TPTP Problem File: SLH0379^1.p
View Solutions
- 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 : ResiduatedTransitionSystem/0000_ResiduatedTransitionSystem/prob_07394_302618__14315436_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1437 ( 399 unt; 155 typ; 0 def)
% Number of atoms : 6354 (1296 equ; 0 cnn)
% Maximal formula atoms : 31 ( 4 avg)
% Number of connectives : 24400 ( 449 ~; 15 |; 436 &;21695 @)
% ( 0 <=>;1805 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 875 ( 875 >; 0 *; 0 +; 0 <<)
% Number of symbols : 149 ( 146 usr; 10 con; 0-4 aty)
% Number of variables : 3168 ( 74 ^;3001 !; 93 ?;3168 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:26.545
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
list_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (146)
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
inf_inf_set_list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
inf_in4657809108759609906list_a: set_set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
cons_set_list_a: set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
nil_set_list_a: list_set_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_M_Eo_J,type,
bot_bot_set_list_a_o: set_list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
bot_bo3186585308812441520list_a: set_set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
ord_le8877086941679407844list_a: set_set_list_a > set_set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__List__Olist_Itf__a_J,type,
cohere6429906645900029933list_a: ( list_a > list_a > list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
cohere4318586772758246349list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001tf__a,type,
cohere6072184133013167079_rts_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocomposite__completion_001t__List__Olist_Itf__a_J,type,
compos5005676353122392380list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocomposite__completion_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
compos522909563978211612list_a: ( set_list_a > set_list_a > set_list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocomposite__completion_001tf__a,type,
compos1915005101613447350tion_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ocomp_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
extens4686118995507326689list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ojoin_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
extens431601257083675878list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
normal2844150749108697120list_a: ( list_list_a > list_list_a > list_list_a ) > set_list_list_a > list_list_a > list_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
normal1724791161856338944list_a: ( list_set_list_a > list_set_list_a > list_set_list_a ) > set_list_set_list_a > list_set_list_a > list_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__List__Olist_Itf__a_J,type,
normal4889798360446511898list_a: ( list_a > list_a > list_a ) > set_list_a > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
normal3869076228660116218list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001tf__a,type,
normal_sub_Cong_a: ( a > a > a ) > set_a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_H_001t__List__Olist_Itf__a_J,type,
normal4820288888589421745list_a: ( list_a > list_a > list_a ) > set_list_a > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_H_001tf__a,type,
normal_sub_Cong_a2: ( a > a > a ) > set_a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001t__List__Olist_Itf__a_J,type,
normal8738828432243163028list_a: ( list_a > list_a > list_a ) > set_list_a > list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
normal8303701782564906868list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > set_list_a > set_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001tf__a,type,
normal7408713899360725774lass_a: ( a > a > a ) > set_a > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001t__List__Olist_Itf__a_J,type,
normal5825812266234710965list_a: set_list_a > list_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001tf__a,type,
normal3259722184653208495_rep_a: set_a > a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__List__Olist_Itf__a_J,type,
normal4947954984078133662list_a: ( list_a > list_a > list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
normal4686252266493823358list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > set_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001tf__a,type,
normal8595587647932138008lass_a: ( a > a > a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001t__List__Olist_Itf__a_J,type,
partial_magma_list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
partia4872338157591926529list_a: ( set_list_a > set_list_a > set_list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001tf__a,type,
partial_magma_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001t__List__Olist_Itf__a_J,type,
partial_null_list_a: ( list_a > list_a > list_a ) > list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
partia8392230974233693834list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001tf__a,type,
partial_null_a: ( a > a > a ) > a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__confluent__rts_001t__List__Olist_Itf__a_J,type,
paths_2227358642439567705list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__confluent__rts_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
paths_3506353789733915961list_a: ( set_list_a > set_list_a > set_list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__confluent__rts_001tf__a,type,
paths_8083246678931295059_rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OArr_001t__List__Olist_Itf__a_J,type,
paths_in_Arr_list_a: ( list_a > list_a > list_a ) > list_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OArr_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
paths_3862341395584752492list_a: ( set_list_a > set_list_a > set_list_a ) > list_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OArr_001tf__a,type,
paths_in_Arr_a: ( a > a > a ) > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OIde_001tf__a,type,
paths_in_Ide_a: ( a > a > a ) > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid1x_001tf__a,type,
paths_in_Resid1x_a: ( a > a > a ) > a > list_a > a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid_001t__List__Olist_Itf__a_J,type,
paths_8620460302779588466list_a: ( list_a > list_a > list_a ) > list_list_a > list_list_a > list_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
paths_6591532302923812690list_a: ( set_list_a > set_list_a > set_list_a ) > list_set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid_001tf__a,type,
paths_in_Resid_a: ( a > a > a ) > list_a > list_a > list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OSrcs_001t__List__Olist_Itf__a_J,type,
paths_in_Srcs_list_a: ( list_a > list_a > list_a ) > list_list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OSrcs_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
paths_5622510820459842474list_a: ( set_list_a > set_list_a > set_list_a ) > list_set_list_a > set_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OSrcs_001tf__a,type,
paths_in_Srcs_a: ( a > a > a ) > list_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OTrgs_001t__List__Olist_Itf__a_J,type,
paths_in_Trgs_list_a: ( list_a > list_a > list_a ) > list_list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OTrgs_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
paths_6697778058162308143list_a: ( set_list_a > set_list_a > set_list_a ) > list_set_list_a > set_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OTrgs_001tf__a,type,
paths_in_Trgs_a: ( a > a > a ) > list_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__coherent__normal_OPCong_001tf__a,type,
paths_4972237869229803137Cong_a: ( a > a > a ) > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__normal_001t__List__Olist_Itf__a_J,type,
paths_8755422415569132860list_a: ( list_a > list_a > list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__normal_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
paths_2496957160577180956list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__normal_001tf__a,type,
paths_3682460814909619382rmal_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__normal_ONPath_001t__List__Olist_Itf__a_J,type,
paths_8553768783572469479list_a: ( list_a > list_a > list_a ) > set_list_a > list_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__normal_ONPath_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
paths_8943339929420292807list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > list_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__normal_ONPath_001tf__a,type,
paths_3208254328100665313Path_a: ( a > a > a ) > set_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001t__List__Olist_Itf__a_J,type,
quotie5013437406004430748list_a: ( list_a > list_a > list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
quotie4778059898687049596list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001tf__a,type,
quotie3282664541148387094rmal_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001t__List__Olist_Itf__a_J,type,
quotie715552529030613631list_a: ( list_a > list_a > list_a ) > set_list_a > set_list_a > set_list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
quotie750419831440180831list_a: ( set_list_a > set_list_a > set_list_a ) > set_set_list_a > set_set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001tf__a,type,
quotie8165075472272353145esid_a: ( a > a > a ) > set_a > set_a > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001t__List__Olist_Itf__a_J,type,
residuation_list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
residu1267353442355242628list_a: ( set_list_a > set_list_a > set_list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001tf__a,type,
residuation_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__List__Olist_Itf__a_J,type,
arr_list_a: ( list_a > list_a > list_a ) > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
arr_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
arr_set_set_list_a: ( set_set_list_a > set_set_list_a > set_set_list_a ) > set_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__Set__Oset_Itf__a_J,type,
arr_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001tf__a,type,
arr_a: ( a > a > a ) > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__List__Olist_Itf__a_J,type,
con_list_a: ( list_a > list_a > list_a ) > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
con_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
con_set_set_list_a: ( set_set_list_a > set_set_list_a > set_set_list_a ) > set_set_list_a > set_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_Itf__a_J,type,
con_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001tf__a,type,
con_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__List__Olist_Itf__a_J,type,
ide_list_a: ( list_a > list_a > list_a ) > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ide_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
ide_set_set_list_a: ( set_set_list_a > set_set_list_a > set_set_list_a ) > set_set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Set__Oset_Itf__a_J,type,
ide_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001tf__a,type,
ide_a: ( a > a > a ) > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__List__Olist_Itf__a_J,type,
trg_list_a: ( list_a > list_a > list_a ) > list_a > list_a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
trg_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001tf__a,type,
trg_a: ( a > a > a ) > a > a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__List__Olist_Itf__a_J,type,
coinitial_list_a: ( list_a > list_a > list_a ) > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
coinitial_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001tf__a,type,
coinitial_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__List__Olist_Itf__a_J,type,
composable_list_a: ( list_a > list_a > list_a ) > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
compos1473025410594674033list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001tf__a,type,
composable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001t__List__Olist_Itf__a_J,type,
coterminal_list_a: ( list_a > list_a > list_a ) > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
coterm239225051077876726list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001tf__a,type,
coterminal_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__List__Olist_Itf__a_J,type,
join_of_list_a: ( list_a > list_a > list_a ) > list_a > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
join_of_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001tf__a,type,
join_of_a: ( a > a > a ) > a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__List__Olist_Itf__a_J,type,
joinable_list_a: ( list_a > list_a > list_a ) > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
joinable_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001tf__a,type,
joinable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__List__Olist_Itf__a_J,type,
seq_list_a: ( list_a > list_a > list_a ) > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
seq_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001tf__a,type,
seq_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sources_list_list_a: ( list_list_a > list_list_a > list_list_a ) > list_list_a > set_list_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
source3381003665373322262list_a: ( list_set_list_a > list_set_list_a > list_set_list_a ) > list_set_list_a > set_list_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__List__Olist_Itf__a_J,type,
sources_list_a: ( list_a > list_a > list_a ) > list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
sources_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001tf__a,type,
sources_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__List__Olist_Itf__a_J,type,
targets_list_a: ( list_a > list_a > list_a ) > list_a > set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
targets_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_set_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001tf__a,type,
targets_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__List__Olist_Itf__a_J,type,
rts_wi1113109597627081724list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
rts_wi8505854437279755228list_a: ( set_list_a > set_list_a > set_list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001tf__a,type,
rts_wi3777564303360811894ites_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
weakly6189092243557426935list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
collec5381118732811369429list_a: ( list_set_list_a > $o ) > set_list_set_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_T,type,
t: list_a ).
thf(sy_v_T_H,type,
t2: list_a ).
thf(sy_v_resid,type,
resid: a > a > a ).
% Relevant facts (1278)
thf(fact_0_P_Ocon__char,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a ) ) ).
% P.con_char
thf(fact_1_R_Ocube,axiom,
! [V: a,T2: a,U2: a] :
( ( resid @ ( resid @ V @ T2 ) @ ( resid @ U2 @ T2 ) )
= ( resid @ ( resid @ V @ U2 ) @ ( resid @ T2 @ U2 ) ) ) ).
% R.cube
thf(fact_2_R_Oex__un__null,axiom,
? [X: a] :
( ! [T3: a] :
( ( ( resid @ X @ T3 )
= X )
& ( ( resid @ T3 @ X )
= X ) )
& ! [Y: a] :
( ! [T4: a] :
( ( ( resid @ Y @ T4 )
= Y )
& ( ( resid @ T4 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% R.ex_un_null
thf(fact_3_composite__completion__axioms,axiom,
compos1915005101613447350tion_a @ resid ).
% composite_completion_axioms
thf(fact_4_P_OCong_092_060_094sub_0620__imp__con,axiom,
! [T2: list_a,T5: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T5 ) ) ).
% P.Cong\<^sub>0_imp_con
thf(fact_5_Cong__eq__Cong_092_060_094sub_0620,axiom,
! [T: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T @ T6 )
= ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% Cong_eq_Cong\<^sub>0
thf(fact_6_P_Ocube,axiom,
! [V: list_a,T2: list_a,U2: list_a] :
( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T2 ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ).
% P.cube
thf(fact_7_P_Oex__un__null,axiom,
? [X: list_a] :
( ! [T3: list_a] :
( ( ( paths_in_Resid_a @ resid @ X @ T3 )
= X )
& ( ( paths_in_Resid_a @ resid @ T3 @ X )
= X ) )
& ! [Y: list_a] :
( ! [T4: list_a] :
( ( ( paths_in_Resid_a @ resid @ Y @ T4 )
= Y )
& ( ( paths_in_Resid_a @ resid @ T4 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% P.ex_un_null
thf(fact_8_N_OCong_092_060_094sub_0620__symmetric,axiom,
! [T2: a,T5: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% N.Cong\<^sub>0_symmetric
thf(fact_9_N_OCong_092_060_094sub_0620__transitive,axiom,
! [T2: a,T5: a,T7: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( ( member_a @ ( resid @ T5 @ T7 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T7 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( member_a @ ( resid @ T2 @ T7 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T7 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ).
% N.Cong\<^sub>0_transitive
thf(fact_10_N_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [T2: a,U2: a,T5: a] :
( ( ( member_a @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ U2 ) @ ( resid @ T2 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ).
% N.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_11_N_Obackward__stable,axiom,
! [U2: a,T2: a] :
( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ ( resid @ T2 @ U2 ) @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( member_a @ T2 @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% N.backward_stable
thf(fact_12_N_Oide__closed,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( member_a @ A @ ( collect_a @ ( ide_a @ resid ) ) ) ) ).
% N.ide_closed
thf(fact_13_N_Oprfx__closed,axiom,
! [U2: a,T2: a] :
( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
=> ( member_a @ T2 @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% N.prfx_closed
thf(fact_14_R_Ocong__symmetric,axiom,
! [T2: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T2 ) ) )
=> ( ( ide_a @ resid @ ( resid @ U2 @ T2 ) )
& ( ide_a @ resid @ ( resid @ T2 @ U2 ) ) ) ) ).
% R.cong_symmetric
thf(fact_15_R_Ocong__transitive,axiom,
! [T2: a,U2: a,V: a] :
( ( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T2 ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U2 @ V ) )
& ( ide_a @ resid @ ( resid @ V @ U2 ) ) )
=> ( ( ide_a @ resid @ ( resid @ T2 @ V ) )
& ( ide_a @ resid @ ( resid @ V @ T2 ) ) ) ) ) ).
% R.cong_transitive
thf(fact_16_R_Oide__backward__stable,axiom,
! [A: a,T2: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ ( resid @ T2 @ A ) )
=> ( ide_a @ resid @ T2 ) ) ) ).
% R.ide_backward_stable
thf(fact_17_R_Oprfx__transitive,axiom,
! [T2: a,U2: a,V: a] :
( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
=> ( ( ide_a @ resid @ ( resid @ U2 @ V ) )
=> ( ide_a @ resid @ ( resid @ T2 @ V ) ) ) ) ).
% R.prfx_transitive
thf(fact_18_P_OCube_I2_J,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ V2 @ U ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ V2 @ U ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ V2 ) @ ( paths_in_Resid_a @ resid @ U @ V2 ) ) ) ) ).
% P.Cube(2)
thf(fact_19_P_OCube_I1_J,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ V2 @ U ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ V2 ) @ ( paths_in_Resid_a @ resid @ U @ V2 ) )
!= nil_a ) ) ).
% P.Cube(1)
thf(fact_20_P_OResid_Osimps_I1_J,axiom,
! [Uu: list_a] :
( ( paths_in_Resid_a @ resid @ nil_a @ Uu )
= nil_a ) ).
% P.Resid.simps(1)
thf(fact_21_P_OCon__sym,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ U @ T )
!= nil_a ) ) ).
% P.Con_sym
thf(fact_22_P_Ocon__sym,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T2 ) ) ).
% P.con_sym
thf(fact_23_P_Oresid__reflects__con,axiom,
! [T2: list_a,V: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ V )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ V )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ V ) @ ( paths_in_Resid_a @ resid @ U2 @ V ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ) ) ).
% P.resid_reflects_con
thf(fact_24_assms,axiom,
normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ t @ t2 ).
% assms
thf(fact_25_P_OCon__append_I2_J,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( V2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T @ ( append_a @ U @ V2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ V2 )
!= nil_a ) ) ) ) ) ) ).
% P.Con_append(2)
thf(fact_26_P_OCon__append_I1_J,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( V2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ V2 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T @ V2 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ U @ ( paths_in_Resid_a @ resid @ V2 @ T ) )
!= nil_a ) ) ) ) ) ) ).
% P.Con_append(1)
thf(fact_27_P_OResid__append__ind,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( V2 != nil_a )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( append_a @ V2 @ T ) @ U )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ V2 @ U )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ U @ V2 ) )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T @ ( append_a @ V2 @ U ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T @ V2 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ V2 ) @ U )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ ( append_a @ V2 @ T ) @ U )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ V2 @ T ) @ U )
= ( append_a @ ( paths_in_Resid_a @ resid @ V2 @ U ) @ ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ U @ V2 ) ) ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T @ ( append_a @ V2 @ U ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ ( append_a @ V2 @ U ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ V2 ) @ U ) ) ) ) ) ) ) ).
% P.Resid_append_ind
thf(fact_28_calculation,axiom,
( ( paths_in_Trgs_a @ resid @ t )
= ( paths_in_Trgs_a @ resid @ ( append_a @ t @ ( paths_in_Resid_a @ resid @ t2 @ t ) ) ) ) ).
% calculation
thf(fact_29_P_OBackward__stable,axiom,
! [U: list_a,T: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
=> ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T ) ) ) ).
% P.Backward_stable
thf(fact_30_P_OCong_092_060_094sub_0620__symmetric,axiom,
! [T2: list_a,T5: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_symmetric
thf(fact_31_P_OCong_092_060_094sub_0620__transitive,axiom,
! [T2: list_a,T5: list_a,T7: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T7 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T7 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T7 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T7 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_transitive
thf(fact_32_P_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [T2: list_a,U2: list_a,T5: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% P.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_33_P_Obackward__stable,axiom,
! [U2: list_a,T2: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( member_list_a @ T2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% P.backward_stable
thf(fact_34_P_OCong_092_060_094sub_0620__append__resid__NPath,axiom,
! [T: list_a,U: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ ( paths_in_Resid_a @ resid @ U @ T ) ) @ U ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U @ ( append_a @ T @ ( paths_in_Resid_a @ resid @ U @ T ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_append_resid_NPath
thf(fact_35_paths__in__rts_OTrgs_Ocong,axiom,
paths_in_Trgs_a = paths_in_Trgs_a ).
% paths_in_rts.Trgs.cong
thf(fact_36_paths__in__rts_OResid_Ocong,axiom,
paths_in_Resid_a = paths_in_Resid_a ).
% paths_in_rts.Resid.cong
thf(fact_37_R_Opartial__magma__axioms,axiom,
partial_magma_a @ resid ).
% R.partial_magma_axioms
thf(fact_38_P_Opartial__magma__axioms,axiom,
partial_magma_list_a @ ( paths_in_Resid_a @ resid ) ).
% P.partial_magma_axioms
thf(fact_39_P_OTrgs__append,axiom,
! [U: list_a,T: list_a] :
( ( U != nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( append_a @ T @ U ) )
= ( paths_in_Trgs_a @ resid @ U ) ) ) ).
% P.Trgs_append
thf(fact_40_P_OTrgs__Resid__sym,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) )
= ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ U @ T ) ) ) ) ).
% P.Trgs_Resid_sym
thf(fact_41_P_OCon__appendI_I2_J,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ V2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ ( append_a @ U @ V2 ) )
!= nil_a ) ) ) ).
% P.Con_appendI(2)
thf(fact_42_P_OCon__appendI_I1_J,axiom,
! [T: list_a,V2: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ V2 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ U @ ( paths_in_Resid_a @ resid @ V2 @ T ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ V2 )
!= nil_a ) ) ) ).
% P.Con_appendI(1)
thf(fact_43_P_OResid__append_I2_J,axiom,
! [U: list_a,V2: list_a,T: list_a] :
( ( U != nil_a )
=> ( ( V2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T @ ( append_a @ U @ V2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ ( append_a @ U @ V2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ V2 ) ) ) ) ) ).
% P.Resid_append(2)
thf(fact_44_P_OResid__append_I1_J,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( T != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ V2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ V2 )
= ( append_a @ ( paths_in_Resid_a @ resid @ T @ V2 ) @ ( paths_in_Resid_a @ resid @ U @ ( paths_in_Resid_a @ resid @ V2 @ T ) ) ) ) ) ) ).
% P.Resid_append(1)
thf(fact_45_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: set_list_a,P: set_list_a > $o] :
( ( member_set_list_a @ A @ ( collect_set_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X2: set_list_a] : ( member_set_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_52_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_53_Collect__cong,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ! [X: set_list_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_list_a @ P )
= ( collect_set_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_54_P_OResid__append2,axiom,
! [T: list_a,U: list_a,V2: list_a,W: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( V2 != nil_a )
=> ( ( W != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ ( append_a @ V2 @ W ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ ( append_a @ V2 @ W ) )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ V2 ) @ W ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ U @ ( paths_in_Resid_a @ resid @ V2 @ T ) ) @ ( paths_in_Resid_a @ resid @ W @ ( paths_in_Resid_a @ resid @ T @ V2 ) ) ) ) ) ) ) ) ) ) ).
% P.Resid_append2
thf(fact_55_P_OCong_092_060_094sub_0620__cancel__left_092_060_094sub_062C_092_060_094sub_062S,axiom,
! [T: list_a,U: list_a,U3: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ ( append_a @ T @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U3 ) @ ( append_a @ T @ U ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( U3 != nil_a )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U @ U3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U3 @ U ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_cancel_left\<^sub>C\<^sub>S
thf(fact_56_P_Orts__with__composites__axioms,axiom,
rts_wi1113109597627081724list_a @ ( paths_in_Resid_a @ resid ) ).
% P.rts_with_composites_axioms
thf(fact_57_P_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [T2: list_a,T5: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_subst_left(2)
thf(fact_58_P_OCong_092_060_094sub_0620__subst__Con,axiom,
! [T2: list_a,T5: list_a,U2: list_a,U4: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T5 @ U4 ) ) ) ) ).
% P.Cong\<^sub>0_subst_Con
thf(fact_59_P_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [U2: list_a,U4: list_a,T2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U4 ) ) ) ).
% P.Cong\<^sub>0_subst_right(1)
thf(fact_60_P_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [U2: list_a,U4: list_a,T2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U4 ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U4 ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_subst_right(2)
thf(fact_61_P_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [T2: list_a,T5: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T5 @ U2 ) ) ) ).
% P.Cong\<^sub>0_subst_left(1)
thf(fact_62_P_Onormal__is__Cong__closed,axiom,
! [T2: list_a,T5: list_a] :
( ( member_list_a @ T2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( member_list_a @ T5 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% P.normal_is_Cong_closed
thf(fact_63_P_OCong__transitive,axiom,
! [T2: list_a,T7: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T7 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T7 @ T5 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) ) ) ).
% P.Cong_transitive
thf(fact_64_P_OCong__symmetric,axiom,
! [T2: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T5 @ T2 ) ) ).
% P.Cong_symmetric
thf(fact_65_P_OCong__def,axiom,
! [T2: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
= ( ? [U5: list_a,U6: list_a] :
( ( member_list_a @ U5 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ U6 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U5 ) @ ( paths_in_Resid_a @ resid @ T5 @ U6 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U6 ) @ ( paths_in_Resid_a @ resid @ T2 @ U5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% P.Cong_def
thf(fact_66_P_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [T2: list_a,T5: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) ) ).
% P.Cong\<^sub>0_implies_Cong
thf(fact_67_P_OCongE,axiom,
! [T2: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ~ ! [U7: list_a] :
( ( member_list_a @ U7 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ! [U8: list_a] :
( ( member_list_a @ U8 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ~ ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U7 ) @ ( paths_in_Resid_a @ resid @ T5 @ U8 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U8 ) @ ( paths_in_Resid_a @ resid @ T2 @ U7 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ).
% P.CongE
thf(fact_68_P_OCong__closure__props_I1_J,axiom,
! [T2: list_a,U2: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ).
% P.Cong_closure_props(1)
thf(fact_69_P_OCong__closure__props_I2_J,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) ) ) ).
% P.Cong_closure_props(2)
thf(fact_70_P_OCong__closure__props_I3_J,axiom,
! [T2: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ).
% P.Cong_closure_props(3)
thf(fact_71_P_OconI_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% P.conI\<^sub>P
thf(fact_72_P_OCongI,axiom,
! [U2: list_a,U4: list_a,T2: list_a,T5: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ U4 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U4 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U4 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) ) ) ) ).
% P.CongI
thf(fact_73_partial__magma__def,axiom,
( partial_magma_a
= ( ^ [OP: a > a > a] :
? [X2: a] :
( ! [T8: a] :
( ( ( OP @ X2 @ T8 )
= X2 )
& ( ( OP @ T8 @ X2 )
= X2 ) )
& ! [Y2: a] :
( ! [T8: a] :
( ( ( OP @ Y2 @ T8 )
= Y2 )
& ( ( OP @ T8 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_74_partial__magma__def,axiom,
( partial_magma_list_a
= ( ^ [OP: list_a > list_a > list_a] :
? [X2: list_a] :
( ! [T8: list_a] :
( ( ( OP @ X2 @ T8 )
= X2 )
& ( ( OP @ T8 @ X2 )
= X2 ) )
& ! [Y2: list_a] :
( ! [T8: list_a] :
( ( ( OP @ Y2 @ T8 )
= Y2 )
& ( ( OP @ T8 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_75_partial__magma__def,axiom,
( partia4872338157591926529list_a
= ( ^ [OP: set_list_a > set_list_a > set_list_a] :
? [X2: set_list_a] :
( ! [T8: set_list_a] :
( ( ( OP @ X2 @ T8 )
= X2 )
& ( ( OP @ T8 @ X2 )
= X2 ) )
& ! [Y2: set_list_a] :
( ! [T8: set_list_a] :
( ( ( OP @ Y2 @ T8 )
= Y2 )
& ( ( OP @ T8 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_76_partial__magma_Oex__un__null,axiom,
! [OP2: a > a > a] :
( ( partial_magma_a @ OP2 )
=> ? [X: a] :
( ! [T3: a] :
( ( ( OP2 @ X @ T3 )
= X )
& ( ( OP2 @ T3 @ X )
= X ) )
& ! [Y: a] :
( ! [T4: a] :
( ( ( OP2 @ Y @ T4 )
= Y )
& ( ( OP2 @ T4 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_77_partial__magma_Oex__un__null,axiom,
! [OP2: list_a > list_a > list_a] :
( ( partial_magma_list_a @ OP2 )
=> ? [X: list_a] :
( ! [T3: list_a] :
( ( ( OP2 @ X @ T3 )
= X )
& ( ( OP2 @ T3 @ X )
= X ) )
& ! [Y: list_a] :
( ! [T4: list_a] :
( ( ( OP2 @ Y @ T4 )
= Y )
& ( ( OP2 @ T4 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_78_partial__magma_Oex__un__null,axiom,
! [OP2: set_list_a > set_list_a > set_list_a] :
( ( partia4872338157591926529list_a @ OP2 )
=> ? [X: set_list_a] :
( ! [T3: set_list_a] :
( ( ( OP2 @ X @ T3 )
= X )
& ( ( OP2 @ T3 @ X )
= X ) )
& ! [Y: set_list_a] :
( ! [T4: set_list_a] :
( ( ( OP2 @ Y @ T4 )
= Y )
& ( ( OP2 @ T4 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_79_partial__magma_Ointro,axiom,
! [OP2: a > a > a] :
( ? [X3: a] :
( ! [T4: a] :
( ( ( OP2 @ X3 @ T4 )
= X3 )
& ( ( OP2 @ T4 @ X3 )
= X3 ) )
& ! [Y3: a] :
( ! [T3: a] :
( ( ( OP2 @ Y3 @ T3 )
= Y3 )
& ( ( OP2 @ T3 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) )
=> ( partial_magma_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_80_partial__magma_Ointro,axiom,
! [OP2: list_a > list_a > list_a] :
( ? [X3: list_a] :
( ! [T4: list_a] :
( ( ( OP2 @ X3 @ T4 )
= X3 )
& ( ( OP2 @ T4 @ X3 )
= X3 ) )
& ! [Y3: list_a] :
( ! [T3: list_a] :
( ( ( OP2 @ Y3 @ T3 )
= Y3 )
& ( ( OP2 @ T3 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) )
=> ( partial_magma_list_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_81_partial__magma_Ointro,axiom,
! [OP2: set_list_a > set_list_a > set_list_a] :
( ? [X3: set_list_a] :
( ! [T4: set_list_a] :
( ( ( OP2 @ X3 @ T4 )
= X3 )
& ( ( OP2 @ T4 @ X3 )
= X3 ) )
& ! [Y3: set_list_a] :
( ! [T3: set_list_a] :
( ( ( OP2 @ Y3 @ T3 )
= Y3 )
& ( ( OP2 @ T3 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) )
=> ( partia4872338157591926529list_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_82_composite__completion_OCong__eq__Cong_092_060_094sub_0620,axiom,
! [Resid: list_a > list_a > list_a,T: list_list_a,T6: list_list_a] :
( ( compos5005676353122392380list_a @ Resid )
=> ( ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) @ T @ T6 )
= ( ( member_list_list_a @ ( paths_8620460302779588466list_a @ Resid @ T @ T6 ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) )
& ( member_list_list_a @ ( paths_8620460302779588466list_a @ Resid @ T6 @ T ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) ) ) ) ) ).
% composite_completion.Cong_eq_Cong\<^sub>0
thf(fact_83_composite__completion_OCong__eq__Cong_092_060_094sub_0620,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T: list_set_list_a,T6: list_set_list_a] :
( ( compos522909563978211612list_a @ Resid )
=> ( ( normal1724791161856338944list_a @ ( paths_6591532302923812690list_a @ Resid ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ ( collect_set_list_a @ ( ide_set_list_a @ Resid ) ) ) ) @ T @ T6 )
= ( ( member5524387281408368019list_a @ ( paths_6591532302923812690list_a @ Resid @ T @ T6 ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ ( collect_set_list_a @ ( ide_set_list_a @ Resid ) ) ) ) )
& ( member5524387281408368019list_a @ ( paths_6591532302923812690list_a @ Resid @ T6 @ T ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ ( collect_set_list_a @ ( ide_set_list_a @ Resid ) ) ) ) ) ) ) ) ).
% composite_completion.Cong_eq_Cong\<^sub>0
thf(fact_84_composite__completion_OCong__eq__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,T: list_a,T6: list_a] :
( ( compos1915005101613447350tion_a @ Resid )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ ( collect_a @ ( ide_a @ Resid ) ) ) ) @ T @ T6 )
= ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ T @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ ( collect_a @ ( ide_a @ Resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ T6 @ T ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ ( collect_a @ ( ide_a @ Resid ) ) ) ) ) ) ) ) ).
% composite_completion.Cong_eq_Cong\<^sub>0
thf(fact_85_residuation_Ocon_Ocong,axiom,
con_list_a = con_list_a ).
% residuation.con.cong
thf(fact_86_residuation_Ocon_Ocong,axiom,
con_a = con_a ).
% residuation.con.cong
thf(fact_87_residuation_Ocon_Ocong,axiom,
con_set_list_a = con_set_list_a ).
% residuation.con.cong
thf(fact_88_residuation_Oide_Ocong,axiom,
ide_a = ide_a ).
% residuation.ide.cong
thf(fact_89_residuation_Oide_Ocong,axiom,
ide_list_a = ide_list_a ).
% residuation.ide.cong
thf(fact_90_residuation_Oide_Ocong,axiom,
ide_set_list_a = ide_set_list_a ).
% residuation.ide.cong
thf(fact_91_paths__in__rts__with__normal_ONPath_Ocong,axiom,
paths_3208254328100665313Path_a = paths_3208254328100665313Path_a ).
% paths_in_rts_with_normal.NPath.cong
thf(fact_92_normal__sub__rts_OCong_Ocong,axiom,
normal4889798360446511898list_a = normal4889798360446511898list_a ).
% normal_sub_rts.Cong.cong
thf(fact_93_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_a = normal_sub_Cong_a ).
% normal_sub_rts.Cong.cong
thf(fact_94_normal__sub__rts_OCong_Ocong,axiom,
normal3869076228660116218list_a = normal3869076228660116218list_a ).
% normal_sub_rts.Cong.cong
thf(fact_95_P_Ocoherent__normal__sub__rts__axioms,axiom,
cohere6429906645900029933list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ).
% P.coherent_normal_sub_rts_axioms
thf(fact_96_R_Ocong__implies__coterminal,axiom,
! [U2: a,U4: a] :
( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( coterminal_a @ resid @ U2 @ U4 ) ) ).
% R.cong_implies_coterminal
thf(fact_97_P_OTrgs__respects__Cong,axiom,
! [T: list_a,T6: list_a,B: a,B2: a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T @ T6 )
=> ( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( paths_in_Trgs_a @ resid @ T6 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( cons_a @ B @ nil_a ) @ ( cons_a @ B2 @ nil_a ) ) ) ) ) ).
% P.Trgs_respects_Cong
thf(fact_98_P_Oresid__along__elem__preserves__con,axiom,
! [T2: list_a,T5: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T5 )
=> ( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) ) ) ) ) ).
% P.resid_along_elem_preserves_con
thf(fact_99_quotient__by__coherent__normal__axioms,axiom,
quotie5013437406004430748list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ).
% quotient_by_coherent_normal_axioms
thf(fact_100_R_Oidentities__form__coherent__normal__sub__rts,axiom,
cohere6072184133013167079_rts_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ).
% R.identities_form_coherent_normal_sub_rts
thf(fact_101_P_Ojoinable__implies__con,axiom,
! [T2: list_a,U2: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.joinable_implies_con
thf(fact_102_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_103_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_104_self__append__conv2,axiom,
! [Y4: list_a,Xs: list_a] :
( ( Y4
= ( append_a @ Xs @ Y4 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_105_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_106_R_Ocon__sym,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( con_a @ resid @ U2 @ T2 ) ) ).
% R.con_sym
thf(fact_107_R_Oresid__reflects__con,axiom,
! [T2: a,V: a,U2: a] :
( ( con_a @ resid @ T2 @ V )
=> ( ( con_a @ resid @ U2 @ V )
=> ( ( con_a @ resid @ ( resid @ T2 @ V ) @ ( resid @ U2 @ V ) )
=> ( con_a @ resid @ T2 @ U2 ) ) ) ) ).
% R.resid_reflects_con
thf(fact_108_P_OTrgs_Ocases,axiom,
! [X4: list_a] :
( ( X4 != nil_a )
=> ( ! [T4: a] :
( X4
!= ( cons_a @ T4 @ nil_a ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( X4
!= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) ) ) ) ).
% P.Trgs.cases
thf(fact_109_P_Ocong__symmetric,axiom,
! [T2: list_a,U2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ) ).
% P.cong_symmetric
thf(fact_110_P_Ocong__transitive,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ V ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ U2 ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ V ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ T2 ) ) ) ) ) ).
% P.cong_transitive
thf(fact_111_P_Oide__backward__stable,axiom,
! [A: list_a,T2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ A ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ).
% P.ide_backward_stable
thf(fact_112_P_Oprfx__transitive,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ V ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ V ) ) ) ) ).
% P.prfx_transitive
thf(fact_113_N_OCong__closure__props_I3_J,axiom,
! [T2: a,U2: a] :
( ( ( member_a @ ( resid @ T2 @ U2 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ U2 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ U2 ) ) ).
% N.Cong_closure_props(3)
thf(fact_114_N_OCong__closure__props_I2_J,axiom,
! [T2: a,U2: a,V: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ U2 )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U2 @ V )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ V ) ) ) ).
% N.Cong_closure_props(2)
thf(fact_115_N_OCong__closure__props_I1_J,axiom,
! [T2: a,U2: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ U2 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U2 @ T2 ) ) ).
% N.Cong_closure_props(1)
thf(fact_116_N_OCongE,axiom,
! [T2: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ~ ! [U7: a] :
( ( member_a @ U7 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ! [U8: a] :
( ( member_a @ U8 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ~ ( ( member_a @ ( resid @ ( resid @ T2 @ U7 ) @ ( resid @ T5 @ U8 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ U8 ) @ ( resid @ T2 @ U7 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% N.CongE
thf(fact_117_N_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [T2: a,T5: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 ) ) ).
% N.Cong\<^sub>0_implies_Cong
thf(fact_118_N_OCong__def,axiom,
! [T2: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
= ( ? [U5: a,U6: a] :
( ( member_a @ U5 @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ U6 @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T2 @ U5 ) @ ( resid @ T5 @ U6 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ U6 ) @ ( resid @ T2 @ U5 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ).
% N.Cong_def
thf(fact_119_N_OCong__symmetric,axiom,
! [T2: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T5 @ T2 ) ) ).
% N.Cong_symmetric
thf(fact_120_N_OCong__transitive,axiom,
! [T2: a,T7: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T7 )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T7 @ T5 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 ) ) ) ).
% N.Cong_transitive
thf(fact_121_N_Onormal__is__Cong__closed,axiom,
! [T2: a,T5: a] :
( ( member_a @ T2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( member_a @ T5 @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% N.normal_is_Cong_closed
thf(fact_122_R_OCong__iff__cong,axiom,
! [T2: a,U2: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ U2 )
= ( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T2 ) ) ) ) ).
% R.Cong_iff_cong
thf(fact_123_N_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [T2: a,T5: a,U2: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( ( member_a @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ U2 ) @ ( resid @ T2 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ).
% N.Cong\<^sub>0_subst_left(2)
thf(fact_124_N_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [T2: a,T5: a,U2: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( con_a @ resid @ T5 @ U2 ) ) ) ).
% N.Cong\<^sub>0_subst_left(1)
thf(fact_125_N_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [U2: a,U4: a,T2: a] :
( ( ( member_a @ ( resid @ U2 @ U4 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ U4 @ U2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( ( member_a @ ( resid @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ U4 @ U2 ) ) @ ( resid @ ( resid @ T2 @ U4 ) @ ( resid @ U2 @ U4 ) ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ ( resid @ T2 @ U4 ) @ ( resid @ U2 @ U4 ) ) @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ U4 @ U2 ) ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ).
% N.Cong\<^sub>0_subst_right(2)
thf(fact_126_N_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [U2: a,U4: a,T2: a] :
( ( ( member_a @ ( resid @ U2 @ U4 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ U4 @ U2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( con_a @ resid @ T2 @ U4 ) ) ) ).
% N.Cong\<^sub>0_subst_right(1)
thf(fact_127_R_Ocong__subst__left_I2_J,axiom,
! [T2: a,T5: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ T2 @ T5 ) )
& ( ide_a @ resid @ ( resid @ T5 @ T2 ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U2 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T5 @ U2 ) @ ( resid @ T2 @ U2 ) ) ) ) ) ) ).
% R.cong_subst_left(2)
thf(fact_128_R_Ocong__subst__left_I1_J,axiom,
! [T2: a,T5: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ T2 @ T5 ) )
& ( ide_a @ resid @ ( resid @ T5 @ T2 ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( con_a @ resid @ T5 @ U2 ) ) ) ).
% R.cong_subst_left(1)
thf(fact_129_R_Ocong__subst__right_I2_J,axiom,
! [U2: a,U4: a,T2: a] :
( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T2 @ U4 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T2 @ U4 ) @ ( resid @ T2 @ U2 ) ) ) ) ) ) ).
% R.cong_subst_right(2)
thf(fact_130_R_Ocong__subst__right_I1_J,axiom,
! [U2: a,U4: a,T2: a] :
( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( con_a @ resid @ T2 @ U4 ) ) ) ).
% R.cong_subst_right(1)
thf(fact_131_N_OCong_092_060_094sub_0620__imp__con,axiom,
! [T2: a,T5: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( con_a @ resid @ T2 @ T5 ) ) ).
% N.Cong\<^sub>0_imp_con
thf(fact_132_N_OCong_092_060_094sub_0620__subst__Con,axiom,
! [T2: a,T5: a,U2: a,U4: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( ( member_a @ ( resid @ U2 @ U4 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ U4 @ U2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( con_a @ resid @ T2 @ U2 )
= ( con_a @ resid @ T5 @ U4 ) ) ) ) ).
% N.Cong\<^sub>0_subst_Con
thf(fact_133_R_Ocon__imp__coinitial__ax,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ? [A3: a] :
( ( ide_a @ resid @ A3 )
& ( con_a @ resid @ A3 @ T2 )
& ( con_a @ resid @ A3 @ U2 ) ) ) ).
% R.con_imp_coinitial_ax
thf(fact_134_R_Ocon__target,axiom,
! [T2: a,U2: a,V: a] :
( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
=> ( ( con_a @ resid @ U2 @ V )
=> ( con_a @ resid @ ( resid @ T2 @ U2 ) @ ( resid @ V @ U2 ) ) ) ) ).
% R.con_target
thf(fact_135_R_Ocon__transitive__on__ide,axiom,
! [A: a,B: a,C: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ B )
=> ( ( ide_a @ resid @ C )
=> ( ( con_a @ resid @ A @ B )
=> ( ( con_a @ resid @ B @ C )
=> ( con_a @ resid @ A @ C ) ) ) ) ) ) ).
% R.con_transitive_on_ide
thf(fact_136_R_OideE,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ~ ( ( con_a @ resid @ A @ A )
=> ( ( resid @ A @ A )
!= A ) ) ) ).
% R.ideE
thf(fact_137_R_Oide__def,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
= ( ( con_a @ resid @ A @ A )
& ( ( resid @ A @ A )
= A ) ) ) ).
% R.ide_def
thf(fact_138_R_Oide__imp__con__iff__cong,axiom,
! [T2: a,U2: a] :
( ( ide_a @ resid @ T2 )
=> ( ( ide_a @ resid @ U2 )
=> ( ( con_a @ resid @ T2 @ U2 )
= ( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T2 ) ) ) ) ) ) ).
% R.ide_imp_con_iff_cong
thf(fact_139_R_Oprfx__implies__con,axiom,
! [T2: a,U2: a] :
( ( ide_a @ resid @ ( resid @ T2 @ U2 ) )
=> ( con_a @ resid @ T2 @ U2 ) ) ).
% R.prfx_implies_con
thf(fact_140_R_Oresid__arr__ide,axiom,
! [A: a,T2: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T2 @ A )
=> ( ( resid @ T2 @ A )
= T2 ) ) ) ).
% R.resid_arr_ide
thf(fact_141_R_Oresid__ide__arr,axiom,
! [A: a,T2: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ A @ T2 )
=> ( ide_a @ resid @ ( resid @ A @ T2 ) ) ) ) ).
% R.resid_ide_arr
thf(fact_142_P_OTrgs__are__con,axiom,
! [B: a,T: list_a,B2: a] :
( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( paths_in_Trgs_a @ resid @ T ) )
=> ( con_a @ resid @ B @ B2 ) ) ) ).
% P.Trgs_are_con
thf(fact_143_P_Ocong__subst__left_I2_J,axiom,
! [T2: list_a,T5: list_a,U2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ) ) ) ).
% P.cong_subst_left(2)
thf(fact_144_P_Ocong__subst__left_I1_J,axiom,
! [T2: list_a,T5: list_a,U2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T5 @ U2 ) ) ) ).
% P.cong_subst_left(1)
thf(fact_145_P_Ocong__subst__right_I2_J,axiom,
! [U2: list_a,U4: list_a,T2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U4 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U4 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ) ) ) ).
% P.cong_subst_right(2)
thf(fact_146_P_Ocong__subst__right_I1_J,axiom,
! [U2: list_a,U4: list_a,T2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U4 ) ) ) ).
% P.cong_subst_right(1)
thf(fact_147_P_Ocon__imp__coinitial__ax,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ? [A3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A3 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A3 @ T2 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A3 @ U2 ) ) ) ).
% P.con_imp_coinitial_ax
thf(fact_148_P_Ocon__target,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ V )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ V @ U2 ) ) ) ) ).
% P.con_target
thf(fact_149_P_Ocon__transitive__on__ide,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ C )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ B )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ B @ C )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ C ) ) ) ) ) ) ).
% P.con_transitive_on_ide
thf(fact_150_P_OideE,axiom,
! [A: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ~ ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A )
=> ( ( paths_in_Resid_a @ resid @ A @ A )
!= A ) ) ) ).
% P.ideE
thf(fact_151_P_Oide__def,axiom,
! [A: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
= ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A )
& ( ( paths_in_Resid_a @ resid @ A @ A )
= A ) ) ) ).
% P.ide_def
thf(fact_152_P_Oide__imp__con__iff__cong,axiom,
! [T2: list_a,U2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) ) ) ) ) ) ).
% P.ide_imp_con_iff_cong
thf(fact_153_P_Oprfx__implies__con,axiom,
! [T2: list_a,U2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.prfx_implies_con
thf(fact_154_P_Oresid__arr__ide,axiom,
! [A: list_a,T2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ A )
=> ( ( paths_in_Resid_a @ resid @ T2 @ A )
= T2 ) ) ) ).
% P.resid_arr_ide
thf(fact_155_P_Oresid__ide__arr,axiom,
! [A: list_a,T2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T2 )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ T2 ) ) ) ) ).
% P.resid_ide_arr
thf(fact_156_P_OCong__iff__cong,axiom,
! [T2: list_a,U2: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( ide_list_a @ ( paths_in_Resid_a @ resid ) ) ) @ T2 @ U2 )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) ) ) ) ).
% P.Cong_iff_cong
thf(fact_157_P_OTrgs__con__closed,axiom,
! [B: a,T: list_a,B2: a] :
( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T ) )
=> ( ( ide_a @ resid @ B2 )
=> ( ( con_a @ resid @ B @ B2 )
=> ( member_a @ B2 @ ( paths_in_Trgs_a @ resid @ T ) ) ) ) ) ).
% P.Trgs_con_closed
thf(fact_158_P_OTrgs_Osimps_I3_J,axiom,
! [T2: a,V: a,Va2: list_a] :
( ( paths_in_Trgs_a @ resid @ ( cons_a @ T2 @ ( cons_a @ V @ Va2 ) ) )
= ( paths_in_Trgs_a @ resid @ ( cons_a @ V @ Va2 ) ) ) ).
% P.Trgs.simps(3)
thf(fact_159_P_Ocoinitial__ide__are__cong,axiom,
! [A: list_a,A4: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A4 )
=> ( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A4 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ A4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A4 @ A ) ) ) ) ) ) ).
% P.coinitial_ide_are_cong
thf(fact_160_P_Ocong__implies__coinitial,axiom,
! [U2: list_a,U4: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ U4 ) ) ).
% P.cong_implies_coinitial
thf(fact_161_P_Oide__closed,axiom,
! [A: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( member_list_a @ A @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ).
% P.ide_closed
thf(fact_162_P_Oprfx__closed,axiom,
! [U2: list_a,T2: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
=> ( member_list_a @ T2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% P.prfx_closed
thf(fact_163_P_OCon__rec_I4_J,axiom,
! [T: list_a,U: list_a,T2: a,U2: a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ ( cons_a @ U2 @ U ) )
!= nil_a )
= ( ( con_a @ resid @ T2 @ U2 )
& ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) @ U )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) ) )
!= nil_a ) ) ) ) ) ).
% P.Con_rec(4)
thf(fact_164_P_OCon__rec_I3_J,axiom,
! [U: list_a,T2: a,U2: a] :
( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ U ) )
!= nil_a )
= ( ( con_a @ resid @ T2 @ U2 )
& ( ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) @ U )
!= nil_a ) ) ) ) ).
% P.Con_rec(3)
thf(fact_165_P_OCon__rec_I2_J,axiom,
! [T: list_a,T2: a,U2: a] :
( ( T != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
= ( ( con_a @ resid @ T2 @ U2 )
& ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) )
!= nil_a ) ) ) ) ).
% P.Con_rec(2)
thf(fact_166_P_OCon__rec_I1_J,axiom,
! [T2: a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
= ( con_a @ resid @ T2 @ U2 ) ) ).
% P.Con_rec(1)
thf(fact_167_P_OResid_Osimps_I3_J,axiom,
! [T2: a,U2: a] :
( ( ( con_a @ resid @ T2 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) ) )
& ( ~ ( con_a @ resid @ T2 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= nil_a ) ) ) ).
% P.Resid.simps(3)
thf(fact_168_P_OCon__cons_I2_J,axiom,
! [T: list_a,U: list_a,U2: a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ U ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) ) @ U )
!= nil_a ) ) ) ) ) ).
% P.Con_cons(2)
thf(fact_169_P_OCon__cons_I1_J,axiom,
! [T: list_a,U: list_a,T2: a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ U )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ T2 @ nil_a ) ) )
!= nil_a ) ) ) ) ) ).
% P.Con_cons(1)
thf(fact_170_P_OResid_Osimps_I2_J,axiom,
! [V: a,Va2: list_a] :
( ( paths_in_Resid_a @ resid @ ( cons_a @ V @ Va2 ) @ nil_a )
= nil_a ) ).
% P.Resid.simps(2)
thf(fact_171_P_OResid__cons_I2_J,axiom,
! [U: list_a,T: list_a,U2: a] :
( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ U ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ U ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) ) @ U ) ) ) ) ).
% P.Resid_cons(2)
thf(fact_172_P_OResid__rec_I3_J,axiom,
! [U: list_a,T2: a,U2: a] :
( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ U ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ U ) )
= ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) @ U ) ) ) ) ).
% P.Resid_rec(3)
thf(fact_173_P_OResid__rec_I2_J,axiom,
! [T: list_a,T2: a,U2: a] :
( ( T != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) ) ) ) ) ) ).
% P.Resid_rec(2)
thf(fact_174_P_OCon__initial__left,axiom,
! [T2: a,T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ U )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U )
!= nil_a ) ) ).
% P.Con_initial_left
thf(fact_175_P_OCon__initial__right,axiom,
! [T: list_a,U2: a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ U ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) )
!= nil_a ) ) ).
% P.Con_initial_right
thf(fact_176_P_Oidentities__form__coherent__normal__sub__rts,axiom,
cohere6429906645900029933list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( ide_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% P.identities_form_coherent_normal_sub_rts
thf(fact_177_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_178_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_179_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_180_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_181_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_182_P_Ocon__imp__coinitial,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.con_imp_coinitial
thf(fact_183_P_Oprfx__decomp,axiom,
! [T2: a,U2: a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) ) @ ( cons_a @ U2 @ nil_a ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ U2 @ nil_a ) @ ( append_a @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) ) ) ) ) ) ).
% P.prfx_decomp
thf(fact_184_P_OResid__cons_I1_J,axiom,
! [U: list_a,T2: a,T: list_a] :
( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ U )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ U )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U ) @ ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ T2 @ nil_a ) ) ) ) ) ) ) ).
% P.Resid_cons(1)
thf(fact_185_P_OResid__rec_I4_J,axiom,
! [T: list_a,U: list_a,T2: a,U2: a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ ( cons_a @ U2 @ U ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ ( cons_a @ U2 @ U ) )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) @ U ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) ) ) ) ) ) ) ) ).
% P.Resid_rec(4)
thf(fact_186_P_OTrgs__Resid__sym__Arr__single,axiom,
! [T: list_a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) ) )
= ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ U2 @ nil_a ) @ T ) ) ) ) ).
% P.Trgs_Resid_sym_Arr_single
thf(fact_187_P_Ojoinable__implies__coinitial,axiom,
! [T2: list_a,U2: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.joinable_implies_coinitial
thf(fact_188_P_Oforward__stable,axiom,
! [U2: list_a,T2: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% P.forward_stable
thf(fact_189_P_OBackward__stable__single,axiom,
! [U: list_a,T2: a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U ) )
=> ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( cons_a @ T2 @ nil_a ) ) ) ) ).
% P.Backward_stable_single
thf(fact_190_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_191_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_192_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_193_self__append__conv,axiom,
! [Y4: list_a,Ys: list_a] :
( ( Y4
= ( append_a @ Y4 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_194_N_OCongI,axiom,
! [U2: a,U4: a,T2: a,T5: a] :
( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ U4 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( member_a @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U4 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ U4 ) @ ( resid @ T2 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 ) ) ) ) ).
% N.CongI
thf(fact_195_R_OideI,axiom,
! [A: a] :
( ( con_a @ resid @ A @ A )
=> ( ( ( resid @ A @ A )
= A )
=> ( ide_a @ resid @ A ) ) ) ).
% R.ideI
thf(fact_196_append1__eq__conv,axiom,
! [Xs: list_a,X4: a,Ys: list_a,Y4: a] :
( ( ( append_a @ Xs @ ( cons_a @ X4 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y4 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X4 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_197_P_OideI,axiom,
! [A: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A )
=> ( ( ( paths_in_Resid_a @ resid @ A @ A )
= A )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ) ).
% P.ideI
thf(fact_198_P_OCon__consI_I2_J,axiom,
! [T: list_a,U: list_a,U2: a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) ) @ U )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ U ) )
!= nil_a ) ) ) ) ) ).
% P.Con_consI(2)
thf(fact_199_P_OCon__consI_I1_J,axiom,
! [T: list_a,U: list_a,T2: a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ T2 @ nil_a ) ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ U )
!= nil_a ) ) ) ) ) ).
% P.Con_consI(1)
thf(fact_200_P_OResid__rec_I1_J,axiom,
! [T2: a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) ) ) ).
% P.Resid_rec(1)
thf(fact_201_P_OResid__single__ide_I2_J,axiom,
! [A: a,T: list_a] :
( ( ide_a @ resid @ A )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ A @ nil_a ) @ T )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ A @ nil_a ) )
= T ) ) ) ).
% P.Resid_single_ide(2)
thf(fact_202_transpose_Ocases,axiom,
! [X4: list_list_a] :
( ( X4 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X4
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X: a,Xs2: list_a,Xss: list_list_a] :
( X4
!= ( cons_list_a @ ( cons_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_203_rts_Ocoterminal_Ocong,axiom,
coterminal_a = coterminal_a ).
% rts.coterminal.cong
thf(fact_204_rts_Ocoterminal_Ocong,axiom,
coterminal_list_a = coterminal_list_a ).
% rts.coterminal.cong
thf(fact_205_rts_Ocoterminal_Ocong,axiom,
coterm239225051077876726list_a = coterm239225051077876726list_a ).
% rts.coterminal.cong
thf(fact_206_not__Cons__self2,axiom,
! [X4: a,Xs: list_a] :
( ( cons_a @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_207_rts_Ocoinitial_Ocong,axiom,
coinitial_list_a = coinitial_list_a ).
% rts.coinitial.cong
thf(fact_208_rts_Ocoinitial_Ocong,axiom,
coinitial_a = coinitial_a ).
% rts.coinitial.cong
thf(fact_209_rts_Ocoinitial_Ocong,axiom,
coinitial_set_list_a = coinitial_set_list_a ).
% rts.coinitial.cong
thf(fact_210_rts_Ojoinable_Ocong,axiom,
joinable_list_a = joinable_list_a ).
% rts.joinable.cong
thf(fact_211_rts_Ojoinable_Ocong,axiom,
joinable_a = joinable_a ).
% rts.joinable.cong
thf(fact_212_rts_Ojoinable_Ocong,axiom,
joinable_set_list_a = joinable_set_list_a ).
% rts.joinable.cong
thf(fact_213_quotient__by__coherent__normal_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( cohere6072184133013167079_rts_a @ Resid @ NN ) ) ).
% quotient_by_coherent_normal.axioms(2)
thf(fact_214_quotient__by__coherent__normal_Oaxioms_I2_J,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( cohere4318586772758246349list_a @ Resid @ NN ) ) ).
% quotient_by_coherent_normal.axioms(2)
thf(fact_215_quotient__by__coherent__normal_Oaxioms_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( cohere6429906645900029933list_a @ Resid @ NN ) ) ).
% quotient_by_coherent_normal.axioms(2)
thf(fact_216_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_217_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a] : ( P @ ( cons_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_218_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y2: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_219_remdups__adj_Ocases,axiom,
! [X4: list_a] :
( ( X4 != nil_a )
=> ( ! [X: a] :
( X4
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y3: a,Xs2: list_a] :
( X4
!= ( cons_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_220_list_Oexhaust,axiom,
! [Y4: list_a] :
( ( Y4 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y4
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_221_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_222_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_223_append__Cons,axiom,
! [X4: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X4 @ Xs ) @ Ys )
= ( cons_a @ X4 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_224_Cons__eq__appendI,axiom,
! [X4: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X4 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X4 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_225_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_226_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_227_Cons__eq__append__conv,axiom,
! [X4: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X4 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X4 @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X4 @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_228_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X4: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X4 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X4 @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X4 @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_229_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_230_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_231_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_232_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_233_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_234_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_235_P_Ocong__implies__coterminal,axiom,
! [U2: list_a,U4: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) )
=> ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ U4 ) ) ).
% P.cong_implies_coterminal
thf(fact_236_N_OCong__class__eqI,axiom,
! [T2: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( ( normal7408713899360725774lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 )
= ( normal7408713899360725774lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T5 ) ) ) ).
% N.Cong_class_eqI
thf(fact_237_P_OCong__char,axiom,
! [T2: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
= ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) ) ).
% P.Cong_char
thf(fact_238_P_OCong__class__membs__are__Cong,axiom,
! [T9: set_list_a,T2: list_a,T5: list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( ( member_list_a @ T2 @ T9 )
=> ( ( member_list_a @ T5 @ T9 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) ) ) ) ).
% P.Cong_class_membs_are_Cong
thf(fact_239_P_OCong__append__NPath__Arr,axiom,
! [T: list_a,U: list_a] :
( ( T != nil_a )
=> ( ( paths_in_Arr_a @ resid @ ( append_a @ U @ T ) )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( append_a @ U @ T ) @ T ) ) ) ) ).
% P.Cong_append_NPath_Arr
thf(fact_240_P_OSrcs__respects__Cong,axiom,
! [T: list_a,T6: list_a,A: a,A4: a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T @ T6 )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T6 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( cons_a @ A @ nil_a ) @ ( cons_a @ A4 @ nil_a ) ) ) ) ) ).
% P.Srcs_respects_Cong
thf(fact_241_P_Opaths__in__rts__with__normal__axioms,axiom,
paths_3682460814909619382rmal_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ).
% P.paths_in_rts_with_normal_axioms
thf(fact_242_P_OCong__class__eqI,axiom,
! [T2: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 )
= ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T5 ) ) ) ).
% P.Cong_class_eqI
thf(fact_243_P_OCong__subst_I2_J,axiom,
! [T2: list_a,T5: list_a,U2: list_a,U4: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U4 ) ) ) ) ) ) ).
% P.Cong_subst(2)
thf(fact_244_P_OCong__subst_I1_J,axiom,
! [T2: list_a,T5: list_a,U2: list_a,U4: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T5 @ U4 ) ) ) ) ) ).
% P.Cong_subst(1)
thf(fact_245_R_Ojoinable__implies__coinitial,axiom,
! [T2: a,U2: a] :
( ( joinable_a @ resid @ T2 @ U2 )
=> ( coinitial_a @ resid @ T2 @ U2 ) ) ).
% R.joinable_implies_coinitial
thf(fact_246_R_Ocong__implies__coinitial,axiom,
! [U2: a,U4: a] :
( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( coinitial_a @ resid @ U2 @ U4 ) ) ).
% R.cong_implies_coinitial
thf(fact_247_R_Ocoinitial__ide__are__cong,axiom,
! [A: a,A4: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ A4 )
=> ( ( coinitial_a @ resid @ A @ A4 )
=> ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) ) ) ) ) ).
% R.coinitial_ide_are_cong
thf(fact_248_N_Oforward__stable,axiom,
! [U2: a,T2: a] :
( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( coinitial_a @ resid @ T2 @ U2 )
=> ( member_a @ ( resid @ U2 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% N.forward_stable
thf(fact_249_R_Ocon__imp__coinitial,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( coinitial_a @ resid @ T2 @ U2 ) ) ).
% R.con_imp_coinitial
thf(fact_250_R_Ojoinable__implies__con,axiom,
! [T2: a,U2: a] :
( ( joinable_a @ resid @ T2 @ U2 )
=> ( con_a @ resid @ T2 @ U2 ) ) ).
% R.joinable_implies_con
thf(fact_251_N_Oresid__along__elem__preserves__con,axiom,
! [T2: a,T5: a,U2: a] :
( ( con_a @ resid @ T2 @ T5 )
=> ( ( coinitial_a @ resid @ T2 @ U2 )
=> ( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( con_a @ resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U2 ) ) ) ) ) ).
% N.resid_along_elem_preserves_con
thf(fact_252_P_OArr_Osimps_I1_J,axiom,
~ ( paths_in_Arr_a @ resid @ nil_a ) ).
% P.Arr.simps(1)
thf(fact_253_P_OSrcs__are__con,axiom,
! [A: a,T: list_a,A4: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T ) )
=> ( con_a @ resid @ A @ A4 ) ) ) ).
% P.Srcs_are_con
thf(fact_254_P_ONPath__implies__Arr,axiom,
! [T: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T )
=> ( paths_in_Arr_a @ resid @ T ) ) ).
% P.NPath_implies_Arr
thf(fact_255_P_OCon__imp__eq__Srcs,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ U ) ) ) ).
% P.Con_imp_eq_Srcs
thf(fact_256_P_OCon__imp__Arr__Resid,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ).
% P.Con_imp_Arr_Resid
thf(fact_257_P_OCon__Arr__self,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Resid_a @ resid @ T @ T )
!= nil_a ) ) ).
% P.Con_Arr_self
thf(fact_258_P_OArr__iff__Con__self,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
= ( ( paths_in_Resid_a @ resid @ T @ T )
!= nil_a ) ) ).
% P.Arr_iff_Con_self
thf(fact_259_P_OCon__implies__Arr_I1_J,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ T ) ) ).
% P.Con_implies_Arr(1)
thf(fact_260_P_OCon__implies__Arr_I2_J,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ U ) ) ).
% P.Con_implies_Arr(2)
thf(fact_261_P_Osources__cong__closed,axiom,
! [A: list_a,T2: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ A4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A4 @ A ) ) )
=> ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ) ).
% P.sources_cong_closed
thf(fact_262_P_Osources__are__cong,axiom,
! [A: list_a,T2: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ A4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A4 @ A ) ) ) ) ) ).
% P.sources_are_cong
thf(fact_263_P_Osource__is__ide,axiom,
! [A: list_a,T2: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% P.source_is_ide
thf(fact_264_P_Osources__are__con,axiom,
! [A: list_a,T2: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A4 ) ) ) ).
% P.sources_are_con
thf(fact_265_P_OSrcs__con__closed,axiom,
! [A: a,T: list_a,A4: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T ) )
=> ( ( ide_a @ resid @ A4 )
=> ( ( con_a @ resid @ A @ A4 )
=> ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T ) ) ) ) ) ).
% P.Srcs_con_closed
thf(fact_266_P_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [T2: list_a,T5: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ).
% P.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_267_P_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [T2: list_a,T5: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) ) ) ).
% P.Cong\<^sub>0_imp_coinitial
thf(fact_268_P_OArr__Resid__single,axiom,
! [T: list_a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) ) ) ) ).
% P.Arr_Resid_single
thf(fact_269_P_OCong_H__if,axiom,
! [U2: list_a,U4: list_a,T2: list_a,T5: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ U4 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U4 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ U4 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) ) ) ) ).
% P.Cong'_if
thf(fact_270_P_OCong_H_Ointros_I1_J,axiom,
! [T2: list_a,U2: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ).
% P.Cong'.intros(1)
thf(fact_271_P_OCong_H_Ointros_I2_J,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
=> ( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) ) ) ).
% P.Cong'.intros(2)
thf(fact_272_P_OCong_H_Ointros_I3_J,axiom,
! [T2: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ).
% P.Cong'.intros(3)
thf(fact_273_P_Osources__con__closed,axiom,
! [A: list_a,T2: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A4 )
=> ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ) ) ).
% P.sources_con_closed
thf(fact_274_P_Oin__sourcesE,axiom,
! [A: list_a,T2: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ~ ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ A ) ) ) ).
% P.in_sourcesE
thf(fact_275_P_OResid__NPath__preserves__reflects__Con,axiom,
! [U: list_a,T: list_a,T6: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ U ) )
=> ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ T6 @ U ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ T @ T6 )
!= nil_a ) ) ) ) ).
% P.Resid_NPath_preserves_reflects_Con
thf(fact_276_P_OResid__along__normal__preserves__reflects__con,axiom,
! [U2: list_a,T2: list_a,T5: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T5 @ U2 ) )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T5 ) ) ) ) ).
% P.Resid_along_normal_preserves_reflects_con
thf(fact_277_P_ONPath__Resid,axiom,
! [T: list_a,U: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T )
=> ( ( paths_in_Arr_a @ resid @ U )
=> ( ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ U ) )
=> ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ) ) ).
% P.NPath_Resid
thf(fact_278_sources__respects__Cong,axiom,
! [T: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T @ T6 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ).
% sources_respects_Cong
thf(fact_279_P_Osources__are__Cong,axiom,
! [A: list_a,T2: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A4 ) ) ) ).
% P.sources_are_Cong
thf(fact_280_P_Oin__sources__respects__Cong,axiom,
! [T2: list_a,T5: list_a,A: list_a,A4: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A4 ) ) ) ) ).
% P.in_sources_respects_Cong
thf(fact_281_P_OCong__closure__props_I4_J,axiom,
! [U2: list_a,T2: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ) ).
% P.Cong_closure_props(4)
thf(fact_282_Srcs__respects__Cong,axiom,
! [T: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T @ T6 )
=> ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ T6 ) ) ) ).
% Srcs_respects_Cong
thf(fact_283_P_Osources__cons,axiom,
! [T2: a,T: list_a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T2 @ T ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T2 @ T ) )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T2 @ nil_a ) ) ) ) ).
% P.sources_cons
thf(fact_284_P_OResid__Arr__Src,axiom,
! [T: list_a,A: a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T ) )
=> ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ A @ nil_a ) )
= T ) ) ) ).
% P.Resid_Arr_Src
thf(fact_285_P_OSrcs__Resid__single__Arr,axiom,
! [U2: a,T: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ U2 @ nil_a ) @ T )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ U2 @ nil_a ) @ T ) )
= ( paths_in_Trgs_a @ resid @ T ) ) ) ).
% P.Srcs_Resid_single_Arr
thf(fact_286_P_Ois__Cong__class__def,axiom,
! [T9: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
= ( ? [T8: list_a] :
( ( member_list_a @ T8 @ T9 )
& ( T9
= ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T8 ) ) ) ) ) ).
% P.is_Cong_class_def
thf(fact_287_P_OArr__appendE_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a] :
( ( paths_in_Arr_a @ resid @ ( append_a @ T @ U ) )
=> ( ( T != nil_a )
=> ( ( U != nil_a )
=> ~ ( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Arr_a @ resid @ U )
=> ( ( paths_in_Trgs_a @ resid @ T )
!= ( paths_in_Srcs_a @ resid @ U ) ) ) ) ) ) ) ).
% P.Arr_appendE\<^sub>P
thf(fact_288_P_OCong_092_060_094sub_0620__append__Arr__NPath,axiom,
! [T: list_a,U: list_a] :
( ( T != nil_a )
=> ( ( paths_in_Arr_a @ resid @ ( append_a @ T @ U ) )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ T ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T @ ( append_a @ T @ U ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_append_Arr_NPath
thf(fact_289_P_OCon__single__ide__ind,axiom,
! [A: a,T: list_a] :
( ( ide_a @ resid @ A )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ A @ nil_a ) @ T )
!= nil_a )
= ( ( paths_in_Arr_a @ resid @ T )
& ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T ) ) ) ) ) ).
% P.Con_single_ide_ind
thf(fact_290_P_OCong__subst__con,axiom,
! [T2: list_a,U2: list_a,T5: list_a,U4: list_a] :
( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T5 @ U4 ) ) ) ) ) ) ).
% P.Cong_subst_con
thf(fact_291_P_OCoherent,axiom,
! [T: list_a,U: list_a,U3: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U3 )
=> ( ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ U ) )
=> ( ( ( paths_in_Srcs_a @ resid @ U )
= ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( ( ( paths_in_Trgs_a @ resid @ U )
= ( paths_in_Trgs_a @ resid @ U3 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ T @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U3 ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ) ) ) ).
% P.Coherent
thf(fact_292_P_OSrcs__append,axiom,
! [T: list_a,U: list_a] :
( ( T != nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( append_a @ T @ U ) )
= ( paths_in_Srcs_a @ resid @ T ) ) ) ).
% P.Srcs_append
thf(fact_293_P_Oin__sourcesI,axiom,
! [A: list_a,T2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ A )
=> ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ) ).
% P.in_sourcesI
thf(fact_294_P_OSrcs__Resid,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) )
= ( paths_in_Trgs_a @ resid @ U ) ) ) ).
% P.Srcs_Resid
thf(fact_295_P_OCon__single__ideI_I1_J,axiom,
! [A: a,T: list_a] :
( ( ide_a @ resid @ A )
=> ( ( paths_in_Arr_a @ resid @ T )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ A @ nil_a ) @ T )
!= nil_a ) ) ) ) ).
% P.Con_single_ideI(1)
thf(fact_296_P_OCon__single__ideI_I2_J,axiom,
! [A: a,T: list_a] :
( ( ide_a @ resid @ A )
=> ( ( paths_in_Arr_a @ resid @ T )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T ) )
=> ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ A @ nil_a ) )
!= nil_a ) ) ) ) ).
% P.Con_single_ideI(2)
thf(fact_297_P_Orep__in__Cong__class,axiom,
! [T9: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( member_list_a @ ( normal5825812266234710965list_a @ T9 ) @ T9 ) ) ).
% P.rep_in_Cong_class
thf(fact_298_P_OCong__class__memb__Cong__rep,axiom,
! [T9: set_list_a,T2: list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( ( member_list_a @ T2 @ T9 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ ( normal5825812266234710965list_a @ T9 ) ) ) ) ).
% P.Cong_class_memb_Cong_rep
thf(fact_299_paths__in__rts__with__normal_ONPath__implies__Arr,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T )
=> ( paths_in_Arr_a @ Resid @ T ) ) ) ).
% paths_in_rts_with_normal.NPath_implies_Arr
thf(fact_300_rts_Osources_Ocong,axiom,
sources_list_a = sources_list_a ).
% rts.sources.cong
thf(fact_301_rts_Osources_Ocong,axiom,
sources_a = sources_a ).
% rts.sources.cong
thf(fact_302_rts_Osources_Ocong,axiom,
sources_set_list_a = sources_set_list_a ).
% rts.sources.cong
thf(fact_303_paths__in__rts_OArr_Ocong,axiom,
paths_in_Arr_a = paths_in_Arr_a ).
% paths_in_rts.Arr.cong
thf(fact_304_paths__in__rts_OSrcs_Ocong,axiom,
paths_in_Srcs_a = paths_in_Srcs_a ).
% paths_in_rts.Srcs.cong
thf(fact_305_normal__sub__rts_OCong_H_Ocong,axiom,
normal4820288888589421745list_a = normal4820288888589421745list_a ).
% normal_sub_rts.Cong'.cong
thf(fact_306_normal__sub__rts_OCong_H_Ocong,axiom,
normal_sub_Cong_a2 = normal_sub_Cong_a2 ).
% normal_sub_rts.Cong'.cong
thf(fact_307_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal4947954984078133662list_a = normal4947954984078133662list_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_308_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal8595587647932138008lass_a = normal8595587647932138008lass_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_309_normal__sub__rts_OCong__class_Ocong,axiom,
normal7408713899360725774lass_a = normal7408713899360725774lass_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_310_normal__sub__rts_OCong__class_Ocong,axiom,
normal8738828432243163028list_a = normal8738828432243163028list_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_311_paths__in__rts__with__normal_ONPath__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,U: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T )
=> ( ( paths_in_Arr_a @ Resid @ U )
=> ( ( ( paths_in_Srcs_a @ Resid @ T )
= ( paths_in_Srcs_a @ Resid @ U ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T @ U ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid
thf(fact_312_paths__in__rts__with__normal_ONPath__Resid__single__Arr,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T2: list_a,U: list_list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( member_list_a @ T2 @ NN )
=> ( ( paths_in_Arr_list_a @ Resid @ U )
=> ( ( ( sources_list_a @ Resid @ T2 )
= ( paths_in_Srcs_list_a @ Resid @ U ) )
=> ( paths_8553768783572469479list_a @ Resid @ NN @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ nil_list_a ) @ U ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_single_Arr
thf(fact_313_paths__in__rts__with__normal_ONPath__Resid__single__Arr,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T2: set_list_a,U: list_set_list_a] :
( ( paths_2496957160577180956list_a @ Resid @ NN )
=> ( ( member_set_list_a @ T2 @ NN )
=> ( ( paths_3862341395584752492list_a @ Resid @ U )
=> ( ( ( sources_set_list_a @ Resid @ T2 )
= ( paths_5622510820459842474list_a @ Resid @ U ) )
=> ( paths_8943339929420292807list_a @ Resid @ NN @ ( paths_6591532302923812690list_a @ Resid @ ( cons_set_list_a @ T2 @ nil_set_list_a ) @ U ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_single_Arr
thf(fact_314_paths__in__rts__with__normal_ONPath__Resid__single__Arr,axiom,
! [Resid: a > a > a,NN: set_a,T2: a,U: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( member_a @ T2 @ NN )
=> ( ( paths_in_Arr_a @ Resid @ U )
=> ( ( ( sources_a @ Resid @ T2 )
= ( paths_in_Srcs_a @ Resid @ U ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ nil_a ) @ U ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_single_Arr
thf(fact_315_paths__in__rts__with__normal_OResid__NPath__preserves__reflects__Con,axiom,
! [Resid: a > a > a,NN: set_a,U: list_a,T: list_a,T6: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U )
=> ( ( ( paths_in_Srcs_a @ Resid @ T )
= ( paths_in_Srcs_a @ Resid @ U ) )
=> ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T @ U ) @ ( paths_in_Resid_a @ Resid @ T6 @ U ) )
!= nil_a )
= ( ( paths_in_Resid_a @ Resid @ T @ T6 )
!= nil_a ) ) ) ) ) ).
% paths_in_rts_with_normal.Resid_NPath_preserves_reflects_Con
thf(fact_316_paths__in__rts__with__normal_OSrcs__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_list_a,T6: list_list_a,A: list_a,A4: list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) @ T @ T6 )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T ) )
=> ( ( member_list_a @ A4 @ ( paths_in_Srcs_list_a @ Resid @ T6 ) )
=> ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) @ ( cons_list_a @ A @ nil_list_a ) @ ( cons_list_a @ A4 @ nil_list_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Srcs_respects_Cong
thf(fact_317_paths__in__rts__with__normal_OSrcs__respects__Cong,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T: list_set_list_a,T6: list_set_list_a,A: set_list_a,A4: set_list_a] :
( ( paths_2496957160577180956list_a @ Resid @ NN )
=> ( ( normal1724791161856338944list_a @ ( paths_6591532302923812690list_a @ Resid ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ NN ) ) @ T @ T6 )
=> ( ( member_set_list_a @ A @ ( paths_5622510820459842474list_a @ Resid @ T ) )
=> ( ( member_set_list_a @ A4 @ ( paths_5622510820459842474list_a @ Resid @ T6 ) )
=> ( normal1724791161856338944list_a @ ( paths_6591532302923812690list_a @ Resid ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ NN ) ) @ ( cons_set_list_a @ A @ nil_set_list_a ) @ ( cons_set_list_a @ A4 @ nil_set_list_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Srcs_respects_Cong
thf(fact_318_paths__in__rts__with__normal_OSrcs__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,T6: list_a,A: a,A4: a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) @ T @ T6 )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ Resid @ T ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ Resid @ T6 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) @ ( cons_a @ A @ nil_a ) @ ( cons_a @ A4 @ nil_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Srcs_respects_Cong
thf(fact_319_paths__in__rts__with__normal_OBackward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U: list_a,T: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T @ U ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ T ) ) ) ) ).
% paths_in_rts_with_normal.Backward_stable
thf(fact_320_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T2: list_a,T5: list_a,U2: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
=> ( ( ( sources_list_a @ Resid @ T5 )
= ( sources_list_a @ Resid @ U4 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ ( Resid @ T2 @ U2 ) @ ( Resid @ T5 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_321_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T2: a,T5: a,U2: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_a @ Resid @ T2 @ U2 )
=> ( ( ( sources_a @ Resid @ T5 )
= ( sources_a @ Resid @ U4 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ ( Resid @ T2 @ U2 ) @ ( Resid @ T5 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_322_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T2: set_list_a,T5: set_list_a,U2: set_list_a,U4: set_list_a] :
( ( cohere4318586772758246349list_a @ Resid @ NN )
=> ( ( normal3869076228660116218list_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal3869076228660116218list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
=> ( ( ( sources_set_list_a @ Resid @ T5 )
= ( sources_set_list_a @ Resid @ U4 ) )
=> ( normal3869076228660116218list_a @ Resid @ NN @ ( Resid @ T2 @ U2 ) @ ( Resid @ T5 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_323_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T2: list_a,T5: list_a,U2: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
=> ( ( ( sources_list_a @ Resid @ T5 )
= ( sources_list_a @ Resid @ U4 ) )
=> ( con_list_a @ Resid @ T5 @ U4 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_324_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T2: a,T5: a,U2: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_a @ Resid @ T2 @ U2 )
=> ( ( ( sources_a @ Resid @ T5 )
= ( sources_a @ Resid @ U4 ) )
=> ( con_a @ Resid @ T5 @ U4 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_325_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T2: set_list_a,T5: set_list_a,U2: set_list_a,U4: set_list_a] :
( ( cohere4318586772758246349list_a @ Resid @ NN )
=> ( ( normal3869076228660116218list_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal3869076228660116218list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
=> ( ( ( sources_set_list_a @ Resid @ T5 )
= ( sources_set_list_a @ Resid @ U4 ) )
=> ( con_set_list_a @ Resid @ T5 @ U4 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_326_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T2: list_a,U2: list_a,T5: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( ( sources_list_a @ Resid @ T2 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( ( ( sources_list_a @ Resid @ T5 )
= ( sources_list_a @ Resid @ U4 ) )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
= ( con_list_a @ Resid @ T5 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_327_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: a > a > a,NN: set_a,T2: a,U2: a,T5: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( ( sources_a @ Resid @ T2 )
= ( sources_a @ Resid @ U2 ) )
=> ( ( ( sources_a @ Resid @ T5 )
= ( sources_a @ Resid @ U4 ) )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_a @ Resid @ T2 @ U2 )
= ( con_a @ Resid @ T5 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_328_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T2: set_list_a,U2: set_list_a,T5: set_list_a,U4: set_list_a] :
( ( cohere4318586772758246349list_a @ Resid @ NN )
=> ( ( ( sources_set_list_a @ Resid @ T2 )
= ( sources_set_list_a @ Resid @ U2 ) )
=> ( ( ( sources_set_list_a @ Resid @ T5 )
= ( sources_set_list_a @ Resid @ U4 ) )
=> ( ( normal3869076228660116218list_a @ Resid @ NN @ T2 @ T5 )
=> ( ( normal3869076228660116218list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
= ( con_set_list_a @ Resid @ T5 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_329_paths__in__rts__with__normal_OCong_092_060_094sub_0620__append__resid__NPath,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,U: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T @ U ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( append_a @ T @ ( paths_in_Resid_a @ Resid @ U @ T ) ) @ U ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ U @ ( append_a @ T @ ( paths_in_Resid_a @ Resid @ U @ T ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Cong\<^sub>0_append_resid_NPath
thf(fact_330_composite__completion_Osources__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,T: list_list_a,T6: list_list_a] :
( ( compos5005676353122392380list_a @ Resid )
=> ( ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) @ T @ T6 )
=> ( ( sources_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T )
= ( sources_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T6 ) ) ) ) ).
% composite_completion.sources_respects_Cong
thf(fact_331_composite__completion_Osources__respects__Cong,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T: list_set_list_a,T6: list_set_list_a] :
( ( compos522909563978211612list_a @ Resid )
=> ( ( normal1724791161856338944list_a @ ( paths_6591532302923812690list_a @ Resid ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ ( collect_set_list_a @ ( ide_set_list_a @ Resid ) ) ) ) @ T @ T6 )
=> ( ( source3381003665373322262list_a @ ( paths_6591532302923812690list_a @ Resid ) @ T )
= ( source3381003665373322262list_a @ ( paths_6591532302923812690list_a @ Resid ) @ T6 ) ) ) ) ).
% composite_completion.sources_respects_Cong
thf(fact_332_composite__completion_Osources__respects__Cong,axiom,
! [Resid: a > a > a,T: list_a,T6: list_a] :
( ( compos1915005101613447350tion_a @ Resid )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ ( collect_a @ ( ide_a @ Resid ) ) ) ) @ T @ T6 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ Resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ Resid ) @ T6 ) ) ) ) ).
% composite_completion.sources_respects_Cong
thf(fact_333_composite__completion_OSrcs__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,T: list_list_a,T6: list_list_a] :
( ( compos5005676353122392380list_a @ Resid )
=> ( ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) @ T @ T6 )
=> ( ( paths_in_Srcs_list_a @ Resid @ T )
= ( paths_in_Srcs_list_a @ Resid @ T6 ) ) ) ) ).
% composite_completion.Srcs_respects_Cong
thf(fact_334_composite__completion_OSrcs__respects__Cong,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T: list_set_list_a,T6: list_set_list_a] :
( ( compos522909563978211612list_a @ Resid )
=> ( ( normal1724791161856338944list_a @ ( paths_6591532302923812690list_a @ Resid ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ ( collect_set_list_a @ ( ide_set_list_a @ Resid ) ) ) ) @ T @ T6 )
=> ( ( paths_5622510820459842474list_a @ Resid @ T )
= ( paths_5622510820459842474list_a @ Resid @ T6 ) ) ) ) ).
% composite_completion.Srcs_respects_Cong
thf(fact_335_composite__completion_OSrcs__respects__Cong,axiom,
! [Resid: a > a > a,T: list_a,T6: list_a] :
( ( compos1915005101613447350tion_a @ Resid )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ ( collect_a @ ( ide_a @ Resid ) ) ) ) @ T @ T6 )
=> ( ( paths_in_Srcs_a @ Resid @ T )
= ( paths_in_Srcs_a @ Resid @ T6 ) ) ) ) ).
% composite_completion.Srcs_respects_Cong
thf(fact_336_paths__in__rts__with__normal_OBackward__stable__single,axiom,
! [Resid: a > a > a,NN: set_a,U: list_a,T2: a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ nil_a ) @ U ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( cons_a @ T2 @ nil_a ) ) ) ) ) ).
% paths_in_rts_with_normal.Backward_stable_single
thf(fact_337_paths__in__rts__with__normal_OCong_092_060_094sub_0620__cancel__left_092_060_094sub_062C_092_060_094sub_062S,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,U: list_a,U3: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( append_a @ T @ U ) @ ( append_a @ T @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( append_a @ T @ U3 ) @ ( append_a @ T @ U ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) )
=> ( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( U3 != nil_a )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ U @ U3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ U3 @ U ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Cong\<^sub>0_cancel_left\<^sub>C\<^sub>S
thf(fact_338_paths__in__rts__with__normal_OTrgs__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_list_a,T6: list_list_a,B: list_a,B2: list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) @ T @ T6 )
=> ( ( member_list_a @ B @ ( paths_in_Trgs_list_a @ Resid @ T ) )
=> ( ( member_list_a @ B2 @ ( paths_in_Trgs_list_a @ Resid @ T6 ) )
=> ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) @ ( cons_list_a @ B @ nil_list_a ) @ ( cons_list_a @ B2 @ nil_list_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Trgs_respects_Cong
thf(fact_339_paths__in__rts__with__normal_OTrgs__respects__Cong,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T: list_set_list_a,T6: list_set_list_a,B: set_list_a,B2: set_list_a] :
( ( paths_2496957160577180956list_a @ Resid @ NN )
=> ( ( normal1724791161856338944list_a @ ( paths_6591532302923812690list_a @ Resid ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ NN ) ) @ T @ T6 )
=> ( ( member_set_list_a @ B @ ( paths_6697778058162308143list_a @ Resid @ T ) )
=> ( ( member_set_list_a @ B2 @ ( paths_6697778058162308143list_a @ Resid @ T6 ) )
=> ( normal1724791161856338944list_a @ ( paths_6591532302923812690list_a @ Resid ) @ ( collec5381118732811369429list_a @ ( paths_8943339929420292807list_a @ Resid @ NN ) ) @ ( cons_set_list_a @ B @ nil_set_list_a ) @ ( cons_set_list_a @ B2 @ nil_set_list_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Trgs_respects_Cong
thf(fact_340_paths__in__rts__with__normal_OTrgs__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,T6: list_a,B: a,B2: a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) @ T @ T6 )
=> ( ( member_a @ B @ ( paths_in_Trgs_a @ Resid @ T ) )
=> ( ( member_a @ B2 @ ( paths_in_Trgs_a @ Resid @ T6 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) @ ( cons_a @ B @ nil_a ) @ ( cons_a @ B2 @ nil_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Trgs_respects_Cong
thf(fact_341_P_ONPath__append,axiom,
! [T: list_a,U: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( append_a @ T @ U ) )
= ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T )
& ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) ) ) ) ) ) ).
% P.NPath_append
thf(fact_342_P_Ois__Cong__classE,axiom,
! [T9: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ~ ( ( T9 != bot_bot_set_list_a )
=> ( ! [T3: list_a] :
( ( member_list_a @ T3 @ T9 )
=> ! [T10: list_a] :
( ( member_list_a @ T10 @ T9 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T3 @ T10 ) ) )
=> ~ ! [T3: list_a] :
( ( member_list_a @ T3 @ T9 )
=> ! [T10: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T10 @ T3 )
=> ( member_list_a @ T10 @ T9 ) ) ) ) ) ) ).
% P.is_Cong_classE
thf(fact_343_P_OArr__append__iff_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( paths_in_Arr_a @ resid @ ( append_a @ T @ U ) )
= ( ( paths_in_Arr_a @ resid @ T )
& ( paths_in_Arr_a @ resid @ U )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) ) ) ) ) ) ).
% P.Arr_append_iff\<^sub>P
thf(fact_344_P_OCong__class__rep,axiom,
! [T9: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( normal5825812266234710965list_a @ T9 ) )
= T9 ) ) ).
% P.Cong_class_rep
thf(fact_345_Resid__by__members,axiom,
! [T9: set_list_a,U9: set_list_a,T2: list_a,U2: list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U9 )
=> ( ( member_list_a @ T2 @ T9 )
=> ( ( member_list_a @ U2 @ U9 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 )
= ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ) ) ) ) ) ).
% Resid_by_members
thf(fact_346_P_OCong__class__is__nonempty,axiom,
! [T9: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( T9 != bot_bot_set_list_a ) ) ).
% P.Cong_class_is_nonempty
thf(fact_347_P_ONPath__Resid__single__Arr,axiom,
! [T2: a,U: list_a] :
( ( member_a @ T2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( paths_in_Arr_a @ resid @ U )
=> ( ( ( sources_a @ resid @ T2 )
= ( paths_in_Srcs_a @ resid @ U ) )
=> ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U ) ) ) ) ) ).
% P.NPath_Resid_single_Arr
thf(fact_348_N_OCong_H_Ointros_I3_J,axiom,
! [T2: a,U2: a] :
( ( ( member_a @ ( resid @ T2 @ U2 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ U2 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ U2 ) ) ).
% N.Cong'.intros(3)
thf(fact_349_N_OCong_H_Ointros_I2_J,axiom,
! [T2: a,U2: a,V: a] :
( ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ U2 )
=> ( ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U2 @ V )
=> ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ V ) ) ) ).
% N.Cong'.intros(2)
thf(fact_350_N_OCong_H_Ointros_I1_J,axiom,
! [T2: a,U2: a] :
( ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ U2 )
=> ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U2 @ T2 ) ) ).
% N.Cong'.intros(1)
thf(fact_351_N_OCong_H__if,axiom,
! [U2: a,U4: a,T2: a,T5: a] :
( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ U4 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( member_a @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U4 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ U4 ) @ ( resid @ T2 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 ) ) ) ) ).
% N.Cong'_if
thf(fact_352_N_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [T2: a,T5: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ T5 ) ) ) ).
% N.Cong\<^sub>0_imp_coinitial
thf(fact_353_N_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [T2: a,T5: a,U2: a] :
( ( ( member_a @ ( resid @ T2 @ T5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T5 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( ( member_a @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ U2 ) @ ( resid @ T2 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% N.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_354_R_Osource__is__ide,axiom,
! [A: a,T2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( ide_a @ resid @ A ) ) ).
% R.source_is_ide
thf(fact_355_R_Osources__are__cong,axiom,
! [A: a,T2: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T2 ) )
=> ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) ) ) ) ).
% R.sources_are_cong
thf(fact_356_R_Osources__cong__closed,axiom,
! [A: a,T2: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) )
=> ( member_a @ A4 @ ( sources_a @ resid @ T2 ) ) ) ) ).
% R.sources_cong_closed
thf(fact_357_N_OCong__char,axiom,
! [T2: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
= ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 ) ) ).
% N.Cong_char
thf(fact_358_N_OCong__class__membs__are__Cong,axiom,
! [T9: set_a,T2: a,T5: a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ( ( member_a @ T2 @ T9 )
=> ( ( member_a @ T5 @ T9 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 ) ) ) ) ).
% N.Cong_class_membs_are_Cong
thf(fact_359_R_Osources__are__con,axiom,
! [A: a,T2: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T2 ) )
=> ( con_a @ resid @ A @ A4 ) ) ) ).
% R.sources_are_con
thf(fact_360_N_Ois__Cong__class__def,axiom,
! [T9: set_a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
= ( ? [T8: a] :
( ( member_a @ T8 @ T9 )
& ( T9
= ( normal7408713899360725774lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T8 ) ) ) ) ) ).
% N.is_Cong_class_def
thf(fact_361_N_OResid__along__normal__preserves__reflects__con,axiom,
! [U2: a,T2: a,T5: a] :
( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( ( con_a @ resid @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U2 ) )
= ( con_a @ resid @ T2 @ T5 ) ) ) ) ).
% N.Resid_along_normal_preserves_reflects_con
thf(fact_362_R_Oin__sourcesE,axiom,
! [A: a,T2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ~ ( ( ide_a @ resid @ A )
=> ~ ( con_a @ resid @ T2 @ A ) ) ) ).
% R.in_sourcesE
thf(fact_363_R_Osources__con__closed,axiom,
! [A: a,T2: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( ( ide_a @ resid @ A4 )
=> ( ( con_a @ resid @ A @ A4 )
=> ( member_a @ A4 @ ( sources_a @ resid @ T2 ) ) ) ) ) ).
% R.sources_con_closed
thf(fact_364_P_OSrcs_Osimps_I3_J,axiom,
! [T2: a,V: a,Va2: list_a] :
( ( paths_in_Srcs_a @ resid @ ( cons_a @ T2 @ ( cons_a @ V @ Va2 ) ) )
= ( sources_a @ resid @ T2 ) ) ).
% P.Srcs.simps(3)
thf(fact_365_N_OCong__closure__props_I4_J,axiom,
! [U2: a,T2: a] :
( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ ( resid @ T2 @ U2 ) ) ) ) ).
% N.Cong_closure_props(4)
thf(fact_366_N_Oin__sources__respects__Cong,axiom,
! [T2: a,T5: a,A: a,A4: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T5 ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ A @ A4 ) ) ) ) ).
% N.in_sources_respects_Cong
thf(fact_367_N_Osources__are__Cong,axiom,
! [A: a,T2: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T2 ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ A @ A4 ) ) ) ).
% N.sources_are_Cong
thf(fact_368_P_OSrcs__are__ide,axiom,
! [T: list_a] : ( ord_less_eq_set_a @ ( paths_in_Srcs_a @ resid @ T ) @ ( collect_a @ ( ide_a @ resid ) ) ) ).
% P.Srcs_are_ide
thf(fact_369_P_OTrgs__are__ide,axiom,
! [T: list_a] : ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( collect_a @ ( ide_a @ resid ) ) ) ).
% P.Trgs_are_ide
thf(fact_370_in__sourcesE,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ~ ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ~ ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ A ) ) ) ).
% in_sourcesE
thf(fact_371_sources__con__closed,axiom,
! [A: set_list_a,T2: set_list_a,A4: set_list_a] :
( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A4 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A4 )
=> ( member_set_list_a @ A4 @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ) ) ).
% sources_con_closed
thf(fact_372_sources__are__con,axiom,
! [A: set_list_a,T2: set_list_a,A4: set_list_a] :
( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( member_set_list_a @ A4 @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A4 ) ) ) ).
% sources_are_con
thf(fact_373_source__is__ide,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A ) ) ).
% source_is_ide
thf(fact_374_sources__are__cong,axiom,
! [A: set_list_a,T2: set_list_a,A4: set_list_a] :
( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( member_set_list_a @ A4 @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A4 @ A ) ) ) ) ) ).
% sources_are_cong
thf(fact_375_sources__cong__closed,axiom,
! [A: set_list_a,T2: set_list_a,A4: set_list_a] :
( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A4 @ A ) ) )
=> ( member_set_list_a @ A4 @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ) ).
% sources_cong_closed
thf(fact_376_cong__subst__left_I2_J,axiom,
! [T2: set_list_a,T5: set_list_a,U2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T5 @ T2 ) ) )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T5 @ U2 ) ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T5 @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_377_cong__subst__left_I1_J,axiom,
! [T2: set_list_a,T5: set_list_a,U2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T5 @ T2 ) ) )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 @ U2 ) ) ) ).
% cong_subst_left(1)
thf(fact_378_cong__subst__right_I2_J,axiom,
! [U2: set_list_a,U4: set_list_a,T2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U4 @ U2 ) ) )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U4 ) ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U4 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_379_cong__subst__right_I1_J,axiom,
! [U2: set_list_a,U4: set_list_a,T2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U4 @ U2 ) ) )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U4 ) ) ) ).
% cong_subst_right(1)
thf(fact_380_con__ide__are__eq,axiom,
! [A: set_list_a,A4: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A4 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A4 )
=> ( A = A4 ) ) ) ) ).
% con_ide_are_eq
thf(fact_381_con__imp__coinitial__ax,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ? [A3: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A3 )
& ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A3 @ T2 )
& ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A3 @ U2 ) ) ) ).
% con_imp_coinitial_ax
thf(fact_382_con__target,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ V )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ U2 ) ) ) ) ).
% con_target
thf(fact_383_con__transitive__on__ide,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ B )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ C )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ B )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ B @ C )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_384_ideE,axiom,
! [A: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ~ ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A )
!= A ) ) ) ).
% ideE
thf(fact_385_ide__def,axiom,
! [A: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
= ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A )
& ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A )
= A ) ) ) ).
% ide_def
thf(fact_386_ide__imp__con__iff__cong,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_387_prfx__implies__con,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ).
% prfx_implies_con
thf(fact_388_resid__arr__ide,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ A )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ A )
= T2 ) ) ) ).
% resid_arr_ide
thf(fact_389_resid__ide__arr,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ T2 )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ T2 ) ) ) ) ).
% resid_ide_arr
thf(fact_390_Cong__iff__cong,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( normal3869076228660116218list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( collect_set_list_a @ ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) @ T2 @ U2 )
= ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ) ) ).
% Cong_iff_cong
thf(fact_391_con__sym,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ T2 ) ) ).
% con_sym
thf(fact_392_resid__reflects__con,axiom,
! [T2: set_list_a,V: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ V )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ V )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V ) )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% resid_reflects_con
thf(fact_393_cong__symmetric,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% cong_symmetric
thf(fact_394_cong__transitive,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) )
=> ( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ U2 ) ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 ) ) ) ) ) ).
% cong_transitive
thf(fact_395_extensional,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) )
=> ( T2 = U2 ) ) ).
% extensional
thf(fact_396_ide__backward__stable,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ A ) )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% ide_backward_stable
thf(fact_397_prfx__transitive,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V ) )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) ) ) ) ).
% prfx_transitive
thf(fact_398_weak__extensionality,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( T2 = U2 ) ) ) ) ).
% weak_extensionality
thf(fact_399_partial__magma__axioms,axiom,
partia4872338157591926529list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% partial_magma_axioms
thf(fact_400_cube,axiom,
! [V: set_list_a,T2: set_list_a,U2: set_list_a] :
( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ).
% cube
thf(fact_401_ex__un__null,axiom,
? [X: set_list_a] :
( ! [T3: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ X @ T3 )
= X )
& ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T3 @ X )
= X ) )
& ! [Y: set_list_a] :
( ! [T4: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ Y @ T4 )
= Y )
& ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T4 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_402_identities__form__coherent__normal__sub__rts,axiom,
cohere4318586772758246349list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( collect_set_list_a @ ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% identities_form_coherent_normal_sub_rts
thf(fact_403_cong__implies__coterminal,axiom,
! [U2: set_list_a,U4: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U4 @ U2 ) ) )
=> ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ U4 ) ) ).
% cong_implies_coterminal
thf(fact_404_joinable__implies__con,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ).
% joinable_implies_con
thf(fact_405_con__imp__coinitial,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ).
% con_imp_coinitial
thf(fact_406_resid__ide_I1_J,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ T2 )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ A )
= T2 ) ) ) ).
% resid_ide(1)
thf(fact_407_coinitial__ide__are__cong,axiom,
! [A: set_list_a,A4: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A4 )
=> ( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A4 )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A4 @ A ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_408_coinitial__ide__are__eq,axiom,
! [A: set_list_a,A4: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A4 )
=> ( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A4 )
=> ( A = A4 ) ) ) ) ).
% coinitial_ide_are_eq
thf(fact_409_cong__implies__coinitial,axiom,
! [U2: set_list_a,U4: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U4 @ U2 ) ) )
=> ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ U4 ) ) ).
% cong_implies_coinitial
thf(fact_410_joinable__implies__coinitial,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ).
% joinable_implies_coinitial
thf(fact_411_P_OSrcs_Osimps_I2_J,axiom,
! [T2: a] :
( ( paths_in_Srcs_a @ resid @ ( cons_a @ T2 @ nil_a ) )
= ( sources_a @ resid @ T2 ) ) ).
% P.Srcs.simps(2)
thf(fact_412_N_OCong__subst_I2_J,axiom,
! [T2: a,T5: a,U2: a,U4: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U2 @ U4 )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( ( ( sources_a @ resid @ T5 )
= ( sources_a @ resid @ U4 ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( resid @ T2 @ U2 ) @ ( resid @ T5 @ U4 ) ) ) ) ) ) ).
% N.Cong_subst(2)
thf(fact_413_N_OCong__subst_I1_J,axiom,
! [T2: a,T5: a,U2: a,U4: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U2 @ U4 )
=> ( ( con_a @ resid @ T2 @ U2 )
=> ( ( ( sources_a @ resid @ T5 )
= ( sources_a @ resid @ U4 ) )
=> ( con_a @ resid @ T5 @ U4 ) ) ) ) ) ).
% N.Cong_subst(1)
thf(fact_414_N_OCong__subst__con,axiom,
! [T2: a,U2: a,T5: a,U4: a] :
( ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( ( ( sources_a @ resid @ T5 )
= ( sources_a @ resid @ U4 ) )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U2 @ U4 )
=> ( ( con_a @ resid @ T2 @ U2 )
= ( con_a @ resid @ T5 @ U4 ) ) ) ) ) ) ).
% N.Cong_subst_con
thf(fact_415_P_Oseq__implies__Trgs__eq__Srcs,axiom,
! [T: list_a,U: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Arr_a @ resid @ U )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) )
=> ( ( paths_in_Trgs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ U ) ) ) ) ) ).
% P.seq_implies_Trgs_eq_Srcs
thf(fact_416_Arr__Resid,axiom,
! [T9: set_list_a,U9: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 ) )
!= bot_bot_set_list_a ) ) ).
% Arr_Resid
thf(fact_417_Con__sym,axiom,
! [T9: set_list_a,U9: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U9 @ T9 )
!= bot_bot_set_list_a ) ) ).
% Con_sym
thf(fact_418_Cube,axiom,
! [V4: set_list_a,T9: set_list_a,U9: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V4 @ T9 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U9 @ T9 ) )
!= bot_bot_set_list_a )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V4 @ T9 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U9 @ T9 ) )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V4 @ U9 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 ) ) ) ) ).
% Cube
thf(fact_419_con__imp__coinitial__members__are__con,axiom,
! [T9: set_list_a,U9: set_list_a,T2: list_a,U2: list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T9 @ U9 )
=> ( ( member_list_a @ T2 @ T9 )
=> ( ( member_list_a @ U2 @ U9 )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ) ) ) ).
% con_imp_coinitial_members_are_con
thf(fact_420_quot_Opreserves__cong,axiom,
! [T2: list_a,U2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 ) ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) ) ) ) ) ).
% quot.preserves_cong
thf(fact_421_quot_Opreserves__ide,axiom,
! [A: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A ) ) ) ).
% quot.preserves_ide
thf(fact_422_quot_Opreserves__prfx,axiom,
! [T2: list_a,U2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 ) ) ) ) ).
% quot.preserves_prfx
thf(fact_423_quot_Opreserves__con,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 ) ) ) ).
% quot.preserves_con
thf(fact_424_Con__witnesses,axiom,
! [T9: set_list_a,U9: set_list_a,T2: list_a,U2: list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( ( member_list_a @ T2 @ T9 )
=> ( ( member_list_a @ U2 @ U9 )
=> ? [V3: list_a,W2: list_a] :
( ( member_list_a @ V3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ W2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ V3 ) @ ( paths_in_Resid_a @ resid @ U2 @ W2 ) ) ) ) ) ) ).
% Con_witnesses
thf(fact_425_con__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [T9: set_list_a,U9: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T9 @ U9 )
= ( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
& ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U9 )
& ? [T8: list_a,U5: list_a] :
( ( member_list_a @ T8 @ T9 )
& ( member_list_a @ U5 @ U9 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T8 @ U5 ) ) ) ) ).
% con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_426_is__Cong__class__Resid,axiom,
! [T9: set_list_a,U9: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 ) ) ) ).
% is_Cong_class_Resid
thf(fact_427_Con__char,axiom,
! [T9: set_list_a,U9: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ U9 )
!= bot_bot_set_list_a )
= ( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
& ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U9 )
& ? [T8: list_a,U5: list_a] :
( ( member_list_a @ T8 @ T9 )
& ( member_list_a @ U5 @ U9 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T8 @ U5 ) ) ) ) ).
% Con_char
thf(fact_428_R_Oin__sourcesI,axiom,
! [A: a,T2: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T2 @ A )
=> ( member_a @ A @ ( sources_a @ resid @ T2 ) ) ) ) ).
% R.in_sourcesI
thf(fact_429_N_Orep__in__Cong__class,axiom,
! [T9: set_a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ( member_a @ ( normal3259722184653208495_rep_a @ T9 ) @ T9 ) ) ).
% N.rep_in_Cong_class
thf(fact_430_in__sourcesI,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ A )
=> ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ) ).
% in_sourcesI
thf(fact_431_ideI,axiom,
! [A: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A )
=> ( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ A )
= A )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A ) ) ) ).
% ideI
thf(fact_432_P_OArr__appendI_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Arr_a @ resid @ U )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) )
=> ( paths_in_Arr_a @ resid @ ( append_a @ T @ U ) ) ) ) ) ).
% P.Arr_appendI\<^sub>P
thf(fact_433_quot_Opreserves__resid,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 ) ) ) ) ).
% quot.preserves_resid
thf(fact_434_N_OCong__class__memb__Cong__rep,axiom,
! [T9: set_a,T2: a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ( ( member_a @ T2 @ T9 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ ( normal3259722184653208495_rep_a @ T9 ) ) ) ) ).
% N.Cong_class_memb_Cong_rep
thf(fact_435_P_ONPath__appendI,axiom,
! [T: list_a,U: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) )
=> ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( append_a @ T @ U ) ) ) ) ) ).
% P.NPath_appendI
thf(fact_436_P_Ois__Cong__classI_H,axiom,
! [T9: set_list_a] :
( ( T9 != bot_bot_set_list_a )
=> ( ! [T4: list_a,T11: list_a] :
( ( member_list_a @ T4 @ T9 )
=> ( ( member_list_a @ T11 @ T9 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T4 @ T11 ) ) )
=> ( ! [T4: list_a,T11: list_a] :
( ( member_list_a @ T4 @ T9 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T11 @ T4 )
=> ( member_list_a @ T11 @ T9 ) ) )
=> ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 ) ) ) ) ).
% P.is_Cong_classI'
thf(fact_437_quotient__by__coherent__normal_OCube,axiom,
! [Resid: a > a > a,NN: set_a,V4: set_a,T9: set_a,U9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V4 @ T9 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ U9 @ T9 ) )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V4 @ T9 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ U9 @ T9 ) )
= ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V4 @ U9 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_438_quotient__by__coherent__normal_OCube,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,V4: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( ( quotie750419831440180831list_a @ Resid @ NN @ ( quotie750419831440180831list_a @ Resid @ NN @ V4 @ T9 ) @ ( quotie750419831440180831list_a @ Resid @ NN @ U9 @ T9 ) )
!= bot_bo3186585308812441520list_a )
=> ( ( quotie750419831440180831list_a @ Resid @ NN @ ( quotie750419831440180831list_a @ Resid @ NN @ V4 @ T9 ) @ ( quotie750419831440180831list_a @ Resid @ NN @ U9 @ T9 ) )
= ( quotie750419831440180831list_a @ Resid @ NN @ ( quotie750419831440180831list_a @ Resid @ NN @ V4 @ U9 ) @ ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_439_quotient__by__coherent__normal_OCube,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,V4: set_list_a,T9: set_list_a,U9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ( quotie715552529030613631list_a @ Resid @ NN @ ( quotie715552529030613631list_a @ Resid @ NN @ V4 @ T9 ) @ ( quotie715552529030613631list_a @ Resid @ NN @ U9 @ T9 ) )
!= bot_bot_set_list_a )
=> ( ( quotie715552529030613631list_a @ Resid @ NN @ ( quotie715552529030613631list_a @ Resid @ NN @ V4 @ T9 ) @ ( quotie715552529030613631list_a @ Resid @ NN @ U9 @ T9 ) )
= ( quotie715552529030613631list_a @ Resid @ NN @ ( quotie715552529030613631list_a @ Resid @ NN @ V4 @ U9 ) @ ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_440_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ U9 @ T9 )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_441_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bo3186585308812441520list_a )
=> ( ( quotie750419831440180831list_a @ Resid @ NN @ U9 @ T9 )
!= bot_bo3186585308812441520list_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_442_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( ( quotie715552529030613631list_a @ Resid @ NN @ U9 @ T9 )
!= bot_bot_set_list_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_443_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 ) )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_444_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bo3186585308812441520list_a )
=> ( ( quotie750419831440180831list_a @ Resid @ NN @ ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 ) @ ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 ) )
!= bot_bo3186585308812441520list_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_445_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( ( quotie715552529030613631list_a @ Resid @ NN @ ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 ) @ ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 ) )
!= bot_bot_set_list_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_446_quotient__by__coherent__normal_OResid_Ocong,axiom,
quotie715552529030613631list_a = quotie715552529030613631list_a ).
% quotient_by_coherent_normal.Resid.cong
thf(fact_447_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a,T2: a,U2: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_a )
=> ( ( member_a @ T2 @ T9 )
=> ( ( member_a @ U2 @ U9 )
=> ? [V3: a,W2: a] :
( ( member_a @ V3 @ NN )
& ( member_a @ W2 @ NN )
& ( con_a @ Resid @ ( Resid @ T2 @ V3 ) @ ( Resid @ U2 @ W2 ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_448_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a,T2: set_list_a,U2: set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bo3186585308812441520list_a )
=> ( ( member_set_list_a @ T2 @ T9 )
=> ( ( member_set_list_a @ U2 @ U9 )
=> ? [V3: set_list_a,W2: set_list_a] :
( ( member_set_list_a @ V3 @ NN )
& ( member_set_list_a @ W2 @ NN )
& ( con_set_list_a @ Resid @ ( Resid @ T2 @ V3 ) @ ( Resid @ U2 @ W2 ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_449_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a,T2: list_a,U2: list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( ( member_list_a @ T2 @ T9 )
=> ( ( member_list_a @ U2 @ U9 )
=> ? [V3: list_a,W2: list_a] :
( ( member_list_a @ V3 @ NN )
& ( member_list_a @ W2 @ NN )
& ( con_list_a @ Resid @ ( Resid @ T2 @ V3 ) @ ( Resid @ U2 @ W2 ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_450_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bo3186585308812441520list_a )
=> ( normal4686252266493823358list_a @ Resid @ NN @ ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_451_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_a )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_452_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( normal4947954984078133662list_a @ Resid @ NN @ ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_453_quotient__by__coherent__normal_Ois__partial__magma,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( partia4872338157591926529list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_partial_magma
thf(fact_454_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bo3186585308812441520list_a )
= ( ( normal4686252266493823358list_a @ Resid @ NN @ T9 )
& ( normal4686252266493823358list_a @ Resid @ NN @ U9 )
& ? [T8: set_list_a,U5: set_list_a] :
( ( member_set_list_a @ T8 @ T9 )
& ( member_set_list_a @ U5 @ U9 )
& ( con_set_list_a @ Resid @ T8 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_455_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_a )
= ( ( normal8595587647932138008lass_a @ Resid @ NN @ T9 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U9 )
& ? [T8: a,U5: a] :
( ( member_a @ T8 @ T9 )
& ( member_a @ U5 @ U9 )
& ( con_a @ Resid @ T8 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_456_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 )
!= bot_bot_set_list_a )
= ( ( normal4947954984078133662list_a @ Resid @ NN @ T9 )
& ( normal4947954984078133662list_a @ Resid @ NN @ U9 )
& ? [T8: list_a,U5: list_a] :
( ( member_list_a @ T8 @ T9 )
& ( member_list_a @ U5 @ U9 )
& ( con_list_a @ Resid @ T8 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_457_quotient__by__coherent__normal_Ocon__imp__coinitial__members__are__con,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a,T2: a,U2: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T9 @ U9 )
=> ( ( member_a @ T2 @ T9 )
=> ( ( member_a @ U2 @ U9 )
=> ( ( ( sources_a @ Resid @ T2 )
= ( sources_a @ Resid @ U2 ) )
=> ( con_a @ Resid @ T2 @ U2 ) ) ) ) ) ) ).
% quotient_by_coherent_normal.con_imp_coinitial_members_are_con
thf(fact_458_quotient__by__coherent__normal_Ocon__imp__coinitial__members__are__con,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a,T2: set_list_a,U2: set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( con_set_set_list_a @ ( quotie750419831440180831list_a @ Resid @ NN ) @ T9 @ U9 )
=> ( ( member_set_list_a @ T2 @ T9 )
=> ( ( member_set_list_a @ U2 @ U9 )
=> ( ( ( sources_set_list_a @ Resid @ T2 )
= ( sources_set_list_a @ Resid @ U2 ) )
=> ( con_set_list_a @ Resid @ T2 @ U2 ) ) ) ) ) ) ).
% quotient_by_coherent_normal.con_imp_coinitial_members_are_con
thf(fact_459_quotient__by__coherent__normal_Ocon__imp__coinitial__members__are__con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a,T2: list_a,U2: list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) @ T9 @ U9 )
=> ( ( member_list_a @ T2 @ T9 )
=> ( ( member_list_a @ U2 @ U9 )
=> ( ( ( sources_list_a @ Resid @ T2 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( con_list_a @ Resid @ T2 @ U2 ) ) ) ) ) ) ).
% quotient_by_coherent_normal.con_imp_coinitial_members_are_con
thf(fact_460_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( con_set_set_list_a @ ( quotie750419831440180831list_a @ Resid @ NN ) @ T9 @ U9 )
= ( ( normal4686252266493823358list_a @ Resid @ NN @ T9 )
& ( normal4686252266493823358list_a @ Resid @ NN @ U9 )
& ? [T8: set_list_a,U5: set_list_a] :
( ( member_set_list_a @ T8 @ T9 )
& ( member_set_list_a @ U5 @ U9 )
& ( con_set_list_a @ Resid @ T8 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_461_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T9 @ U9 )
= ( ( normal8595587647932138008lass_a @ Resid @ NN @ T9 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U9 )
& ? [T8: a,U5: a] :
( ( member_a @ T8 @ T9 )
& ( member_a @ U5 @ U9 )
& ( con_a @ Resid @ T8 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_462_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) @ T9 @ U9 )
= ( ( normal4947954984078133662list_a @ Resid @ NN @ T9 )
& ( normal4947954984078133662list_a @ Resid @ NN @ U9 )
& ? [T8: list_a,U5: list_a] :
( ( member_list_a @ T8 @ T9 )
& ( member_list_a @ U5 @ U9 )
& ( con_list_a @ Resid @ T8 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_463_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T9: set_set_list_a,U9: set_set_list_a,T2: set_list_a,U2: set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( normal4686252266493823358list_a @ Resid @ NN @ T9 )
=> ( ( normal4686252266493823358list_a @ Resid @ NN @ U9 )
=> ( ( member_set_list_a @ T2 @ T9 )
=> ( ( member_set_list_a @ U2 @ U9 )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
=> ( ( quotie750419831440180831list_a @ Resid @ NN @ T9 @ U9 )
= ( normal8303701782564906868list_a @ Resid @ NN @ ( Resid @ T2 @ U2 ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_464_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a,U9: set_a,T2: a,U2: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T9 )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ U9 )
=> ( ( member_a @ T2 @ T9 )
=> ( ( member_a @ U2 @ U9 )
=> ( ( con_a @ Resid @ T2 @ U2 )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T9 @ U9 )
= ( normal7408713899360725774lass_a @ Resid @ NN @ ( Resid @ T2 @ U2 ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_465_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a,U9: set_list_a,T2: list_a,U2: list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T9 )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ U9 )
=> ( ( member_list_a @ T2 @ T9 )
=> ( ( member_list_a @ U2 @ U9 )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
=> ( ( quotie715552529030613631list_a @ Resid @ NN @ T9 @ U9 )
= ( normal8738828432243163028list_a @ Resid @ NN @ ( Resid @ T2 @ U2 ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_466_paths__in__rts__with__normal_ONPath__appendI,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_list_a,U: list_list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ T )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ U )
=> ( ( ord_le8861187494160871172list_a @ ( paths_in_Trgs_list_a @ Resid @ T ) @ ( paths_in_Srcs_list_a @ Resid @ U ) )
=> ( paths_8553768783572469479list_a @ Resid @ NN @ ( append_list_a @ T @ U ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_appendI
thf(fact_467_paths__in__rts__with__normal_ONPath__appendI,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,U: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T ) @ ( paths_in_Srcs_a @ Resid @ U ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( append_a @ T @ U ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_appendI
thf(fact_468_paths__in__rts__with__normal_ONPath__append,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_list_a,U: list_list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( T != nil_list_a )
=> ( ( U != nil_list_a )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ ( append_list_a @ T @ U ) )
= ( ( paths_8553768783572469479list_a @ Resid @ NN @ T )
& ( paths_8553768783572469479list_a @ Resid @ NN @ U )
& ( ord_le8861187494160871172list_a @ ( paths_in_Trgs_list_a @ Resid @ T ) @ ( paths_in_Srcs_list_a @ Resid @ U ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_append
thf(fact_469_paths__in__rts__with__normal_ONPath__append,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,U: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( append_a @ T @ U ) )
= ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T )
& ( paths_3208254328100665313Path_a @ Resid @ NN @ U )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T ) @ ( paths_in_Srcs_a @ Resid @ U ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_append
thf(fact_470_residuation__axioms,axiom,
residu1267353442355242628list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% residuation_axioms
thf(fact_471_P_ONPath__folding,axiom,
! [U: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ? [U7: a] :
( ( member_a @ U7 @ ( collect_a @ ( ide_a @ resid ) ) )
& ( ( sources_a @ resid @ U7 )
= ( paths_in_Srcs_a @ resid @ U ) )
& ( ( targets_a @ resid @ U7 )
= ( paths_in_Trgs_a @ resid @ U ) )
& ! [T3: a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T3 @ nil_a ) @ U )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U ) @ ( cons_a @ ( resid @ T3 @ U7 ) @ nil_a ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T3 @ U7 ) @ nil_a ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ).
% P.NPath_folding
thf(fact_472_P_OCoherent__single,axiom,
! [T2: a,U: list_a,U3: list_a] :
( ( arr_a @ resid @ T2 )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U )
=> ( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U3 )
=> ( ( ( sources_a @ resid @ T2 )
= ( paths_in_Srcs_a @ resid @ U ) )
=> ( ( ( paths_in_Srcs_a @ resid @ U )
= ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( ( ( paths_in_Trgs_a @ resid @ U )
= ( paths_in_Trgs_a @ resid @ U3 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U3 ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ) ) ) ).
% P.Coherent_single
thf(fact_473_P_OSrcs_Oelims,axiom,
! [X4: list_a,Y4: set_a] :
( ( ( paths_in_Srcs_a @ resid @ X4 )
= Y4 )
=> ( ( ( X4 = nil_a )
=> ( Y4 != bot_bot_set_a ) )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ( Y4
!= ( sources_a @ resid @ T4 ) ) )
=> ~ ! [T4: a] :
( ? [V3: a,Va: list_a] :
( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ( Y4
!= ( sources_a @ resid @ T4 ) ) ) ) ) ) ).
% P.Srcs.elims
thf(fact_474_P_ONPath__Resid__Arr__single,axiom,
! [T: list_a,U2: a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T )
=> ( ( arr_a @ resid @ U2 )
=> ( ( ( paths_in_Srcs_a @ resid @ T )
= ( sources_a @ resid @ U2 ) )
=> ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) ) ) ) ) ) ).
% P.NPath_Resid_Arr_single
thf(fact_475_P_OCong__class__eqI_H,axiom,
! [T9: set_list_a,U9: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U9 )
=> ( ( ( inf_inf_set_list_a @ T9 @ U9 )
!= bot_bot_set_list_a )
=> ( T9 = U9 ) ) ) ) ).
% P.Cong_class_eqI'
thf(fact_476_R_Oprfx__reflexive,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( ide_a @ resid @ ( resid @ T2 @ T2 ) ) ) ).
% R.prfx_reflexive
thf(fact_477_R_Oide__implies__arr,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( arr_a @ resid @ A ) ) ).
% R.ide_implies_arr
thf(fact_478_R_Ocong__reflexive,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( ( ide_a @ resid @ ( resid @ T2 @ T2 ) )
& ( ide_a @ resid @ ( resid @ T2 @ T2 ) ) ) ) ).
% R.cong_reflexive
thf(fact_479_N_Oelements__are__arr,axiom,
! [T2: a] :
( ( member_a @ T2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( arr_a @ resid @ T2 ) ) ).
% N.elements_are_arr
thf(fact_480_N_OCong_092_060_094sub_0620__reflexive,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( ( member_a @ ( resid @ T2 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ T2 @ T2 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% N.Cong\<^sub>0_reflexive
thf(fact_481_R_Otargets__cong__closed,axiom,
! [B: a,T2: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T2 ) )
=> ( ( ( ide_a @ resid @ ( resid @ B @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ B ) ) )
=> ( member_a @ B2 @ ( targets_a @ resid @ T2 ) ) ) ) ).
% R.targets_cong_closed
thf(fact_482_R_Otargets__are__cong,axiom,
! [B: a,T2: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T2 ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T2 ) )
=> ( ( ide_a @ resid @ ( resid @ B @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ B ) ) ) ) ) ).
% R.targets_are_cong
thf(fact_483_R_Otarget__is__ide,axiom,
! [A: a,T2: a] :
( ( member_a @ A @ ( targets_a @ resid @ T2 ) )
=> ( ide_a @ resid @ A ) ) ).
% R.target_is_ide
thf(fact_484_R_Oarr__resid__iff__con,axiom,
! [T2: a,U2: a] :
( ( arr_a @ resid @ ( resid @ T2 @ U2 ) )
= ( con_a @ resid @ T2 @ U2 ) ) ).
% R.arr_resid_iff_con
thf(fact_485_R_Oarr__resid,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( arr_a @ resid @ ( resid @ T2 @ U2 ) ) ) ).
% R.arr_resid
thf(fact_486_R_Oarr__def,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
= ( con_a @ resid @ T2 @ T2 ) ) ).
% R.arr_def
thf(fact_487_R_OarrE,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( con_a @ resid @ T2 @ T2 ) ) ).
% R.arrE
thf(fact_488_R_Ocon__implies__arr_I1_J,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( arr_a @ resid @ T2 ) ) ).
% R.con_implies_arr(1)
thf(fact_489_R_Ocon__implies__arr_I2_J,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( arr_a @ resid @ U2 ) ) ).
% R.con_implies_arr(2)
thf(fact_490_R_Otargets__resid__sym,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( ( targets_a @ resid @ ( resid @ T2 @ U2 ) )
= ( targets_a @ resid @ ( resid @ U2 @ T2 ) ) ) ) ).
% R.targets_resid_sym
thf(fact_491_R_Otargets__are__con,axiom,
! [B: a,T2: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T2 ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T2 ) )
=> ( con_a @ resid @ B @ B2 ) ) ) ).
% R.targets_are_con
thf(fact_492_R_Oresid__source__in__targets,axiom,
! [A: a,T2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( member_a @ ( resid @ A @ T2 ) @ ( targets_a @ resid @ T2 ) ) ) ).
% R.resid_source_in_targets
thf(fact_493_R_Otargets__con__closed,axiom,
! [B: a,T2: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T2 ) )
=> ( ( ide_a @ resid @ B2 )
=> ( ( con_a @ resid @ B @ B2 )
=> ( member_a @ B2 @ ( targets_a @ resid @ T2 ) ) ) ) ) ).
% R.targets_con_closed
thf(fact_494_N_Ocoherent_H,axiom,
! [V: a,V5: a,W3: a,W4: a,T2: a,T5: a] :
( ( member_a @ V @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ V5 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ W3 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ W4 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( sources_a @ resid @ V )
= ( sources_a @ resid @ W3 ) )
=> ( ( ( sources_a @ resid @ V5 )
= ( sources_a @ resid @ W4 ) )
=> ( ( ( targets_a @ resid @ W3 )
= ( targets_a @ resid @ W4 ) )
=> ( ( ( member_a @ ( resid @ ( resid @ T2 @ V ) @ ( resid @ T5 @ V5 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ V5 ) @ ( resid @ T2 @ V ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ( ( member_a @ ( resid @ ( resid @ T2 @ W3 ) @ ( resid @ T5 @ W4 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T5 @ W4 ) @ ( resid @ T2 @ W3 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ) ) ) ).
% N.coherent'
thf(fact_495_N_OCong__reflexive,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T2 ) ) ).
% N.Cong_reflexive
thf(fact_496_N_OCong__imp__arr_I1_J,axiom,
! [T2: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( arr_a @ resid @ T2 ) ) ).
% N.Cong_imp_arr(1)
thf(fact_497_N_OCong__imp__arr_I2_J,axiom,
! [T2: a,T5: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( arr_a @ resid @ T5 ) ) ).
% N.Cong_imp_arr(2)
thf(fact_498_N_Otargets__are__Cong,axiom,
! [B: a,T2: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T2 ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T2 ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ B @ B2 ) ) ) ).
% N.targets_are_Cong
thf(fact_499_N_Oin__targets__respects__Cong,axiom,
! [T2: a,T5: a,B: a,B2: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ T5 )
=> ( ( member_a @ B @ ( targets_a @ resid @ T2 ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T5 ) )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ B @ B2 ) ) ) ) ).
% N.in_targets_respects_Cong
thf(fact_500_P_OSrcs_Osimps_I1_J,axiom,
( ( paths_in_Srcs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% P.Srcs.simps(1)
thf(fact_501_P_OTrgs_Osimps_I1_J,axiom,
( ( paths_in_Trgs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% P.Trgs.simps(1)
thf(fact_502_N_OCong__class__memb__is__arr,axiom,
! [T9: set_a,T2: a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ( ( member_a @ T2 @ T9 )
=> ( arr_a @ resid @ T2 ) ) ) ).
% N.Cong_class_memb_is_arr
thf(fact_503_R_Oarr__iff__has__source,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
= ( ( sources_a @ resid @ T2 )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_source
thf(fact_504_P_OArr__has__Src,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Srcs_a @ resid @ T )
!= bot_bot_set_a ) ) ).
% P.Arr_has_Src
thf(fact_505_R_Oarr__iff__has__target,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
= ( ( targets_a @ resid @ T2 )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_target
thf(fact_506_P_OArr__has__Trg,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Trgs_a @ resid @ T )
!= bot_bot_set_a ) ) ).
% P.Arr_has_Trg
thf(fact_507_N_OCong__class__is__nonempty,axiom,
! [T9: set_a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ( T9 != bot_bot_set_a ) ) ).
% N.Cong_class_is_nonempty
thf(fact_508_N_Oarr__in__Cong__class,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( member_a @ T2 @ ( normal7408713899360725774lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 ) ) ) ).
% N.arr_in_Cong_class
thf(fact_509_R_Ocoinitial__iff,axiom,
! [T2: a,T5: a] :
( ( coinitial_a @ resid @ T2 @ T5 )
= ( ( arr_a @ resid @ T2 )
& ( arr_a @ resid @ T5 )
& ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ T5 ) ) ) ) ).
% R.coinitial_iff
thf(fact_510_R_OcoinitialE,axiom,
! [T2: a,U2: a] :
( ( coinitial_a @ resid @ T2 @ U2 )
=> ~ ( ( arr_a @ resid @ T2 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( sources_a @ resid @ T2 )
!= ( sources_a @ resid @ U2 ) ) ) ) ) ).
% R.coinitialE
thf(fact_511_R_Ocoterminal__iff,axiom,
! [T2: a,T5: a] :
( ( coterminal_a @ resid @ T2 @ T5 )
= ( ( arr_a @ resid @ T2 )
& ( arr_a @ resid @ T5 )
& ( ( targets_a @ resid @ T2 )
= ( targets_a @ resid @ T5 ) ) ) ) ).
% R.coterminal_iff
thf(fact_512_R_OcoterminalE,axiom,
! [T2: a,U2: a] :
( ( coterminal_a @ resid @ T2 @ U2 )
=> ~ ( ( arr_a @ resid @ T2 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( targets_a @ resid @ T2 )
!= ( targets_a @ resid @ U2 ) ) ) ) ) ).
% R.coterminalE
thf(fact_513_P_OArr_Osimps_I2_J,axiom,
! [T2: a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T2 @ nil_a ) )
= ( arr_a @ resid @ T2 ) ) ).
% P.Arr.simps(2)
thf(fact_514_P_OTrgs_Osimps_I2_J,axiom,
! [T2: a] :
( ( paths_in_Trgs_a @ resid @ ( cons_a @ T2 @ nil_a ) )
= ( targets_a @ resid @ T2 ) ) ).
% P.Trgs.simps(2)
thf(fact_515_N_Ocoherent,axiom,
! [T2: a,U2: a,U4: a] :
( ( arr_a @ resid @ T2 )
=> ( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( member_a @ U4 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( sources_a @ resid @ U2 )
= ( sources_a @ resid @ U4 ) )
=> ( ( ( targets_a @ resid @ U2 )
= ( targets_a @ resid @ U4 ) )
=> ( ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( ( member_a @ ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T2 @ U4 ) ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ ( resid @ T2 @ U4 ) @ ( resid @ T2 @ U2 ) ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ) ).
% N.coherent
thf(fact_516_N_OCong_H_Osimps,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ A1 @ A22 )
= ( ? [T8: a,U5: a] :
( ( A1 = U5 )
& ( A22 = T8 )
& ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T8 @ U5 ) )
| ? [T8: a,U5: a,V6: a] :
( ( A1 = T8 )
& ( A22 = V6 )
& ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T8 @ U5 )
& ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U5 @ V6 ) )
| ? [T8: a,U5: a] :
( ( A1 = T8 )
& ( A22 = U5 )
& ( member_a @ ( resid @ T8 @ U5 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ U5 @ T8 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
| ? [T8: a,U5: a] :
( ( A1 = T8 )
& ( A22
= ( resid @ T8 @ U5 ) )
& ( arr_a @ resid @ T8 )
& ( member_a @ U5 @ ( collect_a @ ( ide_a @ resid ) ) )
& ( ( sources_a @ resid @ T8 )
= ( sources_a @ resid @ U5 ) ) ) ) ) ).
% N.Cong'.simps
thf(fact_517_N_OCong_H_Ocases,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ A1 @ A22 )
=> ( ~ ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ A22 @ A1 )
=> ( ! [U7: a] :
( ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ A1 @ U7 )
=> ~ ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U7 @ A22 ) )
=> ( ~ ( ( member_a @ ( resid @ A1 @ A22 ) @ ( collect_a @ ( ide_a @ resid ) ) )
& ( member_a @ ( resid @ A22 @ A1 ) @ ( collect_a @ ( ide_a @ resid ) ) ) )
=> ~ ! [U7: a] :
( ( A22
= ( resid @ A1 @ U7 ) )
=> ( ( arr_a @ resid @ A1 )
=> ( ( member_a @ U7 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( sources_a @ resid @ A1 )
!= ( sources_a @ resid @ U7 ) ) ) ) ) ) ) ) ) ).
% N.Cong'.cases
thf(fact_518_N_OCong_H_Ointros_I4_J,axiom,
! [T2: a,U2: a] :
( ( arr_a @ resid @ T2 )
=> ( ( member_a @ U2 @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( normal_sub_Cong_a2 @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 @ ( resid @ T2 @ U2 ) ) ) ) ) ).
% N.Cong'.intros(4)
thf(fact_519_N_Ois__Cong__classE,axiom,
! [T9: set_a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ~ ( ( T9 != bot_bot_set_a )
=> ( ! [T3: a] :
( ( member_a @ T3 @ T9 )
=> ! [T10: a] :
( ( member_a @ T10 @ T9 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T3 @ T10 ) ) )
=> ~ ! [T3: a] :
( ( member_a @ T3 @ T9 )
=> ! [T10: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T10 @ T3 )
=> ( member_a @ T10 @ T9 ) ) ) ) ) ) ).
% N.is_Cong_classE
thf(fact_520_P_Osources__eqI,axiom,
! [T2: list_a,T5: list_a] :
( ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) )
!= bot_bot_set_list_a )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) ) ) ).
% P.sources_eqI
thf(fact_521_N_Ois__Cong__classI,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( normal7408713899360725774lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T2 ) ) ) ).
% N.is_Cong_classI
thf(fact_522_P_OSrcs__Resid__Arr__single,axiom,
! [T: list_a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ T @ ( cons_a @ U2 @ nil_a ) ) )
= ( targets_a @ resid @ U2 ) ) ) ).
% P.Srcs_Resid_Arr_single
thf(fact_523_P_OTrgs_Oelims,axiom,
! [X4: list_a,Y4: set_a] :
( ( ( paths_in_Trgs_a @ resid @ X4 )
= Y4 )
=> ( ( ( X4 = nil_a )
=> ( Y4 != bot_bot_set_a ) )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ( Y4
!= ( targets_a @ resid @ T4 ) ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( ( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ( Y4
!= ( paths_in_Trgs_a @ resid @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ).
% P.Trgs.elims
thf(fact_524_P_Ocon__imp__common__source,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
!= bot_bot_set_list_a ) ) ).
% P.con_imp_common_source
thf(fact_525_P_Ocoinitial__def,axiom,
! [T2: list_a,U2: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
!= bot_bot_set_list_a ) ) ).
% P.coinitial_def
thf(fact_526_P_OArr_Osimps_I3_J,axiom,
! [T2: a,V: a,Va2: list_a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T2 @ ( cons_a @ V @ Va2 ) ) )
= ( ( arr_a @ resid @ T2 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T2 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va2 ) ) ) ) ) ).
% P.Arr.simps(3)
thf(fact_527_P_OArr_Oelims_I1_J,axiom,
! [X4: list_a,Y4: $o] :
( ( ( paths_in_Arr_a @ resid @ X4 )
= Y4 )
=> ( ( ( X4 = nil_a )
=> Y4 )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ( Y4
= ( ~ ( arr_a @ resid @ T4 ) ) ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( ( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ( Y4
= ( ~ ( ( arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V3 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ).
% P.Arr.elims(1)
thf(fact_528_P_OArr_Oelims_I2_J,axiom,
! [X4: list_a] :
( ( paths_in_Arr_a @ resid @ X4 )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ~ ( arr_a @ resid @ T4 ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( ( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ~ ( ( arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V3 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ).
% P.Arr.elims(2)
thf(fact_529_P_OArr_Oelims_I3_J,axiom,
! [X4: list_a] :
( ~ ( paths_in_Arr_a @ resid @ X4 )
=> ( ( X4 != nil_a )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ( arr_a @ resid @ T4 ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( ( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ( ( arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V3 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ) ).
% P.Arr.elims(3)
thf(fact_530_R_OarrI,axiom,
! [T2: a] :
( ( con_a @ resid @ T2 @ T2 )
=> ( arr_a @ resid @ T2 ) ) ).
% R.arrI
thf(fact_531_R_Osources__resid,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( ( sources_a @ resid @ ( resid @ T2 @ U2 ) )
= ( targets_a @ resid @ U2 ) ) ) ).
% R.sources_resid
thf(fact_532_R_OcoinitialI,axiom,
! [T2: a,U2: a] :
( ( arr_a @ resid @ T2 )
=> ( ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( coinitial_a @ resid @ T2 @ U2 ) ) ) ).
% R.coinitialI
thf(fact_533_R_OcoterminalI,axiom,
! [T2: a,U2: a] :
( ( arr_a @ resid @ T2 )
=> ( ( ( targets_a @ resid @ T2 )
= ( targets_a @ resid @ U2 ) )
=> ( coterminal_a @ resid @ T2 @ U2 ) ) ) ).
% R.coterminalI
thf(fact_534_N_Ois__Cong__classI_H,axiom,
! [T9: set_a] :
( ( T9 != bot_bot_set_a )
=> ( ! [T4: a,T11: a] :
( ( member_a @ T4 @ T9 )
=> ( ( member_a @ T11 @ T9 )
=> ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T4 @ T11 ) ) )
=> ( ! [T4: a,T11: a] :
( ( member_a @ T4 @ T9 )
=> ( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T11 @ T4 )
=> ( member_a @ T11 @ T9 ) ) )
=> ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 ) ) ) ) ).
% N.is_Cong_classI'
thf(fact_535_N_OCong__class__rep,axiom,
! [T9: set_a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ( ( normal7408713899360725774lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ ( normal3259722184653208495_rep_a @ T9 ) )
= T9 ) ) ).
% N.Cong_class_rep
thf(fact_536_P_OArr__consI_092_060_094sub_062P,axiom,
! [T2: a,U: list_a] :
( ( arr_a @ resid @ T2 )
=> ( ( paths_in_Arr_a @ resid @ U )
=> ( ( ord_less_eq_set_a @ ( targets_a @ resid @ T2 ) @ ( paths_in_Srcs_a @ resid @ U ) )
=> ( paths_in_Arr_a @ resid @ ( cons_a @ T2 @ U ) ) ) ) ) ).
% P.Arr_consI\<^sub>P
thf(fact_537_residuation_Ocube,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,V: set_list_a,T2: set_list_a,U2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( Resid @ ( Resid @ V @ T2 ) @ ( Resid @ U2 @ T2 ) )
= ( Resid @ ( Resid @ V @ U2 ) @ ( Resid @ T2 @ U2 ) ) ) ) ).
% residuation.cube
thf(fact_538_residuation_Ocube,axiom,
! [Resid: a > a > a,V: a,T2: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( Resid @ ( Resid @ V @ T2 ) @ ( Resid @ U2 @ T2 ) )
= ( Resid @ ( Resid @ V @ U2 ) @ ( Resid @ T2 @ U2 ) ) ) ) ).
% residuation.cube
thf(fact_539_residuation_Ocube,axiom,
! [Resid: list_a > list_a > list_a,V: list_a,T2: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( Resid @ ( Resid @ V @ T2 ) @ ( Resid @ U2 @ T2 ) )
= ( Resid @ ( Resid @ V @ U2 ) @ ( Resid @ T2 @ U2 ) ) ) ) ).
% residuation.cube
thf(fact_540_rts_Otargets_Ocong,axiom,
targets_a = targets_a ).
% rts.targets.cong
thf(fact_541_rts_Otargets_Ocong,axiom,
targets_list_a = targets_list_a ).
% rts.targets.cong
thf(fact_542_rts_Otargets_Ocong,axiom,
targets_set_list_a = targets_set_list_a ).
% rts.targets.cong
thf(fact_543_residuation_Oarr_Ocong,axiom,
arr_a = arr_a ).
% residuation.arr.cong
thf(fact_544_residuation_Oarr_Ocong,axiom,
arr_set_list_a = arr_set_list_a ).
% residuation.arr.cong
thf(fact_545_residuation_Oarr_Ocong,axiom,
arr_list_a = arr_list_a ).
% residuation.arr.cong
thf(fact_546_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
=> ( arr_list_a @ Resid @ U2 ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_547_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: a > a > a,T2: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T2 @ U2 )
=> ( arr_a @ Resid @ U2 ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_548_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a,U2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
=> ( arr_set_list_a @ Resid @ U2 ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_549_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
=> ( arr_list_a @ Resid @ T2 ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_550_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: a > a > a,T2: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T2 @ U2 )
=> ( arr_a @ Resid @ T2 ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_551_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a,U2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
=> ( arr_set_list_a @ Resid @ T2 ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_552_residuation_OarrE,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ T2 )
=> ( con_list_a @ Resid @ T2 @ T2 ) ) ) ).
% residuation.arrE
thf(fact_553_residuation_OarrE,axiom,
! [Resid: a > a > a,T2: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T2 )
=> ( con_a @ Resid @ T2 @ T2 ) ) ) ).
% residuation.arrE
thf(fact_554_residuation_OarrE,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( arr_set_list_a @ Resid @ T2 )
=> ( con_set_list_a @ Resid @ T2 @ T2 ) ) ) ).
% residuation.arrE
thf(fact_555_residuation_OarrI,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T2 @ T2 )
=> ( arr_list_a @ Resid @ T2 ) ) ) ).
% residuation.arrI
thf(fact_556_residuation_OarrI,axiom,
! [Resid: a > a > a,T2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T2 @ T2 )
=> ( arr_a @ Resid @ T2 ) ) ) ).
% residuation.arrI
thf(fact_557_residuation_OarrI,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( con_set_list_a @ Resid @ T2 @ T2 )
=> ( arr_set_list_a @ Resid @ T2 ) ) ) ).
% residuation.arrI
thf(fact_558_residuation_Oarr__def,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ T2 )
= ( con_list_a @ Resid @ T2 @ T2 ) ) ) ).
% residuation.arr_def
thf(fact_559_residuation_Oarr__def,axiom,
! [Resid: a > a > a,T2: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T2 )
= ( con_a @ Resid @ T2 @ T2 ) ) ) ).
% residuation.arr_def
thf(fact_560_residuation_Oarr__def,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( arr_set_list_a @ Resid @ T2 )
= ( con_set_list_a @ Resid @ T2 @ T2 ) ) ) ).
% residuation.arr_def
thf(fact_561_residuation_Oarr__resid,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
=> ( arr_list_a @ Resid @ ( Resid @ T2 @ U2 ) ) ) ) ).
% residuation.arr_resid
thf(fact_562_residuation_Oarr__resid,axiom,
! [Resid: a > a > a,T2: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T2 @ U2 )
=> ( arr_a @ Resid @ ( Resid @ T2 @ U2 ) ) ) ) ).
% residuation.arr_resid
thf(fact_563_residuation_Oarr__resid,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a,U2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
=> ( arr_set_list_a @ Resid @ ( Resid @ T2 @ U2 ) ) ) ) ).
% residuation.arr_resid
thf(fact_564_residuation_Oarr__resid__iff__con,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ ( Resid @ T2 @ U2 ) )
= ( con_list_a @ Resid @ T2 @ U2 ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_565_residuation_Oarr__resid__iff__con,axiom,
! [Resid: a > a > a,T2: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ ( Resid @ T2 @ U2 ) )
= ( con_a @ Resid @ T2 @ U2 ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_566_residuation_Oarr__resid__iff__con,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a,U2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( arr_set_list_a @ Resid @ ( Resid @ T2 @ U2 ) )
= ( con_set_list_a @ Resid @ T2 @ U2 ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_567_residuation_Oide__implies__arr,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( arr_a @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_568_residuation_Oide__implies__arr,axiom,
! [Resid: list_a > list_a > list_a,A: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ide_list_a @ Resid @ A )
=> ( arr_list_a @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_569_residuation_Oide__implies__arr,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,A: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( ide_set_list_a @ Resid @ A )
=> ( arr_set_list_a @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_570_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T2: list_a,U2: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T2 )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( member_list_a @ U4 @ NN )
=> ( ( ( sources_list_a @ Resid @ U2 )
= ( sources_list_a @ Resid @ U4 ) )
=> ( ( ( targets_list_a @ Resid @ U2 )
= ( targets_list_a @ Resid @ U4 ) )
=> ( ( ( sources_list_a @ Resid @ T2 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T2 @ U2 ) @ ( Resid @ T2 @ U4 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T2 @ U4 ) @ ( Resid @ T2 @ U2 ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_571_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: a > a > a,NN: set_a,T2: a,U2: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T2 )
=> ( ( member_a @ U2 @ NN )
=> ( ( member_a @ U4 @ NN )
=> ( ( ( sources_a @ Resid @ U2 )
= ( sources_a @ Resid @ U4 ) )
=> ( ( ( targets_a @ Resid @ U2 )
= ( targets_a @ Resid @ U4 ) )
=> ( ( ( sources_a @ Resid @ T2 )
= ( sources_a @ Resid @ U2 ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T2 @ U2 ) @ ( Resid @ T2 @ U4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T2 @ U4 ) @ ( Resid @ T2 @ U2 ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_572_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T2: set_list_a,U2: set_list_a,U4: set_list_a] :
( ( cohere4318586772758246349list_a @ Resid @ NN )
=> ( ( arr_set_list_a @ Resid @ T2 )
=> ( ( member_set_list_a @ U2 @ NN )
=> ( ( member_set_list_a @ U4 @ NN )
=> ( ( ( sources_set_list_a @ Resid @ U2 )
= ( sources_set_list_a @ Resid @ U4 ) )
=> ( ( ( targets_set_list_a @ Resid @ U2 )
= ( targets_set_list_a @ Resid @ U4 ) )
=> ( ( ( sources_set_list_a @ Resid @ T2 )
= ( sources_set_list_a @ Resid @ U2 ) )
=> ( ( member_set_list_a @ ( Resid @ ( Resid @ T2 @ U2 ) @ ( Resid @ T2 @ U4 ) ) @ NN )
& ( member_set_list_a @ ( Resid @ ( Resid @ T2 @ U4 ) @ ( Resid @ T2 @ U2 ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_573_residuation_Ocon__sym,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T2 @ U2 )
=> ( con_list_a @ Resid @ U2 @ T2 ) ) ) ).
% residuation.con_sym
thf(fact_574_residuation_Ocon__sym,axiom,
! [Resid: a > a > a,T2: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T2 @ U2 )
=> ( con_a @ Resid @ U2 @ T2 ) ) ) ).
% residuation.con_sym
thf(fact_575_residuation_Ocon__sym,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a,U2: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( con_set_list_a @ Resid @ T2 @ U2 )
=> ( con_set_list_a @ Resid @ U2 @ T2 ) ) ) ).
% residuation.con_sym
thf(fact_576_residuation_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( partial_magma_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_577_residuation_Oaxioms_I1_J,axiom,
! [Resid: list_a > list_a > list_a] :
( ( residuation_list_a @ Resid )
=> ( partial_magma_list_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_578_residuation_Oaxioms_I1_J,axiom,
! [Resid: set_list_a > set_list_a > set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( partia4872338157591926529list_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_579_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,V: list_a,V5: list_a,W3: list_a,W4: list_a,T2: list_a,T5: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( member_list_a @ V @ NN )
=> ( ( member_list_a @ V5 @ NN )
=> ( ( member_list_a @ W3 @ NN )
=> ( ( member_list_a @ W4 @ NN )
=> ( ( ( sources_list_a @ Resid @ V )
= ( sources_list_a @ Resid @ W3 ) )
=> ( ( ( sources_list_a @ Resid @ V5 )
= ( sources_list_a @ Resid @ W4 ) )
=> ( ( ( targets_list_a @ Resid @ W3 )
= ( targets_list_a @ Resid @ W4 ) )
=> ( ( ( member_list_a @ ( Resid @ ( Resid @ T2 @ V ) @ ( Resid @ T5 @ V5 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T5 @ V5 ) @ ( Resid @ T2 @ V ) ) @ NN ) )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T2 @ W3 ) @ ( Resid @ T5 @ W4 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T5 @ W4 ) @ ( Resid @ T2 @ W3 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_580_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: a > a > a,NN: set_a,V: a,V5: a,W3: a,W4: a,T2: a,T5: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( member_a @ V @ NN )
=> ( ( member_a @ V5 @ NN )
=> ( ( member_a @ W3 @ NN )
=> ( ( member_a @ W4 @ NN )
=> ( ( ( sources_a @ Resid @ V )
= ( sources_a @ Resid @ W3 ) )
=> ( ( ( sources_a @ Resid @ V5 )
= ( sources_a @ Resid @ W4 ) )
=> ( ( ( targets_a @ Resid @ W3 )
= ( targets_a @ Resid @ W4 ) )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T2 @ V ) @ ( Resid @ T5 @ V5 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T5 @ V5 ) @ ( Resid @ T2 @ V ) ) @ NN ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T2 @ W3 ) @ ( Resid @ T5 @ W4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T5 @ W4 ) @ ( Resid @ T2 @ W3 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_581_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,V: set_list_a,V5: set_list_a,W3: set_list_a,W4: set_list_a,T2: set_list_a,T5: set_list_a] :
( ( cohere4318586772758246349list_a @ Resid @ NN )
=> ( ( member_set_list_a @ V @ NN )
=> ( ( member_set_list_a @ V5 @ NN )
=> ( ( member_set_list_a @ W3 @ NN )
=> ( ( member_set_list_a @ W4 @ NN )
=> ( ( ( sources_set_list_a @ Resid @ V )
= ( sources_set_list_a @ Resid @ W3 ) )
=> ( ( ( sources_set_list_a @ Resid @ V5 )
= ( sources_set_list_a @ Resid @ W4 ) )
=> ( ( ( targets_set_list_a @ Resid @ W3 )
= ( targets_set_list_a @ Resid @ W4 ) )
=> ( ( ( member_set_list_a @ ( Resid @ ( Resid @ T2 @ V ) @ ( Resid @ T5 @ V5 ) ) @ NN )
& ( member_set_list_a @ ( Resid @ ( Resid @ T5 @ V5 ) @ ( Resid @ T2 @ V ) ) @ NN ) )
=> ( ( member_set_list_a @ ( Resid @ ( Resid @ T2 @ W3 ) @ ( Resid @ T5 @ W4 ) ) @ NN )
& ( member_set_list_a @ ( Resid @ ( Resid @ T5 @ W4 ) @ ( Resid @ T2 @ W3 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_582_residuation_OideE,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ~ ( ( con_a @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_583_residuation_OideE,axiom,
! [Resid: list_a > list_a > list_a,A: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ide_list_a @ Resid @ A )
=> ~ ( ( con_list_a @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_584_residuation_OideE,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,A: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( ide_set_list_a @ Resid @ A )
=> ~ ( ( con_set_list_a @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_585_residuation_OideI,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_a @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_586_residuation_OideI,axiom,
! [Resid: list_a > list_a > list_a,A: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_list_a @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_587_residuation_OideI,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,A: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( con_set_list_a @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_set_list_a @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_588_residuation_Oide__def,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A )
= ( ( con_a @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_589_residuation_Oide__def,axiom,
! [Resid: list_a > list_a > list_a,A: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ide_list_a @ Resid @ A )
= ( ( con_list_a @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_590_residuation_Oide__def,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,A: set_list_a] :
( ( residu1267353442355242628list_a @ Resid )
=> ( ( ide_set_list_a @ Resid @ A )
= ( ( con_set_list_a @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_591_quotient__by__coherent__normal_Ois__residuation,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( residu1267353442355242628list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_residuation
thf(fact_592_paths__in__rts__with__normal_ONPath__Resid__Arr__single,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,T: list_set_list_a,U2: set_list_a] :
( ( paths_2496957160577180956list_a @ Resid @ NN )
=> ( ( paths_8943339929420292807list_a @ Resid @ NN @ T )
=> ( ( arr_set_list_a @ Resid @ U2 )
=> ( ( ( paths_5622510820459842474list_a @ Resid @ T )
= ( sources_set_list_a @ Resid @ U2 ) )
=> ( paths_8943339929420292807list_a @ Resid @ NN @ ( paths_6591532302923812690list_a @ Resid @ T @ ( cons_set_list_a @ U2 @ nil_set_list_a ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_Arr_single
thf(fact_593_paths__in__rts__with__normal_ONPath__Resid__Arr__single,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_list_a,U2: list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ T )
=> ( ( arr_list_a @ Resid @ U2 )
=> ( ( ( paths_in_Srcs_list_a @ Resid @ T )
= ( sources_list_a @ Resid @ U2 ) )
=> ( paths_8553768783572469479list_a @ Resid @ NN @ ( paths_8620460302779588466list_a @ Resid @ T @ ( cons_list_a @ U2 @ nil_list_a ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_Arr_single
thf(fact_594_paths__in__rts__with__normal_ONPath__Resid__Arr__single,axiom,
! [Resid: a > a > a,NN: set_a,T: list_a,U2: a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T )
=> ( ( arr_a @ Resid @ U2 )
=> ( ( ( paths_in_Srcs_a @ Resid @ T )
= ( sources_a @ Resid @ U2 ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T @ ( cons_a @ U2 @ nil_a ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_Arr_single
thf(fact_595_cong__respects__seq,axiom,
! [T2: set_list_a,U2: set_list_a,T5: set_list_a,U4: set_list_a] :
( ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T5 @ T2 ) ) )
=> ( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ U4 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U4 @ U2 ) ) )
=> ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 @ U4 ) ) ) ) ).
% cong_respects_seq
thf(fact_596_coinitial__def,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( inf_in4657809108759609906list_a @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
!= bot_bo3186585308812441520list_a ) ) ).
% coinitial_def
thf(fact_597_con__imp__common__source,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( inf_in4657809108759609906list_a @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
!= bot_bo3186585308812441520list_a ) ) ).
% con_imp_common_source
thf(fact_598_ide__char,axiom,
! [U9: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U9 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U9 )
& ( ( inf_inf_set_list_a @ U9 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
!= bot_bot_set_list_a ) ) ) ).
% ide_char
thf(fact_599_Int__subset__iff,axiom,
! [C2: set_set_list_a,A2: set_set_list_a,B3: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ C2 @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) )
= ( ( ord_le8877086941679407844list_a @ C2 @ A2 )
& ( ord_le8877086941679407844list_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_600_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_601_Int__subset__iff,axiom,
! [C2: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( ( ord_le8861187494160871172list_a @ C2 @ A2 )
& ( ord_le8861187494160871172list_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_602_R_Oresiduation__axioms,axiom,
residuation_a @ resid ).
% R.residuation_axioms
thf(fact_603_P_Oresiduation__axioms,axiom,
residuation_list_a @ ( paths_in_Resid_a @ resid ) ).
% P.residuation_axioms
thf(fact_604_P_Otarget__is__ide,axiom,
! [A: list_a,T2: list_a] :
( ( member_list_a @ A @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% P.target_is_ide
thf(fact_605_P_Otargets__are__cong,axiom,
! [B: list_a,T2: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B @ B2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B2 @ B ) ) ) ) ) ).
% P.targets_are_cong
thf(fact_606_P_Otargets__cong__closed,axiom,
! [B: list_a,T2: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B @ B2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B2 @ B ) ) )
=> ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ) ).
% P.targets_cong_closed
thf(fact_607_P_Ocong__reflexive,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ T2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ T2 ) ) ) ) ).
% P.cong_reflexive
thf(fact_608_P_Oide__implies__arr,axiom,
! [A: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% P.ide_implies_arr
thf(fact_609_P_Oprfx__reflexive,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ T2 ) ) ) ).
% P.prfx_reflexive
thf(fact_610_P_Otargets__are__con,axiom,
! [B: list_a,T2: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ B @ B2 ) ) ) ).
% P.targets_are_con
thf(fact_611_P_Otargets__resid__sym,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) ) ) ) ).
% P.targets_resid_sym
thf(fact_612_P_Ocon__implies__arr_I2_J,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ).
% P.con_implies_arr(2)
thf(fact_613_P_Ocon__implies__arr_I1_J,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ).
% P.con_implies_arr(1)
thf(fact_614_P_OarrE,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T2 ) ) ).
% P.arrE
thf(fact_615_P_Oarr__def,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T2 ) ) ).
% P.arr_def
thf(fact_616_P_Oarr__resid,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ).
% P.arr_resid
thf(fact_617_P_Oarr__resid__iff__con,axiom,
! [T2: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.arr_resid_iff_con
thf(fact_618_P_Oresid__source__in__targets,axiom,
! [A: list_a,T2: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( member_list_a @ ( paths_in_Resid_a @ resid @ A @ T2 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ).
% P.resid_source_in_targets
thf(fact_619_P_Oarr__char,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( paths_in_Arr_a @ resid @ T ) ) ).
% P.arr_char
thf(fact_620_P_Oarr__iff__has__target,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
!= bot_bot_set_list_a ) ) ).
% P.arr_iff_has_target
thf(fact_621_R_Osources__eqI,axiom,
! [T2: a,T5: a] :
( ( ( inf_inf_set_a @ ( sources_a @ resid @ T2 ) @ ( sources_a @ resid @ T5 ) )
!= bot_bot_set_a )
=> ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ T5 ) ) ) ).
% R.sources_eqI
thf(fact_622_P_OSrcs__eqI,axiom,
! [T: list_a,T6: list_a] :
( ( ( inf_inf_set_a @ ( paths_in_Srcs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ T6 ) )
!= bot_bot_set_a )
=> ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ T6 ) ) ) ).
% P.Srcs_eqI
thf(fact_623_R_Otargets__eqI,axiom,
! [T2: a,T5: a] :
( ( ( inf_inf_set_a @ ( targets_a @ resid @ T2 ) @ ( targets_a @ resid @ T5 ) )
!= bot_bot_set_a )
=> ( ( targets_a @ resid @ T2 )
= ( targets_a @ resid @ T5 ) ) ) ).
% R.targets_eqI
thf(fact_624_P_OTrgs__eqI,axiom,
! [T: list_a,T6: list_a] :
( ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Trgs_a @ resid @ T6 ) )
!= bot_bot_set_a )
=> ( ( paths_in_Trgs_a @ resid @ T )
= ( paths_in_Trgs_a @ resid @ T6 ) ) ) ).
% P.Trgs_eqI
thf(fact_625_P_OcoterminalE,axiom,
! [T2: list_a,U2: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
!= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ) ).
% P.coterminalE
thf(fact_626_P_Ocoterminal__iff,axiom,
! [T2: list_a,T5: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T5 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T5 )
& ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) ) ) ) ).
% P.coterminal_iff
thf(fact_627_P_OCong_092_060_094sub_0620__reflexive,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T2 @ T2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% P.Cong\<^sub>0_reflexive
thf(fact_628_P_Oelements__are__arr,axiom,
! [T2: list_a] :
( ( member_list_a @ T2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ).
% P.elements_are_arr
thf(fact_629_P_Otargets__con__closed,axiom,
! [B: list_a,T2: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ B @ B2 )
=> ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ) ) ).
% P.targets_con_closed
thf(fact_630_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X2: list_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_631_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_632_empty__Collect__eq,axiom,
! [P: set_list_a > $o] :
( ( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ P ) )
= ( ! [X2: set_list_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_633_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X2: list_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_634_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_635_Collect__empty__eq,axiom,
! [P: set_list_a > $o] :
( ( ( collect_set_list_a @ P )
= bot_bo3186585308812441520list_a )
= ( ! [X2: set_list_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_636_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X2: list_a] :
~ ( member_list_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_637_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_638_all__not__in__conv,axiom,
! [A2: set_set_list_a] :
( ( ! [X2: set_list_a] :
~ ( member_set_list_a @ X2 @ A2 ) )
= ( A2 = bot_bo3186585308812441520list_a ) ) ).
% all_not_in_conv
thf(fact_639_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_640_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_641_empty__iff,axiom,
! [C: set_list_a] :
~ ( member_set_list_a @ C @ bot_bo3186585308812441520list_a ) ).
% empty_iff
thf(fact_642_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_643_subset__antisym,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_644_subsetI,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ! [X: set_list_a] :
( ( member_set_list_a @ X @ A2 )
=> ( member_set_list_a @ X @ B3 ) )
=> ( ord_le8877086941679407844list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_645_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_646_subsetI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_647_P_Oarr__iff__has__source,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
!= bot_bot_set_list_a ) ) ).
% P.arr_iff_has_source
thf(fact_648_Int__iff,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( ( member_list_a @ C @ A2 )
& ( member_list_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_649_Int__iff,axiom,
! [C: set_list_a,A2: set_set_list_a,B3: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) )
= ( ( member_set_list_a @ C @ A2 )
& ( member_set_list_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_650_Int__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_651_IntI,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ( member_list_a @ C @ B3 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_652_IntI,axiom,
! [C: set_list_a,A2: set_set_list_a,B3: set_set_list_a] :
( ( member_set_list_a @ C @ A2 )
=> ( ( member_set_list_a @ C @ B3 )
=> ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_653_IntI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_654_R_Ocon__imp__common__source,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( ( inf_inf_set_a @ ( sources_a @ resid @ T2 ) @ ( sources_a @ resid @ U2 ) )
!= bot_bot_set_a ) ) ).
% R.con_imp_common_source
thf(fact_655_P_OcoinitialE,axiom,
! [T2: list_a,U2: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ) ).
% P.coinitialE
thf(fact_656_P_Ocoinitial__iff,axiom,
! [T2: list_a,T5: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T5 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T5 )
& ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) ) ) ) ).
% P.coinitial_iff
thf(fact_657_N_OCong__class__eqI_H,axiom,
! [T9: set_a,U9: set_a] :
( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T9 )
=> ( ( normal8595587647932138008lass_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ U9 )
=> ( ( ( inf_inf_set_a @ T9 @ U9 )
!= bot_bot_set_a )
=> ( T9 = U9 ) ) ) ) ).
% N.Cong_class_eqI'
thf(fact_658_P_Otargets__eqI,axiom,
! [T2: list_a,T5: list_a] :
( ( ( inf_inf_set_list_a @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) )
!= bot_bot_set_list_a )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) ) ) ).
% P.targets_eqI
thf(fact_659_R_Ocoinitial__def,axiom,
! [T2: a,U2: a] :
( ( coinitial_a @ resid @ T2 @ U2 )
= ( ( inf_inf_set_a @ ( sources_a @ resid @ T2 ) @ ( sources_a @ resid @ U2 ) )
!= bot_bot_set_a ) ) ).
% R.coinitial_def
thf(fact_660_R_Ocoterminal__def,axiom,
! [T2: a,U2: a] :
( ( coterminal_a @ resid @ T2 @ U2 )
= ( ( inf_inf_set_a @ ( targets_a @ resid @ T2 ) @ ( targets_a @ resid @ U2 ) )
!= bot_bot_set_a ) ) ).
% R.coterminal_def
thf(fact_661_P_Ocoherent,axiom,
! [T2: list_a,U2: list_a,U4: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ U4 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U4 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U4 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ) ) ) ).
% P.coherent
thf(fact_662_P_Ocoherent_H,axiom,
! [V: list_a,V5: list_a,W3: list_a,W4: list_a,T2: list_a,T5: list_a] :
( ( member_list_a @ V @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ V5 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ W3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( member_list_a @ W4 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ W3 ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V5 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ W4 ) )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ W3 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ W4 ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ V ) @ ( paths_in_Resid_a @ resid @ T5 @ V5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ V5 ) @ ( paths_in_Resid_a @ resid @ T2 @ V ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ W3 ) @ ( paths_in_Resid_a @ resid @ T5 @ W4 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T5 @ W4 ) @ ( paths_in_Resid_a @ resid @ T2 @ W3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% P.coherent'
thf(fact_663_P_Oin__targets__respects__Cong,axiom,
! [T2: list_a,T5: list_a,B: list_a,B2: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ B @ B2 ) ) ) ) ).
% P.in_targets_respects_Cong
thf(fact_664_P_Otargets__are__Cong,axiom,
! [B: list_a,T2: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ B @ B2 ) ) ) ).
% P.targets_are_Cong
thf(fact_665_P_OCong__imp__arr_I2_J,axiom,
! [T2: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T5 ) ) ).
% P.Cong_imp_arr(2)
thf(fact_666_P_OCong__imp__arr_I1_J,axiom,
! [T2: list_a,T5: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T5 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ).
% P.Cong_imp_arr(1)
thf(fact_667_P_OCong__reflexive,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T2 ) ) ).
% P.Cong_reflexive
thf(fact_668_P_Oarr__in__Cong__class,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( member_list_a @ T2 @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) ) ) ).
% P.arr_in_Cong_class
thf(fact_669_P_OCong__class__memb__is__arr,axiom,
! [T9: set_list_a,T2: list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 )
=> ( ( member_list_a @ T2 @ T9 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ).
% P.Cong_class_memb_is_arr
thf(fact_670_target__is__ide,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( member_set_list_a @ A @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A ) ) ).
% target_is_ide
thf(fact_671_targets__are__cong,axiom,
! [B: set_list_a,T2: set_list_a,B2: set_list_a] :
( ( member_set_list_a @ B @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( member_set_list_a @ B2 @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ B @ B2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ B2 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_672_targets__cong__closed,axiom,
! [B: set_list_a,T2: set_list_a,B2: set_list_a] :
( ( member_set_list_a @ B @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ B @ B2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ B2 @ B ) ) )
=> ( member_set_list_a @ B2 @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ) ).
% targets_cong_closed
thf(fact_673_targets__are__con,axiom,
! [B: set_list_a,T2: set_list_a,B2: set_list_a] :
( ( member_set_list_a @ B @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( member_set_list_a @ B2 @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ B @ B2 ) ) ) ).
% targets_are_con
thf(fact_674_targets__resid__sym,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ) ) ).
% targets_resid_sym
thf(fact_675_arr__has__un__target,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ? [X: set_list_a] :
( ( member_set_list_a @ X @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
& ! [Y: set_list_a] :
( ( member_set_list_a @ Y @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( Y = X ) ) ) ) ).
% arr_has_un_target
thf(fact_676_resid__source__in__targets,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( member_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ T2 ) @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% resid_source_in_targets
thf(fact_677_P_Ocoterminal__def,axiom,
! [T2: list_a,U2: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( ( inf_inf_set_list_a @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
!= bot_bot_set_list_a ) ) ).
% P.coterminal_def
thf(fact_678_P_OCong_H_Ointros_I4_J,axiom,
! [T2: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ) ) ).
% P.Cong'.intros(4)
thf(fact_679_P_OCong_H_Ocases,axiom,
! [A1: list_a,A22: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A1 @ A22 )
=> ( ~ ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A22 @ A1 )
=> ( ! [U7: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A1 @ U7 )
=> ~ ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U7 @ A22 ) )
=> ( ~ ( ( member_list_a @ ( paths_in_Resid_a @ resid @ A1 @ A22 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ A22 @ A1 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
=> ~ ! [U7: list_a] :
( ( A22
= ( paths_in_Resid_a @ resid @ A1 @ U7 ) )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ A1 )
=> ( ( member_list_a @ U7 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ A1 )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U7 ) ) ) ) ) ) ) ) ) ).
% P.Cong'.cases
thf(fact_680_P_OCong_H_Osimps,axiom,
! [A1: list_a,A22: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A1 @ A22 )
= ( ? [T8: list_a,U5: list_a] :
( ( A1 = U5 )
& ( A22 = T8 )
& ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T8 @ U5 ) )
| ? [T8: list_a,U5: list_a,V6: list_a] :
( ( A1 = T8 )
& ( A22 = V6 )
& ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T8 @ U5 )
& ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U5 @ V6 ) )
| ? [T8: list_a,U5: list_a] :
( ( A1 = T8 )
& ( A22 = U5 )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T8 @ U5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U5 @ T8 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
| ? [T8: list_a,U5: list_a] :
( ( A1 = T8 )
& ( A22
= ( paths_in_Resid_a @ resid @ T8 @ U5 ) )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T8 )
& ( member_list_a @ U5 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) )
& ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T8 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U5 ) ) ) ) ) ).
% P.Cong'.simps
thf(fact_681_P_Ois__Cong__classI,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) ) ) ).
% P.is_Cong_classI
thf(fact_682_targets__con__closed,axiom,
! [B: set_list_a,T2: set_list_a,B2: set_list_a] :
( ( member_set_list_a @ B @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ B2 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ B @ B2 )
=> ( member_set_list_a @ B2 @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ) ) ).
% targets_con_closed
thf(fact_683_cong__char,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
& ( T2 = U2 ) ) ) ).
% cong_char
thf(fact_684_cong__reflexive,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T2 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T2 ) ) ) ) ).
% cong_reflexive
thf(fact_685_ide__implies__arr,axiom,
! [A: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A ) ) ).
% ide_implies_arr
thf(fact_686_prfx__reflexive,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T2 ) ) ) ).
% prfx_reflexive
thf(fact_687_ide__char_H,axiom,
! [A5: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A5 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A5 )
& ( ord_le8861187494160871172list_a @ A5 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% ide_char'
thf(fact_688_con__implies__arr_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ).
% con_implies_arr(2)
thf(fact_689_con__implies__arr_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% con_implies_arr(1)
thf(fact_690_arrE,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T2 ) ) ).
% arrE
thf(fact_691_arr__def,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T2 ) ) ).
% arr_def
thf(fact_692_arr__resid,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ).
% arr_resid
thf(fact_693_arr__resid__iff__con,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ).
% arr_resid_iff_con
thf(fact_694_arr__has__un__source,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ? [X: set_list_a] :
( ( member_set_list_a @ X @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
& ! [Y: set_list_a] :
( ( member_set_list_a @ Y @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( Y = X ) ) ) ) ).
% arr_has_un_source
thf(fact_695_arr__char,axiom,
! [T9: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T9 )
= ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 ) ) ).
% arr_char
thf(fact_696_coterminalE,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ~ ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ) ).
% coterminalE
thf(fact_697_coterminal__iff,axiom,
! [T2: set_list_a,T5: set_list_a] :
( ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T5 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
& ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 )
& ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 ) ) ) ) ).
% coterminal_iff
thf(fact_698_arr__iff__has__target,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= bot_bo3186585308812441520list_a ) ) ).
% arr_iff_has_target
thf(fact_699_empty__subsetI,axiom,
! [A2: set_set_list_a] : ( ord_le8877086941679407844list_a @ bot_bo3186585308812441520list_a @ A2 ) ).
% empty_subsetI
thf(fact_700_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_701_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_702_subset__empty,axiom,
! [A2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A2 @ bot_bo3186585308812441520list_a )
= ( A2 = bot_bo3186585308812441520list_a ) ) ).
% subset_empty
thf(fact_703_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_704_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_705_targets__eqI,axiom,
! [T2: set_list_a,T5: set_list_a] :
( ( ( inf_in4657809108759609906list_a @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 ) )
!= bot_bo3186585308812441520list_a )
=> ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 ) ) ) ).
% targets_eqI
thf(fact_706_coinitialE,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ~ ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ) ).
% coinitialE
thf(fact_707_coinitial__iff,axiom,
! [T2: set_list_a,T5: set_list_a] :
( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T5 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
& ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 )
& ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 ) ) ) ) ).
% coinitial_iff
thf(fact_708_seq__def,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
& ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
& ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ).
% seq_def
thf(fact_709_seqE,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ~ ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ) ).
% seqE
thf(fact_710_arr__iff__has__source,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= bot_bo3186585308812441520list_a ) ) ).
% arr_iff_has_source
thf(fact_711_sources__eqI,axiom,
! [T2: set_list_a,T5: set_list_a] :
( ( ( inf_in4657809108759609906list_a @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 ) )
!= bot_bo3186585308812441520list_a )
=> ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 ) ) ) ).
% sources_eqI
thf(fact_712_coterminal__def,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( inf_in4657809108759609906list_a @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
!= bot_bo3186585308812441520list_a ) ) ).
% coterminal_def
thf(fact_713_P_OarrI,axiom,
! [T2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ).
% P.arrI
thf(fact_714_P_OarrI_092_060_094sub_062P,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ).
% P.arrI\<^sub>P
thf(fact_715_P_OcoterminalI,axiom,
! [T2: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ) ).
% P.coterminalI
thf(fact_716_P_Osources__resid,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ).
% P.sources_resid
thf(fact_717_P_OcoinitialI,axiom,
! [T2: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ) ).
% P.coinitialI
thf(fact_718_arrI,axiom,
! [T2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T2 )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% arrI
thf(fact_719_sources__resid,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ).
% sources_resid
thf(fact_720_quot_Opreserves__reflects__arr,axiom,
! [T2: list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) )
= ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ).
% quot.preserves_reflects_arr
thf(fact_721_coterminalI,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
=> ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% coterminalI
thf(fact_722_coinitialI,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
=> ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% coinitialI
thf(fact_723_seqI,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
=> ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% seqI
thf(fact_724_rts_Oseq_Ocong,axiom,
seq_set_list_a = seq_set_list_a ).
% rts.seq.cong
thf(fact_725_rts_Oseq_Ocong,axiom,
seq_a = seq_a ).
% rts.seq.cong
thf(fact_726_rts_Oseq_Ocong,axiom,
seq_list_a = seq_list_a ).
% rts.seq.cong
thf(fact_727_quotient__by__coherent__normal_Oarr__char,axiom,
! [Resid: a > a > a,NN: set_a,T9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T9 )
= ( normal8595587647932138008lass_a @ Resid @ NN @ T9 ) ) ) ).
% quotient_by_coherent_normal.arr_char
thf(fact_728_quotient__by__coherent__normal_Oarr__char,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) @ T9 )
= ( normal4947954984078133662list_a @ Resid @ NN @ T9 ) ) ) ).
% quotient_by_coherent_normal.arr_char
thf(fact_729_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_730_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_731_ex__in__conv,axiom,
! [A2: set_set_list_a] :
( ( ? [X2: set_list_a] : ( member_set_list_a @ X2 @ A2 ) )
= ( A2 != bot_bo3186585308812441520list_a ) ) ).
% ex_in_conv
thf(fact_732_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_733_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_734_equals0I,axiom,
! [A2: set_set_list_a] :
( ! [Y3: set_list_a] :
~ ( member_set_list_a @ Y3 @ A2 )
=> ( A2 = bot_bo3186585308812441520list_a ) ) ).
% equals0I
thf(fact_735_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_736_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_737_equals0D,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( A2 = bot_bo3186585308812441520list_a )
=> ~ ( member_set_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_738_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_739_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_740_emptyE,axiom,
! [A: set_list_a] :
~ ( member_set_list_a @ A @ bot_bo3186585308812441520list_a ) ).
% emptyE
thf(fact_741_Collect__mono__iff,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) )
= ( ! [X2: set_list_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_742_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_743_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X2: list_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_744_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A6 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_745_set__eq__subset,axiom,
( ( ^ [Y5: set_list_a,Z: set_list_a] : ( Y5 = Z ) )
= ( ^ [A6: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_746_subset__trans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_747_subset__trans,axiom,
! [A2: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ord_le8861187494160871172list_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_748_Collect__mono,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ! [X: set_list_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_749_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_750_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_751_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_752_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_753_subset__iff,axiom,
( ord_le8877086941679407844list_a
= ( ^ [A6: set_set_list_a,B4: set_set_list_a] :
! [T8: set_list_a] :
( ( member_set_list_a @ T8 @ A6 )
=> ( member_set_list_a @ T8 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_754_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
! [T8: a] :
( ( member_a @ T8 @ A6 )
=> ( member_a @ T8 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_755_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B4: set_list_a] :
! [T8: list_a] :
( ( member_list_a @ T8 @ A6 )
=> ( member_list_a @ T8 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_756_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_757_equalityD2,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_758_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_759_equalityD1,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_760_subset__eq,axiom,
( ord_le8877086941679407844list_a
= ( ^ [A6: set_set_list_a,B4: set_set_list_a] :
! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A6 )
=> ( member_set_list_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_761_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_762_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B4: set_list_a] :
! [X2: list_a] :
( ( member_list_a @ X2 @ A6 )
=> ( member_list_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_763_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_764_equalityE,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ~ ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_765_subsetD,axiom,
! [A2: set_set_list_a,B3: set_set_list_a,C: set_list_a] :
( ( ord_le8877086941679407844list_a @ A2 @ B3 )
=> ( ( member_set_list_a @ C @ A2 )
=> ( member_set_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_766_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_767_subsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_768_in__mono,axiom,
! [A2: set_set_list_a,B3: set_set_list_a,X4: set_list_a] :
( ( ord_le8877086941679407844list_a @ A2 @ B3 )
=> ( ( member_set_list_a @ X4 @ A2 )
=> ( member_set_list_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_769_in__mono,axiom,
! [A2: set_a,B3: set_a,X4: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_770_in__mono,axiom,
! [A2: set_list_a,B3: set_list_a,X4: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ X4 @ A2 )
=> ( member_list_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_771_Int__left__commute,axiom,
! [A2: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C2 ) )
= ( inf_inf_set_list_a @ B3 @ ( inf_inf_set_list_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_772_Int__left__commute,axiom,
! [A2: set_set_list_a,B3: set_set_list_a,C2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A2 @ ( inf_in4657809108759609906list_a @ B3 @ C2 ) )
= ( inf_in4657809108759609906list_a @ B3 @ ( inf_in4657809108759609906list_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_773_Int__left__commute,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_774_Int__left__absorb,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_775_Int__left__absorb,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A2 @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) )
= ( inf_in4657809108759609906list_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_776_Int__left__absorb,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_777_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B4: set_list_a] : ( inf_inf_set_list_a @ B4 @ A6 ) ) ) ).
% Int_commute
thf(fact_778_Int__commute,axiom,
( inf_in4657809108759609906list_a
= ( ^ [A6: set_set_list_a,B4: set_set_list_a] : ( inf_in4657809108759609906list_a @ B4 @ A6 ) ) ) ).
% Int_commute
thf(fact_779_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A6 ) ) ) ).
% Int_commute
thf(fact_780_Int__absorb,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_781_Int__absorb,axiom,
! [A2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_782_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_783_Int__assoc,axiom,
! [A2: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ C2 )
= ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_784_Int__assoc,axiom,
! [A2: set_set_list_a,B3: set_set_list_a,C2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) @ C2 )
= ( inf_in4657809108759609906list_a @ A2 @ ( inf_in4657809108759609906list_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_785_Int__assoc,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_786_IntD2,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ B3 ) ) ).
% IntD2
thf(fact_787_IntD2,axiom,
! [C: set_list_a,A2: set_set_list_a,B3: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) )
=> ( member_set_list_a @ C @ B3 ) ) ).
% IntD2
thf(fact_788_IntD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_789_IntD1,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% IntD1
thf(fact_790_IntD1,axiom,
! [C: set_list_a,A2: set_set_list_a,B3: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) )
=> ( member_set_list_a @ C @ A2 ) ) ).
% IntD1
thf(fact_791_IntD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_792_IntE,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ~ ( member_list_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_793_IntE,axiom,
! [C: set_list_a,A2: set_set_list_a,B3: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) )
=> ~ ( ( member_set_list_a @ C @ A2 )
=> ~ ( member_set_list_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_794_IntE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_795_quotient__by__coherent__normal_Oide__char_H,axiom,
! [Resid: a > a > a,NN: set_a,A5: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ A5 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ A5 )
& ( ord_less_eq_set_a @ A5 @ NN ) ) ) ) ).
% quotient_by_coherent_normal.ide_char'
thf(fact_796_quotient__by__coherent__normal_Oide__char_H,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,A5: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) @ A5 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) @ A5 )
& ( ord_le8861187494160871172list_a @ A5 @ NN ) ) ) ) ).
% quotient_by_coherent_normal.ide_char'
thf(fact_797_quotient__by__coherent__normal_Oide__char,axiom,
! [Resid: a > a > a,NN: set_a,U9: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ U9 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ U9 )
& ( ( inf_inf_set_a @ U9 @ NN )
!= bot_bot_set_a ) ) ) ) ).
% quotient_by_coherent_normal.ide_char
thf(fact_798_quotient__by__coherent__normal_Oide__char,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,NN: set_set_list_a,U9: set_set_list_a] :
( ( quotie4778059898687049596list_a @ Resid @ NN )
=> ( ( ide_set_set_list_a @ ( quotie750419831440180831list_a @ Resid @ NN ) @ U9 )
= ( ( arr_set_set_list_a @ ( quotie750419831440180831list_a @ Resid @ NN ) @ U9 )
& ( ( inf_in4657809108759609906list_a @ U9 @ NN )
!= bot_bo3186585308812441520list_a ) ) ) ) ).
% quotient_by_coherent_normal.ide_char
thf(fact_799_quotient__by__coherent__normal_Oide__char,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U9: set_list_a] :
( ( quotie5013437406004430748list_a @ Resid @ NN )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) @ U9 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ Resid @ NN ) @ U9 )
& ( ( inf_inf_set_list_a @ U9 @ NN )
!= bot_bot_set_list_a ) ) ) ) ).
% quotient_by_coherent_normal.ide_char
thf(fact_800_Int__emptyI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ( member_list_a @ X @ B3 ) )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_801_Int__emptyI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ~ ( member_a @ X @ B3 ) )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_802_Int__emptyI,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ! [X: set_list_a] :
( ( member_set_list_a @ X @ A2 )
=> ~ ( member_set_list_a @ X @ B3 ) )
=> ( ( inf_in4657809108759609906list_a @ A2 @ B3 )
= bot_bo3186585308812441520list_a ) ) ).
% Int_emptyI
thf(fact_803_disjoint__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ~ ( member_list_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_804_disjoint__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_805_disjoint__iff,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A2 @ B3 )
= bot_bo3186585308812441520list_a )
= ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
=> ~ ( member_set_list_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_806_Int__empty__left,axiom,
! [B3: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B3 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_807_Int__empty__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_808_Int__empty__left,axiom,
! [B3: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ B3 )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_left
thf(fact_809_Int__empty__right,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_810_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_811_Int__empty__right,axiom,
! [A2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A2 @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_right
thf(fact_812_disjoint__iff__not__equal,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_813_disjoint__iff__not__equal,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_814_disjoint__iff__not__equal,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A2 @ B3 )
= bot_bo3186585308812441520list_a )
= ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
=> ! [Y2: set_list_a] :
( ( member_set_list_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_815_Int__mono,axiom,
! [A2: set_set_list_a,C2: set_set_list_a,B3: set_set_list_a,D: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A2 @ C2 )
=> ( ( ord_le8877086941679407844list_a @ B3 @ D )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) @ ( inf_in4657809108759609906list_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_816_Int__mono,axiom,
! [A2: set_a,C2: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_817_Int__mono,axiom,
! [A2: set_list_a,C2: set_list_a,B3: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ ( inf_inf_set_list_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_818_Int__lower1,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_819_Int__lower1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_820_Int__lower1,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_821_Int__lower2,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_822_Int__lower2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_823_Int__lower2,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_824_Int__absorb1,axiom,
! [B3: set_set_list_a,A2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ B3 @ A2 )
=> ( ( inf_in4657809108759609906list_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_825_Int__absorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_826_Int__absorb1,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_827_Int__absorb2,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A2 @ B3 )
=> ( ( inf_in4657809108759609906list_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_828_Int__absorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_829_Int__absorb2,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_830_Int__greatest,axiom,
! [C2: set_set_list_a,A2: set_set_list_a,B3: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ C2 @ A2 )
=> ( ( ord_le8877086941679407844list_a @ C2 @ B3 )
=> ( ord_le8877086941679407844list_a @ C2 @ ( inf_in4657809108759609906list_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_831_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_832_Int__greatest,axiom,
! [C2: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C2 @ B3 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_833_Int__Collect__mono,axiom,
! [A2: set_set_list_a,B3: set_set_list_a,P: set_list_a > $o,Q: set_list_a > $o] :
( ( ord_le8877086941679407844list_a @ A2 @ B3 )
=> ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A2 @ ( collect_set_list_a @ P ) ) @ ( inf_in4657809108759609906list_a @ B3 @ ( collect_set_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_834_Int__Collect__mono,axiom,
! [A2: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_835_Int__Collect__mono,axiom,
! [A2: set_list_a,B3: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B3 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_836_composable__imp__seq,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ).
% composable_imp_seq
thf(fact_837_composableD_I3_J,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ).
% composableD(3)
thf(fact_838_composableD_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ).
% composableD(2)
thf(fact_839_composableD_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% composableD(1)
thf(fact_840_R_Ocong__respects__seq,axiom,
! [T2: a,U2: a,T5: a,U4: a] :
( ( seq_a @ resid @ T2 @ U2 )
=> ( ( ( ide_a @ resid @ ( resid @ T2 @ T5 ) )
& ( ide_a @ resid @ ( resid @ T5 @ T2 ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( seq_a @ resid @ T5 @ U4 ) ) ) ) ).
% R.cong_respects_seq
thf(fact_841_P_Ocong__respects__seq_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a,T6: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ U ) ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U3 ) ) ) ) ).
% P.cong_respects_seq\<^sub>P
thf(fact_842_P_Ocong__respects__seq,axiom,
! [T2: list_a,U2: list_a,T5: list_a,U4: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ T5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T5 @ T2 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T5 @ U4 ) ) ) ) ).
% P.cong_respects_seq
thf(fact_843_R_OseqE,axiom,
! [T2: a,U2: a] :
( ( seq_a @ resid @ T2 @ U2 )
=> ~ ( ( arr_a @ resid @ T2 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( targets_a @ resid @ T2 )
!= ( sources_a @ resid @ U2 ) ) ) ) ) ).
% R.seqE
thf(fact_844_R_Oseq__def,axiom,
! [T2: a,U2: a] :
( ( seq_a @ resid @ T2 @ U2 )
= ( ( arr_a @ resid @ T2 )
& ( arr_a @ resid @ U2 )
& ( ( targets_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) ) ) ) ).
% R.seq_def
thf(fact_845_P_Oarr__append__imp__seq,axiom,
! [T: list_a,U: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T @ U ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ) ) ).
% P.arr_append_imp_seq
thf(fact_846_P_Oseq__char,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( paths_in_Arr_a @ resid @ T )
& ( paths_in_Arr_a @ resid @ U )
& ( ( paths_in_Trgs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ U ) ) ) ) ).
% P.seq_char
thf(fact_847_P_OseqE,axiom,
! [T2: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ) ).
% P.seqE
thf(fact_848_P_Oseq__def,axiom,
! [T2: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
& ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ).
% P.seq_def
thf(fact_849_P_Oseq__char_H,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( paths_in_Arr_a @ resid @ T )
& ( paths_in_Arr_a @ resid @ U )
& ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) )
!= bot_bot_set_a ) ) ) ).
% P.seq_char'
thf(fact_850_P_Ocong__append,axiom,
! [T: list_a,U: list_a,T6: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ U ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ ( append_a @ T6 @ U3 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T6 @ U3 ) @ ( append_a @ T @ U ) ) ) ) ) ) ) ).
% P.cong_append
thf(fact_851_P_Ocong__append__ideI_I1_J,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ ( append_a @ T @ U ) ) ) ) ) ) ).
% P.cong_append_ideI(1)
thf(fact_852_P_Ocong__append__ideI_I2_J,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ T ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ ( append_a @ T @ U ) ) ) ) ) ) ).
% P.cong_append_ideI(2)
thf(fact_853_P_Ocong__append__ideI_I3_J,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ ( append_a @ T @ U ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ U ) ) ) ) ) ).
% P.cong_append_ideI(3)
thf(fact_854_P_Ocong__append__ideI_I4_J,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ ( append_a @ T @ U ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T @ U ) @ T ) ) ) ) ) ).
% P.cong_append_ideI(4)
thf(fact_855_P_Osources__append,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T @ U ) )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ).
% P.sources_append
thf(fact_856_P_Oarr__append,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T @ U ) ) ) ).
% P.arr_append
thf(fact_857_P_Otargets__append,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T @ U ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ).
% P.targets_append
thf(fact_858_R_OseqI,axiom,
! [T2: a,U2: a] :
( ( arr_a @ resid @ T2 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( ( targets_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) )
=> ( seq_a @ resid @ T2 @ U2 ) ) ) ) ).
% R.seqI
thf(fact_859_P_OseqI,axiom,
! [T2: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ) ) ).
% P.seqI
thf(fact_860_P_Ocong__cons,axiom,
! [T2: a,U: list_a,T5: a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T2 @ nil_a ) @ U )
=> ( ( ( ide_a @ resid @ ( resid @ T2 @ T5 ) )
& ( ide_a @ resid @ ( resid @ T5 @ T2 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ U ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ U ) @ ( cons_a @ T5 @ U3 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T5 @ U3 ) @ ( cons_a @ T2 @ U ) ) ) ) ) ) ) ).
% P.cong_cons
thf(fact_861_P_Ocong__cons__ideI_I1_J,axiom,
! [T2: a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T2 @ nil_a ) @ U )
=> ( ( ide_a @ resid @ T2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ U ) @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ T2 @ U ) ) ) ) ) ) ).
% P.cong_cons_ideI(1)
thf(fact_862_P_Ocong__cons__ideI_I2_J,axiom,
! [T2: a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T2 @ nil_a ) @ U )
=> ( ( ide_a @ resid @ T2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ T2 @ U ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ U ) @ U ) ) ) ) ) ).
% P.cong_cons_ideI(2)
thf(fact_863_P_OseqI_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Arr_a @ resid @ U )
=> ( ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) )
!= bot_bot_set_a )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ) ) ).
% P.seqI\<^sub>P
thf(fact_864_rts_Ocomposable_Ocong,axiom,
compos1473025410594674033list_a = compos1473025410594674033list_a ).
% rts.composable.cong
thf(fact_865_rts_Ocomposable_Ocong,axiom,
composable_list_a = composable_list_a ).
% rts.composable.cong
thf(fact_866_rts_Ocomposable_Ocong,axiom,
composable_a = composable_a ).
% rts.composable.cong
thf(fact_867_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,G: set_list_a,F: set_list_a] :
( ( rts_wi8505854437279755228list_a @ Resid )
=> ( ( compos1473025410594674033list_a @ Resid @ G @ F )
= ( seq_set_list_a @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_868_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: a > a > a,G: a,F: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( composable_a @ Resid @ G @ F )
= ( seq_a @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_869_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: list_a > list_a > list_a,G: list_a,F: list_a] :
( ( rts_wi1113109597627081724list_a @ Resid )
=> ( ( composable_list_a @ Resid @ G @ F )
= ( seq_list_a @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_870_rts__with__composites_Ohas__composites,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a,U2: set_list_a] :
( ( rts_wi8505854437279755228list_a @ Resid )
=> ( ( seq_set_list_a @ Resid @ T2 @ U2 )
=> ( compos1473025410594674033list_a @ Resid @ T2 @ U2 ) ) ) ).
% rts_with_composites.has_composites
thf(fact_871_rts__with__composites_Ohas__composites,axiom,
! [Resid: a > a > a,T2: a,U2: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( seq_a @ Resid @ T2 @ U2 )
=> ( composable_a @ Resid @ T2 @ U2 ) ) ) ).
% rts_with_composites.has_composites
thf(fact_872_rts__with__composites_Ohas__composites,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a,U2: list_a] :
( ( rts_wi1113109597627081724list_a @ Resid )
=> ( ( seq_list_a @ Resid @ T2 @ U2 )
=> ( composable_list_a @ Resid @ T2 @ U2 ) ) ) ).
% rts_with_composites.has_composites
thf(fact_873_P_OcomposableD_I3_J,axiom,
! [T2: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ).
% P.composableD(3)
thf(fact_874_P_OPCong_Osimps,axiom,
! [A1: list_a,A22: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ A1 @ A22 )
= ( ? [T12: list_a] :
( ( A1 = T12 )
& ( A22 = T12 )
& ( paths_in_Arr_a @ resid @ T12 ) )
| ? [T12: list_a,U10: list_a] :
( ( A1 = U10 )
& ( A22 = T12 )
& ( paths_4972237869229803137Cong_a @ resid @ T12 @ U10 ) )
| ? [T12: list_a,U10: list_a,V7: list_a] :
( ( A1 = T12 )
& ( A22 = V7 )
& ( paths_4972237869229803137Cong_a @ resid @ T12 @ U10 )
& ( paths_4972237869229803137Cong_a @ resid @ U10 @ V7 ) )
| ? [T12: list_a,U10: list_a,T13: list_a,U11: list_a] :
( ( A1
= ( append_a @ T12 @ U10 ) )
& ( A22
= ( append_a @ T13 @ U11 ) )
& ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T12 @ U10 )
& ( paths_4972237869229803137Cong_a @ resid @ T12 @ T13 )
& ( paths_4972237869229803137Cong_a @ resid @ U10 @ U11 ) )
| ? [T12: list_a,B5: a] :
( ( A1
= ( append_a @ T12 @ ( cons_a @ B5 @ nil_a ) ) )
& ( A22 = T12 )
& ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T12 @ ( cons_a @ B5 @ nil_a ) )
& ( ide_a @ resid @ B5 ) )
| ? [T8: a,U5: a] :
( ( A1
= ( cons_a @ T8 @ ( cons_a @ ( resid @ U5 @ T8 ) @ nil_a ) ) )
& ( A22
= ( cons_a @ U5 @ ( cons_a @ ( resid @ T8 @ U5 ) @ nil_a ) ) )
& ( con_a @ resid @ T8 @ U5 ) ) ) ) ).
% P.PCong.simps
thf(fact_875_R_OcomposableD_I2_J,axiom,
! [T2: a,U2: a] :
( ( composable_a @ resid @ T2 @ U2 )
=> ( arr_a @ resid @ U2 ) ) ).
% R.composableD(2)
thf(fact_876_R_OcomposableD_I1_J,axiom,
! [T2: a,U2: a] :
( ( composable_a @ resid @ T2 @ U2 )
=> ( arr_a @ resid @ T2 ) ) ).
% R.composableD(1)
thf(fact_877_P_OPCong_Ointros_I3_J,axiom,
! [T: list_a,U: list_a,V2: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ T @ U )
=> ( ( paths_4972237869229803137Cong_a @ resid @ U @ V2 )
=> ( paths_4972237869229803137Cong_a @ resid @ T @ V2 ) ) ) ).
% P.PCong.intros(3)
thf(fact_878_P_OPCong_Ointros_I2_J,axiom,
! [T: list_a,U: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ T @ U )
=> ( paths_4972237869229803137Cong_a @ resid @ U @ T ) ) ).
% P.PCong.intros(2)
thf(fact_879_R_Ocomposable__imp__seq,axiom,
! [T2: a,U2: a] :
( ( composable_a @ resid @ T2 @ U2 )
=> ( seq_a @ resid @ T2 @ U2 ) ) ).
% R.composable_imp_seq
thf(fact_880_R_OcomposableD_I3_J,axiom,
! [T2: a,U2: a] :
( ( composable_a @ resid @ T2 @ U2 )
=> ( ( targets_a @ resid @ T2 )
= ( sources_a @ resid @ U2 ) ) ) ).
% R.composableD(3)
thf(fact_881_P_OPCong_Ointros_I1_J,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( paths_4972237869229803137Cong_a @ resid @ T @ T ) ) ).
% P.PCong.intros(1)
thf(fact_882_P_OCong__iff__PCong,axiom,
! [T: list_a,U: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) )
= ( paths_4972237869229803137Cong_a @ resid @ T @ U ) ) ).
% P.Cong_iff_PCong
thf(fact_883_P_OCong__imp__PCong,axiom,
! [T: list_a,U: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) )
=> ( paths_4972237869229803137Cong_a @ resid @ T @ U ) ) ).
% P.Cong_imp_PCong
thf(fact_884_P_OPCong__imp__Cong,axiom,
! [T: list_a,U: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ T @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) ) ) ).
% P.PCong_imp_Cong
thf(fact_885_P_OcomposableD_I2_J,axiom,
! [T2: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ).
% P.composableD(2)
thf(fact_886_P_OcomposableD_I1_J,axiom,
! [T2: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ).
% P.composableD(1)
thf(fact_887_P_Ocomposable__iff__seq,axiom,
! [G: list_a,F: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ G @ F )
= ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ G @ F ) ) ).
% P.composable_iff_seq
thf(fact_888_P_Ocomposable__imp__seq,axiom,
! [T2: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.composable_imp_seq
thf(fact_889_P_Ohas__composites,axiom,
! [T2: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.has_composites
thf(fact_890_P_OPCong__permute,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= nil_a )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ T @ ( paths_in_Resid_a @ resid @ U @ T ) ) @ ( append_a @ U @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ) ).
% P.PCong_permute
thf(fact_891_P_OPCong_Ointros_I6_J,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( paths_4972237869229803137Cong_a @ resid @ ( cons_a @ T2 @ ( cons_a @ ( resid @ U2 @ T2 ) @ nil_a ) ) @ ( cons_a @ U2 @ ( cons_a @ ( resid @ T2 @ U2 ) @ nil_a ) ) ) ) ).
% P.PCong.intros(6)
thf(fact_892_P_OPCong_Ointros_I4_J,axiom,
! [T: list_a,U: list_a,T6: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( paths_4972237869229803137Cong_a @ resid @ T @ T6 )
=> ( ( paths_4972237869229803137Cong_a @ resid @ U @ U3 )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ T @ U ) @ ( append_a @ T6 @ U3 ) ) ) ) ) ).
% P.PCong.intros(4)
thf(fact_893_P_OPCong__permute__single,axiom,
! [T2: a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U )
!= nil_a )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ ( cons_a @ T2 @ nil_a ) @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ T2 @ nil_a ) ) ) @ ( append_a @ U @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U ) ) ) ) ).
% P.PCong_permute_single
thf(fact_894_P_OPCong_Ointros_I5_J,axiom,
! [T: list_a,B: a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ ( cons_a @ B @ nil_a ) )
=> ( ( ide_a @ resid @ B )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ T @ ( cons_a @ B @ nil_a ) ) @ T ) ) ) ).
% P.PCong.intros(5)
thf(fact_895_P_OPCong_Ocases,axiom,
! [A1: list_a,A22: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ A1 @ A22 )
=> ( ( ( A22 = A1 )
=> ~ ( paths_in_Arr_a @ resid @ A1 ) )
=> ( ~ ( paths_4972237869229803137Cong_a @ resid @ A22 @ A1 )
=> ( ! [U12: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ A1 @ U12 )
=> ~ ( paths_4972237869229803137Cong_a @ resid @ U12 @ A22 ) )
=> ( ! [T14: list_a,U12: list_a] :
( ( A1
= ( append_a @ T14 @ U12 ) )
=> ! [T15: list_a,U13: list_a] :
( ( A22
= ( append_a @ T15 @ U13 ) )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T14 @ U12 )
=> ( ( paths_4972237869229803137Cong_a @ resid @ T14 @ T15 )
=> ~ ( paths_4972237869229803137Cong_a @ resid @ U12 @ U13 ) ) ) ) )
=> ( ! [T14: list_a,B6: a] :
( ( A1
= ( append_a @ T14 @ ( cons_a @ B6 @ nil_a ) ) )
=> ( ( A22 = T14 )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T14 @ ( cons_a @ B6 @ nil_a ) )
=> ~ ( ide_a @ resid @ B6 ) ) ) )
=> ~ ! [T4: a,U7: a] :
( ( A1
= ( cons_a @ T4 @ ( cons_a @ ( resid @ U7 @ T4 ) @ nil_a ) ) )
=> ( ( A22
= ( cons_a @ U7 @ ( cons_a @ ( resid @ T4 @ U7 ) @ nil_a ) ) )
=> ~ ( con_a @ resid @ T4 @ U7 ) ) ) ) ) ) ) ) ) ).
% P.PCong.cases
thf(fact_896_paths__in__rts__with__coherent__normal_OPCong_Ocong,axiom,
paths_4972237869229803137Cong_a = paths_4972237869229803137Cong_a ).
% paths_in_rts_with_coherent_normal.PCong.cong
thf(fact_897_join__of__arr__src_I2_J,axiom,
! [T2: set_list_a,A: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ A @ T2 ) ) ) ).
% join_of_arr_src(2)
thf(fact_898_join__of__arr__src_I1_J,axiom,
! [T2: set_list_a,A: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( member_set_list_a @ A @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ T2 @ T2 ) ) ) ).
% join_of_arr_src(1)
thf(fact_899_joinable__iff__arr__join,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% joinable_iff_arr_join
thf(fact_900_join__of__unique,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a,V5: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V5 )
=> ( V = V5 ) ) ) ).
% join_of_unique
thf(fact_901_join__of__symmetric,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ T2 @ V ) ) ).
% join_of_symmetric
thf(fact_902_join__of__un__upto__cong,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a,V5: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V5 )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ V5 ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V5 @ V ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_903_join__eqI,axiom,
! [T2: set_list_a,V: set_list_a,U2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V ) )
=> ( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ U2 )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
=> ( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
=> ( ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= V ) ) ) ) ) ).
% join_eqI
thf(fact_904_join__of__resid,axiom,
! [T2: set_list_a,U2: set_list_a,W3: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ W3 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V @ W3 )
=> ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ W3 @ V ) ) ) ) ).
% join_of_resid
thf(fact_905_con__with__join__of__iff_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,W3: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ W3 )
=> ( ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ V )
& ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ W3 @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_906_con__with__join__of__iff_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,W3: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ W3 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ W3 @ V )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ V )
& ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_907_join__of__arr__self,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T2 @ T2 ) ) ).
% join_of_arr_self
thf(fact_908_join__self,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T2 )
= T2 ) ) ).
% join_self
thf(fact_909_sources__join__of_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% sources_join_of(1)
thf(fact_910_sources__join__of_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
= ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% sources_join_of(2)
thf(fact_911_targets__join__of_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% targets_join_of(1)
thf(fact_912_targets__join__of_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
= ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% targets_join_of(2)
thf(fact_913_joinable__def,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ? [X5: set_list_a] : ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ X5 ) ) ) ).
% joinable_def
thf(fact_914_arr__prfx__join__self,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% arr_prfx_join_self
thf(fact_915_join__is__join__of,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% join_is_join_of
thf(fact_916_resid__join_092_060_094sub_062E_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ) ) ).
% resid_join\<^sub>E(1)
thf(fact_917_resid__join_092_060_094sub_062E_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ) ) ) ).
% resid_join\<^sub>E(2)
thf(fact_918_resid__join_092_060_094sub_062E_I3_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ V )
= ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ V ) ) ) ) ) ).
% resid_join\<^sub>E(3)
thf(fact_919_rts_Ojoin__of_Ocong,axiom,
join_of_set_list_a = join_of_set_list_a ).
% rts.join_of.cong
thf(fact_920_rts_Ojoin__of_Ocong,axiom,
join_of_list_a = join_of_list_a ).
% rts.join_of.cong
thf(fact_921_rts_Ojoin__of_Ocong,axiom,
join_of_a = join_of_a ).
% rts.join_of.cong
thf(fact_922_extensional__rts_Ojoin_Ocong,axiom,
extens431601257083675878list_a = extens431601257083675878list_a ).
% extensional_rts.join.cong
thf(fact_923_quot_Opreserves__joins,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V )
=> ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V ) ) ) ).
% quot.preserves_joins
thf(fact_924_comp__join_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,U4: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U4 ) )
=> ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ U4 ) ) ) ).
% comp_join(1)
thf(fact_925_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_926_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_927_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X4 @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_928_R_Ojoin__of__symmetric,axiom,
! [T2: a,U2: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ V )
=> ( join_of_a @ resid @ U2 @ T2 @ V ) ) ).
% R.join_of_symmetric
thf(fact_929_R_Ojoin__of__un__upto__cong,axiom,
! [T2: a,U2: a,V: a,V5: a] :
( ( join_of_a @ resid @ T2 @ U2 @ V )
=> ( ( join_of_a @ resid @ T2 @ U2 @ V5 )
=> ( ( ide_a @ resid @ ( resid @ V @ V5 ) )
& ( ide_a @ resid @ ( resid @ V5 @ V ) ) ) ) ) ).
% R.join_of_un_upto_cong
thf(fact_930_R_Ocon__with__join__of__iff_I2_J,axiom,
! [T2: a,U2: a,W3: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ W3 )
=> ( ( con_a @ resid @ W3 @ V )
=> ( ( con_a @ resid @ T2 @ V )
& ( con_a @ resid @ ( resid @ V @ T2 ) @ ( resid @ U2 @ T2 ) ) ) ) ) ).
% R.con_with_join_of_iff(2)
thf(fact_931_R_Ocon__with__join__of__iff_I1_J,axiom,
! [T2: a,U2: a,W3: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ W3 )
=> ( ( ( con_a @ resid @ U2 @ V )
& ( con_a @ resid @ ( resid @ V @ U2 ) @ ( resid @ T2 @ U2 ) ) )
=> ( con_a @ resid @ W3 @ V ) ) ) ).
% R.con_with_join_of_iff(1)
thf(fact_932_R_Ojoin__of__resid,axiom,
! [T2: a,U2: a,W3: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ W3 )
=> ( ( con_a @ resid @ V @ W3 )
=> ( join_of_a @ resid @ ( resid @ T2 @ V ) @ ( resid @ U2 @ V ) @ ( resid @ W3 @ V ) ) ) ) ).
% R.join_of_resid
thf(fact_933_R_Osources__join__of_I2_J,axiom,
! [T2: a,U2: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ V )
=> ( ( sources_a @ resid @ U2 )
= ( sources_a @ resid @ V ) ) ) ).
% R.sources_join_of(2)
thf(fact_934_R_Osources__join__of_I1_J,axiom,
! [T2: a,U2: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ V )
=> ( ( sources_a @ resid @ T2 )
= ( sources_a @ resid @ V ) ) ) ).
% R.sources_join_of(1)
thf(fact_935_R_Ojoin__of__arr__self,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( join_of_a @ resid @ T2 @ T2 @ T2 ) ) ).
% R.join_of_arr_self
thf(fact_936_R_Otargets__join__of_I2_J,axiom,
! [T2: a,U2: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ V )
=> ( ( targets_a @ resid @ ( resid @ U2 @ T2 ) )
= ( targets_a @ resid @ V ) ) ) ).
% R.targets_join_of(2)
thf(fact_937_R_Otargets__join__of_I1_J,axiom,
! [T2: a,U2: a,V: a] :
( ( join_of_a @ resid @ T2 @ U2 @ V )
=> ( ( targets_a @ resid @ ( resid @ T2 @ U2 ) )
= ( targets_a @ resid @ V ) ) ) ).
% R.targets_join_of(1)
thf(fact_938_R_Ojoinable__def,axiom,
! [T2: a,U2: a] :
( ( joinable_a @ resid @ T2 @ U2 )
= ( ? [X5: a] : ( join_of_a @ resid @ T2 @ U2 @ X5 ) ) ) ).
% R.joinable_def
thf(fact_939_R_Ojoin__of__arr__src_I2_J,axiom,
! [T2: a,A: a] :
( ( arr_a @ resid @ T2 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( join_of_a @ resid @ T2 @ A @ T2 ) ) ) ).
% R.join_of_arr_src(2)
thf(fact_940_R_Ojoin__of__arr__src_I1_J,axiom,
! [T2: a,A: a] :
( ( arr_a @ resid @ T2 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T2 ) )
=> ( join_of_a @ resid @ A @ T2 @ T2 ) ) ) ).
% R.join_of_arr_src(1)
thf(fact_941_P_Ojoin__of__symmetric,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T2 @ V ) ) ).
% P.join_of_symmetric
thf(fact_942_P_Ojoin__of__un__upto__cong,axiom,
! [T2: list_a,U2: list_a,V: list_a,V5: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V )
=> ( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V5 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ V5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V5 @ V ) ) ) ) ) ).
% P.join_of_un_upto_cong
thf(fact_943_P_Ocon__with__join__of__iff_I2_J,axiom,
! [T2: list_a,U2: list_a,W3: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ W3 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W3 @ V )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ V )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ T2 ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) ) ) ) ) ).
% P.con_with_join_of_iff(2)
thf(fact_944_P_Ocon__with__join__of__iff_I1_J,axiom,
! [T2: list_a,U2: list_a,W3: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ W3 )
=> ( ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ V )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W3 @ V ) ) ) ).
% P.con_with_join_of_iff(1)
thf(fact_945_P_Ojoin__of__resid,axiom,
! [T2: list_a,U2: list_a,W3: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ W3 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ V @ W3 )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ V ) @ ( paths_in_Resid_a @ resid @ U2 @ V ) @ ( paths_in_Resid_a @ resid @ W3 @ V ) ) ) ) ).
% P.join_of_resid
thf(fact_946_P_Osources__join__of_I2_J,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% P.sources_join_of(2)
thf(fact_947_P_Osources__join__of_I1_J,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% P.sources_join_of(1)
thf(fact_948_P_Ojoin__of__arr__self,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ T2 @ T2 ) ) ).
% P.join_of_arr_self
thf(fact_949_P_Otargets__join__of_I2_J,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% P.targets_join_of(2)
thf(fact_950_P_Otargets__join__of_I1_J,axiom,
! [T2: list_a,U2: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ V )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% P.targets_join_of(1)
thf(fact_951_P_Ojoinable__def,axiom,
! [T2: list_a,U2: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( ? [X5: list_a] : ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 @ X5 ) ) ) ).
% P.joinable_def
thf(fact_952_P_Ojoin__of__arr__src_I2_J,axiom,
! [T2: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ A @ T2 ) ) ) ).
% P.join_of_arr_src(2)
thf(fact_953_P_Ojoin__of__arr__src_I1_J,axiom,
! [T2: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T2 @ T2 ) ) ) ).
% P.join_of_arr_src(1)
thf(fact_954_comp__eqI,axiom,
! [T2: set_list_a,V: set_list_a,U2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ V ) )
=> ( ( U2
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 ) )
=> ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= V ) ) ) ).
% comp_eqI
thf(fact_955_comp__ide__self,axiom,
! [A: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ A )
= A ) ) ).
% comp_ide_self
thf(fact_956_prfx__decomp,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
=> ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
= U2 ) ) ).
% prfx_decomp
thf(fact_957_resid__comp_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,W3: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ W3 )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ W3 )
= ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ W3 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ W3 @ T2 ) ) ) ) ) ).
% resid_comp(2)
thf(fact_958_resid__comp_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,W3: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ W3 )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ W3 @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ W3 @ T2 ) @ U2 ) ) ) ).
% resid_comp(1)
thf(fact_959_comp__cancel__left,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
=> ( ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ V ) )
=> ( U2 = V ) ) ) ).
% comp_cancel_left
thf(fact_960_comp__assoc,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ V )
=> ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ V ) )
= ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ V ) ) ) ).
% comp_assoc
thf(fact_961_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X4 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_962_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_963_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ X4 )
= bot_bo3186585308812441520list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_964_bounded__imp__con_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a,T5: set_list_a,U4: set_list_a] :
( ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 @ U4 ) ) )
& ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T5 @ U4 ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ T5 ) ) ).
% bounded_imp_con\<^sub>E
thf(fact_965_prfx__comp,axiom,
! [U2: set_list_a,T2: set_list_a,V: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ V )
= U2 )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% prfx_comp
thf(fact_966_con__comp__iff,axiom,
! [W3: set_list_a,T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ W3 @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
& ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ W3 @ T2 ) @ U2 ) ) ) ).
% con_comp_iff
thf(fact_967_composable__iff__arr__comp,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% composable_iff_arr_comp
thf(fact_968_comp__join_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,U4: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U4 ) )
=> ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ U4 ) )
= ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U4 ) ) ) ) ).
% comp_join(2)
thf(fact_969_comp__resid__prfx,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ T2 )
= U2 ) ) ).
% comp_resid_prfx
thf(fact_970_con__compI_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,W3: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ W3 @ T2 ) @ U2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) @ W3 ) ) ) ).
% con_compI(2)
thf(fact_971_con__compI_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,W3: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ W3 @ T2 ) @ U2 )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ W3 @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% con_compI(1)
thf(fact_972_arr__comp,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% arr_comp
thf(fact_973_extensional__rts_Ocomp_Ocong,axiom,
extens4686118995507326689list_a = extens4686118995507326689list_a ).
% extensional_rts.comp.cong
thf(fact_974_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_975_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_976_bot__set__def,axiom,
( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ bot_bot_set_list_a_o ) ) ).
% bot_set_def
thf(fact_977_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_list_a,K: set_list_a,B: set_list_a,A: set_list_a] :
( ( B3
= ( inf_inf_set_list_a @ K @ B ) )
=> ( ( inf_inf_set_list_a @ A @ B3 )
= ( inf_inf_set_list_a @ K @ ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_978_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_set_list_a,K: set_set_list_a,B: set_set_list_a,A: set_set_list_a] :
( ( B3
= ( inf_in4657809108759609906list_a @ K @ B ) )
=> ( ( inf_in4657809108759609906list_a @ A @ B3 )
= ( inf_in4657809108759609906list_a @ K @ ( inf_in4657809108759609906list_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_979_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_980_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_list_a,K: set_list_a,A: set_list_a,B: set_list_a] :
( ( A2
= ( inf_inf_set_list_a @ K @ A ) )
=> ( ( inf_inf_set_list_a @ A2 @ B )
= ( inf_inf_set_list_a @ K @ ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_981_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_set_list_a,K: set_set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( A2
= ( inf_in4657809108759609906list_a @ K @ A ) )
=> ( ( inf_in4657809108759609906list_a @ A2 @ B )
= ( inf_in4657809108759609906list_a @ K @ ( inf_in4657809108759609906list_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_982_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_983_inf__bot__left,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X4 )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_984_inf__bot__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_985_inf__bot__left,axiom,
! [X4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ X4 )
= bot_bo3186585308812441520list_a ) ).
% inf_bot_left
thf(fact_986_inf__bot__right,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_987_inf__bot__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_988_inf__bot__right,axiom,
! [X4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X4 @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% inf_bot_right
thf(fact_989_le__inf__iff,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a,Z2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ X4 @ ( inf_in4657809108759609906list_a @ Y4 @ Z2 ) )
= ( ( ord_le8877086941679407844list_a @ X4 @ Y4 )
& ( ord_le8877086941679407844list_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_990_le__inf__iff,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_991_le__inf__iff,axiom,
! [X4: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( inf_inf_set_list_a @ Y4 @ Z2 ) )
= ( ( ord_le8861187494160871172list_a @ X4 @ Y4 )
& ( ord_le8861187494160871172list_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_992_inf_Oidem,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_993_inf_Oidem,axiom,
! [A: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_994_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_995_inf__idem,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_996_inf__idem,axiom,
! [X4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_997_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_998_inf_Oleft__idem,axiom,
! [A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B ) )
= ( inf_inf_set_list_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_999_inf_Oleft__idem,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A @ ( inf_in4657809108759609906list_a @ A @ B ) )
= ( inf_in4657809108759609906list_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_1000_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_1001_inf__left__idem,axiom,
! [X4: set_list_a,Y4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ X4 @ Y4 ) )
= ( inf_inf_set_list_a @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_1002_inf__left__idem,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X4 @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) )
= ( inf_in4657809108759609906list_a @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_1003_inf__left__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_1004_inf_Oright__idem,axiom,
! [A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ B )
= ( inf_inf_set_list_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_1005_inf_Oright__idem,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ B )
= ( inf_in4657809108759609906list_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_1006_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_1007_inf__right__idem,axiom,
! [X4: set_list_a,Y4: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X4 @ Y4 ) @ Y4 )
= ( inf_inf_set_list_a @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_1008_inf__right__idem,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) @ Y4 )
= ( inf_in4657809108759609906list_a @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_1009_inf__right__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_1010_inf_Obounded__iff,axiom,
! [A: set_set_list_a,B: set_set_list_a,C: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ ( inf_in4657809108759609906list_a @ B @ C ) )
= ( ( ord_le8877086941679407844list_a @ A @ B )
& ( ord_le8877086941679407844list_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1011_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1012_inf_Obounded__iff,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B @ C ) )
= ( ( ord_le8861187494160871172list_a @ A @ B )
& ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1013_inf__sup__aci_I4_J,axiom,
! [X4: set_list_a,Y4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ X4 @ Y4 ) )
= ( inf_inf_set_list_a @ X4 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_1014_inf__sup__aci_I4_J,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X4 @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) )
= ( inf_in4657809108759609906list_a @ X4 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_1015_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_1016_inf__sup__aci_I3_J,axiom,
! [X4: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y4 @ Z2 ) )
= ( inf_inf_set_list_a @ Y4 @ ( inf_inf_set_list_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_1017_inf__sup__aci_I3_J,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a,Z2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X4 @ ( inf_in4657809108759609906list_a @ Y4 @ Z2 ) )
= ( inf_in4657809108759609906list_a @ Y4 @ ( inf_in4657809108759609906list_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_1018_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z2 ) )
= ( inf_inf_set_a @ Y4 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_1019_inf__sup__aci_I2_J,axiom,
! [X4: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X4 @ Y4 ) @ Z2 )
= ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_1020_inf__sup__aci_I2_J,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a,Z2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) @ Z2 )
= ( inf_in4657809108759609906list_a @ X4 @ ( inf_in4657809108759609906list_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_1021_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_1022_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X2: set_list_a,Y2: set_list_a] : ( inf_inf_set_list_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1023_inf__sup__aci_I1_J,axiom,
( inf_in4657809108759609906list_a
= ( ^ [X2: set_set_list_a,Y2: set_set_list_a] : ( inf_in4657809108759609906list_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1024_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1025_inf_Oassoc,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ C )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_1026_inf_Oassoc,axiom,
! [A: set_set_list_a,B: set_set_list_a,C: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ C )
= ( inf_in4657809108759609906list_a @ A @ ( inf_in4657809108759609906list_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_1027_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_1028_inf__assoc,axiom,
! [X4: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X4 @ Y4 ) @ Z2 )
= ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y4 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_1029_inf__assoc,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a,Z2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) @ Z2 )
= ( inf_in4657809108759609906list_a @ X4 @ ( inf_in4657809108759609906list_a @ Y4 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_1030_inf__assoc,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_1031_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A7: set_list_a,B5: set_list_a] : ( inf_inf_set_list_a @ B5 @ A7 ) ) ) ).
% inf.commute
thf(fact_1032_inf_Ocommute,axiom,
( inf_in4657809108759609906list_a
= ( ^ [A7: set_set_list_a,B5: set_set_list_a] : ( inf_in4657809108759609906list_a @ B5 @ A7 ) ) ) ).
% inf.commute
thf(fact_1033_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A7 ) ) ) ).
% inf.commute
thf(fact_1034_inf__commute,axiom,
( inf_inf_set_list_a
= ( ^ [X2: set_list_a,Y2: set_list_a] : ( inf_inf_set_list_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_1035_inf__commute,axiom,
( inf_in4657809108759609906list_a
= ( ^ [X2: set_set_list_a,Y2: set_set_list_a] : ( inf_in4657809108759609906list_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_1036_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_1037_inf_Oleft__commute,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ B @ ( inf_inf_set_list_a @ A @ C ) )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_1038_inf_Oleft__commute,axiom,
! [B: set_set_list_a,A: set_set_list_a,C: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ B @ ( inf_in4657809108759609906list_a @ A @ C ) )
= ( inf_in4657809108759609906list_a @ A @ ( inf_in4657809108759609906list_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_1039_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_1040_inf__left__commute,axiom,
! [X4: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y4 @ Z2 ) )
= ( inf_inf_set_list_a @ Y4 @ ( inf_inf_set_list_a @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_1041_inf__left__commute,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a,Z2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X4 @ ( inf_in4657809108759609906list_a @ Y4 @ Z2 ) )
= ( inf_in4657809108759609906list_a @ Y4 @ ( inf_in4657809108759609906list_a @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_1042_inf__left__commute,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z2 ) )
= ( inf_inf_set_a @ Y4 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_1043_inf_OcoboundedI2,axiom,
! [B: set_set_list_a,C: set_set_list_a,A: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ B @ C )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1044_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1045_inf_OcoboundedI2,axiom,
! [B: set_list_a,C: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1046_inf_OcoboundedI1,axiom,
! [A: set_set_list_a,C: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ C )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1047_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1048_inf_OcoboundedI1,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1049_inf_Oabsorb__iff2,axiom,
( ord_le8877086941679407844list_a
= ( ^ [B5: set_set_list_a,A7: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A7 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_1050_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A7: set_a] :
( ( inf_inf_set_a @ A7 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_1051_inf_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A7: set_list_a] :
( ( inf_inf_set_list_a @ A7 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_1052_inf_Oabsorb__iff1,axiom,
( ord_le8877086941679407844list_a
= ( ^ [A7: set_set_list_a,B5: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A7 @ B5 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_1053_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B5: set_a] :
( ( inf_inf_set_a @ A7 @ B5 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_1054_inf_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B5: set_list_a] :
( ( inf_inf_set_list_a @ A7 @ B5 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_1055_inf_Ocobounded2,axiom,
! [A: set_set_list_a,B: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1056_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1057_inf_Ocobounded2,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1058_inf_Ocobounded1,axiom,
! [A: set_set_list_a,B: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1059_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1060_inf_Ocobounded1,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1061_inf_Oorder__iff,axiom,
( ord_le8877086941679407844list_a
= ( ^ [A7: set_set_list_a,B5: set_set_list_a] :
( A7
= ( inf_in4657809108759609906list_a @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1062_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B5: set_a] :
( A7
= ( inf_inf_set_a @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1063_inf_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B5: set_list_a] :
( A7
= ( inf_inf_set_list_a @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1064_inf__greatest,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a,Z2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ X4 @ Y4 )
=> ( ( ord_le8877086941679407844list_a @ X4 @ Z2 )
=> ( ord_le8877086941679407844list_a @ X4 @ ( inf_in4657809108759609906list_a @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1065_inf__greatest,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ X4 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1066_inf__greatest,axiom,
! [X4: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y4 )
=> ( ( ord_le8861187494160871172list_a @ X4 @ Z2 )
=> ( ord_le8861187494160871172list_a @ X4 @ ( inf_inf_set_list_a @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1067_inf_OboundedI,axiom,
! [A: set_set_list_a,B: set_set_list_a,C: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ( ord_le8877086941679407844list_a @ A @ C )
=> ( ord_le8877086941679407844list_a @ A @ ( inf_in4657809108759609906list_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1068_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1069_inf_OboundedI,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1070_inf_OboundedE,axiom,
! [A: set_set_list_a,B: set_set_list_a,C: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ ( inf_in4657809108759609906list_a @ B @ C ) )
=> ~ ( ( ord_le8877086941679407844list_a @ A @ B )
=> ~ ( ord_le8877086941679407844list_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1071_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1072_inf_OboundedE,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B @ C ) )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ~ ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1073_inf__absorb2,axiom,
! [Y4: set_set_list_a,X4: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ Y4 @ X4 )
=> ( ( inf_in4657809108759609906list_a @ X4 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1074_inf__absorb2,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1075_inf__absorb2,axiom,
! [Y4: set_list_a,X4: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y4 @ X4 )
=> ( ( inf_inf_set_list_a @ X4 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1076_inf__absorb1,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ X4 @ Y4 )
=> ( ( inf_in4657809108759609906list_a @ X4 @ Y4 )
= X4 ) ) ).
% inf_absorb1
thf(fact_1077_inf__absorb1,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= X4 ) ) ).
% inf_absorb1
thf(fact_1078_inf__absorb1,axiom,
! [X4: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y4 )
=> ( ( inf_inf_set_list_a @ X4 @ Y4 )
= X4 ) ) ).
% inf_absorb1
thf(fact_1079_inf_Oabsorb2,axiom,
! [B: set_set_list_a,A: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ B @ A )
=> ( ( inf_in4657809108759609906list_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1080_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1081_inf_Oabsorb2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( inf_inf_set_list_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1082_inf_Oabsorb1,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ( inf_in4657809108759609906list_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1083_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1084_inf_Oabsorb1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( inf_inf_set_list_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1085_le__iff__inf,axiom,
( ord_le8877086941679407844list_a
= ( ^ [X2: set_set_list_a,Y2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1086_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1087_le__iff__inf,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X2: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1088_inf__unique,axiom,
! [F: set_set_list_a > set_set_list_a > set_set_list_a,X4: set_set_list_a,Y4: set_set_list_a] :
( ! [X: set_set_list_a,Y3: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_set_list_a,Y3: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_set_list_a,Y3: set_set_list_a,Z3: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ X @ Y3 )
=> ( ( ord_le8877086941679407844list_a @ X @ Z3 )
=> ( ord_le8877086941679407844list_a @ X @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in4657809108759609906list_a @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1089_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y4: set_a] :
( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Z3 )
=> ( ord_less_eq_set_a @ X @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1090_inf__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X4: set_list_a,Y4: set_list_a] :
( ! [X: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_list_a,Y3: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ X @ Z3 )
=> ( ord_le8861187494160871172list_a @ X @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_list_a @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1091_inf_OorderI,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ( A
= ( inf_in4657809108759609906list_a @ A @ B ) )
=> ( ord_le8877086941679407844list_a @ A @ B ) ) ).
% inf.orderI
thf(fact_1092_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_1093_inf_OorderI,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A
= ( inf_inf_set_list_a @ A @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% inf.orderI
thf(fact_1094_inf_OorderE,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ B )
=> ( A
= ( inf_in4657809108759609906list_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1095_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1096_inf_OorderE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( A
= ( inf_inf_set_list_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1097_le__infI2,axiom,
! [B: set_set_list_a,X4: set_set_list_a,A: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ B @ X4 )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_1098_le__infI2,axiom,
! [B: set_a,X4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_1099_le__infI2,axiom,
! [B: set_list_a,X4: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ X4 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_1100_le__infI1,axiom,
! [A: set_set_list_a,X4: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ X4 )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_1101_le__infI1,axiom,
! [A: set_a,X4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_1102_le__infI1,axiom,
! [A: set_list_a,X4: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ X4 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_1103_inf__mono,axiom,
! [A: set_set_list_a,C: set_set_list_a,B: set_set_list_a,D2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ C )
=> ( ( ord_le8877086941679407844list_a @ B @ D2 )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ B ) @ ( inf_in4657809108759609906list_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1104_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1105_inf__mono,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ ( inf_inf_set_list_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1106_le__infI,axiom,
! [X4: set_set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ X4 @ A )
=> ( ( ord_le8877086941679407844list_a @ X4 @ B )
=> ( ord_le8877086941679407844list_a @ X4 @ ( inf_in4657809108759609906list_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1107_le__infI,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ X4 @ B )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1108_le__infI,axiom,
! [X4: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ A )
=> ( ( ord_le8861187494160871172list_a @ X4 @ B )
=> ( ord_le8861187494160871172list_a @ X4 @ ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1109_le__infE,axiom,
! [X4: set_set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ X4 @ ( inf_in4657809108759609906list_a @ A @ B ) )
=> ~ ( ( ord_le8877086941679407844list_a @ X4 @ A )
=> ~ ( ord_le8877086941679407844list_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_1110_le__infE,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A )
=> ~ ( ord_less_eq_set_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_1111_le__infE,axiom,
! [X4: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( inf_inf_set_list_a @ A @ B ) )
=> ~ ( ( ord_le8861187494160871172list_a @ X4 @ A )
=> ~ ( ord_le8861187494160871172list_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_1112_inf__le2,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1113_inf__le2,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1114_inf__le2,axiom,
! [X4: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1115_inf__le1,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) @ X4 ) ).
% inf_le1
thf(fact_1116_inf__le1,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_le1
thf(fact_1117_inf__le1,axiom,
! [X4: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X4 @ Y4 ) @ X4 ) ).
% inf_le1
thf(fact_1118_inf__sup__ord_I1_J,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_1119_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_1120_inf__sup__ord_I1_J,axiom,
! [X4: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X4 @ Y4 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_1121_inf__sup__ord_I2_J,axiom,
! [X4: set_set_list_a,Y4: set_set_list_a] : ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1122_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1123_inf__sup__ord_I2_J,axiom,
! [X4: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1124_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: list_a > list_a > list_a,T2: list_a,U: list_list_a] :
( ( paths_2227358642439567705list_a @ Resid )
=> ( ! [T4: list_a,U7: list_a] :
( ( coinitial_list_a @ Resid @ T4 @ U7 )
=> ( con_list_a @ Resid @ T4 @ U7 ) )
=> ( ( arr_list_a @ Resid @ T2 )
=> ( ( paths_in_Arr_list_a @ Resid @ U )
=> ( ( ( sources_list_a @ Resid @ T2 )
= ( paths_in_Srcs_list_a @ Resid @ U ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ nil_list_a ) @ U )
!= nil_list_a ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_1125_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: set_list_a > set_list_a > set_list_a,T2: set_list_a,U: list_set_list_a] :
( ( paths_3506353789733915961list_a @ Resid )
=> ( ! [T4: set_list_a,U7: set_list_a] :
( ( coinitial_set_list_a @ Resid @ T4 @ U7 )
=> ( con_set_list_a @ Resid @ T4 @ U7 ) )
=> ( ( arr_set_list_a @ Resid @ T2 )
=> ( ( paths_3862341395584752492list_a @ Resid @ U )
=> ( ( ( sources_set_list_a @ Resid @ T2 )
= ( paths_5622510820459842474list_a @ Resid @ U ) )
=> ( ( paths_6591532302923812690list_a @ Resid @ ( cons_set_list_a @ T2 @ nil_set_list_a ) @ U )
!= nil_set_list_a ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_1126_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: a > a > a,T2: a,U: list_a] :
( ( paths_8083246678931295059_rts_a @ Resid )
=> ( ! [T4: a,U7: a] :
( ( coinitial_a @ Resid @ T4 @ U7 )
=> ( con_a @ Resid @ T4 @ U7 ) )
=> ( ( arr_a @ Resid @ T2 )
=> ( ( paths_in_Arr_a @ Resid @ U )
=> ( ( ( sources_a @ Resid @ T2 )
= ( paths_in_Srcs_a @ Resid @ U ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ nil_a ) @ U )
!= nil_a ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_1127_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_1128_dual__order_Orefl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% dual_order.refl
thf(fact_1129_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_1130_order__refl,axiom,
! [X4: set_list_a] : ( ord_le8861187494160871172list_a @ X4 @ X4 ) ).
% order_refl
thf(fact_1131_order__antisym__conv,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_1132_order__antisym__conv,axiom,
! [Y4: set_list_a,X4: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y4 @ X4 )
=> ( ( ord_le8861187494160871172list_a @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_1133_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1134_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1135_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1136_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1137_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1138_ord__eq__le__subst,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1139_ord__eq__le__subst,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1140_ord__eq__le__subst,axiom,
! [A: set_list_a,F: set_list_a > set_list_a,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1141_order__eq__refl,axiom,
! [X4: set_a,Y4: set_a] :
( ( X4 = Y4 )
=> ( ord_less_eq_set_a @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_1142_order__eq__refl,axiom,
! [X4: set_list_a,Y4: set_list_a] :
( ( X4 = Y4 )
=> ( ord_le8861187494160871172list_a @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_1143_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1144_in__targetsE,axiom,
! [B: set_list_a,T2: set_list_a] :
( ( member_set_list_a @ B @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
=> ~ ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ B )
=> ~ ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ B ) ) ) ).
% in_targetsE
thf(fact_1145_joinable__iff__join__not__null,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% joinable_iff_join_not_null
thf(fact_1146_trg__def,axiom,
! [T2: set_list_a] :
( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ T2 ) ) ).
% trg_def
thf(fact_1147_trg__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [T9: set_list_a] :
( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T9 )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T9 @ T9 ) ) ).
% trg_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_1148_apex__sym,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ) ).
% apex_sym
thf(fact_1149_null__eqI,axiom,
! [N: set_list_a] :
( ! [T4: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ N @ T4 )
= N )
& ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T4 @ N )
= N ) )
=> ( N
= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% null_eqI
thf(fact_1150_cube__ax,axiom,
! [V: set_list_a,T2: set_list_a,U2: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ T2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
= ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ V @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% cube_ax
thf(fact_1151_con__sym__ax,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% con_sym_ax
thf(fact_1152_con__imp__arr__resid,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% con_imp_arr_resid
thf(fact_1153_trg__resid__sym,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) ) ) ) ).
% trg_resid_sym
thf(fact_1154_con__def,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% con_def
thf(fact_1155_conE,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% conE
thf(fact_1156_not__arr__null,axiom,
~ ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% not_arr_null
thf(fact_1157_trg__join__of_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% trg_join_of(1)
thf(fact_1158_trg__join__of_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ U2 @ T2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% trg_join_of(2)
thf(fact_1159_join__sym,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) )
=> ( ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 @ T2 ) ) ) ).
% join_sym
thf(fact_1160_null__char,axiom,
( ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) )
= bot_bot_set_list_a ) ).
% null_char
thf(fact_1161_ide__trg,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% ide_trg
thf(fact_1162_ide__iff__trg__self,axiom,
! [A: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
= ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
= A ) ) ) ).
% ide_iff_trg_self
thf(fact_1163_trg__in__targets,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( member_set_list_a @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% trg_in_targets
thf(fact_1164_quot_Oextensional,axiom,
! [T2: list_a] :
( ~ ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 )
= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% quot.extensional
thf(fact_1165_resid__ide_I2_J,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ T2 )
=> ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ A @ T2 )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ) ).
% resid_ide(2)
thf(fact_1166_composable__iff__comp__not__null,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ) ).
% composable_iff_comp_not_null
thf(fact_1167_coterminal__iff__con__trg,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ).
% coterminal_iff_con_trg
thf(fact_1168_coterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
& ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
& ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ).
% coterminal_iff\<^sub>W\<^sub>E
thf(fact_1169_coterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ~ ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ) ).
% coterminalE\<^sub>W\<^sub>E
thf(fact_1170_trg__join,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% trg_join
thf(fact_1171_trg__trg,axiom,
! [T2: set_list_a] :
( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% trg_trg
thf(fact_1172_null__is__zero_I1_J,axiom,
! [T2: set_list_a] :
( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) @ T2 )
= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% null_is_zero(1)
thf(fact_1173_null__is__zero_I2_J,axiom,
! [T2: set_list_a] :
( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) )
= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% null_is_zero(2)
thf(fact_1174_trg__ide,axiom,
! [A: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
= A ) ) ).
% trg_ide
thf(fact_1175_arr__trg__iff__arr,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
= ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% arr_trg_iff_arr
thf(fact_1176_conI,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 )
!= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) )
=> ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ).
% conI
thf(fact_1177_comp__null_I1_J,axiom,
! [T2: set_list_a] :
( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) @ T2 )
= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% comp_null(1)
thf(fact_1178_comp__null_I2_J,axiom,
! [T2: set_list_a] :
( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) )
= ( partia8392230974233693834list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ) ) ).
% comp_null(2)
thf(fact_1179_comp__arr__trg,axiom,
! [T2: set_list_a,B: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= B )
=> ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ B )
= T2 ) ) ) ).
% comp_arr_trg
thf(fact_1180_trg__comp,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ).
% trg_comp
thf(fact_1181_coterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
=> ( coterm239225051077876726list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% coterminalI\<^sub>W\<^sub>E
thf(fact_1182_in__targetsI,axiom,
! [B: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ B )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ B )
=> ( member_set_list_a @ B @ ( targets_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ) ).
% in_targetsI
thf(fact_1183_quot_Opreserves__trg,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 ) ) ) ) ).
% quot.preserves_trg
thf(fact_1184_seqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ~ ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ) ).
% seqE\<^sub>W\<^sub>E
thf(fact_1185_R_Ocon__imp__arr__resid,axiom,
! [T2: a,U2: a] :
( ( ( resid @ T2 @ U2 )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ T2 @ U2 ) @ ( resid @ T2 @ U2 ) )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_imp_arr_resid
thf(fact_1186_R_Ocon__sym__ax,axiom,
! [T2: a,U2: a] :
( ( ( resid @ T2 @ U2 )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ U2 @ T2 )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_sym_ax
thf(fact_1187_R_Ocube__ax,axiom,
! [V: a,T2: a,U2: a] :
( ( ( resid @ ( resid @ V @ T2 ) @ ( resid @ U2 @ T2 ) )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ V @ T2 ) @ ( resid @ U2 @ T2 ) )
= ( resid @ ( resid @ V @ U2 ) @ ( resid @ T2 @ U2 ) ) ) ) ).
% R.cube_ax
thf(fact_1188_R_Onull__eqI,axiom,
! [N: a] :
( ! [T4: a] :
( ( ( resid @ N @ T4 )
= N )
& ( ( resid @ T4 @ N )
= N ) )
=> ( N
= ( partial_null_a @ resid ) ) ) ).
% R.null_eqI
thf(fact_1189_R_Otrg__def,axiom,
! [T2: a] :
( ( trg_a @ resid @ T2 )
= ( resid @ T2 @ T2 ) ) ).
% R.trg_def
thf(fact_1190_P_Ocon__imp__arr__resid,axiom,
! [T2: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T2 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% P.con_imp_arr_resid
thf(fact_1191_P_Ocon__sym__ax,axiom,
! [T2: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T2 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ U2 @ T2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% P.con_sym_ax
thf(fact_1192_P_Ocube__ax,axiom,
! [V: list_a,T2: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T2 ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T2 ) @ ( paths_in_Resid_a @ resid @ U2 @ T2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ U2 ) @ ( paths_in_Resid_a @ resid @ T2 @ U2 ) ) ) ) ).
% P.cube_ax
thf(fact_1193_P_Onull__eqI,axiom,
! [N: list_a] :
( ! [T4: list_a] :
( ( ( paths_in_Resid_a @ resid @ N @ T4 )
= N )
& ( ( paths_in_Resid_a @ resid @ T4 @ N )
= N ) )
=> ( N
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% P.null_eqI
thf(fact_1194_R_OconE,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
=> ( ( resid @ T2 @ U2 )
!= ( partial_null_a @ resid ) ) ) ).
% R.conE
thf(fact_1195_R_Ocon__def,axiom,
! [T2: a,U2: a] :
( ( con_a @ resid @ T2 @ U2 )
= ( ( resid @ T2 @ U2 )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_def
thf(fact_1196_R_Onot__arr__null,axiom,
~ ( arr_a @ resid @ ( partial_null_a @ resid ) ) ).
% R.not_arr_null
thf(fact_1197_P_Onull__char,axiom,
( ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) )
= nil_a ) ).
% P.null_char
thf(fact_1198_P_OconE,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ T2 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% P.conE
thf(fact_1199_P_Ocon__def,axiom,
! [T2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( ( paths_in_Resid_a @ resid @ T2 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% P.con_def
thf(fact_1200_R_Oide__trg,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( ide_a @ resid @ ( trg_a @ resid @ T2 ) ) ) ).
% R.ide_trg
thf(fact_1201_P_Onot__arr__null,axiom,
~ ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% P.not_arr_null
thf(fact_1202_R_Otrg__in__targets,axiom,
! [T2: a] :
( ( arr_a @ resid @ T2 )
=> ( member_a @ ( trg_a @ resid @ T2 ) @ ( targets_a @ resid @ T2 ) ) ) ).
% R.trg_in_targets
thf(fact_1203_P_Otrg__def,axiom,
! [T2: list_a] :
( ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
= ( paths_in_Resid_a @ resid @ T2 @ T2 ) ) ).
% P.trg_def
thf(fact_1204_R_Ocoterminal__iff__con__trg,axiom,
! [T2: a,U2: a] :
( ( coterminal_a @ resid @ T2 @ U2 )
= ( con_a @ resid @ ( trg_a @ resid @ T2 ) @ ( trg_a @ resid @ U2 ) ) ) ).
% R.coterminal_iff_con_trg
thf(fact_1205_R_Oin__targetsE,axiom,
! [B: a,T2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T2 ) )
=> ~ ( ( ide_a @ resid @ B )
=> ~ ( con_a @ resid @ ( trg_a @ resid @ T2 ) @ B ) ) ) ).
% R.in_targetsE
thf(fact_1206_P_Oide__trg,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ).
% P.ide_trg
thf(fact_1207_P_Otrg__in__targets,axiom,
! [T2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T2 )
=> ( member_list_a @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ).
% P.trg_in_targets
thf(fact_1208_R_Onull__is__zero_I2_J,axiom,
! [T2: a] :
( ( resid @ T2 @ ( partial_null_a @ resid ) )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(2)
thf(fact_1209_R_Onull__is__zero_I1_J,axiom,
! [T2: a] :
( ( resid @ ( partial_null_a @ resid ) @ T2 )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(1)
thf(fact_1210_P_Ocoterminal__iff__con__trg,axiom,
! [T2: list_a,U2: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ).
% P.coterminal_iff_con_trg
thf(fact_1211_P_Oreflects__con,axiom,
! [T2: a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ ( if_list_a @ ( arr_a @ resid @ T2 ) @ ( cons_a @ T2 @ nil_a ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) @ ( if_list_a @ ( arr_a @ resid @ U2 ) @ ( cons_a @ U2 @ nil_a ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) )
!= nil_a )
=> ( con_a @ resid @ T2 @ U2 ) ) ).
% P.reflects_con
thf(fact_1212_P_Oin__targetsE,axiom,
! [B: list_a,T2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B )
=> ~ ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ B ) ) ) ).
% P.in_targetsE
thf(fact_1213_con__imp__eq__src,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ).
% con_imp_eq_src
thf(fact_1214_src__join__of_I2_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% src_join_of(2)
thf(fact_1215_src__join__of_I1_J,axiom,
! [T2: set_list_a,U2: set_list_a,V: set_list_a] :
( ( join_of_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 @ V )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ V ) ) ) ).
% src_join_of(1)
thf(fact_1216_src__eqI,axiom,
! [A: set_list_a,T2: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ T2 )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= A ) ) ) ).
% src_eqI
thf(fact_1217_ide__iff__src__self,axiom,
! [A: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
= ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
= A ) ) ) ).
% ide_iff_src_self
thf(fact_1218_ide__src,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% ide_src
thf(fact_1219_src__in__sources,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( member_set_list_a @ ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ ( sources_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% src_in_sources
thf(fact_1220_join__src,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) @ T2 )
= T2 ) ) ).
% join_src
thf(fact_1221_coinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ~ ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
!= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ) ).
% coinitialE\<^sub>W\<^sub>E
thf(fact_1222_coinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
= ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
& ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
& ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ) ).
% coinitial_iff\<^sub>W\<^sub>E
thf(fact_1223_src__join,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( joinable_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens431601257083675878list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% src_join
thf(fact_1224_P_Onull__is__zero_I1_J,axiom,
! [T2: list_a] :
( ( paths_in_Resid_a @ resid @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) @ T2 )
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% P.null_is_zero(1)
thf(fact_1225_P_Onull__is__zero_I2_J,axiom,
! [T2: list_a] :
( ( paths_in_Resid_a @ resid @ T2 @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% P.null_is_zero(2)
thf(fact_1226_R_OconI,axiom,
! [T2: a,U2: a] :
( ( ( resid @ T2 @ U2 )
!= ( partial_null_a @ resid ) )
=> ( con_a @ resid @ T2 @ U2 ) ) ).
% R.conI
thf(fact_1227_P_OconI,axiom,
! [T2: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T2 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T2 @ U2 ) ) ).
% P.conI
thf(fact_1228_R_Oin__targetsI,axiom,
! [B: a,T2: a] :
( ( ide_a @ resid @ B )
=> ( ( con_a @ resid @ ( trg_a @ resid @ T2 ) @ B )
=> ( member_a @ B @ ( targets_a @ resid @ T2 ) ) ) ) ).
% R.in_targetsI
thf(fact_1229_src__src,axiom,
! [T2: set_list_a] :
( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% src_src
thf(fact_1230_P_Oin__targetsI,axiom,
! [B: list_a,T2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) @ B )
=> ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T2 ) ) ) ) ).
% P.in_targetsI
thf(fact_1231_src__ide,axiom,
! [A: set_list_a] :
( ( ide_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A )
= A ) ) ).
% src_ide
thf(fact_1232_arr__src__iff__arr,axiom,
! [T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
= ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% arr_src_iff_arr
thf(fact_1233_trg__src,axiom,
! [T2: set_list_a] :
( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% trg_src
thf(fact_1234_src__trg,axiom,
! [T2: set_list_a] :
( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ).
% src_trg
thf(fact_1235_src__resid,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( con_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) @ T2 @ U2 ) )
= ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) ) ) ).
% src_resid
thf(fact_1236_comp__src__arr,axiom,
! [T2: set_list_a,A: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= A )
=> ( ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ A @ T2 )
= T2 ) ) ) ).
% comp_src_arr
thf(fact_1237_coinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
=> ( coinitial_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ).
% coinitialI\<^sub>W\<^sub>E
thf(fact_1238_src__comp,axiom,
! [T2: set_list_a,U2: set_list_a] :
( ( compos1473025410594674033list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 )
=> ( ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ ( extens4686118995507326689list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 ) ) ) ).
% src_comp
thf(fact_1239_seqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [U2: set_list_a,T2: set_list_a] :
( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 )
=> ( ( arr_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
=> ( ( ( trg_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 )
= ( weakly6189092243557426935list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ U2 ) )
=> ( seq_set_list_a @ ( quotie715552529030613631list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) @ T2 @ U2 ) ) ) ) ).
% seqI\<^sub>W\<^sub>E
thf(fact_1240_P_OTrgs__simp_092_060_094sub_062P,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( paths_in_Trgs_a @ resid @ T )
= ( targets_a @ resid @ ( last_a @ T ) ) ) ) ).
% P.Trgs_simp\<^sub>P
thf(fact_1241_P_OIde_Oelims_I1_J,axiom,
! [X4: list_a,Y4: $o] :
( ( ( paths_in_Ide_a @ resid @ X4 )
= Y4 )
=> ( ( ( X4 = nil_a )
=> Y4 )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ( Y4
= ( ~ ( ide_a @ resid @ T4 ) ) ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( ( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ( Y4
= ( ~ ( ( ide_a @ resid @ T4 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V3 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ).
% P.Ide.elims(1)
thf(fact_1242_P_OIde_Osimps_I1_J,axiom,
~ ( paths_in_Ide_a @ resid @ nil_a ) ).
% P.Ide.simps(1)
thf(fact_1243_P_OIde__implies__Arr,axiom,
! [T: list_a] :
( ( paths_in_Ide_a @ resid @ T )
=> ( paths_in_Arr_a @ resid @ T ) ) ).
% P.Ide_implies_Arr
thf(fact_1244_NPath__char,axiom,
! [T: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T )
= ( paths_in_Ide_a @ resid @ T ) ) ).
% NPath_char
thf(fact_1245_P_OIde__implies__NPath,axiom,
! [T: list_a] :
( ( paths_in_Ide_a @ resid @ T )
=> ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T ) ) ).
% P.Ide_implies_NPath
thf(fact_1246_P_OResid__Ide__Arr__ind,axiom,
! [A2: list_a,T: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ A2 @ T )
!= nil_a )
=> ( paths_in_Ide_a @ resid @ ( paths_in_Resid_a @ resid @ A2 @ T ) ) ) ) ).
% P.Resid_Ide_Arr_ind
thf(fact_1247_P_OResid__Arr__Ide__ind,axiom,
! [A2: list_a,T: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ T @ A2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ A2 )
= T ) ) ) ).
% P.Resid_Arr_Ide_ind
thf(fact_1248_P_OResid__Ide_I1_J,axiom,
! [A2: list_a,T: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ A2 @ T )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T @ A2 )
= T ) ) ) ).
% P.Resid_Ide(1)
thf(fact_1249_P_Oide__char,axiom,
! [T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( paths_in_Ide_a @ resid @ T ) ) ).
% P.ide_char
thf(fact_1250_P_OResid__Arr__self,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( paths_in_Ide_a @ resid @ ( paths_in_Resid_a @ resid @ T @ T ) ) ) ).
% P.Resid_Arr_self
thf(fact_1251_P_OIde__imp__Ide__last,axiom,
! [T: list_a] :
( ( paths_in_Ide_a @ resid @ T )
=> ( ide_a @ resid @ ( last_a @ T ) ) ) ).
% P.Ide_imp_Ide_last
thf(fact_1252_P_OArr__imp__arr__last,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( arr_a @ resid @ ( last_a @ T ) ) ) ).
% P.Arr_imp_arr_last
thf(fact_1253_P_OIde_Osimps_I2_J,axiom,
! [T2: a] :
( ( paths_in_Ide_a @ resid @ ( cons_a @ T2 @ nil_a ) )
= ( ide_a @ resid @ T2 ) ) ).
% P.Ide.simps(2)
thf(fact_1254_P_OIde__imp__sources__eq__targets,axiom,
! [T: list_a] :
( ( paths_in_Ide_a @ resid @ T )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ).
% P.Ide_imp_sources_eq_targets
thf(fact_1255_P_OCon__Ide__iff,axiom,
! [A2: list_a,T: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ A2 @ T )
!= nil_a )
= ( ( paths_in_Arr_a @ resid @ T )
& ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ A2 ) ) ) ) ) ).
% P.Con_Ide_iff
thf(fact_1256_P_OCon__IdeI_I1_J,axiom,
! [A2: list_a,T: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( paths_in_Arr_a @ resid @ T )
=> ( ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ A2 ) )
=> ( ( paths_in_Resid_a @ resid @ A2 @ T )
!= nil_a ) ) ) ) ).
% P.Con_IdeI(1)
thf(fact_1257_P_OCon__IdeI_I2_J,axiom,
! [A2: list_a,T: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( paths_in_Arr_a @ resid @ T )
=> ( ( ( paths_in_Srcs_a @ resid @ T )
= ( paths_in_Srcs_a @ resid @ A2 ) )
=> ( ( paths_in_Resid_a @ resid @ T @ A2 )
!= nil_a ) ) ) ) ).
% P.Con_IdeI(2)
thf(fact_1258_P_OPCong__append__Ide,axiom,
! [T: list_a,B3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ B3 )
=> ( ( paths_in_Ide_a @ resid @ B3 )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ T @ B3 ) @ T ) ) ) ).
% P.PCong_append_Ide
thf(fact_1259_P_OIde_Osimps_I3_J,axiom,
! [T2: a,V: a,Va2: list_a] :
( ( paths_in_Ide_a @ resid @ ( cons_a @ T2 @ ( cons_a @ V @ Va2 ) ) )
= ( ( ide_a @ resid @ T2 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T2 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va2 ) ) ) ) ) ).
% P.Ide.simps(3)
thf(fact_1260_P_OIde__append__iff_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a] :
( ( T != nil_a )
=> ( ( U != nil_a )
=> ( ( paths_in_Ide_a @ resid @ ( append_a @ T @ U ) )
= ( ( paths_in_Ide_a @ resid @ T )
& ( paths_in_Ide_a @ resid @ U )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) ) ) ) ) ) ).
% P.Ide_append_iff\<^sub>P
thf(fact_1261_P_OIde_Oelims_I3_J,axiom,
! [X4: list_a] :
( ~ ( paths_in_Ide_a @ resid @ X4 )
=> ( ( X4 != nil_a )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ( ide_a @ resid @ T4 ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( ( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ( ( ide_a @ resid @ T4 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V3 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ) ).
% P.Ide.elims(3)
thf(fact_1262_P_OIde_Oelims_I2_J,axiom,
! [X4: list_a] :
( ( paths_in_Ide_a @ resid @ X4 )
=> ( ! [T4: a] :
( ( X4
= ( cons_a @ T4 @ nil_a ) )
=> ~ ( ide_a @ resid @ T4 ) )
=> ~ ! [T4: a,V3: a,Va: list_a] :
( ( X4
= ( cons_a @ T4 @ ( cons_a @ V3 @ Va ) ) )
=> ~ ( ( ide_a @ resid @ T4 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V3 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ).
% P.Ide.elims(2)
thf(fact_1263_P_OIde__appendI_092_060_094sub_062P,axiom,
! [T: list_a,U: list_a] :
( ( paths_in_Ide_a @ resid @ T )
=> ( ( paths_in_Ide_a @ resid @ U )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U ) )
=> ( paths_in_Ide_a @ resid @ ( append_a @ T @ U ) ) ) ) ) ).
% P.Ide_appendI\<^sub>P
thf(fact_1264_P_OResid_Osimps_I4_J,axiom,
! [T2: a,U2: a,V: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T2 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T2 @ U2 ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ ( cons_a @ V @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T2 @ U2 ) @ ( cons_a @ V @ Va2 ) ) @ nil_a ) ) )
& ( ~ ( ( con_a @ resid @ T2 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T2 @ U2 ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ ( cons_a @ U2 @ ( cons_a @ V @ Va2 ) ) )
= nil_a ) ) ) ).
% P.Resid.simps(4)
thf(fact_1265_P_OIde__char,axiom,
! [T: list_a] :
( ( paths_in_Ide_a @ resid @ T )
= ( ( paths_in_Arr_a @ resid @ T )
& ( ord_less_eq_set_a @ ( set_a2 @ T ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% P.Ide_char
thf(fact_1266_P_OResid1x_Osimps_I3_J,axiom,
! [T2: a,U2: a,V: a,Va2: list_a] :
( ( paths_in_Resid1x_a @ resid @ T2 @ ( cons_a @ U2 @ ( cons_a @ V @ Va2 ) ) )
= ( paths_in_Resid1x_a @ resid @ ( resid @ T2 @ U2 ) @ ( cons_a @ V @ Va2 ) ) ) ).
% P.Resid1x.simps(3)
thf(fact_1267_P_OResid1x__null,axiom,
! [T: list_a] :
( ( paths_in_Resid1x_a @ resid @ ( partial_null_a @ resid ) @ T )
= ( partial_null_a @ resid ) ) ).
% P.Resid1x_null
thf(fact_1268_P_OResid1x_Osimps_I2_J,axiom,
! [T2: a,U2: a] :
( ( paths_in_Resid1x_a @ resid @ T2 @ ( cons_a @ U2 @ nil_a ) )
= ( resid @ T2 @ U2 ) ) ).
% P.Resid1x.simps(2)
thf(fact_1269_P_OResid1x__ide,axiom,
! [A: a,T: list_a] :
( ( ide_a @ resid @ A )
=> ( ( ( paths_in_Resid1x_a @ resid @ A @ T )
!= ( partial_null_a @ resid ) )
=> ( ide_a @ resid @ ( paths_in_Resid1x_a @ resid @ A @ T ) ) ) ) ).
% P.Resid1x_ide
thf(fact_1270_P_OResid1x_Osimps_I1_J,axiom,
! [T2: a] :
( ( paths_in_Resid1x_a @ resid @ T2 @ nil_a )
= ( partial_null_a @ resid ) ) ).
% P.Resid1x.simps(1)
thf(fact_1271_P_OResid1x__as__Resid,axiom,
! [T2: a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ T2 @ U ) @ nil_a ) ) ) ).
% P.Resid1x_as_Resid
thf(fact_1272_P_OResid__cons_H,axiom,
! [T: list_a,T2: a,U: list_a] :
( ( T != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ U )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T ) @ U )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ T2 @ U ) @ ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ U @ ( cons_a @ T2 @ nil_a ) ) ) ) ) ) ) ).
% P.Resid_cons'
thf(fact_1273_P_OResid1x_Oelims,axiom,
! [X4: a,Xa: list_a,Y4: a] :
( ( ( paths_in_Resid1x_a @ resid @ X4 @ Xa )
= Y4 )
=> ( ( ( Xa = nil_a )
=> ( Y4
!= ( partial_null_a @ resid ) ) )
=> ( ! [U7: a] :
( ( Xa
= ( cons_a @ U7 @ nil_a ) )
=> ( Y4
!= ( resid @ X4 @ U7 ) ) )
=> ~ ! [U7: a,V3: a,Va: list_a] :
( ( Xa
= ( cons_a @ U7 @ ( cons_a @ V3 @ Va ) ) )
=> ( Y4
!= ( paths_in_Resid1x_a @ resid @ ( resid @ X4 @ U7 ) @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ).
% P.Resid1x.elims
thf(fact_1274_P_Oset__Ide__subset__ide,axiom,
! [T: list_a] :
( ( paths_in_Ide_a @ resid @ T )
=> ( ord_less_eq_set_a @ ( set_a2 @ T ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ).
% P.set_Ide_subset_ide
thf(fact_1275_P_Oset__Arr__subset__arr,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ord_less_eq_set_a @ ( set_a2 @ T ) @ ( collect_a @ ( arr_a @ resid ) ) ) ) ).
% P.set_Arr_subset_arr
thf(fact_1276_P_ONPath__def,axiom,
! [T: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T )
= ( ( paths_in_Arr_a @ resid @ T )
& ( ord_less_eq_set_a @ ( set_a2 @ T ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% P.NPath_def
thf(fact_1277_P_OIdeI,axiom,
! [T: list_a] :
( ( paths_in_Arr_a @ resid @ T )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ T ) @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( paths_in_Ide_a @ resid @ T ) ) ) ).
% P.IdeI
% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X4: list_a,Y4: list_a] :
( ( if_list_a @ $false @ X4 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X4: list_a,Y4: list_a] :
( ( if_list_a @ $true @ X4 @ Y4 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( paths_in_Trgs_a @ resid @ ( append_a @ t @ ( paths_in_Resid_a @ resid @ t2 @ t ) ) )
= ( paths_in_Trgs_a @ resid @ ( append_a @ t2 @ ( paths_in_Resid_a @ resid @ t @ t2 ) ) ) ) ).
%------------------------------------------------------------------------------