_nat] :
( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( enumerate_nat @ N @ Xs3 ) ) )
= ( ( ord_less_eq_nat @ N @ ( product_fst_nat_nat @ P2 ) )
& ( ord_less_nat @ ( product_fst_nat_nat @ P2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs3 ) @ N ) )
& ( ( nth_nat @ Xs3 @ ( minus_minus_nat @ ( product_fst_nat_nat @ P2 ) @ N ) )
= ( product_snd_nat_nat @ P2 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_1038_length__enumerate,axiom,
! [N: nat,Xs3: list_Ground_gterm_a] :
( ( size_s2378411291002718200term_a @ ( enumer8595146184022354964term_a @ N @ Xs3 ) )
= ( size_s1893836384466666913term_a @ Xs3 ) ) ).
% length_enumerate
thf(fact_1039_nth__enumerate__eq,axiom,
! [M2: nat,Xs3: list_nat,N: nat] :
( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs3 ) @ M2 )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs3 @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1040_nth__enumerate__eq,axiom,
! [M2: nat,Xs3: list_Ground_gterm_a,N: nat] :
( ( ord_less_nat @ M2 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( nth_Pr5718682320234703373term_a @ ( enumer8595146184022354964term_a @ N @ Xs3 ) @ M2 )
= ( produc4697074808646628614term_a @ ( plus_plus_nat @ N @ M2 ) @ ( nth_Ground_gterm_a @ Xs3 @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1041_add__le__imp__le__diff,axiom,
! [I4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N )
=> ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1042_add__le__add__imp__diff__le,axiom,
! [I4: nat,K: nat,N: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1043_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1044_exE__realizer,axiom,
! [P: nat > nat > $o,P2: product_prod_nat_nat,Q: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat] :
( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
=> ( ! [X5: nat,Y4: nat] :
( ( P @ Y4 @ X5 )
=> ( Q @ ( F @ X5 @ Y4 ) ) )
=> ( Q @ ( produc2626176000494625587at_nat @ F @ P2 ) ) ) ) ).
% exE_realizer
thf(fact_1045_exI__realizer,axiom,
! [P: ground_gterm_a > ground_gterm_a > $o,Y: ground_gterm_a,X: ground_gterm_a] :
( ( P @ Y @ X )
=> ( P @ ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ X @ Y ) ) @ ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1046_exI__realizer,axiom,
! [P: ground_gmctxt_a > ground_gmctxt_a > $o,Y: ground_gmctxt_a,X: ground_gmctxt_a] :
( ( P @ Y @ X )
=> ( P @ ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) ) @ ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1047_exI__realizer,axiom,
! [P: nat > ( nat > nat ) > $o,Y: nat,X: nat > nat] :
( ( P @ Y @ X )
=> ( P @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X @ Y ) ) @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1048_exI__realizer,axiom,
! [P: nat > nat > $o,Y: nat,X: nat] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1049_conjI__realizer,axiom,
! [P: ground_gterm_a > $o,P2: ground_gterm_a,Q: ground_gterm_a > $o,Q2: ground_gterm_a] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ P2 @ Q2 ) ) )
& ( Q @ ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1050_conjI__realizer,axiom,
! [P: ground_gmctxt_a > $o,P2: ground_gmctxt_a,Q: ground_gmctxt_a > $o,Q2: ground_gmctxt_a] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ P2 @ Q2 ) ) )
& ( Q @ ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1051_conjI__realizer,axiom,
! [P: ( nat > nat ) > $o,P2: nat > nat,Q: nat > $o,Q2: nat] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ P2 @ Q2 ) ) )
& ( Q @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1052_conjI__realizer,axiom,
! [P: nat > $o,P2: nat,Q: nat > $o,Q2: nat] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1053_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ground_gterm_a > ground_gterm_a > $o,X: ground_gterm_a,Y: ground_gterm_a,A: produc1525979690750518759term_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc7419039783125785943term_a @ X @ Y ) )
=> ( P @ ( produc5097907526672912195term_a @ A ) @ ( produc5323275678763266181term_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1054_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ground_gmctxt_a > ground_gmctxt_a > $o,X: ground_gmctxt_a,Y: ground_gmctxt_a,A: produc8175205341597284509ctxt_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc6347872285926497493ctxt_a @ X @ Y ) )
=> ( P @ ( produc844137399493384385ctxt_a @ A ) @ ( produc3308216634374912259ctxt_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1055_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( nat > nat ) > nat > $o,X: nat > nat,Y: nat,A: produc8199716216217303280at_nat] :
( ( P @ X @ Y )
=> ( ( A
= ( produc72220940542539688at_nat @ X @ Y ) )
=> ( P @ ( produc6156676138143019412at_nat @ A ) @ ( produc1852801350702243542at_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1056_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: nat > nat > $o,X: nat,Y: nat,A: product_prod_nat_nat] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_nat_nat @ X @ Y ) )
=> ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1057_subset__Collect__iff,axiom,
! [B5: set_list_nat,A3: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B5 @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ B5
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1058_subset__Collect__iff,axiom,
! [B5: set_Pr8705490787052945405ctxt_a,A3: set_Pr8705490787052945405ctxt_a,P: produc8175205341597284509ctxt_a > $o] :
( ( ord_le2690539916662146973ctxt_a @ B5 @ A3 )
=> ( ( ord_le2690539916662146973ctxt_a @ B5
@ ( collec5218855173738699400ctxt_a
@ ^ [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1059_subset__Collect__iff,axiom,
! [B5: set_Pr8282532413760598727term_a,A3: set_Pr8282532413760598727term_a,P: produc1525979690750518759term_a > $o] :
( ( ord_le7556488698670074471term_a @ B5 @ A3 )
=> ( ( ord_le7556488698670074471term_a @ B5
@ ( collec6713184107424615762term_a
@ ^ [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1060_subset__CollectI,axiom,
! [B5: set_list_nat,A3: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B5 @ A3 )
=> ( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ B5 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ B5 )
& ( Q @ X2 ) ) )
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1061_subset__CollectI,axiom,
! [B5: set_Pr8705490787052945405ctxt_a,A3: set_Pr8705490787052945405ctxt_a,Q: produc8175205341597284509ctxt_a > $o,P: produc8175205341597284509ctxt_a > $o] :
( ( ord_le2690539916662146973ctxt_a @ B5 @ A3 )
=> ( ! [X5: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X5 @ B5 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_le2690539916662146973ctxt_a
@ ( collec5218855173738699400ctxt_a
@ ^ [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ B5 )
& ( Q @ X2 ) ) )
@ ( collec5218855173738699400ctxt_a
@ ^ [X2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1062_subset__CollectI,axiom,
! [B5: set_Pr8282532413760598727term_a,A3: set_Pr8282532413760598727term_a,Q: produc1525979690750518759term_a > $o,P: produc1525979690750518759term_a > $o] :
( ( ord_le7556488698670074471term_a @ B5 @ A3 )
=> ( ! [X5: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X5 @ B5 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_le7556488698670074471term_a
@ ( collec6713184107424615762term_a
@ ^ [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ B5 )
& ( Q @ X2 ) ) )
@ ( collec6713184107424615762term_a
@ ^ [X2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1063_restrict__def,axiom,
( abstra4205430435060868312st_nat
= ( ^ [R6: set_Pr3451248702717554689st_nat,S5: set_list_nat] :
( collec1570431334306492044st_nat
@ ( produc6382697403170970154_nat_o
@ ^ [X2: list_nat,Y2: list_nat] :
( ( member_list_nat @ X2 @ S5 )
& ( member_list_nat @ Y2 @ S5 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ).
% restrict_def
thf(fact_1064_restrict__def,axiom,
( abstra6312477298767733231term_a
= ( ^ [R6: set_Pr5156918327056800967term_a,S5: set_Pr8282532413760598727term_a] :
( collec7632642491840594258term_a
@ ( produc8204158254750672172rm_a_o
@ ^ [X2: produc1525979690750518759term_a,Y2: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ X2 @ S5 )
& ( member4351891425000572176term_a @ Y2 @ S5 )
& ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ).
% restrict_def
thf(fact_1065_restrict__def,axiom,
( abstra4135781082980823205ctxt_a
= ( ^ [R6: set_Pr490164551564945991ctxt_a,S5: set_Pr8705490787052945405ctxt_a] :
( collec3796792608423614674ctxt_a
@ ( produc6872319198995859628xt_a_o
@ ^ [X2: produc8175205341597284509ctxt_a,Y2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X2 @ S5 )
& ( member461722496487281478ctxt_a @ Y2 @ S5 )
& ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ).
% restrict_def
thf(fact_1066_restrict__def,axiom,
( abstra354823185397780407term_a
= ( ^ [R6: set_Pr8282532413760598727term_a,S5: set_Ground_gterm_a] :
( collec6713184107424615762term_a
@ ( produc7608081248467820332rm_a_o
@ ^ [X2: ground_gterm_a,Y2: ground_gterm_a] :
( ( member5261315023172567896term_a @ X2 @ S5 )
& ( member5261315023172567896term_a @ Y2 @ S5 )
& ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ).
% restrict_def
thf(fact_1067_restrict__def,axiom,
( abstra1138918260165758998ctxt_a
= ( ^ [R6: set_Pr8705490787052945405ctxt_a,S5: set_Ground_gmctxt_a] :
( collec5218855173738699400ctxt_a
@ ( produc4116768209347204654xt_a_o
@ ^ [X2: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X2 @ S5 )
& ( member3306386364193653941ctxt_a @ Y2 @ S5 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ).
% restrict_def
thf(fact_1068_restrict__def,axiom,
( abstra1219450721576609224ct_nat
= ( ^ [R6: set_Pr1261947904930325089at_nat,S5: set_nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ S5 )
& ( member_nat @ Y2 @ S5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ).
% restrict_def
thf(fact_1069_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( F @ K2 @ I ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K3 @ K5 )
& ( F @ K5 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1070_sndI,axiom,
! [X: produc1525979690750518759term_a,Y: ground_gterm_a,Z: ground_gterm_a] :
( ( X
= ( produc7419039783125785943term_a @ Y @ Z ) )
=> ( ( produc5323275678763266181term_a @ X )
= Z ) ) ).
% sndI
thf(fact_1071_sndI,axiom,
! [X: produc8175205341597284509ctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( X
= ( produc6347872285926497493ctxt_a @ Y @ Z ) )
=> ( ( produc3308216634374912259ctxt_a @ X )
= Z ) ) ).
% sndI
thf(fact_1072_sndI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_snd_nat_nat @ X )
= Z ) ) ).
% sndI
thf(fact_1073_sndI,axiom,
! [X: produc8199716216217303280at_nat,Y: nat > nat,Z: nat] :
( ( X
= ( produc72220940542539688at_nat @ Y @ Z ) )
=> ( ( produc1852801350702243542at_nat @ X )
= Z ) ) ).
% sndI
thf(fact_1074_fstI,axiom,
! [X: produc1525979690750518759term_a,Y: ground_gterm_a,Z: ground_gterm_a] :
( ( X
= ( produc7419039783125785943term_a @ Y @ Z ) )
=> ( ( produc5097907526672912195term_a @ X )
= Y ) ) ).
% fstI
thf(fact_1075_fstI,axiom,
! [X: produc8175205341597284509ctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( X
= ( produc6347872285926497493ctxt_a @ Y @ Z ) )
=> ( ( produc844137399493384385ctxt_a @ X )
= Y ) ) ).
% fstI
thf(fact_1076_fstI,axiom,
! [X: produc8199716216217303280at_nat,Y: nat > nat,Z: nat] :
( ( X
= ( produc72220940542539688at_nat @ Y @ Z ) )
=> ( ( produc6156676138143019412at_nat @ X )
= Y ) ) ).
% fstI
thf(fact_1077_fstI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_fst_nat_nat @ X )
= Y ) ) ).
% fstI
thf(fact_1078_eq__snd__iff,axiom,
! [B: ground_gterm_a,P2: produc1525979690750518759term_a] :
( ( B
= ( produc5323275678763266181term_a @ P2 ) )
= ( ? [A2: ground_gterm_a] :
( P2
= ( produc7419039783125785943term_a @ A2 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1079_eq__snd__iff,axiom,
! [B: ground_gmctxt_a,P2: produc8175205341597284509ctxt_a] :
( ( B
= ( produc3308216634374912259ctxt_a @ P2 ) )
= ( ? [A2: ground_gmctxt_a] :
( P2
= ( produc6347872285926497493ctxt_a @ A2 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1080_eq__snd__iff,axiom,
! [B: nat,P2: product_prod_nat_nat] :
( ( B
= ( product_snd_nat_nat @ P2 ) )
= ( ? [A2: nat] :
( P2
= ( product_Pair_nat_nat @ A2 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1081_eq__snd__iff,axiom,
! [B: nat,P2: produc8199716216217303280at_nat] :
( ( B
= ( produc1852801350702243542at_nat @ P2 ) )
= ( ? [A2: nat > nat] :
( P2
= ( produc72220940542539688at_nat @ A2 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1082_eq__fst__iff,axiom,
! [A: ground_gterm_a,P2: produc1525979690750518759term_a] :
( ( A
= ( produc5097907526672912195term_a @ P2 ) )
= ( ? [B2: ground_gterm_a] :
( P2
= ( produc7419039783125785943term_a @ A @ B2 ) ) ) ) ).
% eq_fst_iff
thf(fact_1083_eq__fst__iff,axiom,
! [A: ground_gmctxt_a,P2: produc8175205341597284509ctxt_a] :
( ( A
= ( produc844137399493384385ctxt_a @ P2 ) )
= ( ? [B2: ground_gmctxt_a] :
( P2
= ( produc6347872285926497493ctxt_a @ A @ B2 ) ) ) ) ).
% eq_fst_iff
thf(fact_1084_eq__fst__iff,axiom,
! [A: nat > nat,P2: produc8199716216217303280at_nat] :
( ( A
= ( produc6156676138143019412at_nat @ P2 ) )
= ( ? [B2: nat] :
( P2
= ( produc72220940542539688at_nat @ A @ B2 ) ) ) ) ).
% eq_fst_iff
thf(fact_1085_eq__fst__iff,axiom,
! [A: nat,P2: product_prod_nat_nat] :
( ( A
= ( product_fst_nat_nat @ P2 ) )
= ( ? [B2: nat] :
( P2
= ( product_Pair_nat_nat @ A @ B2 ) ) ) ) ).
% eq_fst_iff
thf(fact_1086_fill__gholes__GMHole,axiom,
! [Ss2: list_Ground_gterm_a] :
( ( ( size_s1893836384466666913term_a @ Ss2 )
= ( suc @ zero_zero_nat ) )
=> ( ( ground_fill_gholes_a @ ground_GMHole_a @ Ss2 )
= ( nth_Ground_gterm_a @ Ss2 @ zero_zero_nat ) ) ) ).
% fill_gholes_GMHole
thf(fact_1087_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1088_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1089_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_1090_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_1091_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1092_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1093_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_1094_diff__Suc__diff__eq2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) @ I4 )
= ( minus_minus_nat @ ( suc @ J3 ) @ ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1095_diff__Suc__diff__eq1,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ ( suc @ J3 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1096_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1097_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_1098_lift__Suc__antimono__le,axiom,
! [F: nat > ground_gmctxt_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le1940079914304802430ctxt_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1099_lift__Suc__antimono__le,axiom,
! [F: nat > set_Pr8282532413760598727term_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le7556488698670074471term_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le7556488698670074471term_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1100_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1101_lift__Suc__mono__le,axiom,
! [F: nat > ground_gmctxt_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le1940079914304802430ctxt_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le1940079914304802430ctxt_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1102_lift__Suc__mono__le,axiom,
! [F: nat > set_Pr8282532413760598727term_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le7556488698670074471term_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le7556488698670074471term_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1103_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1104_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1105_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1106_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1107_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_1108_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1109_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1110_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1111_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_1112_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1113_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1114_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1115_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1116_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_1117_less__trans__Suc,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1118_less__Suc__induct,axiom,
! [I4: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_1119_strict__inc__induct,axiom,
! [I4: nat,J3: nat,P: nat > $o] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_1120_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_1121_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X5: nat] : ( R2 @ X5 @ X5 )
=> ( ! [X5: nat,Y4: nat,Z3: nat] :
( ( R2 @ X5 @ Y4 )
=> ( ( R2 @ Y4 @ Z3 )
=> ( R2 @ X5 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1122_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1123_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1124_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1125_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1126_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1127_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1128_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1129_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1130_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1131_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1132_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1133_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J4: nat] :
( ( M2
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1134_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1135_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1136_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1137_Abstract__Rewriting_Ochain__mono,axiom,
! [R4: set_Pr8705490787052945405ctxt_a,R2: set_Pr8705490787052945405ctxt_a,Seq: nat > ground_gmctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ R4 @ R2 )
=> ( ! [I2: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 )
=> ! [I: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R2 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1138_Abstract__Rewriting_Ochain__mono,axiom,
! [R4: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
( ( ord_le3146513528884898305at_nat @ R4 @ R2 )
=> ( ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 )
=> ! [I: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R2 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1139_Abstract__Rewriting_Ochain__mono,axiom,
! [R4: set_Pr8282532413760598727term_a,R2: set_Pr8282532413760598727term_a,Seq: nat > ground_gterm_a] :
( ( ord_le7556488698670074471term_a @ R4 @ R2 )
=> ( ! [I2: nat] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 )
=> ! [I: nat] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R2 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1140_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1141_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_1142_dec__induct,axiom,
! [I4: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( P @ I4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_1143_inc__induct,axiom,
! [I4: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( P @ J3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% inc_induct
thf(fact_1144_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_1145_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_1146_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_1147_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1148_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_1149_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1150_less__add__Suc1,axiom,
! [I4: nat,M2: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1151_less__add__Suc2,axiom,
! [I4: nat,M2: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M2 @ I4 ) ) ) ).
% less_add_Suc2
thf(fact_1152_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
? [K4: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1153_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_1154_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_1155_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1156_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1157_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1158_diff__Suc__less,axiom,
! [N: nat,I4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1159_num__gholes_Osimps_I1_J,axiom,
( ( ground_num_gholes_a @ ground_GMHole_a )
= ( suc @ zero_zero_nat ) ) ).
% num_gholes.simps(1)
thf(fact_1160_chain__imp__trancl,axiom,
! [S2: nat > ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a,I4: nat,J3: nat] :
( ! [I2: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R3 )
=> ( ( ord_less_nat @ I4 @ J3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( S2 @ I4 ) @ ( S2 @ J3 ) ) @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ) ).
% chain_imp_trancl
thf(fact_1161_chain__imp__trancl,axiom,
! [S2: nat > nat,R3: set_Pr1261947904930325089at_nat,I4: nat,J3: nat] :
( ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R3 )
=> ( ( ord_less_nat @ I4 @ J3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I4 ) @ ( S2 @ J3 ) ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% chain_imp_trancl
thf(fact_1162_chain__imp__trancl,axiom,
! [S2: nat > ground_gterm_a,R3: set_Pr8282532413760598727term_a,I4: nat,J3: nat] :
( ! [I2: nat] : ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R3 )
=> ( ( ord_less_nat @ I4 @ J3 )
=> ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( S2 @ I4 ) @ ( S2 @ J3 ) ) @ ( transi8854406246868658384term_a @ R3 ) ) ) ) ).
% chain_imp_trancl
thf(fact_1163_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K2: nat,M4: nat] :
( X
!= ( product_Pair_nat_nat @ K2 @ M4 ) ) ).
% prod_decode_aux.cases
thf(fact_1164_inf__concat__simple_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [F3: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ F3 @ zero_zero_nat ) )
=> ~ ! [F3: nat > nat,N3: nat] :
( X
!= ( produc72220940542539688at_nat @ F3 @ ( suc @ N3 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_1165_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I3: nat,J4: nat] : ( if_nat @ ( ord_less_nat @ J4 @ I3 ) @ J4 @ ( minus_minus_nat @ J4 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_1166_Euclidean__Division_Odivmod__nat__if,axiom,
( euclidean_divmod_nat
= ( ^ [M: nat,N2: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N2 = zero_zero_nat )
| ( ord_less_nat @ M @ N2 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M )
@ ( produc2626176000494625587at_nat
@ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
@ ( euclidean_divmod_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% Euclidean_Division.divmod_nat_if
thf(fact_1167_shift_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
~ ! [F3: nat > nat,J2: nat] :
( X
!= ( produc72220940542539688at_nat @ F3 @ J2 ) ) ).
% shift.cases
thf(fact_1168_inf__concat__simple_Oelims,axiom,
! [X: nat > nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( inf_concat_simple @ X @ Xa )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y
!= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) )
=> ~ ! [N3: nat] :
( ( Xa
= ( suc @ N3 ) )
=> ( Y
!= ( produc2626176000494625587at_nat
@ ^ [I3: nat,J4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J4 ) @ ( X @ I3 ) ) @ ( product_Pair_nat_nat @ I3 @ ( suc @ J4 ) ) @ ( product_Pair_nat_nat @ ( suc @ I3 ) @ zero_zero_nat ) )
@ ( inf_concat_simple @ X @ N3 ) ) ) ) ) ) ).
% inf_concat_simple.elims
thf(fact_1169_inf__concat__simple_Osimps_I2_J,axiom,
! [F: nat > nat,N: nat] :
( ( inf_concat_simple @ F @ ( suc @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [I3: nat,J4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J4 ) @ ( F @ I3 ) ) @ ( product_Pair_nat_nat @ I3 @ ( suc @ J4 ) ) @ ( product_Pair_nat_nat @ ( suc @ I3 ) @ zero_zero_nat ) )
@ ( inf_concat_simple @ F @ N ) ) ) ).
% inf_concat_simple.simps(2)
thf(fact_1170_inf__concat__simple_Osimps_I1_J,axiom,
! [F: nat > nat] :
( ( inf_concat_simple @ F @ zero_zero_nat )
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% inf_concat_simple.simps(1)
thf(fact_1171_inf__concat__simple__surj__zero,axiom,
! [F: nat > nat,I4: nat] :
? [K2: nat] :
( ( inf_concat_simple @ F @ K2 )
= ( product_Pair_nat_nat @ I4 @ zero_zero_nat ) ) ).
% inf_concat_simple_surj_zero
thf(fact_1172_inf__concat__simple__surj,axiom,
! [J3: nat,F: nat > nat,I4: nat] :
( ( ord_less_nat @ J3 @ ( F @ I4 ) )
=> ? [K2: nat] :
( ( inf_concat_simple @ F @ K2 )
= ( product_Pair_nat_nat @ I4 @ J3 ) ) ) ).
% inf_concat_simple_surj
thf(fact_1173_inf__concat__simple__mono,axiom,
! [K: nat,K6: nat,F: nat > nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ord_less_eq_nat @ ( product_fst_nat_nat @ ( inf_concat_simple @ F @ K ) ) @ ( product_fst_nat_nat @ ( inf_concat_simple @ F @ K6 ) ) ) ) ).
% inf_concat_simple_mono
thf(fact_1174_inf__concat__simple__add,axiom,
! [F: nat > nat,K: nat,I4: nat,J3: nat,L2: nat] :
( ( ( inf_concat_simple @ F @ K )
= ( product_Pair_nat_nat @ I4 @ J3 ) )
=> ( ( ord_less_nat @ ( plus_plus_nat @ J3 @ L2 ) @ ( F @ I4 ) )
=> ( ( inf_concat_simple @ F @ ( plus_plus_nat @ K @ L2 ) )
= ( product_Pair_nat_nat @ I4 @ ( plus_plus_nat @ J3 @ L2 ) ) ) ) ) ).
% inf_concat_simple_add
thf(fact_1175_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K4: nat,M: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M @ K4 ) @ ( product_Pair_nat_nat @ M @ ( minus_minus_nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus_nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_1176_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_1177_remove__nth__sound__r,axiom,
! [N: nat,P2: nat,Xs3: list_Ground_gterm_a] :
( ( ord_less_eq_nat @ N @ P2 )
=> ( ( ord_less_nat @ P2 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( nth_Ground_gterm_a @ ( missin1630609993678954280term_a @ N @ Xs3 ) @ P2 )
= ( nth_Ground_gterm_a @ Xs3 @ ( suc @ P2 ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_1178_gmctxt__of__gctxt__num__gholes,axiom,
! [C: ground_gctxt_a] :
( ( ground_num_gholes_a @ ( ground5442125528501538874ctxt_a @ C ) )
= ( suc @ zero_zero_nat ) ) ).
% gmctxt_of_gctxt_num_gholes
thf(fact_1179_remove__nth__sound__l,axiom,
! [P2: nat,N: nat,Xs3: list_Ground_gterm_a] :
( ( ord_less_nat @ P2 @ N )
=> ( ( nth_Ground_gterm_a @ ( missin1630609993678954280term_a @ N @ Xs3 ) @ P2 )
= ( nth_Ground_gterm_a @ Xs3 @ P2 ) ) ) ).
% remove_nth_sound_l
thf(fact_1180_remove__nth__id,axiom,
! [Xs3: list_Ground_gterm_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s1893836384466666913term_a @ Xs3 ) @ N )
=> ( ( missin1630609993678954280term_a @ N @ Xs3 )
= Xs3 ) ) ).
% remove_nth_id
thf(fact_1181_remove__nth__len,axiom,
! [I4: nat,Xs3: list_Ground_gterm_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( size_s1893836384466666913term_a @ Xs3 )
= ( suc @ ( size_s1893836384466666913term_a @ ( missin1630609993678954280term_a @ I4 @ Xs3 ) ) ) ) ) ).
% remove_nth_len
thf(fact_1182_remove__nth__P__compat,axiom,
! [As: list_Ground_gterm_a,Bs: list_Ground_gterm_a,P: ground_gterm_a > ground_gterm_a > $o,P2: nat] :
( ( ( size_s1893836384466666913term_a @ As )
= ( size_s1893836384466666913term_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ As ) )
=> ( P @ ( nth_Ground_gterm_a @ As @ I2 ) @ ( nth_Ground_gterm_a @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ ( missin1630609993678954280term_a @ P2 @ As ) ) )
=> ( P @ ( nth_Ground_gterm_a @ ( missin1630609993678954280term_a @ P2 @ As ) @ I ) @ ( nth_Ground_gterm_a @ ( missin1630609993678954280term_a @ P2 @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_1183_adjust__idx__rev__length,axiom,
! [I4: nat,Xs3: list_Ground_gterm_a,J3: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( ord_less_nat @ J3 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( J3 != I4 )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J3 ) @ ( size_s1893836384466666913term_a @ ( missin1630609993678954280term_a @ I4 @ Xs3 ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_1184_nth__remove__nth__conv,axiom,
! [I4: nat,N: nat,Xs3: list_Ground_gterm_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( missin1630609993678954280term_a @ N @ Xs3 ) ) )
=> ( ( nth_Ground_gterm_a @ ( missin1630609993678954280term_a @ N @ Xs3 ) @ I4 )
= ( nth_Ground_gterm_a @ Xs3 @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_1185_adjust__idx__rev__nth,axiom,
! [I4: nat,Xs3: list_Ground_gterm_a,J3: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( J3 != I4 )
=> ( ( nth_Ground_gterm_a @ ( missin1630609993678954280term_a @ I4 @ Xs3 ) @ ( missin3815256168798769645dx_rev @ I4 @ J3 ) )
= ( nth_Ground_gterm_a @ Xs3 @ J3 ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_1186_adjust__idx__nth,axiom,
! [I4: nat,Xs3: list_Ground_gterm_a,J3: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( nth_Ground_gterm_a @ ( missin1630609993678954280term_a @ I4 @ Xs3 ) @ J3 )
= ( nth_Ground_gterm_a @ Xs3 @ ( missing_adjust_idx @ I4 @ J3 ) ) ) ) ).
% adjust_idx_nth
thf(fact_1187_adjust__idx__length,axiom,
! [I4: nat,Xs3: list_Ground_gterm_a,J3: nat] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs3 ) )
=> ( ( ord_less_nat @ J3 @ ( size_s1893836384466666913term_a @ ( missin1630609993678954280term_a @ I4 @ Xs3 ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J3 ) @ ( size_s1893836384466666913term_a @ Xs3 ) ) ) ) ).
% adjust_idx_length
thf(fact_1188_adjust__idx__i,axiom,
! [I4: nat,J3: nat] :
( ( missing_adjust_idx @ I4 @ J3 )
!= I4 ) ).
% adjust_idx_i
thf(fact_1189_adjust__idx__rev1,axiom,
! [I4: nat,J3: nat] :
( ( missin3815256168798769645dx_rev @ I4 @ ( missing_adjust_idx @ I4 @ J3 ) )
= J3 ) ).
% adjust_idx_rev1
thf(fact_1190_adjust__idx__rev2,axiom,
! [J3: nat,I4: nat] :
( ( J3 != I4 )
=> ( ( missing_adjust_idx @ I4 @ ( missin3815256168798769645dx_rev @ I4 @ J3 ) )
= J3 ) ) ).
% adjust_idx_rev2
thf(fact_1191_adjust__idx__def,axiom,
( missing_adjust_idx
= ( ^ [I3: nat,J4: nat] : ( if_nat @ ( ord_less_nat @ J4 @ I3 ) @ J4 @ ( suc @ J4 ) ) ) ) ).
% adjust_idx_def
thf(fact_1192_gmctxt__of__gctxt__gctxt__of__gmctxt,axiom,
! [C: ground_gmctxt_a] :
( ( ( ground_num_gholes_a @ C )
= ( suc @ zero_zero_nat ) )
=> ( ( ground5442125528501538874ctxt_a @ ( ground5333112502500237606ctxt_a @ C ) )
= C ) ) ).
% gmctxt_of_gctxt_gctxt_of_gmctxt
thf(fact_1193_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1194_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1195_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_1196_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_1197_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q5 @ X5 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P6 @ X3 )
& ( Q5 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1198_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q5 @ X5 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P6 @ X3 )
| ( Q5 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1199_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_1200_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_1201_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_1202_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_1203_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q5 @ X5 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P6 @ X3 )
& ( Q5 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1204_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q5 @ X5 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P6 @ X3 )
| ( Q5 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1205_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1206_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1207_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_1208_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_1209_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_1210_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_1211_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C6: nat] :
( ( ord_less_eq_nat @ A @ C6 )
& ( ord_less_eq_nat @ C6 @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C6 ) )
=> ( P @ X3 ) )
& ! [D6: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ D6 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D6 @ C6 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1212_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1213_verit__comp__simplify1_I2_J,axiom,
! [A: ground_gmctxt_a] : ( ord_le1940079914304802430ctxt_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1214_verit__comp__simplify1_I2_J,axiom,
! [A: set_Pr8282532413760598727term_a] : ( ord_le7556488698670074471term_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1215_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1216_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1217_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1218_last__nthI,axiom,
! [I4: nat,Ts2: list_Ground_gterm_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ts2 ) )
=> ( ~ ( ord_less_nat @ I4 @ ( minus_minus_nat @ ( size_s1893836384466666913term_a @ Ts2 ) @ ( suc @ zero_zero_nat ) ) )
=> ( ( nth_Ground_gterm_a @ Ts2 @ I4 )
= ( last_Ground_gterm_a @ Ts2 ) ) ) ) ).
% last_nthI
thf(fact_1219_permut__aux__sound,axiom,
! [I4: nat,As: list_Ground_gterm_a,F: nat > nat,Bs: list_Ground_gterm_a] :
( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ As ) )
=> ( ( nth_Ground_gterm_a @ ( missin7316750521486066929term_a @ As @ F @ Bs ) @ I4 )
= ( nth_Ground_gterm_a @ Bs @ ( F @ I4 ) ) ) ) ).
% permut_aux_sound
thf(fact_1220_concat__index__split_Osimps,axiom,
! [I_idx: nat,X: list_Ground_gterm_a,O_idx: nat,Xs3: list_l4678906206682822203term_a] :
( ( ( ord_less_nat @ I_idx @ ( size_s1893836384466666913term_a @ X ) )
=> ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_l7239566772969696491term_a @ X @ Xs3 ) )
= ( product_Pair_nat_nat @ O_idx @ I_idx ) ) )
& ( ~ ( ord_less_nat @ I_idx @ ( size_s1893836384466666913term_a @ X ) )
=> ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_l7239566772969696491term_a @ X @ Xs3 ) )
= ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ ( suc @ O_idx ) @ ( minus_minus_nat @ I_idx @ ( size_s1893836384466666913term_a @ X ) ) ) @ Xs3 ) ) ) ) ).
% concat_index_split.simps
thf(fact_1221_inf__concat__simple_Opelims,axiom,
! [X: nat > nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( inf_concat_simple @ X @ Xa )
= Y )
=> ( ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X @ Xa ) )
=> ( ( ( Xa = zero_zero_nat )
=> ( ( Y
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
=> ~ ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X @ zero_zero_nat ) ) ) )
=> ~ ! [N3: nat] :
( ( Xa
= ( suc @ N3 ) )
=> ( ( Y
= ( produc2626176000494625587at_nat
@ ^ [I3: nat,J4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J4 ) @ ( X @ I3 ) ) @ ( product_Pair_nat_nat @ I3 @ ( suc @ J4 ) ) @ ( product_Pair_nat_nat @ ( suc @ I3 ) @ zero_zero_nat ) )
@ ( inf_concat_simple @ X @ N3 ) ) )
=> ~ ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% inf_concat_simple.pelims
thf(fact_1222_nth__Cons__0,axiom,
! [X: ground_gterm_a,Xs3: list_Ground_gterm_a] :
( ( nth_Ground_gterm_a @ ( cons_Ground_gterm_a @ X @ Xs3 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1223_nth__Cons__Suc,axiom,
! [X: ground_gterm_a,Xs3: list_Ground_gterm_a,N: nat] :
( ( nth_Ground_gterm_a @ ( cons_Ground_gterm_a @ X @ Xs3 ) @ ( suc @ N ) )
= ( nth_Ground_gterm_a @ Xs3 @ N ) ) ).
% nth_Cons_Suc
thf(fact_1224_Ground__Closure_Otrancl__list__appendI,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a,X: ground_gterm_a,Y: ground_gterm_a] :
( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( ground4492235380174378228term_a @ R ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ R )
=> ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( cons_Ground_gterm_a @ X @ Xs3 ) @ ( cons_Ground_gterm_a @ Y @ Ys2 ) ) @ ( ground4492235380174378228term_a @ R ) ) ) ) ).
% Ground_Closure.trancl_list_appendI
thf(fact_1225_Ground__Closure_Otrancl__list__appendI,axiom,
! [Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( ground4295926921573292569ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ R )
=> ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( cons_Ground_gmctxt_a @ X @ Xs3 ) @ ( cons_Ground_gmctxt_a @ Y @ Ys2 ) ) @ ( ground4295926921573292569ctxt_a @ R ) ) ) ) ).
% Ground_Closure.trancl_list_appendI
thf(fact_1226_Ground__Closure_Otrancl__list__appendI,axiom,
! [Xs3: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( ground4984107042736062923st_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( ground4984107042736062923st_nat @ R ) ) ) ) ).
% Ground_Closure.trancl_list_appendI
thf(fact_1227_Utils_Otrancl__list__appendI,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a,X: ground_gterm_a,Y: ground_gterm_a] :
( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( trancl1902929335784340279term_a @ R ) )
=> ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ R )
=> ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( cons_Ground_gterm_a @ X @ Xs3 ) @ ( cons_Ground_gterm_a @ Y @ Ys2 ) ) @ ( trancl1902929335784340279term_a @ R ) ) ) ) ).
% Utils.trancl_list_appendI
thf(fact_1228_Utils_Otrancl__list__appendI,axiom,
! [Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,X: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ R )
=> ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( cons_Ground_gmctxt_a @ X @ Xs3 ) @ ( cons_Ground_gmctxt_a @ Y @ Ys2 ) ) @ ( trancl6028151697025156246ctxt_a @ R ) ) ) ) ).
% Utils.trancl_list_appendI
thf(fact_1229_Utils_Otrancl__list__appendI,axiom,
! [Xs3: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( trancl_list_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( trancl_list_nat @ R ) ) ) ) ).
% Utils.trancl_list_appendI
thf(fact_1230_enumerate__simps_I2_J,axiom,
! [N: nat,X: nat,Xs3: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs3 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs3 ) ) ) ).
% enumerate_simps(2)
thf(fact_1231_Ground__Closure_Otrancl__list__append__tranclI,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( ground4295926921573292569ctxt_a @ R ) )
=> ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( cons_Ground_gmctxt_a @ X @ Xs3 ) @ ( cons_Ground_gmctxt_a @ Y @ Ys2 ) ) @ ( ground4295926921573292569ctxt_a @ R ) ) ) ) ).
% Ground_Closure.trancl_list_append_tranclI
thf(fact_1232_Ground__Closure_Otrancl__list__append__tranclI,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs3: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( ground4984107042736062923st_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( ground4984107042736062923st_nat @ R ) ) ) ) ).
% Ground_Closure.trancl_list_append_tranclI
thf(fact_1233_Ground__Closure_Otrancl__list__append__tranclI,axiom,
! [X: ground_gterm_a,Y: ground_gterm_a,R: set_Pr8282532413760598727term_a,Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ ( transi8854406246868658384term_a @ R ) )
=> ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( ground4492235380174378228term_a @ R ) )
=> ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( cons_Ground_gterm_a @ X @ Xs3 ) @ ( cons_Ground_gterm_a @ Y @ Ys2 ) ) @ ( ground4492235380174378228term_a @ R ) ) ) ) ).
% Ground_Closure.trancl_list_append_tranclI
thf(fact_1234_Utils_Otrancl__list__append__tranclI,axiom,
! [X: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,Xs3: list_Ground_gmctxt_a,Ys2: list_Ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys2 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
=> ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( cons_Ground_gmctxt_a @ X @ Xs3 ) @ ( cons_Ground_gmctxt_a @ Y @ Ys2 ) ) @ ( trancl6028151697025156246ctxt_a @ R ) ) ) ) ).
% Utils.trancl_list_append_tranclI
thf(fact_1235_Utils_Otrancl__list__append__tranclI,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs3: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( trancl_list_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( trancl_list_nat @ R ) ) ) ) ).
% Utils.trancl_list_append_tranclI
thf(fact_1236_Utils_Otrancl__list__append__tranclI,axiom,
! [X: ground_gterm_a,Y: ground_gterm_a,R: set_Pr8282532413760598727term_a,Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a] :
( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y ) @ ( transi8854406246868658384term_a @ R ) )
=> ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs3 @ Ys2 ) @ ( trancl1902929335784340279term_a @ R ) )
=> ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( cons_Ground_gterm_a @ X @ Xs3 ) @ ( cons_Ground_gterm_a @ Y @ Ys2 ) ) @ ( trancl1902929335784340279term_a @ R ) ) ) ) ).
% Utils.trancl_list_append_tranclI
thf(fact_1237_set__ConsD,axiom,
! [Y: list_nat,X: list_nat,Xs3: list_list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs3 ) ) )
=> ( ( Y = X )
| ( member_list_nat @ Y @ ( set_list_nat2 @ Xs3 ) ) ) ) ).
% set_ConsD
thf(fact_1238_set__ConsD,axiom,
! [Y: produc1525979690750518759term_a,X: produc1525979690750518759term_a,Xs3: list_P2433084232546440685term_a] :
( ( member4351891425000572176term_a @ Y @ ( set_Pr5405221470436175740term_a @ ( cons_P5554873272834883997term_a @ X @ Xs3 ) ) )
=> ( ( Y = X )
| ( member4351891425000572176term_a @ Y @ ( set_Pr5405221470436175740term_a @ Xs3 ) ) ) ) ).
% set_ConsD
thf(fact_1239_set__ConsD,axiom,
! [Y: produc8175205341597284509ctxt_a,X: produc8175205341597284509ctxt_a,Xs3: list_P2170180374705713187ctxt_a] :
( ( member461722496487281478ctxt_a @ Y @ ( set_Pr5641390345007089586ctxt_a @ ( cons_P7416119761977153875ctxt_a @ X @ Xs3 ) ) )
=> ( ( Y = X )
| ( member461722496487281478ctxt_a @ Y @ ( set_Pr5641390345007089586ctxt_a @ Xs3 ) ) ) ) ).
% set_ConsD
thf(fact_1240_list_Oset__cases,axiom,
! [E2: list_nat,A: list_list_nat] :
( ( member_list_nat @ E2 @ ( set_list_nat2 @ A ) )
=> ( ! [Z22: list_list_nat] :
( A
!= ( cons_list_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E2 @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1241_list_Oset__cases,axiom,
! [E2: produc1525979690750518759term_a,A: list_P2433084232546440685term_a] :
( ( member4351891425000572176term_a @ E2 @ ( set_Pr5405221470436175740term_a @ A ) )
=> ( ! [Z22: list_P2433084232546440685term_a] :
( A
!= ( cons_P5554873272834883997term_a @ E2 @ Z22 ) )
=> ~ ! [Z1: produc1525979690750518759term_a,Z22: list_P2433084232546440685term_a] :
( ( A
= ( cons_P5554873272834883997term_a @ Z1 @ Z22 ) )
=> ~ ( member4351891425000572176term_a @ E2 @ ( set_Pr5405221470436175740term_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1242_list_Oset__cases,axiom,
! [E2: produc8175205341597284509ctxt_a,A: list_P2170180374705713187ctxt_a] :
( ( member461722496487281478ctxt_a @ E2 @ ( set_Pr5641390345007089586ctxt_a @ A ) )
=> ( ! [Z22: list_P2170180374705713187ctxt_a] :
( A
!= ( cons_P7416119761977153875ctxt_a @ E2 @ Z22 ) )
=> ~ ! [Z1: produc8175205341597284509ctxt_a,Z22: list_P2170180374705713187ctxt_a] :
( ( A
= ( cons_P7416119761977153875ctxt_a @ Z1 @ Z22 ) )
=> ~ ( member461722496487281478ctxt_a @ E2 @ ( set_Pr5641390345007089586ctxt_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1243_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1244_list_Oset__intros_I1_J,axiom,
! [X21: produc1525979690750518759term_a,X222: list_P2433084232546440685term_a] : ( member4351891425000572176term_a @ X21 @ ( set_Pr5405221470436175740term_a @ ( cons_P5554873272834883997term_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1245_list_Oset__intros_I1_J,axiom,
! [X21: produc8175205341597284509ctxt_a,X222: list_P2170180374705713187ctxt_a] : ( member461722496487281478ctxt_a @ X21 @ ( set_Pr5641390345007089586ctxt_a @ ( cons_P7416119761977153875ctxt_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1246_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X222: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ X222 ) )
=> ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1247_list_Oset__intros_I2_J,axiom,
! [Y: produc1525979690750518759term_a,X222: list_P2433084232546440685term_a,X21: produc1525979690750518759term_a] :
( ( member4351891425000572176term_a @ Y @ ( set_Pr5405221470436175740term_a @ X222 ) )
=> ( member4351891425000572176term_a @ Y @ ( set_Pr5405221470436175740term_a @ ( cons_P5554873272834883997term_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1248_list_Oset__intros_I2_J,axiom,
! [Y: produc8175205341597284509ctxt_a,X222: list_P2170180374705713187ctxt_a,X21: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Y @ ( set_Pr5641390345007089586ctxt_a @ X222 ) )
=> ( member461722496487281478ctxt_a @ Y @ ( set_Pr5641390345007089586ctxt_a @ ( cons_P7416119761977153875ctxt_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1249_set__subset__Cons,axiom,
! [Xs3: list_P2433084232546440685term_a,X: produc1525979690750518759term_a] : ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Xs3 ) @ ( set_Pr5405221470436175740term_a @ ( cons_P5554873272834883997term_a @ X @ Xs3 ) ) ) ).
% set_subset_Cons
thf(fact_1250_impossible__Cons,axiom,
! [Xs3: list_Ground_gterm_a,Ys2: list_Ground_gterm_a,X: ground_gterm_a] :
( ( ord_less_eq_nat @ ( size_s1893836384466666913term_a @ Xs3 ) @ ( size_s1893836384466666913term_a @ Ys2 ) )
=> ( Xs3
!= ( cons_Ground_gterm_a @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1251_length__Cons,axiom,
! [X: ground_gterm_a,Xs3: list_Ground_gterm_a] :
( ( size_s1893836384466666913term_a @ ( cons_Ground_gterm_a @ X @ Xs3 ) )
= ( suc @ ( size_s1893836384466666913term_a @ Xs3 ) ) ) ).
% length_Cons
thf(fact_1252_Suc__length__conv,axiom,
! [N: nat,Xs3: list_Ground_gterm_a] :
( ( ( suc @ N )
= ( size_s1893836384466666913term_a @ Xs3 ) )
= ( ? [Y2: ground_gterm_a,Ys: list_Ground_gterm_a] :
( ( Xs3
= ( cons_Ground_gterm_a @ Y2 @ Ys ) )
& ( ( size_s1893836384466666913term_a @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1253_length__Suc__conv,axiom,
! [Xs3: list_Ground_gterm_a,N: nat] :
( ( ( size_s1893836384466666913term_a @ Xs3 )
= ( suc @ N ) )
= ( ? [Y2: ground_gterm_a,Ys: list_Ground_gterm_a] :
( ( Xs3
= ( cons_Ground_gterm_a @ Y2 @ Ys ) )
& ( ( size_s1893836384466666913term_a @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1254_Suc__le__length__iff,axiom,
! [N: nat,Xs3: list_Ground_gterm_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s1893836384466666913term_a @ Xs3 ) )
= ( ? [X2: ground_gterm_a,Ys: list_Ground_gterm_a] :
( ( Xs3
= ( cons_Ground_gterm_a @ X2 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_s1893836384466666913term_a @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1255_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
% 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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [P7: list_nat] :
( ( member_list_nat @ P7 @ ( ground_ghole_poss_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) ) )
=> ( ( ( nth_Ground_gterm_a @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ s ) @ i )
= ( ground_gsubt_at_a @ ( ground_fill_gholes_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ s ) ) @ P7 ) )
=> ( ( ( nth_Ground_gterm_a @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ u ) @ i )
= ( ground_gsubt_at_a @ ( ground_fill_gholes_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ ( ground8214737007138946830oles_a @ ( inf_in8301665843363081392ctxt_a @ c @ d ) @ u ) ) @ P7 ) )
=> thesis ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------